# include # include # include # include # include "square_minimal_rule.h" /******************************************************************************/ double *r8mat_copy_new ( int m, int n, double a1[] ) /******************************************************************************/ /* Purpose: R8MAT_COPY_NEW copies one R8MAT to a "new" R8MAT. Discussion: An R8MAT is a doubly dimensioned array of R8 values, stored as a vector in column-major order. Licensing: This code is distributed under the MIT license. Modified: 26 July 2008 Author: John Burkardt Parameters: Input, int M, N, the number of rows and columns. Input, double A1[M*N], the matrix to be copied. Output, double R8MAT_COPY_NEW[M*N], the copy of A1. */ { double *a2; int i; int j; a2 = ( double * ) malloc ( m * n * sizeof ( double ) ); for ( j = 0; j < n; j++ ) { for ( i = 0; i < m; i++ ) { a2[i+j*m] = a1[i+j*m]; } } return a2; } /******************************************************************************/ double r8vec_sum ( int n, double a[] ) /******************************************************************************/ /* Purpose: R8VEC_SUM returns the sum of an R8VEC. Discussion: An R8VEC is a vector of R8's. Licensing: This code is distributed under the MIT license. Modified: 26 August 2008 Author: John Burkardt Parameters: Input, int N, the number of entries in the vector. Input, double A[N], the vector. Output, double R8VEC_SUM, the sum of the vector. */ { int i; double value; value = 0.0; for ( i = 0; i < n; i++ ) { value = value + a[i]; } return value; } /******************************************************************************/ double *smr00 ( ) /******************************************************************************/ /* Purpose: SMR00 returns the SMR rule of degree 0. Discussion: DEGREE: 0 SYMM.: (X,Y), (-X,-Y) POINTS CARDINALITY: 1 NORM INF MOMS. RESIDUAL: 0.00000e+00, SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR00[3*1], the requested rule. */ { const int degree = 0; int order; double xw[3*1] = { 0.000000000000000e+00, 0.000000000000000e+00, 4.000000000000000e+00 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr01 ( ) /******************************************************************************/ /* Purpose: SMR01 returns the SMR rule of degree 1. Discussion: DEGREE: 1 SYMM.: (X,Y), (-X,-Y) POINTS CARDINALITY: 1 NORM INF MOMS. RESIDUAL: 0.00000e+00, SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 20 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR01[3*1], the requested rule. */ { const int degree = 1; int order; double xw[3*1] = { 0.000000000000000e+00, 0.000000000000000e+00, 4.000000000000000e+00 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr02 ( ) /******************************************************************************/ /* Purpose: SMR02 returns the SMR rule of degree 2. Discussion: DEGREE: 2 POINTS CARDINALITY: 3 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR02[3*3], the requested rule. */ { const int degree = 2; int order; double xw[3*3] = { 6.519542382482019e-01, 3.623444315022428e-01, 1.498681096511322e+00, -8.070876861583226e-01, 5.322309038833022e-01, 1.051530016968792e+00, -8.856086946552445e-02, -7.605904084126465e-01, 1.449788886519887e+00 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr03 ( ) /******************************************************************************/ /* Purpose: SMR03 returns the SMR rule of degree 3. Discussion: DEGREE: 3 SYMMETRY: (X,Y), (-Y,X), (-X,-Y), (Y,-X) POINTS CARDINALITY: 4 NORM INF MOMS. RESIDUAL: 2.22045e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR03[3*4], the requested rule. */ { const int degree = 3; int order; double xw[3*4] = { -5.773502691896257e-01, -5.773502691896257e-01, 1.000000000000000e+00, -5.773502691896257e-01, 5.773502691896257e-01, 1.000000000000000e+00, 5.773502691896257e-01, 5.773502691896257e-01, 1.000000000000000e+00, 5.773502691896257e-01, -5.773502691896257e-01, 1.000000000000000e+00 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr04 ( ) /******************************************************************************/ /* Purpose: SMR04 returns the SMR rule of degree 4. Discussion: DEGREE: 4 POINTS CARDINALITY: 6 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR04[3*6], the requested rule. */ { const int degree = 4; int order; double xw[3*6] = { 9.298664473826397e-01, 6.361197473108544e-02, 4.979283660841867e-01, -7.329012618874027e-01, 5.903145258425608e-01, 6.883081069413867e-01, 3.895446419719248e-01, 8.325323327063485e-01, 6.340849824642651e-01, 5.169294362169509e-01, -8.804002381721473e-01, 4.856847322376568e-01, -5.223159975544114e-02, -1.540167862605936e-01, 1.180646405191258e+00, -7.693563599017555e-01, -6.943687766134327e-01, 5.133474070812475e-01 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr05 ( ) /******************************************************************************/ /* Purpose: SMR05 returns the SMR rule of degree 5. Discussion: DEGREE: 5 SYMMETRY: (X,Y), (-X,-Y) POINTS CARDINALITY: 7 NORM INF MOMS. RESIDUAL: 1.66533e-16 SUM NEGATIVE WEIGHTS: 0 Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR05[3*7], the requested rule. */ { const int degree = 5; int order; double xw[3*7] = { -9.660917830792960e-01, 0.000000000000000e+00, 3.174603174603175e-01, -5.773502691896258e-01, -7.745966692414834e-01, 5.555555555555556e-01, -5.773502691896258e-01, 7.745966692414834e-01, 5.555555555555556e-01, 0.000000000000000e+00, 0.000000000000000e+00, 1.142857142857143e+00, 5.773502691896258e-01, 7.745966692414834e-01, 5.555555555555556e-01, 5.773502691896258e-01, -7.745966692414834e-01, 5.555555555555556e-01, 9.660917830792960e-01, 0.000000000000000e+00, 3.174603174603175e-01 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr06 ( ) /******************************************************************************/ /* Purpose: SMR06 returns the SMR rule of degree 6. Discussion: DEGREE: 6 POINTS CARDINALITY: 10 NORM INF MOMS. RESIDUAL: 9.02056e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR06[3*10], the requested rule. */ { const int degree = 6; int order; double xw[3*10] = { 9.785155476851563e-01, 2.032663758845348e-01, 2.115144101443901e-01, 8.002071174796732e-01, -8.068212789789568e-01, 2.633128596562718e-01, 6.478447271179293e-01, 8.174437151345265e-01, 3.765392584231271e-01, 4.527740405918522e-01, -2.317108088207500e-01, 7.128261121734308e-01, -4.775668731692919e-01, 9.766597077168154e-01, 1.915080888326935e-01, -7.569420164799104e-02, 4.409055401533251e-01, 7.647398979197457e-01, -1.241563248035317e-01, -8.409222499092199e-01, 4.422079284502612e-01, -9.014158913156406e-01, 5.483121720080090e-01, 2.801510395399764e-01, -6.560277628978444e-01, -2.866497529648795e-01, 6.536308666716537e-01, -9.537871517453275e-01, -8.861523430050633e-01, 1.035695381884487e-01 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr07 ( ) /******************************************************************************/ /* Purpose: SMR07 returns the SMR rule of degree 7. Discussion: DEGREE: 7 ROTATIONALLY INVARIANT: (X,Y),(-Y,X),(-X,-Y),(Y,-X). POINTS CARDINALITY: 12 NORM INF MOMS. RESIDUAL: 2.77556e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR07[3*12], the requested rule. */ { const int degree = 7; int order; double xw[3*12] = { 3.938313610187890e-01, -3.669979343404274e-01, 5.209286223023044e-01, 8.019684762065925e-01, 8.099259297580919e-01, 2.374024056137991e-01, -1.832732766424723e-02, 9.259609259310837e-01, 2.416689720838965e-01, 3.669979343404274e-01, 3.938313610187890e-01, 5.209286223023044e-01, -8.099259297580919e-01, 8.019684762065925e-01, 2.374024056137991e-01, -9.259609259310837e-01, -1.832732766424723e-02, 2.416689720838965e-01, -3.938313610187890e-01, 3.669979343404274e-01, 5.209286223023044e-01, -8.019684762065925e-01, -8.099259297580919e-01, 2.374024056137991e-01, 1.832732766424723e-02, -9.259609259310837e-01, 2.416689720838965e-01, -3.669979343404274e-01, -3.938313610187890e-01, 5.209286223023044e-01, 8.099259297580919e-01, -8.019684762065925e-01, 2.374024056137991e-01, 9.259609259310837e-01, 1.832732766424723e-02, 2.416689720838965e-01 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr08 ( ) /******************************************************************************/ /* Purpose: SMR08 returns the SMR rule of degree 8. Discussion: R.COOLS SAYS THERE IS A RULE WITH 15 PTS. DEGREE: 8 POINTS CARDINALITY: 16 NORM INF MOMS. RESIDUAL: 5.27356e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR08[3*16], the requested rule. */ { const int degree = 8; int order; double xw[3*16] = { 9.200233979246099e-01, -1.061811254185043e-01, 2.066851970823297e-01, 8.930854460206108e-01, 7.538456868810103e-01, 1.637084884729354e-01, 6.675369061590991e-01, -6.602769863240182e-01, 3.448725288612837e-01, 9.736841855611369e-01, -9.418906397956112e-01, 4.154990098574050e-02, 4.634059670545806e-01, 9.713146222867728e-01, 1.207701070516287e-01, 5.257251055722159e-01, 3.284384783969826e-01, 4.761612149298055e-01, -3.171906088585705e-01, 8.431651719683217e-01, 8.714564872094989e-02, -6.189113622341841e-02, 6.853227004655966e-01, 3.755078795520529e-01, 6.720825424065331e-02, -2.654137657769545e-01, 5.939493422195995e-01, 1.957478656950412e-01, -9.531380985058940e-01, 1.576823081145121e-01, -7.280591295595631e-01, 9.245623167165428e-01, 1.261748042046440e-01, -5.538775451118830e-01, 1.849451143123379e-01, 4.968422862065046e-01, -4.448446986979669e-01, -7.417922043114903e-01, 3.611024723493892e-01, -9.421081322947572e-01, 5.652760298758271e-01, 1.413572491732567e-01, -8.900106969566213e-01, -3.626326971963237e-01, 2.363188753269229e-01, -8.841558907294489e-01, -9.418511077281306e-01, 7.017169674844463e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr09 ( ) /******************************************************************************/ /* Purpose: SMR09 returns the SMR rule of degree 9. Discussion: DEGREE: 9 ROTATIONALLY INVARIANT: (X,Y),(-Y,X),(-X,-Y),(Y,-X).- POINTS CARDINALITY: 17 NORM INF MOMS. RESIDUAL: 2.77556e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR09[3*17], the requested rule. */ { const int degree = 9; int order; double xw[3*17] = { -6.306801197316687e-01, 9.688499663619777e-01, 8.887937817019871e-02, 4.533398211356471e-01, -5.237358202144294e-01, 3.982824392620701e-01, 8.526157293336624e-01, -7.620832819261733e-02, 2.690513376397807e-01, -7.502770999789007e-01, -9.279616459595696e-01, 1.120996021295965e-01, -9.688499663619777e-01, -6.306801197316687e-01, 8.887937817019871e-02, 5.237358202144294e-01, 4.533398211356471e-01, 3.982824392620701e-01, 7.620832819261733e-02, 8.526157293336624e-01, 2.690513376397807e-01, 9.279616459595696e-01, -7.502770999789007e-01, 1.120996021295965e-01, 6.306801197316687e-01, -9.688499663619777e-01, 8.887937817019871e-02, -4.533398211356471e-01, 5.237358202144294e-01, 3.982824392620701e-01, -8.526157293336624e-01, 7.620832819261733e-02, 2.690513376397807e-01, 7.502770999789007e-01, 9.279616459595696e-01, 1.120996021295965e-01, 9.688499663619777e-01, 6.306801197316687e-01, 8.887937817019871e-02, -5.237358202144294e-01, -4.533398211356471e-01, 3.982824392620701e-01, -7.620832819261733e-02, -8.526157293336624e-01, 2.690513376397807e-01, -9.279616459595696e-01, 7.502770999789007e-01, 1.120996021295965e-01, 0.000000000000000e+00, 0.000000000000000e+00, 5.267489711934157e-01 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr10 ( ) /******************************************************************************/ /* Purpose: SMR10 returns the SMR rule of degree 10. Discussion: DEGREE: 10 POINTS CARDINALITY: 22 NORM INF MOMS. RESIDUAL: 3.88578e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR10[3*22], the requested rule. */ { const int degree = 10; int order; double xw[3*22] = { 9.406185571992117e-01, -3.188453596839293e-01, 1.198415853239125e-01, 9.817119264047970e-01, 6.258661935323968e-01, 6.326927276111106e-02, 8.117151060164873e-01, 9.246684242905354e-01, 8.868620221697536e-02, 7.788254159831852e-01, 2.105273891482154e-01, 2.658711347712608e-01, 6.476354842626755e-01, -6.474946981752544e-01, 2.305700455337005e-01, 9.225481682574119e-01, -8.792348043990323e-01, 6.253794118755233e-02, 1.142951736422380e-01, 9.492191314088700e-01, 1.186176720746597e-01, 4.711392149070169e-01, 6.666473305982111e-01, 2.839584221827897e-01, 3.507267260891899e-01, -1.844717206212197e-01, 3.881144740244089e-01, 6.240243795898468e-01, -9.853833119314599e-01, 3.338712924707296e-02, -3.924748753960961e-01, 7.631114939243835e-01, 2.582793941034283e-01, -3.211002312038658e-02, 3.187935759364070e-01, 4.082419772615457e-01, -1.073322786510871e-01, -6.188661913929927e-01, 3.149887822123112e-01, 2.065013461988724e-01, -9.119588710357345e-01, 1.346097738619806e-01, -7.529656324799602e-01, 9.707183739677747e-01, 5.911019515035110e-02, -7.605065507139738e-01, 3.663139167806791e-01, 2.604239768191683e-01, -4.732489884927658e-01, -1.655785251003833e-01, 3.717176493089616e-01, -5.089131904296067e-01, -9.215290755789827e-01, 1.333773119224012e-01, -9.538019223425510e-01, 7.551269206143554e-01, 7.180589876051692e-02, -9.663420836873585e-01, -1.043123255663638e-01, 9.794042948413186e-02, -8.103749226019182e-01, -6.115967830349249e-01, 2.001832062027751e-01, -9.577495916000752e-01, -9.262105001258389e-01, 3.446752558898364e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr11 ( ) /******************************************************************************/ /* Purpose: SMR11 returns the SMR rule of degree 11. Discussion: DEGREE: 11 ROTATIONALLY INVARIANT: (X,Y),(-Y,X),(-X,-Y),(Y,-X). POINTS CARDINALITY: 24 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR11[3*24], the requested rule. */ { const int degree = 11; int order; double xw[3*24] = { -4.165807191202228e-02, 4.248472488486693e-01, 3.511587183982455e-01, 3.156234329152542e-01, 8.125205483048132e-01, 2.117363499989486e-01, 7.120019130753363e-01, 5.253202503645478e-01, 2.256260617288634e-01, 6.980761045495677e-01, 9.826392235408555e-01, 4.802076335072379e-02, 9.535395282015321e-01, 1.885861387186419e-01, 9.738677735866817e-02, 9.394863828167369e-01, 8.257758359029640e-01, 6.607132916455061e-02, -4.248472488486693e-01, -4.165807191202228e-02, 3.511587183982455e-01, -8.125205483048132e-01, 3.156234329152542e-01, 2.117363499989486e-01, -5.253202503645478e-01, 7.120019130753363e-01, 2.256260617288634e-01, -9.826392235408555e-01, 6.980761045495677e-01, 4.802076335072379e-02, -1.885861387186419e-01, 9.535395282015321e-01, 9.738677735866817e-02, -8.257758359029640e-01, 9.394863828167369e-01, 6.607132916455061e-02, 4.165807191202228e-02, -4.248472488486693e-01, 3.511587183982455e-01, -3.156234329152542e-01, -8.125205483048132e-01, 2.117363499989486e-01, -7.120019130753363e-01, -5.253202503645478e-01, 2.256260617288634e-01, -6.980761045495677e-01, -9.826392235408555e-01, 4.802076335072379e-02, -9.535395282015321e-01, -1.885861387186419e-01, 9.738677735866817e-02, -9.394863828167369e-01, -8.257758359029640e-01, 6.607132916455061e-02, 4.248472488486693e-01, 4.165807191202228e-02, 3.511587183982455e-01, 8.125205483048132e-01, -3.156234329152542e-01, 2.117363499989486e-01, 5.253202503645478e-01, -7.120019130753363e-01, 2.256260617288634e-01, 9.826392235408555e-01, -6.980761045495677e-01, 4.802076335072379e-02, 1.885861387186419e-01, -9.535395282015321e-01, 9.738677735866817e-02, 8.257758359029640e-01, -9.394863828167369e-01, 6.607132916455061e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr12 ( ) /******************************************************************************/ /* Purpose: SMR12 returns the SMR rule of degree 12. Discussion: DEGREE: 12 POINTS CARDINALITY: 31 NORM INF MOMS. RESIDUAL: 5.55112e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR12[3*31], the requested rule. */ { const int degree = 12; int order; double xw[3*31] = { 7.887961133518859e-01, 9.788423074444343e-01, 3.643740653079505e-02, 9.636803326457559e-01, -8.325162377112404e-01, 4.156946367307449e-02, 5.727054392308150e-01, 7.717261293782711e-01, 1.680361793533787e-01, 9.634302613611325e-01, 8.254780463742046e-01, 4.273104217362153e-02, 9.850208941660858e-01, 2.929038490713011e-03, 5.527116827932585e-02, 8.590486289231106e-01, -4.568247023502062e-01, 1.528634492877730e-01, 7.730732984501992e-01, -9.683129045193353e-01, 4.413124147353821e-02, -1.779670698164110e-01, 7.009187390097460e-01, 2.328494912954924e-01, 8.543638116305748e-01, 4.518538584893645e-01, 1.556045127005831e-01, 6.291273683568680e-01, -7.545283511153206e-03, 2.625512570459704e-01, 5.528579662023739e-01, -7.445078276040368e-01, 1.822720395257231e-01, 2.838402579450193e-01, 4.114855681818982e-01, 2.766169541326807e-01, -8.461631704059508e-02, 8.864609533086124e-02, 2.383849367209651e-01, 2.425324152553718e-01, -3.742080386362673e-01, 2.920973373876344e-01, 1.732349372327210e-01, -9.397208645575091e-01, 1.020918275656317e-01, 1.711409536609614e-01, 9.405188030525048e-01, 1.014134103497821e-01, -4.125379277895744e-01, 5.932317296236497e-02, 1.455335918991387e-01, -2.114002132094871e-01, -2.840983635497378e-01, 7.946218537051837e-02, -4.914206209696642e-01, 9.453457692968009e-01, 8.467100517264473e-02, -6.078146688229159e-01, 4.486792064962008e-01, 2.211679207393056e-01, -5.774993705610445e-01, -3.758525621312278e-01, 2.131288659567110e-01, -4.943548938878011e-01, -9.487282033931176e-01, 8.106167025374129e-02, -8.307832892042354e-01, 8.056995002918920e-01, 1.095231158786915e-01, -8.780106307148161e-01, 1.661245295800410e-01, 9.555205419558256e-02, -9.855207994154948e-01, -5.172324935881427e-01, 4.485874655537044e-02, -1.843157675090382e-01, -7.106474367504330e-01, 2.242636308727417e-01, -9.639808262111604e-01, 9.935149440412500e-01, 1.153343223604792e-02, -9.905890478558095e-01, 5.280652625496880e-01, 4.024626264818196e-02, -8.720537162767097e-01, -1.329065631973914e-01, 1.058449657495581e-01, -9.263657835203469e-01, -9.372594643356640e-01, 3.401287490362680e-02, -7.769128911427001e-01, -7.339368334159078e-01, 1.242179600721695e-01 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr13 ( ) /******************************************************************************/ /* Purpose: SMR13 returns the SMR rule of degree 13. Discussion: DEGREE: 13 ROTATIONALLY INVARIANT: (X,Y),(-Y,X),(-X,-Y),(Y,-X). POINTS CARDINALITY: 33 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR13[3*33], the requested rule. */ { const int degree = 13; int order; double xw[3*33] = { 4.758086252182755e-01, 8.500766736997488e-01, 1.188446673005955e-01, 3.427165560404069e-01, 4.093045616940387e-01, 2.568707494819678e-01, 7.558053565720811e-01, 6.478216371870110e-01, 1.297635503700028e-01, 9.413272258729251e-01, 3.907362161294611e-01, 7.749273853310545e-02, 9.572976997863074e-01, 8.595560056416388e-01, 3.817442131708364e-02, 7.788097115544195e-01, 9.834866824398723e-01, 2.999183886449914e-02, -7.074150899644469e-02, 6.962500784917494e-01, 2.133415814571894e-01, 1.381834598624650e-01, 9.589251702875350e-01, 6.042492381774981e-02, -8.500766736997488e-01, 4.758086252182755e-01, 1.188446673005955e-01, -4.093045616940387e-01, 3.427165560404069e-01, 2.568707494819678e-01, -6.478216371870110e-01, 7.558053565720811e-01, 1.297635503700028e-01, -3.907362161294611e-01, 9.413272258729251e-01, 7.749273853310545e-02, -8.595560056416388e-01, 9.572976997863074e-01, 3.817442131708364e-02, -9.834866824398723e-01, 7.788097115544195e-01, 2.999183886449914e-02, -6.962500784917494e-01, -7.074150899644469e-02, 2.133415814571894e-01, -9.589251702875350e-01, 1.381834598624650e-01, 6.042492381774981e-02, -4.758086252182755e-01, -8.500766736997488e-01, 1.188446673005955e-01, -3.427165560404069e-01, -4.093045616940387e-01, 2.568707494819678e-01, -7.558053565720811e-01, -6.478216371870110e-01, 1.297635503700028e-01, -9.413272258729251e-01, -3.907362161294611e-01, 7.749273853310545e-02, -9.572976997863074e-01, -8.595560056416388e-01, 3.817442131708364e-02, -7.788097115544195e-01, -9.834866824398723e-01, 2.999183886449914e-02, 7.074150899644469e-02, -6.962500784917494e-01, 2.133415814571894e-01, -1.381834598624650e-01, -9.589251702875350e-01, 6.042492381774981e-02, 8.500766736997488e-01, -4.758086252182755e-01, 1.188446673005955e-01, 4.093045616940387e-01, -3.427165560404069e-01, 2.568707494819678e-01, 6.478216371870110e-01, -7.558053565720811e-01, 1.297635503700028e-01, 3.907362161294611e-01, -9.413272258729251e-01, 7.749273853310545e-02, 8.595560056416388e-01, -9.572976997863074e-01, 3.817442131708364e-02, 9.834866824398723e-01, -7.788097115544195e-01, 2.999183886449914e-02, 6.962500784917494e-01, 7.074150899644469e-02, 2.133415814571894e-01, 9.589251702875350e-01, -1.381834598624650e-01, 6.042492381774981e-02, 0.000000000000000e+00, 0.000000000000000e+00, 3.003821154312253e-01 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr14 ( ) /******************************************************************************/ /* Purpose: SMR14 returns the SMR rule of degree 14. Discussion: DEGREE: 14 POINTS CARDINALITY: 40 NORM INF MOMS. RESIDUAL: 9.57567e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR14[3*40], the requested rule. */ { const int degree = 14; int order; double xw[3*40] = { 9.784247248091626e-01, 9.849379037345679e-01, 7.042609290974809e-03, 9.696033367128390e-01, -3.331633352358324e-01, 5.127000401619231e-02, 9.635907900433699e-01, -8.235135827696192e-01, 2.835904116834926e-02, 9.609680853689336e-01, -9.665848054209388e-01, 5.802552341290735e-03, 8.759175244824323e-01, 8.440519016975554e-01, 6.406839799781101e-02, 9.850622977861553e-01, 5.927006882272775e-01, 3.059827453183588e-02, 9.132378164262995e-01, 1.686644107446046e-01, 1.015293304177829e-01, 8.210255575251280e-01, -6.312138484114799e-01, 1.070566792196576e-01, 7.253802697646771e-01, 5.271299424251080e-01, 1.491850201682654e-01, 7.129972234617307e-01, -1.746725364206650e-01, 1.672674729613243e-01, 5.551781984544302e-01, -8.617677569495097e-01, 1.035166248728787e-01, 8.047627946516351e-01, -9.674432786379303e-01, 2.787008138269169e-02, 6.424373183439024e-01, 9.660008737161317e-01, 4.265704272492368e-02, 4.202620588036903e-01, 7.809596116887254e-01, 1.425075466703280e-01, 4.547166615470812e-01, 1.927401717576898e-01, 2.105516162791665e-01, 4.259990143599163e-01, -4.962208715768625e-01, 1.867185172918136e-01, 9.710021027954509e-02, 9.536977082453567e-01, 6.972182442589203e-02, 1.114008919498024e-01, 4.923616289896904e-01, 2.043948618313033e-01, 1.209667130758268e-01, -1.503469100876254e-01, 2.335930539280487e-01, 7.524097400070873e-02, -7.452253206810735e-01, 1.601269026668343e-01, -4.518656196438389e-01, 9.427795716582286e-01, 6.465270834286081e-02, -2.040271575380492e-01, 7.591224031418166e-01, 1.466929221365241e-01, -2.427862218874625e-01, 1.736708825032474e-01, 2.280281871073875e-01, -2.459782585741061e-01, -4.449955936922590e-01, 2.055272577034085e-01, 2.124852450722984e-01, -9.810732833242932e-01, 4.316769231929027e-02, -7.568108239381779e-01, 7.901122315570046e-01, 9.576721669303179e-02, -5.448202693114659e-01, 5.034324548840123e-01, 1.739315171493671e-01, -5.908385723762482e-01, -1.268935867110808e-01, 1.941490024297708e-01, -2.895046302710989e-01, -9.116133911949098e-01, 9.663345199603172e-02, -6.417036777526361e-01, -9.886715561256082e-01, 2.266331268661072e-02, -7.921595150813847e-01, 9.955322518565568e-01, 1.393382316893550e-02, -9.527313016559716e-01, 5.688243284550245e-01, 5.657603379663307e-02, -8.300912488610223e-01, 2.262029630794567e-01, 1.327223448878936e-01, -5.758839055540124e-01, -6.756366015563920e-01, 1.369481741372994e-01, -7.954880187581953e-01, -8.506684869408652e-01, 6.033321912862107e-02, -9.562007514872694e-01, 9.139906777934969e-01, 2.419546581425405e-02, -9.839876523259704e-01, -5.381597268716472e-02, 4.116110545021220e-02, -8.552518042944189e-01, -4.002028380944249e-01, 1.164178174263354e-01, -9.694591723778574e-01, -6.823556992833991e-01, 3.472756253802061e-02, -9.468282975367817e-01, -9.506255113908446e-01, 1.793373090014833e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr15 ( ) /******************************************************************************/ /* Purpose: SMR15 returns the SMR rule of degree 15. Discussion: DEGREE: 15 SYMMETRY: (X,Y), (-X,-Y). POINTS CARDINALITY: 43 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR15[3*43], the requested rule. */ { const int degree = 15; int order; double xw[3*43] = { -9.897035809317545e-01, 5.425907314742811e-01, 2.692004637463620e-02, -9.844256804341962e-01, -8.630433434831161e-01, 1.749593663841038e-02, -9.720740612950027e-01, -3.529770957215327e-01, 4.407397330060381e-02, -9.717518222429252e-01, 9.683214017531246e-01, 9.597984117068133e-03, -9.192734438999380e-01, 1.159789159156424e-01, 9.186688636374692e-02, -8.920718007989080e-01, 8.096447739678093e-01, 6.009039847791869e-02, -8.694522415973330e-01, -9.761961093345928e-01, 2.099677952515304e-02, -8.685236892209494e-01, -6.756459657106383e-01, 7.822899155683208e-02, -7.630389610522654e-01, -4.519651830370793e-01, 4.323794116684067e-02, -7.606746138826611e-01, 4.797962173343035e-01, 1.341465202538838e-01, -7.151716310198851e-01, -1.863833474025069e-01, 1.456054555637575e-01, -6.987339962580051e-01, 9.595571077523438e-01, 4.226674647754744e-02, -6.348443155921265e-01, -8.760006464754001e-01, 8.606581576391330e-02, -5.034250302130044e-01, 7.544476347964910e-01, 1.336407358940195e-01, -4.795655397342558e-01, 1.795809740683987e-01, 2.029010677239986e-01, -4.639944974152491e-01, -5.789694673013793e-01, 1.560859171904400e-01, -3.244129251598891e-01, -9.815797807964038e-01, 3.669707979316125e-02, -2.413326601845606e-01, -2.562094364546668e-01, 1.805943476814126e-01, -2.289847857525782e-01, 9.462454193863400e-01, 7.000525056015877e-02, -1.648117901013778e-01, 5.076930665870906e-01, 2.008843851797360e-01, -1.220776156497820e-01, -8.000225811280368e-01, 1.383276061021405e-01, 0.000000000000000e+00, 0.000000000000000e+00, 1.605402685892418e-01, 1.220776156497820e-01, 8.000225811280368e-01, 1.383276061021405e-01, 1.648117901013778e-01, -5.076930665870906e-01, 2.008843851797360e-01, 2.289847857525782e-01, -9.462454193863400e-01, 7.000525056015877e-02, 2.413326601845606e-01, 2.562094364546668e-01, 1.805943476814126e-01, 3.244129251598891e-01, 9.815797807964038e-01, 3.669707979316125e-02, 4.639944974152491e-01, 5.789694673013793e-01, 1.560859171904400e-01, 4.795655397342558e-01, -1.795809740683987e-01, 2.029010677239986e-01, 5.034250302130044e-01, -7.544476347964910e-01, 1.336407358940195e-01, 6.348443155921265e-01, 8.760006464754001e-01, 8.606581576391330e-02, 6.987339962580051e-01, -9.595571077523438e-01, 4.226674647754744e-02, 7.151716310198851e-01, 1.863833474025069e-01, 1.456054555637575e-01, 7.606746138826611e-01, -4.797962173343035e-01, 1.341465202538838e-01, 7.630389610522654e-01, 4.519651830370793e-01, 4.323794116684067e-02, 8.685236892209494e-01, 6.756459657106383e-01, 7.822899155683208e-02, 8.694522415973330e-01, 9.761961093345928e-01, 2.099677952515304e-02, 8.920718007989080e-01, -8.096447739678093e-01, 6.009039847791869e-02, 9.192734438999380e-01, -1.159789159156424e-01, 9.186688636374692e-02, 9.717518222429252e-01, -9.683214017531246e-01, 9.597984117068133e-03, 9.720740612950027e-01, 3.529770957215327e-01, 4.407397330060381e-02, 9.844256804341962e-01, 8.630433434831161e-01, 1.749593663841038e-02, 9.897035809317545e-01, -5.425907314742811e-01, 2.692004637463620e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr16 ( ) /******************************************************************************/ /* Purpose: SMR16 returns the SMR rule of degree 16. Discussion: DEGREE: 16 POINTS CARDINALITY: 52 NORM INF MOMS. RESIDUAL: 9.15934e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR16[3*52], the requested rule. */ { const int degree = 16; int order; double xw[3*52] = { 9.849840927490825e-01, -8.331548011067518e-01, 1.148497840181236e-02, 9.872657987157976e-01, 8.588448788294810e-01, 1.473194969200236e-02, 9.793426944857834e-01, 3.488215241207180e-01, 3.098444037686296e-02, 9.710730712462519e-01, -1.976275486943358e-01, 3.939075140298535e-02, 9.543665314722640e-01, -6.214859681775430e-01, 3.181988181461071e-02, 8.967156835567945e-01, 6.357233965740013e-01, 6.754726963166503e-02, 8.560787402043915e-01, 9.300163636799799e-02, 9.806208102265382e-02, 8.352602163787974e-01, -4.546303275105074e-01, 8.007156600306299e-02, 9.242704749020406e-01, -9.624752146350085e-01, 1.773036206031470e-02, 8.914637296434523e-01, 9.704982463181010e-01, 1.966130427988303e-02, 7.225506553250621e-01, 8.352620376414029e-01, 7.379440929879094e-02, 6.725198105909391e-01, 3.980674436765583e-01, 1.347645483339404e-01, 6.371105617029685e-01, -1.999912055384129e-01, 1.423112416621634e-01, 7.808910471981420e-01, -8.134492991124070e-01, 7.085421872868532e-02, 6.337062775345687e-01, -9.719980099330250e-01, 2.925279318090791e-02, 5.291062928070972e-01, 9.679384351925090e-01, 3.562931685878622e-02, 4.244965002466626e-01, 6.479593280988540e-01, 1.310143500409354e-01, 3.773110188906150e-01, 1.020856639775450e-01, 1.867695133142744e-01, 5.192633407488266e-01, -6.187964471903282e-01, 1.329295627808754e-01, 3.642312437714700e-01, -8.863235986829771e-01, 7.576971496593095e-02, 5.561745660244605e-02, 9.824987512094673e-01, 2.458923517607215e-02, 2.315339247267523e-01, 8.666994462444320e-01, 7.856108542290755e-02, 7.921301678093465e-02, 3.983722019927857e-01, 1.821994092983056e-01, 2.060553068006480e-01, -3.586293462932311e-01, 1.877843167547426e-01, 5.968748053306436e-02, -7.377671587273356e-01, 1.194943798093123e-01, 9.345890458174631e-02, -9.856937685796492e-01, 2.683866060217570e-02, -2.309174536279237e-01, 9.038233934314195e-01, 5.704293324973451e-02, -8.490129042248308e-02, 7.103616948287095e-01, 1.008414737624086e-01, -1.156391657288008e-01, -5.831278342511520e-02, 2.065290952798645e-01, -2.469994669672659e-01, -5.451953129096770e-01, 1.486152719982061e-01, -2.407569095414160e-01, -9.110001765460239e-01, 7.369448825579748e-02, -5.483869894552442e-01, -9.876593748951622e-01, 2.052629805335149e-02, -4.504247208566849e-01, 9.823752262299734e-01, 2.182431901411460e-02, -3.866081747635596e-01, 6.227238096470654e-01, 9.969926465444570e-02, -3.830201431115215e-01, 2.758592684399684e-01, 1.620659760964653e-01, -5.052430403406701e-01, -2.965505634060009e-01, 1.515331269600610e-01, -5.534470349649314e-01, -7.802045621704854e-01, 9.917678556230886e-02, -8.118615876869535e-01, -9.295671133984095e-01, 4.092795912351316e-02, -6.110904286470239e-01, 8.416598154470650e-01, 7.681751936941439e-02, -7.180850315887866e-01, 5.275124303877409e-01, 1.078772215341901e-01, -6.717184597177530e-01, 5.190959173969474e-02, 1.274953461648875e-01, -7.688067943153316e-01, -5.765647129796945e-01, 9.670064900592909e-02, -8.122178392749709e-01, 9.667410754993326e-01, 2.507418689814922e-02, -9.001198519620254e-01, 7.660124377216176e-01, 5.337920648061494e-02, -8.927917900428630e-01, 2.801343969829519e-01, 7.365567660310912e-02, -8.625316023871907e-01, -2.304828850740908e-01, 7.976422028192462e-02, -9.444379963884221e-01, -7.886904781685101e-01, 3.680993075145274e-02, -9.814383394418308e-01, 9.386107932617476e-01, 1.041932477487203e-02, -9.868302403965578e-01, 5.376625218147606e-01, 2.163454704946022e-02, -9.782564506674100e-01, -7.629524842176458e-03, 2.926520573014411e-02, -9.734724345230769e-01, -4.774981225500561e-01, 2.926793258824966e-02, -9.832450869161593e-01, -9.836383787047436e-01, 5.320699842671481e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr17 ( ) /******************************************************************************/ /* Purpose: SMR17 returns the SMR rule of degree 17. Discussion: DEGREE: 17 SYMMETRY (X,Y), (-X,-Y) POINTS CARDINALITY: 54 NORM INF MOMS. RESIDUAL: 4.44089e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR17[3*54], the requested rule. */ { const int degree = 17; int order; double xw[3*54] = { -9.863407162001064e-01, -5.350413962053188e-01, 2.305601467067279e-02, -9.797917920771685e-01, 9.250929106098536e-02, 3.240673241115238e-02, -9.759748695590249e-01, -9.216725954273451e-01, 1.270290274356204e-02, -9.751604610540874e-01, 8.226615823534071e-01, 6.967658416756498e-03, -9.655463590660567e-01, 5.873038194012039e-01, 3.379494459982002e-02, -9.580417051272648e-01, 9.392146008421756e-01, 1.490471492599059e-02, -8.990410418369936e-01, -2.442899840300571e-01, 8.052033202871829e-02, -8.929777462525665e-01, -7.628113086068719e-01, 5.184525239733102e-02, -8.684155248367631e-01, -9.940332573157988e-01, 8.888523934900541e-03, -8.543834155553983e-01, 3.348609125819369e-01, 8.942431131283363e-02, -8.198994002699886e-01, 7.861039744617548e-01, 6.602369125585303e-02, -7.927874398654909e-01, 9.867131971602722e-01, 1.519282623398942e-02, -7.313489231885142e-01, -5.375736491397500e-01, 1.043702032541429e-01, -7.090784726108733e-01, -9.201347632293920e-01, 4.542324896856204e-02, -6.903443278598889e-01, 8.463704695540367e-03, 1.345081852973794e-01, -6.159136297254387e-01, 5.598744500532884e-01, 1.235292068763380e-01, -5.503655196818333e-01, 9.111946838818689e-01, 6.486019261315923e-02, -5.088888961572688e-01, -7.826367523208496e-01, 8.485070415264370e-02, -4.701395263110675e-01, -3.185012886935781e-01, 1.454492206755823e-01, -3.956137656607415e-01, 2.488545706442738e-01, 1.612339647041861e-01, -3.787399827823459e-01, -9.766193765657439e-01, 3.251051105156653e-02, -2.831429683388207e-01, 7.382031830574145e-01, 1.224068303196167e-01, -2.468625326731463e-01, -6.309764352559230e-01, 1.169672902517008e-01, -2.225539938942038e-01, 9.838126768040562e-01, 2.855916848841619e-02, -1.626641413140420e-01, -1.008646063203129e-01, 1.626132038726945e-01, -6.874965210282233e-02, -8.853346747431012e-01, 8.682936644584309e-02, -5.847903009448556e-02, 4.510735833781594e-01, 1.501607980965882e-01, 5.847903009448556e-02, -4.510735833781594e-01, 1.501607980965882e-01, 6.874965210282233e-02, 8.853346747431012e-01, 8.682936644584309e-02, 1.626641413140420e-01, 1.008646063203129e-01, 1.626132038726945e-01, 2.225539938942038e-01, -9.838126768040562e-01, 2.855916848841619e-02, 2.468625326731463e-01, 6.309764352559230e-01, 1.169672902517008e-01, 2.831429683388207e-01, -7.382031830574145e-01, 1.224068303196167e-01, 3.787399827823459e-01, 9.766193765657439e-01, 3.251051105156653e-02, 3.956137656607415e-01, -2.488545706442738e-01, 1.612339647041861e-01, 4.701395263110675e-01, 3.185012886935781e-01, 1.454492206755823e-01, 5.088888961572688e-01, 7.826367523208496e-01, 8.485070415264370e-02, 5.503655196818333e-01, -9.111946838818689e-01, 6.486019261315923e-02, 6.159136297254387e-01, -5.598744500532884e-01, 1.235292068763380e-01, 6.903443278598889e-01, -8.463704695540367e-03, 1.345081852973794e-01, 7.090784726108733e-01, 9.201347632293920e-01, 4.542324896856204e-02, 7.313489231885142e-01, 5.375736491397500e-01, 1.043702032541429e-01, 7.927874398654909e-01, -9.867131971602722e-01, 1.519282623398942e-02, 8.198994002699886e-01, -7.861039744617548e-01, 6.602369125585303e-02, 8.543834155553983e-01, -3.348609125819369e-01, 8.942431131283363e-02, 8.684155248367631e-01, 9.940332573157988e-01, 8.888523934900541e-03, 8.929777462525665e-01, 7.628113086068719e-01, 5.184525239733102e-02, 8.990410418369936e-01, 2.442899840300571e-01, 8.052033202871829e-02, 9.580417051272648e-01, -9.392146008421756e-01, 1.490471492599059e-02, 9.655463590660567e-01, -5.873038194012039e-01, 3.379494459982002e-02, 9.751604610540874e-01, -8.226615823534071e-01, 6.967658416756498e-03, 9.759748695590249e-01, 9.216725954273451e-01, 1.270290274356204e-02, 9.797917920771685e-01, -9.250929106098536e-02, 3.240673241115238e-02, 9.863407162001064e-01, 5.350413962053188e-01, 2.305601467067279e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr18 ( ) /******************************************************************************/ /* Purpose: SMR18 returns the SMR rule of degree 18. Discussion: DEGREE: 18 POINTS CARDINALITY: 64 NORM INF MOMS. RESIDUAL: 1.80411e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR18[3*64], the requested rule. */ { const int degree = 18; int order; double xw[3*64] = { -9.901134796801783e-01, -3.636920739279686e-01, 1.924401162821098e-02, -9.813479194657442e-01, 2.269728889226950e-01, 2.377081803411145e-02, -9.751246202174101e-01, -8.087895536063188e-01, 1.824571677716675e-02, -9.741192020631542e-01, 8.966914512186399e-01, 1.040325999881736e-02, -9.699879010686473e-01, -9.814171856874379e-01, 5.169228082043371e-03, -9.695988425630034e-01, 6.372691271564769e-01, 2.701864190192290e-02, -9.502627815202467e-01, 9.663374399441544e-01, 6.832430188474440e-03, -9.197723058789439e-01, -5.864804338113919e-02, 5.855723621166473e-02, -9.059247381128510e-01, -5.898828597989898e-01, 5.297590163131558e-02, -8.627599153540467e-01, 4.164987829940056e-01, 7.111515467482395e-02, -8.569855990839045e-01, -9.267827643653366e-01, 2.894383892650155e-02, -8.389143776305836e-01, 8.004661957859885e-01, 5.373522067458378e-02, -8.065675356071547e-01, 9.834437446052690e-01, 1.315188692146605e-02, -7.872448206722253e-01, -2.985727024319632e-01, 8.409595266673971e-02, -7.325897102338846e-01, -7.627707340005139e-01, 6.493193699070636e-02, -7.108608115543918e-01, 1.482798927069730e-01, 1.059543276308938e-01, -6.620192350608188e-01, -9.914865774170796e-01, 1.356692677046532e-02, -6.334849400427111e-01, 6.037434357654228e-01, 1.051942339342666e-01, -6.014705998319453e-01, 9.143204709310846e-01, 5.210486977766542e-02, -5.892324845704049e-01, -5.045342974742788e-01, 9.788981223530169e-02, -5.131040457735190e-01, -9.095715109599031e-02, 1.108235515488634e-01, -5.033556139567553e-01, -8.956709590819707e-01, 5.541443510056418e-02, -3.947567656009430e-01, 3.383853875972697e-01, 1.469566677471172e-01, -3.365255254825216e-01, -6.932064968553141e-01, 9.848736935036356e-02, -3.344750516678310e-01, 9.866136919798809e-01, 2.170786884754973e-02, -3.310287403968845e-01, 7.731192382826196e-01, 1.015778188232984e-01, -2.999505641193486e-01, -2.928443015341402e-01, 1.107816184233573e-01, -2.331455368936777e-01, -9.731854050766667e-01, 2.884073562303302e-02, -1.116980244718904e-01, 4.992438237750351e-02, 1.612001429698854e-01, -6.575627688742440e-02, 5.455174655246902e-01, 1.390613750634900e-01, -6.089043840299364e-02, -8.562318836763884e-01, 6.853830062046717e-02, -4.781118177875937e-02, -4.996330916388668e-01, 1.194580692354339e-01, -2.644564050493325e-02, 9.165721602454180e-01, 6.685724630082367e-02, 1.924944168163869e-01, -2.367466035142029e-01, 1.491161554688392e-01, 1.951330272152753e-01, -7.328036825494620e-01, 8.630902364197904e-02, 2.028158693449157e-01, -9.703380537473529e-01, 3.364243858164141e-02, 2.092663468167508e-01, 2.662322825395520e-01, 1.486647112920279e-01, 2.283553896503115e-01, 7.496185156938090e-01, 1.027379035896721e-01, 2.876554348315164e-01, 9.873845509936813e-01, 2.192507210698709e-02, 4.237159887642933e-01, -5.535582535086475e-01, 1.058102193186085e-01, 4.581157947253051e-01, 5.130944233380290e-01, 1.090905648997083e-01, 4.752280776399702e-01, -8.857999997554080e-01, 6.346475038588688e-02, 4.831508023294471e-01, -2.549857293827913e-02, 1.337777655566684e-01, 5.057868104561423e-01, 8.896750734704260e-01, 6.079360942822091e-02, 6.099162517568020e-01, -3.853032051834505e-01, 6.436419858280257e-02, 6.559598836045895e-01, 2.795821152702244e-01, 8.177939511493874e-02, 6.565622107706015e-01, -9.883582556647157e-01, 1.640381218907065e-02, 6.956126158721206e-01, 6.950821057395169e-01, 7.746885102883669e-02, 7.094177605066919e-01, -7.370072009566638e-01, 7.788188331156506e-02, 7.290386435476074e-01, 9.703386432783726e-01, 2.313579562579107e-02, 7.557526341619506e-01, -2.941608525026456e-01, 7.412112987017343e-02, 7.912367791054172e-01, 8.744914899411797e-02, 6.420122545616397e-02, 8.543844628271234e-01, -9.182475827792237e-01, 3.336594260262386e-02, 8.645615995455240e-01, 4.552151496141779e-01, 6.553880387109420e-02, 8.705823452020542e-01, 8.420186142289366e-01, 3.949607694089933e-02, 8.978579920125014e-01, -5.603807009892569e-01, 5.906741413770155e-02, 9.236747719664095e-01, -1.092483343302797e-01, 5.348329279696051e-02, 9.440144649649265e-01, 9.743199890031753e-01, 8.946275110809443e-03, 9.689413373175574e-01, 6.453835143088201e-01, 2.577199176900479e-02, 9.716342103019509e-01, -9.824772013181601e-01, 5.332369479915357e-03, 9.729481716901303e-01, -7.936277261963781e-01, 2.070804573643469e-02, 9.758025156447697e-01, 2.267876624457436e-01, 2.981298882062168e-02, 9.939173935792275e-01, 8.841354641450845e-01, 5.745299494229030e-03, 9.953701290021145e-01, -3.661129988733890e-01, 1.543638847873499e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr19 ( ) /******************************************************************************/ /* Purpose: SMR19 returns the SMR rule of degree 19. Discussion: DEGREE: 19 SYMMETRY: (X,Y), (-X,-Y) POINTS CARDINALITY: 67 NORM INF MOMS. RESIDUAL: 4.44089e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR19[3*67], the requested rule. */ { const int degree = 19; int order; double xw[3*67] = { -9.882811207043626e-01, 5.499680659586386e-02, 2.092785503234627e-02, -9.849643667633750e-01, 6.212597907298936e-01, 1.808240699704512e-02, -9.813999666122014e-01, -4.952365265213790e-01, 2.248743945205191e-02, -9.782104202505005e-01, 9.275801424350509e-01, 1.011830773881234e-02, -9.691016249998836e-01, -9.857365066859964e-01, 4.619328461670566e-03, -9.683241755943918e-01, -8.352830082748723e-01, 1.956137669036511e-02, -9.202706700587867e-01, 3.511851149995373e-01, 5.773555533944277e-02, -9.094971072531579e-01, -2.172757421209814e-01, 6.345799736852624e-02, -8.955283373897494e-01, 7.976174402318889e-01, 3.990477349231469e-02, -8.910382534296201e-01, 9.945369448321910e-01, 6.887461729231832e-03, -8.645217306506207e-01, -6.631996105098346e-01, 5.773686209899585e-02, -8.361991822628865e-01, -9.403589726574770e-01, 2.822764839262689e-02, -7.730414957980034e-01, 8.238831136924889e-02, 1.001138265151715e-01, -7.650572339362595e-01, 5.862972313979000e-01, 7.938451847571938e-02, -7.339165122981658e-01, 9.301444324799357e-01, 3.698855138726674e-02, -7.142591688630336e-01, -4.144687459254282e-01, 9.862108370826694e-02, -6.567593878397442e-01, -8.187456800244232e-01, 6.430400825705974e-02, -5.911965681070577e-01, -9.890895416891479e-01, 1.619432554055879e-02, -5.650612797733545e-01, 3.562777186069158e-01, 1.145600986645687e-01, -5.522558344482561e-01, 7.877180001548971e-01, 7.542302799441891e-02, -5.204497226646213e-01, -1.319110721429938e-01, 1.271745083852966e-01, -4.843104169217222e-01, -6.372481900547796e-01, 7.509621769261574e-02, -4.576710954771651e-01, 9.804344277407575e-01, 2.360353464684936e-02, -3.501819505299977e-01, -9.155258592028724e-01, 5.647374387294429e-02, -3.449909757079617e-01, -5.171112886647338e-01, 6.683630505556533e-02, -3.245616103968320e-01, 6.112729098917052e-01, 1.024013654183961e-01, -2.899888936344285e-01, 1.640917354232377e-01, 1.271636872050067e-01, -2.124604121926062e-01, 8.907682205113933e-01, 6.854209870863227e-02, -2.101863370194021e-01, -2.888406804953518e-01, 1.103357066015591e-01, -2.061044019817992e-01, -7.806540187071476e-01, 3.167739924216664e-02, -6.183281639699825e-02, 4.515274811396188e-01, 1.060858442601201e-01, -5.624698000526564e-02, -7.388749473949220e-01, 8.327955875718944e-02, -4.730013829225772e-02, -9.811237074398366e-01, 2.608831452074966e-02, 0.000000000000000e+00, 0.000000000000000e+00, 1.198105245928960e-01, 4.730013829225772e-02, 9.811237074398366e-01, 2.608831452074966e-02, 5.624698000526564e-02, 7.388749473949220e-01, 8.327955875718944e-02, 6.183281639699825e-02, -4.515274811396188e-01, 1.060858442601201e-01, 2.061044019817992e-01, 7.806540187071476e-01, 3.167739924216664e-02, 2.101863370194021e-01, 2.888406804953518e-01, 1.103357066015591e-01, 2.124604121926062e-01, -8.907682205113933e-01, 6.854209870863227e-02, 2.899888936344285e-01, -1.640917354232377e-01, 1.271636872050067e-01, 3.245616103968320e-01, -6.112729098917052e-01, 1.024013654183961e-01, 3.449909757079617e-01, 5.171112886647338e-01, 6.683630505556533e-02, 3.501819505299977e-01, 9.155258592028724e-01, 5.647374387294429e-02, 4.576710954771651e-01, -9.804344277407575e-01, 2.360353464684936e-02, 4.843104169217222e-01, 6.372481900547796e-01, 7.509621769261574e-02, 5.204497226646213e-01, 1.319110721429938e-01, 1.271745083852966e-01, 5.522558344482561e-01, -7.877180001548971e-01, 7.542302799441891e-02, 5.650612797733545e-01, -3.562777186069158e-01, 1.145600986645687e-01, 5.911965681070577e-01, 9.890895416891479e-01, 1.619432554055879e-02, 6.567593878397442e-01, 8.187456800244232e-01, 6.430400825705974e-02, 7.142591688630336e-01, 4.144687459254282e-01, 9.862108370826694e-02, 7.339165122981658e-01, -9.301444324799357e-01, 3.698855138726674e-02, 7.650572339362595e-01, -5.862972313979000e-01, 7.938451847571938e-02, 7.730414957980034e-01, -8.238831136924889e-02, 1.001138265151715e-01, 8.361991822628865e-01, 9.403589726574770e-01, 2.822764839262689e-02, 8.645217306506207e-01, 6.631996105098346e-01, 5.773686209899585e-02, 8.910382534296201e-01, -9.945369448321910e-01, 6.887461729231832e-03, 8.955283373897494e-01, -7.976174402318889e-01, 3.990477349231469e-02, 9.094971072531579e-01, 2.172757421209814e-01, 6.345799736852624e-02, 9.202706700587867e-01, -3.511851149995373e-01, 5.773555533944277e-02, 9.683241755943918e-01, 8.352830082748723e-01, 1.956137669036511e-02, 9.691016249998836e-01, 9.857365066859964e-01, 4.619328461670566e-03, 9.782104202505005e-01, -9.275801424350509e-01, 1.011830773881234e-02, 9.813999666122014e-01, 4.952365265213790e-01, 2.248743945205191e-02, 9.849643667633750e-01, -6.212597907298936e-01, 1.808240699704512e-02, 9.882811207043626e-01, -5.499680659586386e-02, 2.092785503234627e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr19s ( ) /******************************************************************************/ /* Purpose: SMR19S returns the rotationally invariant SMR rule of degree 19. Discussion: DEGREE: 19 ROTATIONALLY INVARIANT: (X,Y),(-Y,X),(-X,-Y),(Y,-X). POINTS CARDINALITY: 68 NORM INF MOMS. RESIDUAL: 2.63678e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR19S[3*68], the requested rule. */ { const int degree = 19; int order; double xw[3*68] = { 6.772712303303220e-02, 1.091623307294788e-01, 5.494509440087280e-02, 1.911320239476111e-01, 7.684428790753622e-01, 9.924039371581232e-02, 1.293584417049455e-01, 9.843401047067886e-01, 2.298164089064930e-02, 2.653188588974263e-01, 3.187377804093571e-01, 1.356277329376223e-01, 4.805884103058745e-01, 5.883672104526556e-01, 1.092830463202312e-01, 4.353820421144143e-01, 9.253041863610407e-01, 5.332719030254342e-02, 5.327122733511525e-01, 5.307888414194004e-02, 1.342811802398570e-01, 6.817257899596626e-01, 8.006754071171294e-01, 6.668259225153295e-02, 6.897274550636672e-01, 9.904094449990978e-01, 1.399432148751916e-02, 8.960307328244307e-01, 1.256777467976840e-01, 6.752580403681208e-02, 7.189012994269330e-01, 3.612251785232622e-01, 1.014916530440626e-01, 8.636764064784566e-01, 6.273567750914366e-01, 5.639122590523545e-02, 8.689071260438135e-01, 9.214958682077944e-01, 2.840189695206082e-02, 9.851138463638244e-01, 3.506711929883037e-01, 1.440758011223270e-02, 9.604323781459383e-01, 4.868528161195464e-01, 1.812287875530189e-02, 9.757200071559576e-01, 7.955215584636620e-01, 1.826958037918748e-02, 9.734329880957604e-01, 9.798840213938864e-01, 5.026188268466660e-03, -1.091623307294788e-01, 6.772712303303220e-02, 5.494509440087280e-02, -7.684428790753622e-01, 1.911320239476111e-01, 9.924039371581232e-02, -9.843401047067886e-01, 1.293584417049455e-01, 2.298164089064930e-02, -3.187377804093571e-01, 2.653188588974263e-01, 1.356277329376223e-01, -5.883672104526556e-01, 4.805884103058745e-01, 1.092830463202312e-01, -9.253041863610407e-01, 4.353820421144143e-01, 5.332719030254342e-02, -5.307888414194004e-02, 5.327122733511525e-01, 1.342811802398570e-01, -8.006754071171294e-01, 6.817257899596626e-01, 6.668259225153295e-02, -9.904094449990978e-01, 6.897274550636672e-01, 1.399432148751916e-02, -1.256777467976840e-01, 8.960307328244307e-01, 6.752580403681208e-02, -3.612251785232622e-01, 7.189012994269330e-01, 1.014916530440626e-01, -6.273567750914366e-01, 8.636764064784566e-01, 5.639122590523545e-02, -9.214958682077944e-01, 8.689071260438135e-01, 2.840189695206082e-02, -3.506711929883037e-01, 9.851138463638244e-01, 1.440758011223270e-02, -4.868528161195464e-01, 9.604323781459383e-01, 1.812287875530189e-02, -7.955215584636620e-01, 9.757200071559576e-01, 1.826958037918748e-02, -9.798840213938864e-01, 9.734329880957604e-01, 5.026188268466660e-03, -6.772712303303220e-02, -1.091623307294788e-01, 5.494509440087280e-02, -1.911320239476111e-01, -7.684428790753622e-01, 9.924039371581232e-02, -1.293584417049455e-01, -9.843401047067886e-01, 2.298164089064930e-02, -2.653188588974263e-01, -3.187377804093571e-01, 1.356277329376223e-01, -4.805884103058745e-01, -5.883672104526556e-01, 1.092830463202312e-01, -4.353820421144143e-01, -9.253041863610407e-01, 5.332719030254342e-02, -5.327122733511525e-01, -5.307888414194004e-02, 1.342811802398570e-01, -6.817257899596626e-01, -8.006754071171294e-01, 6.668259225153295e-02, -6.897274550636672e-01, -9.904094449990978e-01, 1.399432148751916e-02, -8.960307328244307e-01, -1.256777467976840e-01, 6.752580403681208e-02, -7.189012994269330e-01, -3.612251785232622e-01, 1.014916530440626e-01, -8.636764064784566e-01, -6.273567750914366e-01, 5.639122590523545e-02, -8.689071260438135e-01, -9.214958682077944e-01, 2.840189695206082e-02, -9.851138463638244e-01, -3.506711929883037e-01, 1.440758011223270e-02, -9.604323781459383e-01, -4.868528161195464e-01, 1.812287875530189e-02, -9.757200071559576e-01, -7.955215584636620e-01, 1.826958037918748e-02, -9.734329880957604e-01, -9.798840213938864e-01, 5.026188268466660e-03, 1.091623307294788e-01, -6.772712303303220e-02, 5.494509440087280e-02, 7.684428790753622e-01, -1.911320239476111e-01, 9.924039371581232e-02, 9.843401047067886e-01, -1.293584417049455e-01, 2.298164089064930e-02, 3.187377804093571e-01, -2.653188588974263e-01, 1.356277329376223e-01, 5.883672104526556e-01, -4.805884103058745e-01, 1.092830463202312e-01, 9.253041863610407e-01, -4.353820421144143e-01, 5.332719030254342e-02, 5.307888414194004e-02, -5.327122733511525e-01, 1.342811802398570e-01, 8.006754071171294e-01, -6.817257899596626e-01, 6.668259225153295e-02, 9.904094449990978e-01, -6.897274550636672e-01, 1.399432148751916e-02, 1.256777467976840e-01, -8.960307328244307e-01, 6.752580403681208e-02, 3.612251785232622e-01, -7.189012994269330e-01, 1.014916530440626e-01, 6.273567750914366e-01, -8.636764064784566e-01, 5.639122590523545e-02, 9.214958682077944e-01, -8.689071260438135e-01, 2.840189695206082e-02, 3.506711929883037e-01, -9.851138463638244e-01, 1.440758011223270e-02, 4.868528161195464e-01, -9.604323781459383e-01, 1.812287875530189e-02, 7.955215584636620e-01, -9.757200071559576e-01, 1.826958037918748e-02, 9.798840213938864e-01, -9.734329880957604e-01, 5.026188268466660e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr20 ( ) /******************************************************************************/ /* Purpose: SMR20 returns the SMR rule of degree 20. Discussion: DEGREE: 20 POINTS CARDINALITY: 78 NORM INF MOMS. RESIDUAL: 2.28983e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR20[3*78], the requested rule. */ { const int degree = 20; int order; double xw[3*78] = { -9.915878317634396e-01, -3.766765890053987e-01, 1.499554515729458e-02, -9.848784210298892e-01, 2.360994416770047e-01, 2.023115319845336e-02, -9.825526677091110e-01, -8.035427779709161e-01, 1.282555877962797e-02, -9.823044706287086e-01, 9.079824601518448e-01, 8.347241351837816e-03, -9.779664722705005e-01, 6.550056336206199e-01, 1.862995031692569e-02, -9.749883694415311e-01, -9.764755256224437e-01, 4.686323174122413e-03, -9.385487288532800e-01, 9.860342433350299e-01, 5.797911563640461e-03, -9.382963126427573e-01, -8.801426949855569e-02, 4.706757194633274e-02, -9.249968857981914e-01, -6.085959286922041e-01, 4.012822177863082e-02, -8.999893404040346e-01, 4.529514070355635e-01, 5.107860141818281e-02, -8.920744993132258e-01, -9.198780734342489e-01, 2.159208371061492e-02, -8.839154125809222e-01, 8.143818242625289e-01, 3.594236238111754e-02, -8.255579083847894e-01, -3.654427417348043e-01, 6.695838633949613e-02, -8.150245024284021e-01, 1.358607026009545e-01, 6.745427113661845e-02, -7.845038084549548e-01, 9.550568485953186e-01, 2.204090981614230e-02, -7.824159434580745e-01, -7.878743325718499e-01, 4.748312482965101e-02, -7.695472095330197e-01, -9.958600578952408e-01, 8.095042436189770e-03, -7.297305046678980e-01, 6.478626088426425e-01, 7.099765909663569e-02, -6.917823169027344e-01, 3.281317067480576e-01, 5.185357866003446e-02, -6.707656574849767e-01, -1.664730150689492e-01, 7.925719271981636e-02, -6.531604608468555e-01, -6.061553872260436e-01, 6.838834261564096e-02, -6.077773437763014e-01, 8.732147826609341e-01, 4.367407053165663e-02, -6.066275299755738e-01, -9.326014647185664e-01, 3.595011352210572e-02, -5.368890549246464e-01, 9.928457944663468e-01, 1.126554474822698e-02, -5.316204099083646e-01, 6.576158887431759e-02, 8.069676572542248e-02, -4.990046627082800e-01, 4.435890153276254e-01, 9.122827397472959e-02, -4.796158821615813e-01, -4.431584361510255e-01, 8.384497173647569e-02, -4.453245842342572e-01, -8.121904960479558e-01, 5.866651296620028e-02, -4.339460304187116e-01, 7.546357919895570e-01, 6.052390807315380e-02, -3.360460286836234e-01, -1.779527551281239e-01, 9.606753662405314e-02, -3.163304450629529e-01, -9.789738490980646e-01, 2.421886578475265e-02, -2.973007276058863e-01, 9.384528339104452e-01, 3.945003410423680e-02, -2.583665449486478e-01, -6.746223007740130e-01, 7.803382675359716e-02, -2.518910778242828e-01, 2.047144992617838e-01, 1.190753521874101e-01, -2.350658939126170e-01, 6.176223949714494e-01, 8.152900900964591e-02, -1.249372040555246e-01, -4.012798196143892e-01, 9.737789681157598e-02, -1.071681420632334e-01, 9.911853557002673e-01, 7.840583991172302e-03, -7.990032115620818e-02, -8.881537557485726e-01, 6.417774009554068e-02, -4.818581787472142e-02, 8.368871897464495e-01, 6.958137992100039e-02, -6.302932436761816e-03, 4.427387497778728e-01, 1.046411531707668e-01, 6.282561382377720e-03, -4.735863969094835e-02, 1.283072770780011e-01, 7.678421140871043e-02, -6.515968986198225e-01, 7.386235866198392e-02, 1.223163344818137e-01, -5.049264867963420e-01, 3.280543393245010e-02, 1.476259613184704e-01, 9.693655840355563e-01, 2.767850463294367e-02, 1.518625838305689e-01, -9.802704961935971e-01, 2.416012781979250e-02, 2.026981528589298e-01, 6.904257555809832e-01, 8.975917019163195e-02, 2.329194026833933e-01, 2.327373099667877e-01, 1.156666854710510e-01, 2.594187814838830e-01, -2.856358493697289e-01, 1.124110482665555e-01, 2.798470302423499e-01, -8.056133086386197e-01, 7.131679353184883e-02, 3.814217712474836e-01, 8.883375933116525e-01, 5.600628102613372e-02, 4.444592404902306e-01, 5.043261157258173e-01, 1.006394209690602e-01, 4.558914387037205e-01, -5.329822923030598e-01, 9.411407048103884e-02, 4.614020071813320e-01, 4.486972130091344e-03, 1.101092346172853e-01, 4.685495664357138e-01, -9.368544770988728e-01, 4.082934734266681e-02, 5.560918457946907e-01, 9.862938876987586e-01, 1.658263314370683e-02, 6.138352275991088e-01, 7.500124042821561e-01, 7.207159699616439e-02, 6.168426991815920e-01, -7.309006089346149e-01, 6.388336371014162e-02, 6.496172646332740e-01, -2.465046206462174e-01, 9.502316165822158e-02, 6.597659995735623e-01, 2.810856634851854e-01, 9.635446672099211e-02, 6.758048936769788e-01, -9.975229702582141e-01, 8.901892537462009e-03, 7.526986651007189e-01, -8.790172927450672e-01, 3.744847919043556e-02, 7.612710727469121e-01, 9.187271298549152e-01, 3.541134799086990e-02, 7.917231830960033e-01, -4.936809625913063e-01, 6.733975809456394e-02, 8.045897042890074e-01, 5.573763685706831e-01, 6.859164893126736e-02, 8.267194449217807e-01, 2.268030189899398e-02, 7.832191919270795e-02, 8.905802715963324e-01, -9.579826342974288e-01, 1.558785638304684e-02, 8.926961921705238e-01, -7.092071287453895e-01, 3.889483241199626e-02, 9.008369589898633e-01, 9.865275306057119e-01, 7.998523354461931e-03, 9.055629673123163e-01, 7.845529526667002e-01, 3.438925925800883e-02, 9.323802074714704e-01, -2.645795079832657e-01, 4.838142723685821e-02, 9.345858376931122e-01, 3.200026555755083e-01, 4.824416289256322e-02, 9.738506914356970e-01, -8.469006629251953e-01, 1.394893863211126e-02, 9.755796434358871e-01, 8.802225242154167e-01, 6.292208457909152e-03, 9.843147800221137e-01, -5.383749684018156e-01, 1.737917079753502e-02, 9.849316953313286e-01, 9.464841822891732e-01, 4.298012753035497e-03, 9.851840541929131e-01, 6.021648971571338e-01, 1.660764544264520e-02, 9.870587239914860e-01, -9.834157626132080e-01, 2.426383339927545e-03, 9.921802352398980e-01, 2.608826889643944e-02, 1.626098469219482e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr21 ( ) /******************************************************************************/ /* Purpose: SMR21 returns the SMR rule of degree 21. Discussion: DEGREE: 21 ROTATIONALLY INVARIANT: (X, Y), (-Y, X), (-X, -Y), (Y, -X). POINTS CARDINALITY: 81 NORM INF MOMS. RESIDUAL: 4.44089e-15 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR21[3*81], the requested rule. */ { const int degree = 21; int order; double xw[3*81] = { 2.659786054558319e-01, 2.472943466792205e-01, 1.257081274315271e-01, 2.966299561078732e-01, 6.837659120145181e-01, 9.195620074644660e-02, 3.476456848560975e-01, 9.391305012893792e-01, 3.823913002575823e-02, 5.204899365134467e-01, 4.701056221362075e-01, 9.955659700462254e-02, 5.565331479249288e-01, 8.299687476559192e-01, 5.711233679557120e-02, 5.608625825041144e-01, 9.938235381537796e-01, 8.125409684532186e-03, 7.107545267842089e-01, 2.218786231241980e-01, 8.989795633682901e-02, 7.323028382034139e-01, 6.675759613958541e-01, 6.285712965516167e-02, 7.248337121391071e-01, 9.556008374660931e-01, 2.068885521020362e-02, 8.678958423780866e-01, 4.595394054487673e-01, 5.463376630254965e-02, 8.501142309217191e-01, 8.692220828069680e-01, 2.950403142043507e-02, 9.142859758630335e-01, 9.852274923275850e-01, 6.850217129206400e-03, 9.537223544012101e-01, 2.050108803358328e-01, 3.630414793101230e-02, 9.934425198626209e-01, 5.102048128337472e-01, 1.110556726680624e-02, 9.448573049959050e-01, 7.218738735054576e-01, 2.723130314700101e-02, 9.865716562931908e-01, 8.882998675210052e-01, 6.165546352541271e-03, 9.790570750021207e-01, 9.587134606894607e-01, 2.900715554613127e-03, 1.944817128807972e-02, 4.950400647800665e-01, 1.165426493105556e-01, 5.600805589569329e-02, 8.501230581964183e-01, 6.754741413634147e-02, 1.000588448960449e-01, 9.908677016040155e-01, 1.332984242008183e-02, -2.472943466792205e-01, 2.659786054558319e-01, 1.257081274315271e-01, -6.837659120145181e-01, 2.966299561078732e-01, 9.195620074644660e-02, -9.391305012893792e-01, 3.476456848560975e-01, 3.823913002575823e-02, -4.701056221362075e-01, 5.204899365134467e-01, 9.955659700462254e-02, -8.299687476559192e-01, 5.565331479249288e-01, 5.711233679557120e-02, -9.938235381537796e-01, 5.608625825041144e-01, 8.125409684532186e-03, -2.218786231241980e-01, 7.107545267842089e-01, 8.989795633682901e-02, -6.675759613958541e-01, 7.323028382034139e-01, 6.285712965516167e-02, -9.556008374660931e-01, 7.248337121391071e-01, 2.068885521020362e-02, -4.595394054487673e-01, 8.678958423780866e-01, 5.463376630254965e-02, -8.692220828069680e-01, 8.501142309217191e-01, 2.950403142043507e-02, -9.852274923275850e-01, 9.142859758630335e-01, 6.850217129206400e-03, -2.050108803358328e-01, 9.537223544012101e-01, 3.630414793101230e-02, -5.102048128337472e-01, 9.934425198626209e-01, 1.110556726680624e-02, -7.218738735054576e-01, 9.448573049959050e-01, 2.723130314700101e-02, -8.882998675210052e-01, 9.865716562931908e-01, 6.165546352541271e-03, -9.587134606894607e-01, 9.790570750021207e-01, 2.900715554613127e-03, -4.950400647800665e-01, 1.944817128807972e-02, 1.165426493105556e-01, -8.501230581964183e-01, 5.600805589569329e-02, 6.754741413634147e-02, -9.908677016040155e-01, 1.000588448960449e-01, 1.332984242008183e-02, -2.659786054558319e-01, -2.472943466792205e-01, 1.257081274315271e-01, -2.966299561078732e-01, -6.837659120145181e-01, 9.195620074644660e-02, -3.476456848560975e-01, -9.391305012893792e-01, 3.823913002575823e-02, -5.204899365134467e-01, -4.701056221362075e-01, 9.955659700462254e-02, -5.565331479249288e-01, -8.299687476559192e-01, 5.711233679557120e-02, -5.608625825041144e-01, -9.938235381537796e-01, 8.125409684532186e-03, -7.107545267842089e-01, -2.218786231241980e-01, 8.989795633682901e-02, -7.323028382034139e-01, -6.675759613958541e-01, 6.285712965516167e-02, -7.248337121391071e-01, -9.556008374660931e-01, 2.068885521020362e-02, -8.678958423780866e-01, -4.595394054487673e-01, 5.463376630254965e-02, -8.501142309217191e-01, -8.692220828069680e-01, 2.950403142043507e-02, -9.142859758630335e-01, -9.852274923275850e-01, 6.850217129206400e-03, -9.537223544012101e-01, -2.050108803358328e-01, 3.630414793101230e-02, -9.934425198626209e-01, -5.102048128337472e-01, 1.110556726680624e-02, -9.448573049959050e-01, -7.218738735054576e-01, 2.723130314700101e-02, -9.865716562931908e-01, -8.882998675210052e-01, 6.165546352541271e-03, -9.790570750021207e-01, -9.587134606894607e-01, 2.900715554613127e-03, -1.944817128807972e-02, -4.950400647800665e-01, 1.165426493105556e-01, -5.600805589569329e-02, -8.501230581964183e-01, 6.754741413634147e-02, -1.000588448960449e-01, -9.908677016040155e-01, 1.332984242008183e-02, 2.472943466792205e-01, -2.659786054558319e-01, 1.257081274315271e-01, 6.837659120145181e-01, -2.966299561078732e-01, 9.195620074644660e-02, 9.391305012893792e-01, -3.476456848560975e-01, 3.823913002575823e-02, 4.701056221362075e-01, -5.204899365134467e-01, 9.955659700462254e-02, 8.299687476559192e-01, -5.565331479249288e-01, 5.711233679557120e-02, 9.938235381537796e-01, -5.608625825041144e-01, 8.125409684532186e-03, 2.218786231241980e-01, -7.107545267842089e-01, 8.989795633682901e-02, 6.675759613958541e-01, -7.323028382034139e-01, 6.285712965516167e-02, 9.556008374660931e-01, -7.248337121391071e-01, 2.068885521020362e-02, 4.595394054487673e-01, -8.678958423780866e-01, 5.463376630254965e-02, 8.692220828069680e-01, -8.501142309217191e-01, 2.950403142043507e-02, 9.852274923275850e-01, -9.142859758630335e-01, 6.850217129206400e-03, 2.050108803358328e-01, -9.537223544012101e-01, 3.630414793101230e-02, 5.102048128337472e-01, -9.934425198626209e-01, 1.110556726680624e-02, 7.218738735054576e-01, -9.448573049959050e-01, 2.723130314700101e-02, 8.882998675210052e-01, -9.865716562931908e-01, 6.165546352541271e-03, 9.587134606894607e-01, -9.790570750021207e-01, 2.900715554613127e-03, 4.950400647800665e-01, -1.944817128807972e-02, 1.165426493105556e-01, 8.501230581964183e-01, -5.600805589569329e-02, 6.754741413634147e-02, 9.908677016040155e-01, -1.000588448960449e-01, 1.332984242008183e-02, 0.000000000000000e+00, 0.000000000000000e+00, 1.349722245528155e-01 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr22 ( ) /******************************************************************************/ /* Purpose: SMR22 returns the SMR rule of degree 22. Discussion: DEGREE: 22 POINTS CARDINALITY: 93 NORM INF MOMS. RESIDUAL: 9.10296e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR22[3*93], the requested rule. */ { const int degree = 22; int order; double xw[3*93] = { 9.886329057797582e-01, -1.145674729933659e-01, 1.325842639382102e-02, 9.888026938509622e-01, -5.213303149527411e-01, 1.068979190883013e-02, 9.661816000824490e-01, -7.555070898927371e-01, 1.743977208490352e-02, 9.889031095854282e-01, -9.320977012046493e-01, 5.174381878169184e-03, 9.873701927717461e-01, 9.897035459144938e-01, 1.707304801541134e-03, 9.905941115128364e-01, 8.327302191279294e-01, 6.754790334156039e-03, 9.706393731700478e-01, 5.982438926502055e-01, 2.062095495154808e-02, 9.803136579961507e-01, 2.717042647614646e-01, 1.891617092472248e-02, 9.341083383086605e-01, -3.353056638554749e-01, 3.390112966783673e-02, 8.734893606329978e-01, -6.054983585922311e-01, 3.753341638181854e-02, 8.810582900635474e-01, -8.777017334048872e-01, 2.414667023001881e-02, 9.308712251904757e-01, -9.880767541958921e-01, 5.413462408552709e-03, 9.377374302689546e-01, 9.330744362659844e-01, 1.265810122238417e-02, 8.843781920205257e-01, 7.681591801566370e-01, 3.367838185981432e-02, 8.775411599331090e-01, 4.195677919703100e-01, 4.701250942104521e-02, 9.082100522297584e-01, 4.460741468226298e-02, 4.492517531449552e-02, 7.763303548789647e-01, -4.810624802969050e-01, 3.017348276324992e-02, 7.230162511431557e-01, -7.596607824735281e-01, 4.911947899024260e-02, 7.562342329425361e-01, -9.545271924860271e-01, 1.392058621129393e-02, 8.327763332833239e-01, 9.863527280259253e-01, 8.265982196594024e-03, 7.412680367128046e-01, 8.954212932861624e-01, 3.350811254077402e-02, 7.400126973983402e-01, 6.111125638321177e-01, 5.811729500469508e-02, 7.414564617437679e-01, 1.965720776730482e-01, 7.150247641033709e-02, 7.867408087541262e-01, -1.930389785977610e-01, 6.417378369478929e-02, 5.369437186749637e-01, -6.314307127870589e-01, 4.820720884716240e-02, 5.461961159674565e-01, -8.910958585786000e-01, 4.010297901331633e-02, 7.505636824335776e-01, -9.781952487239193e-01, 5.423665310144266e-03, 5.668739124520142e-01, 9.769610350593898e-01, 1.855890684449772e-02, 5.608631907999050e-01, 7.717573834145061e-01, 5.676854811058375e-02, 5.612115855209996e-01, 4.088149453043886e-01, 8.333374346661079e-02, 5.661336430919435e-01, -4.176475015217607e-02, 9.157138052866115e-02, 6.134724653625782e-01, -4.169917924461408e-01, 6.572843506020827e-02, 3.374627534596129e-01, -7.953338524031655e-01, 4.698932393055661e-02, 5.506194243847290e-01, -9.906600094074560e-01, 6.956589831798293e-03, 3.682032600755504e-01, 8.998189655297543e-01, 4.291338908110833e-02, 3.503173557999109e-01, 6.043160593908300e-01, 8.199842424347994e-02, 3.484934492594987e-01, 1.842187014808652e-01, 1.042545147901201e-01, 3.599239753698877e-01, -2.793900622896136e-01, 1.003220388009486e-01, 3.352794040120309e-01, -6.090983723997480e-01, 5.250936392916155e-02, 1.648479846846835e-01, -9.222811023803827e-01, 3.600095811243217e-02, 3.409639556515009e-01, -9.775579820216516e-01, 1.513630382707296e-02, 2.013308254889806e-01, 9.817828174442371e-01, 1.803413215387369e-02, 1.340901732488610e-01, 7.766562660948982e-01, 6.844087294220280e-02, 1.144967045243097e-01, 4.069075707084039e-01, 1.019382263127832e-01, 1.189485784665626e-01, -5.433922354403713e-02, 1.136907024324531e-01, 1.058758908864689e-01, -4.793787778363424e-01, 9.348793573923811e-02, 2.839855852222449e-02, -7.559768193014593e-01, 6.803236606704954e-02, -4.712493250696739e-02, -9.874476808665996e-01, 1.515020459751181e-02, -1.410549552011679e-01, 9.945136764387909e-01, 7.185771234424889e-03, -5.100979131120797e-02, 9.148649196209385e-01, 4.369402100644239e-02, -1.137013612566373e-01, 6.139356044623114e-01, 8.777317450897354e-02, -1.244170197023402e-01, 1.823653793283892e-01, 1.117684867762398e-01, -1.370122651236654e-01, -2.712575708608372e-01, 1.085208525145446e-01, -2.232031959837765e-01, -6.067425313332153e-01, 8.558870495241622e-02, -1.940829548165339e-01, -8.901639697157221e-01, 4.867603590079570e-02, -4.081653916012992e-01, -9.709368148205605e-01, 2.331260017933040e-02, -3.533981306028999e-01, 9.616410187563108e-01, 2.242566292797548e-02, -3.045652224491460e-01, 7.985802428860417e-01, 6.420737202085385e-02, -3.526743970139553e-01, 4.146812551454919e-01, 9.787271335988859e-02, -3.720004998868196e-01, -3.666778868685774e-02, 1.068882608071387e-01, -4.514888617224385e-01, -4.096349836635951e-01, 9.192684575976928e-02, -4.429908638318878e-01, -7.743729186166529e-01, 6.295500628522636e-02, -6.815740834481019e-01, -9.928214216661736e-01, 7.026776085145282e-03, -5.632109089288005e-01, 9.935896447624268e-01, 6.283490740799424e-03, -5.621512958714884e-01, 8.896399470067906e-01, 3.761225364844008e-02, -5.382057775784931e-01, 6.382212857626252e-01, 7.506638282008458e-02, -5.813167555829912e-01, 2.056862702815186e-01, 9.243697007930982e-02, -6.516405814152544e-01, -1.846901900997915e-01, 8.505041739260445e-02, -6.506998051630787e-01, -6.087053290037806e-01, 6.818365853952568e-02, -6.334805048429584e-01, -8.975976820838455e-01, 3.682517885499488e-02, -7.472275238683430e-01, 9.641740358177714e-01, 1.669151214269935e-02, -7.482067350979565e-01, 7.780755148965470e-01, 4.495260598024266e-02, -7.400821884669309e-01, 4.532491832762973e-01, 6.928645874800578e-02, -8.105044748991020e-01, 5.416962245544283e-02, 6.619505274805781e-02, -8.156792303551951e-01, -4.060746003340047e-01, 6.042829810764130e-02, -8.090136873528386e-01, -7.748312229117293e-01, 4.151899530758620e-02, -8.335513136071991e-01, -9.535072625443444e-01, 1.514946999120562e-02, -8.872174992437346e-01, 8.965769271361460e-01, 2.104433235476698e-02, -8.931086765195151e-01, 6.328025905174818e-01, 3.860028858677801e-02, -8.916333811989803e-01, 3.038140953910366e-01, 2.786266023247349e-02, -9.292789589232818e-01, -1.761834888463487e-01, 4.117008802426377e-02, -9.292197203932411e-01, -6.069529981720342e-01, 3.315912528944302e-02, -9.385590119112288e-01, -8.777667192165588e-01, 1.580481441801968e-02, -9.284304209620261e-01, -9.951824052865140e-01, 2.664793505134054e-03, -9.053874905610783e-01, 9.962748823261235e-01, 3.593987239589643e-03, -9.747952911035821e-01, 7.907984327057133e-01, 1.405694455607466e-02, -9.866861855312228e-01, 5.118586070736004e-01, 1.306184877568784e-02, -9.863703578500000e-01, 6.111749406829102e-02, 1.559138821480461e-02, -9.889270365025400e-01, -3.994782279754348e-01, 1.342812184963877e-02, -9.908021213083021e-01, -7.596541692019284e-01, 7.741415848728735e-03, -9.815692523507022e-01, 9.572922384144454e-01, 5.171764118207254e-03, -9.471376718841760e-01, 2.927695425907414e-01, 2.000641714794302e-02, -9.874787306662429e-01, -9.621211118728035e-01, 3.377128849473962e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr23 ( ) /******************************************************************************/ /* Purpose: SMR23 returns the SMR rule of degree 23. Discussion: DEGREE: 23 ROTATIONALLY INVARIANT: (X, Y), (-Y, X), (-X, -Y), (Y, -X). POINTS CARDINALITY: 96 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR23[3*96], the requested rule. */ { const int degree = 23; int order; double xw[3*96] = { 1.853314713323298e-01, -1.194182324265387e-02, 6.529202325372914e-02, 1.659642315868930e-01, 3.367719887764350e-01, 6.231767492170222e-02, 9.876738163488666e-02, 8.306969615801286e-01, 6.201416548818468e-02, 8.869002322225777e-02, 9.854102218186098e-01, 1.324109920098785e-02, 3.982734759789451e-01, 1.335875115514376e-01, 6.923888037215149e-02, 2.307675985221515e-01, 5.659053189940804e-01, 7.857529731333650e-02, 4.154938603680680e-01, 7.488761930933899e-01, 6.583136220575310e-02, 3.258331163946451e-01, 9.405377066569092e-01, 3.516290651110937e-02, 4.718068156165912e-01, 3.742733716935872e-01, 7.391201672123172e-02, 6.214843667854110e-01, 5.757611415753823e-01, 6.633209783965152e-02, 6.111965358277813e-01, 8.835074910197936e-01, 4.096440841363334e-02, 5.425835924085424e-01, 9.934011708856250e-01, 9.791218112671066e-03, 6.454787887214178e-01, 7.290903205395782e-02, 7.527535021842932e-02, 7.663883133711373e-01, 3.281528217933192e-01, 6.564383307544155e-02, 7.763362157153701e-01, 7.457829318163177e-01, 4.507790595281267e-02, 7.869974305642068e-01, 9.638578123979478e-01, 1.814838304609862e-02, 9.212736070974256e-01, 1.716486762023320e-01, 4.117445213281692e-02, 8.886542125222323e-01, 5.442346685704154e-01, 4.215575969073930e-02, 9.057049801164405e-01, 8.685411402721253e-01, 2.274825956928111e-02, 9.457512810375365e-01, 9.860275528096041e-01, 5.078778817313888e-03, 9.878935301664215e-01, 7.759170798217239e-02, 4.765144456210194e-03, 9.836661500284433e-01, 4.041056452269987e-01, 1.596460029217852e-02, 9.734962304125074e-01, 7.140118690494257e-01, 1.705297096357996e-02, 9.936548069316368e-01, 9.210767826681668e-01, 4.241411430955866e-03, 1.194182324265387e-02, 1.853314713323298e-01, 6.529202325372914e-02, -3.367719887764350e-01, 1.659642315868930e-01, 6.231767492170222e-02, -8.306969615801286e-01, 9.876738163488666e-02, 6.201416548818468e-02, -9.854102218186098e-01, 8.869002322225777e-02, 1.324109920098785e-02, -1.335875115514376e-01, 3.982734759789451e-01, 6.923888037215149e-02, -5.659053189940804e-01, 2.307675985221515e-01, 7.857529731333650e-02, -7.488761930933899e-01, 4.154938603680680e-01, 6.583136220575310e-02, -9.405377066569092e-01, 3.258331163946451e-01, 3.516290651110937e-02, -3.742733716935872e-01, 4.718068156165912e-01, 7.391201672123172e-02, -5.757611415753823e-01, 6.214843667854110e-01, 6.633209783965152e-02, -8.835074910197936e-01, 6.111965358277813e-01, 4.096440841363334e-02, -9.934011708856250e-01, 5.425835924085424e-01, 9.791218112671066e-03, -7.290903205395782e-02, 6.454787887214178e-01, 7.527535021842932e-02, -3.281528217933192e-01, 7.663883133711373e-01, 6.564383307544155e-02, -7.457829318163177e-01, 7.763362157153701e-01, 4.507790595281267e-02, -9.638578123979478e-01, 7.869974305642068e-01, 1.814838304609862e-02, -1.716486762023320e-01, 9.212736070974256e-01, 4.117445213281692e-02, -5.442346685704154e-01, 8.886542125222323e-01, 4.215575969073930e-02, -8.685411402721253e-01, 9.057049801164405e-01, 2.274825956928111e-02, -9.860275528096041e-01, 9.457512810375365e-01, 5.078778817313888e-03, -7.759170798217239e-02, 9.878935301664215e-01, 4.765144456210194e-03, -4.041056452269987e-01, 9.836661500284433e-01, 1.596460029217852e-02, -7.140118690494257e-01, 9.734962304125074e-01, 1.705297096357996e-02, -9.210767826681668e-01, 9.936548069316368e-01, 4.241411430955866e-03, -1.853314713323298e-01, 1.194182324265387e-02, 6.529202325372914e-02, -1.659642315868930e-01, -3.367719887764350e-01, 6.231767492170222e-02, -9.876738163488666e-02, -8.306969615801286e-01, 6.201416548818468e-02, -8.869002322225777e-02, -9.854102218186098e-01, 1.324109920098785e-02, -3.982734759789451e-01, -1.335875115514376e-01, 6.923888037215149e-02, -2.307675985221515e-01, -5.659053189940804e-01, 7.857529731333650e-02, -4.154938603680680e-01, -7.488761930933899e-01, 6.583136220575310e-02, -3.258331163946451e-01, -9.405377066569092e-01, 3.516290651110937e-02, -4.718068156165912e-01, -3.742733716935872e-01, 7.391201672123172e-02, -6.214843667854110e-01, -5.757611415753823e-01, 6.633209783965152e-02, -6.111965358277813e-01, -8.835074910197936e-01, 4.096440841363334e-02, -5.425835924085424e-01, -9.934011708856250e-01, 9.791218112671066e-03, -6.454787887214178e-01, -7.290903205395782e-02, 7.527535021842932e-02, -7.663883133711373e-01, -3.281528217933192e-01, 6.564383307544155e-02, -7.763362157153701e-01, -7.457829318163177e-01, 4.507790595281267e-02, -7.869974305642068e-01, -9.638578123979478e-01, 1.814838304609862e-02, -9.212736070974256e-01, -1.716486762023320e-01, 4.117445213281692e-02, -8.886542125222323e-01, -5.442346685704154e-01, 4.215575969073930e-02, -9.057049801164405e-01, -8.685411402721253e-01, 2.274825956928111e-02, -9.457512810375365e-01, -9.860275528096041e-01, 5.078778817313888e-03, -9.878935301664215e-01, -7.759170798217239e-02, 4.765144456210194e-03, -9.836661500284433e-01, -4.041056452269987e-01, 1.596460029217852e-02, -9.734962304125074e-01, -7.140118690494257e-01, 1.705297096357996e-02, -9.936548069316368e-01, -9.210767826681668e-01, 4.241411430955866e-03, -1.194182324265387e-02, -1.853314713323298e-01, 6.529202325372914e-02, 3.367719887764350e-01, -1.659642315868930e-01, 6.231767492170222e-02, 8.306969615801286e-01, -9.876738163488666e-02, 6.201416548818468e-02, 9.854102218186098e-01, -8.869002322225777e-02, 1.324109920098785e-02, 1.335875115514376e-01, -3.982734759789451e-01, 6.923888037215149e-02, 5.659053189940804e-01, -2.307675985221515e-01, 7.857529731333650e-02, 7.488761930933899e-01, -4.154938603680680e-01, 6.583136220575310e-02, 9.405377066569092e-01, -3.258331163946451e-01, 3.516290651110937e-02, 3.742733716935872e-01, -4.718068156165912e-01, 7.391201672123172e-02, 5.757611415753823e-01, -6.214843667854110e-01, 6.633209783965152e-02, 8.835074910197936e-01, -6.111965358277813e-01, 4.096440841363334e-02, 9.934011708856250e-01, -5.425835924085424e-01, 9.791218112671066e-03, 7.290903205395782e-02, -6.454787887214178e-01, 7.527535021842932e-02, 3.281528217933192e-01, -7.663883133711373e-01, 6.564383307544155e-02, 7.457829318163177e-01, -7.763362157153701e-01, 4.507790595281267e-02, 9.638578123979478e-01, -7.869974305642068e-01, 1.814838304609862e-02, 1.716486762023320e-01, -9.212736070974256e-01, 4.117445213281692e-02, 5.442346685704154e-01, -8.886542125222323e-01, 4.215575969073930e-02, 8.685411402721253e-01, -9.057049801164405e-01, 2.274825956928111e-02, 9.860275528096041e-01, -9.457512810375365e-01, 5.078778817313888e-03, 7.759170798217239e-02, -9.878935301664215e-01, 4.765144456210194e-03, 4.041056452269987e-01, -9.836661500284433e-01, 1.596460029217852e-02, 7.140118690494257e-01, -9.734962304125074e-01, 1.705297096357996e-02, 9.210767826681668e-01, -9.936548069316368e-01, 4.241411430955866e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr24 ( ) /******************************************************************************/ /* Purpose: SMR24 returns the SMR rule of degree 24. Discussion: DEGREE: 24 POINTS CARDINALITY: 109 NORM INF MOMS. RESIDUAL: 1.33227e-15 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR24[3*109], the requested rule. */ { const int degree = 24; int order; double xw[3*109] = { -9.947552488434938e-01, -4.385704490896414e-01, 2.698557791124145e-03, -9.931097629772323e-01, 6.703739358435329e-01, 7.377682774083533e-03, -9.926803856666530e-01, 2.074620875386726e-01, 6.953676686934647e-03, -9.906389564778285e-01, 9.389787594673664e-01, 4.005676169355294e-03, -9.853982980908041e-01, -1.546975310338203e-01, 1.391680440670080e-02, -9.837508975466611e-01, -8.419003469340943e-01, 8.844680842740435e-03, -9.834972384053117e-01, -5.844467591604228e-01, 1.101492580337329e-02, -9.818922098458703e-01, -9.808906804560406e-01, 2.833762301624789e-03, -9.737197710731226e-01, 3.846778957131831e-01, 1.346113092355880e-02, -9.500074961120746e-01, 8.209864656561424e-01, 1.735987104490214e-02, -9.403701834437002e-01, 9.898046509843106e-01, 4.134638994139726e-03, -9.322891009559756e-01, 6.891056205665078e-02, 3.385109951711678e-02, -9.291432442434125e-01, -3.649292941055270e-01, 3.313278955371516e-02, -9.278770514198088e-01, 5.372507215773373e-01, 2.595775803019849e-02, -9.160984294365914e-01, -7.088454133225529e-01, 2.740072723324332e-02, -9.158788280467461e-01, -9.354605631847065e-01, 1.341647825529986e-02, -8.614049341981235e-01, 9.223186488292008e-01, 1.828556090710842e-02, -8.411326781777169e-01, 2.986796133696130e-01, 5.109735452042367e-02, -8.399585109946057e-01, 6.941827653602577e-01, 3.578648643426428e-02, -8.284250818876616e-01, -1.413597089673544e-01, 5.450666609336831e-02, -8.258893393997301e-01, -9.946839974593770e-01, 5.524951523211650e-03, -8.061244501112171e-01, -5.249750016790736e-01, 5.069979449576008e-02, -8.039912747312149e-01, -8.393703760802331e-01, 3.118857585337858e-02, -7.503800600330690e-01, 9.808995135027622e-01, 1.020614373666720e-02, -7.159580396066608e-01, 8.204005009530653e-01, 3.345975676433573e-02, -7.066954614920985e-01, 5.015305614605768e-01, 6.001952691278750e-02, -6.885485793465114e-01, 9.202328849593100e-02, 7.346166114914886e-02, -6.800088501791149e-01, -9.443640836874199e-01, 2.298808927262541e-02, -6.560939841533527e-01, -3.083819563661853e-01, 7.136431171052350e-02, -6.549319709970016e-01, -6.928080924969605e-01, 5.352097766179938e-02, -5.986758659840633e-01, 9.036887631193311e-01, 2.206238875433044e-02, -5.440986421726908e-01, 6.690288689480005e-01, 5.524621622310746e-02, -5.421013331274553e-01, 9.805973671548335e-01, 1.050046021811538e-02, -5.117472121011815e-01, 3.140452735180922e-01, 8.289952810298720e-02, -5.059194733599800e-01, -9.837457937119135e-01, 1.096035229919467e-02, -4.980272781651189e-01, -8.405454275729934e-01, 4.487909208174960e-02, -4.827305043646576e-01, -4.980524078016305e-01, 7.003741329201849e-02, -4.657032616378755e-01, -8.311620500718582e-02, 9.203159768871901e-02, -3.984335431880944e-01, 7.823560336425450e-01, 3.533735180399633e-02, -3.812251842054403e-01, 9.211692018557595e-01, 2.479377740140328e-02, -3.254324649968378e-01, -6.106485217368141e-01, 1.987123465724002e-02, -3.081376572739211e-01, 5.152565145304252e-01, 8.148061878192561e-02, -3.061609471247951e-01, -9.953099499439115e-01, 4.421627578115842e-03, -3.040806099754977e-01, -6.945205873058414e-01, 4.874084869821138e-02, -3.040627168926376e-01, -9.238254614973810e-01, 2.731221830374910e-02, -2.730267547433428e-01, 9.914365779967292e-01, 9.183887978612177e-03, -2.548424890251378e-01, -3.019158326395579e-01, 9.514384875199645e-02, -2.519554856003144e-01, 1.492385558662303e-01, 1.000641506538448e-01, -1.931995209634114e-01, 8.232643617111496e-01, 3.769059900817138e-02, -1.885349353760669e-01, -8.893470388562837e-01, 8.285027518755494e-03, -1.018544959716391e-01, -7.832201508773523e-01, 4.767805838303123e-02, -9.655685106805004e-02, 6.703558020663598e-01, 5.541671674358566e-02, -7.901309787694769e-02, -9.687392227645953e-01, 1.819078482187320e-02, -7.169337674372403e-02, 9.452615682987324e-01, 2.860240343608491e-02, -4.349902330261272e-02, -5.144306594431935e-01, 8.777502677071561e-02, -2.884848343031785e-02, -7.515483993225638e-02, 1.033082821984657e-01, -1.950518268267537e-02, 3.687032706515938e-01, 9.771287132446460e-02, 1.000713405257931e-01, -8.778592595775899e-01, 4.025365288851937e-02, 1.288386576225213e-01, 8.499166400570952e-01, 4.727125786790465e-02, 1.307155265497598e-01, -9.938241813833318e-01, 7.039833852159185e-03, 1.308425147211156e-01, 6.526807421421726e-01, 4.979273601404909e-02, 1.451164406166477e-01, 9.905591807186089e-01, 1.078258330904423e-02, 1.853995153478054e-01, -3.035646033025806e-01, 9.690391758696711e-02, 1.943247761163270e-01, -6.769527305130935e-01, 7.109638809439672e-02, 2.052879132552834e-01, 1.494567879060908e-01, 1.010903998636895e-01, 2.768585369952569e-01, 5.067311062472540e-01, 6.479712852470575e-02, 3.149836876173980e-01, -9.498509878332206e-01, 2.600605597250065e-02, 3.466439434503832e-01, 9.410184803711036e-01, 2.912818672968244e-02, 3.903088921667896e-01, 4.250693093572162e-01, 2.571049499832495e-02, 3.993865541451484e-01, 7.683264787575288e-01, 5.493699586057709e-02, 4.086593369430472e-01, -4.969944946335217e-01, 7.910634640474691e-02, 4.120045473805335e-01, -8.379854233451292e-02, 9.482614431233395e-02, 4.170298041496714e-01, -8.110827678737819e-01, 5.059001095427527e-02, 4.991948804043371e-01, 2.844383580006540e-01, 7.600131857765238e-02, 5.215718300412919e-01, -9.911476542754794e-01, 8.938791600642888e-03, 5.358795456370800e-01, 9.892706596240752e-01, 9.652257973711614e-03, 5.675663650098263e-01, 6.269930547839709e-01, 5.565333636824790e-02, 6.083113501735691e-01, -6.671573460757463e-01, 5.724631842495273e-02, 6.091213803480886e-01, -3.002786277521432e-01, 7.620974249495195e-02, 6.098130539524327e-01, 8.880641734983817e-01, 3.607781947134865e-02, 6.133229089892412e-01, -9.115716813284614e-01, 2.976045072649638e-02, 6.644692975845461e-01, 7.278284636752541e-02, 7.402473800633137e-02, 7.058360022640749e-01, 4.736368348307747e-01, 5.128474162165504e-02, 7.603978094655672e-01, 7.687530787837937e-01, 3.987086504850027e-02, 7.680081930648918e-01, 9.719796486153569e-01, 1.425802911781755e-02, 7.730566152310493e-01, -8.076103665001979e-01, 3.513944563783513e-02, 7.740524973445363e-01, -5.000787609526253e-01, 5.331933422231531e-02, 7.765031539359070e-01, -9.744702466310179e-01, 1.298770336406751e-02, 8.105891537849745e-01, -1.496106096240346e-01, 5.659879783131575e-02, 8.136030860267544e-01, 2.997934729123835e-01, 4.597071032159282e-02, 8.738296548638522e-01, 6.252207344858673e-01, 3.235229209061567e-02, 8.820767009721266e-01, 8.997350185706089e-01, 1.969080467560540e-02, 8.968725800328660e-01, -6.749357059674395e-01, 3.049246549718226e-02, 9.008567820439595e-01, -9.076509501580650e-01, 1.724041843190777e-02, 9.036351991894711e-01, 1.066673951092828e-01, 3.553262749083069e-02, 9.194113498715898e-01, -3.652426921938732e-01, 3.438229944741761e-02, 9.235901436599459e-01, 4.967015317618840e-01, 1.373744804296171e-02, 9.328830195971040e-01, -9.941380940892665e-01, 2.989382978667606e-03, 9.404372628037821e-01, 9.889603686778536e-01, 4.443564861581591e-03, 9.607926125333919e-01, 7.879019305115507e-01, 1.649904214759328e-02, 9.627435059550357e-01, 3.759293384523862e-01, 1.893234461167443e-02, 9.661246051400586e-01, -9.321465960805159e-02, 2.021199306716822e-02, 9.772105063681134e-01, -8.054188902050837e-01, 1.143950510739337e-02, 9.818083790572272e-01, -5.613562549296841e-01, 1.303588980687769e-02, 9.873606864519788e-01, -9.558632391723191e-01, 3.650162385277064e-03, 9.909481727417117e-01, 9.350098107984304e-01, 4.199764168210779e-03, 9.933228865506706e-01, 1.717028973685849e-01, 7.977198513429708e-03, 9.934734927244968e-01, 6.113673375988821e-01, 7.151800672626832e-03, 9.958206097496789e-01, -2.883979781434184e-01, 5.617961121686336e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr25 ( ) /******************************************************************************/ /* Purpose: SMR25 returns the SMR rule of degree 25. Discussion: DEGREE: 25 ROTATIONALLY INVARIANT: (X, Y), (-Y, X), (-X, -Y), (Y, -X). POINTS CARDINALITY: 113 NORM INF MOMS. RESIDUAL: 2.04121e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR25[3*113], the requested rule. */ { const int degree = 25; int order; double xw[3*113] = { 2.408979561244244e-01, 4.202921149345177e-01, 8.649643596540968e-02, 2.718022358106373e-01, 7.547569098063632e-01, 5.930450471454697e-02, 2.949651173549945e-01, 9.557139743958390e-01, 2.492523876506716e-02, 4.390520493777193e-01, 2.026918805813806e-01, 8.650651080976479e-02, 4.662064901266157e-01, 5.942245210433184e-01, 6.800866187330593e-02, 4.830786617878046e-01, 8.732633185547312e-01, 3.820005521117373e-02, 5.045376712354956e-01, 9.940169417092007e-01, 7.120126112016729e-03, 7.757488163218268e-01, 1.779409708250335e-01, 5.827884791218824e-02, 6.369947479572929e-01, 4.005073391747475e-01, 6.761649569671979e-02, 6.535675504807226e-01, 7.533634302760710e-01, 4.601021918794469e-02, 6.595342990537197e-01, 9.508745400630684e-01, 1.723971223901597e-02, 7.906371918772621e-01, 5.934656821653986e-01, 4.568246307970081e-02, 8.055336482199160e-01, 8.773080252379840e-01, 2.484701585804588e-02, 7.535373835857363e-01, 9.795331551151395e-01, 4.313716122504384e-03, 9.564646592216111e-01, 1.571205964662031e-01, 2.535428637556193e-02, 8.902743405153437e-01, 3.914673487680078e-01, 3.753743912008318e-02, 9.033639329394400e-01, 7.595177874407598e-01, 2.457091299936516e-02, 9.372983112686489e-01, 9.385069580562527e-01, 1.007243031836228e-02, 8.636461603424476e-01, 9.896801193034730e-01, 4.961458044877222e-03, 9.938360564492993e-01, 3.750300609078687e-01, 7.598805738926257e-03, 9.619543373713801e-01, 5.836819379362060e-01, 1.776328589308987e-02, 9.918091540820021e-01, 7.176122922535305e-01, 3.747585881528770e-03, 9.847916123916505e-01, 8.571417379883992e-01, 6.289982457940641e-03, 9.885652833157735e-01, 9.860593798999778e-01, 1.697350354991505e-03, 9.233777086225921e-03, 2.208671709154587e-01, 9.688701249182402e-02, 3.041113022131709e-02, 6.137415273447637e-01, 7.643513771616485e-02, 6.708941821053316e-02, 8.786573477860689e-01, 4.373962481400533e-02, 7.276240742442623e-02, 9.928846185998859e-01, 8.791036677341798e-03, -4.202921149345177e-01, 2.408979561244244e-01, 8.649643596540968e-02, -7.547569098063632e-01, 2.718022358106373e-01, 5.930450471454697e-02, -9.557139743958390e-01, 2.949651173549945e-01, 2.492523876506716e-02, -2.026918805813806e-01, 4.390520493777193e-01, 8.650651080976479e-02, -5.942245210433184e-01, 4.662064901266157e-01, 6.800866187330593e-02, -8.732633185547312e-01, 4.830786617878046e-01, 3.820005521117373e-02, -9.940169417092007e-01, 5.045376712354956e-01, 7.120126112016729e-03, -1.779409708250335e-01, 7.757488163218268e-01, 5.827884791218824e-02, -4.005073391747475e-01, 6.369947479572929e-01, 6.761649569671979e-02, -7.533634302760710e-01, 6.535675504807226e-01, 4.601021918794469e-02, -9.508745400630684e-01, 6.595342990537197e-01, 1.723971223901597e-02, -5.934656821653986e-01, 7.906371918772621e-01, 4.568246307970081e-02, -8.773080252379840e-01, 8.055336482199160e-01, 2.484701585804588e-02, -9.795331551151395e-01, 7.535373835857363e-01, 4.313716122504384e-03, -1.571205964662031e-01, 9.564646592216111e-01, 2.535428637556193e-02, -3.914673487680078e-01, 8.902743405153437e-01, 3.753743912008318e-02, -7.595177874407598e-01, 9.033639329394400e-01, 2.457091299936516e-02, -9.385069580562527e-01, 9.372983112686489e-01, 1.007243031836228e-02, -9.896801193034730e-01, 8.636461603424476e-01, 4.961458044877222e-03, -3.750300609078687e-01, 9.938360564492993e-01, 7.598805738926257e-03, -5.836819379362060e-01, 9.619543373713801e-01, 1.776328589308987e-02, -7.176122922535305e-01, 9.918091540820021e-01, 3.747585881528770e-03, -8.571417379883992e-01, 9.847916123916505e-01, 6.289982457940641e-03, -9.860593798999778e-01, 9.885652833157735e-01, 1.697350354991505e-03, -2.208671709154587e-01, 9.233777086225921e-03, 9.688701249182402e-02, -6.137415273447637e-01, 3.041113022131709e-02, 7.643513771616485e-02, -8.786573477860689e-01, 6.708941821053316e-02, 4.373962481400533e-02, -9.928846185998859e-01, 7.276240742442623e-02, 8.791036677341798e-03, -2.408979561244244e-01, -4.202921149345177e-01, 8.649643596540968e-02, -2.718022358106373e-01, -7.547569098063632e-01, 5.930450471454697e-02, -2.949651173549945e-01, -9.557139743958390e-01, 2.492523876506716e-02, -4.390520493777193e-01, -2.026918805813806e-01, 8.650651080976479e-02, -4.662064901266157e-01, -5.942245210433184e-01, 6.800866187330593e-02, -4.830786617878046e-01, -8.732633185547312e-01, 3.820005521117373e-02, -5.045376712354956e-01, -9.940169417092007e-01, 7.120126112016729e-03, -7.757488163218268e-01, -1.779409708250335e-01, 5.827884791218824e-02, -6.369947479572929e-01, -4.005073391747475e-01, 6.761649569671979e-02, -6.535675504807226e-01, -7.533634302760710e-01, 4.601021918794469e-02, -6.595342990537197e-01, -9.508745400630684e-01, 1.723971223901597e-02, -7.906371918772621e-01, -5.934656821653986e-01, 4.568246307970081e-02, -8.055336482199160e-01, -8.773080252379840e-01, 2.484701585804588e-02, -7.535373835857363e-01, -9.795331551151395e-01, 4.313716122504384e-03, -9.564646592216111e-01, -1.571205964662031e-01, 2.535428637556193e-02, -8.902743405153437e-01, -3.914673487680078e-01, 3.753743912008318e-02, -9.033639329394400e-01, -7.595177874407598e-01, 2.457091299936516e-02, -9.372983112686489e-01, -9.385069580562527e-01, 1.007243031836228e-02, -8.636461603424476e-01, -9.896801193034730e-01, 4.961458044877222e-03, -9.938360564492993e-01, -3.750300609078687e-01, 7.598805738926257e-03, -9.619543373713801e-01, -5.836819379362060e-01, 1.776328589308987e-02, -9.918091540820021e-01, -7.176122922535305e-01, 3.747585881528770e-03, -9.847916123916505e-01, -8.571417379883992e-01, 6.289982457940641e-03, -9.885652833157735e-01, -9.860593798999778e-01, 1.697350354991505e-03, -9.233777086225921e-03, -2.208671709154587e-01, 9.688701249182402e-02, -3.041113022131709e-02, -6.137415273447637e-01, 7.643513771616485e-02, -6.708941821053316e-02, -8.786573477860689e-01, 4.373962481400533e-02, -7.276240742442623e-02, -9.928846185998859e-01, 8.791036677341798e-03, 4.202921149345177e-01, -2.408979561244244e-01, 8.649643596540968e-02, 7.547569098063632e-01, -2.718022358106373e-01, 5.930450471454697e-02, 9.557139743958390e-01, -2.949651173549945e-01, 2.492523876506716e-02, 2.026918805813806e-01, -4.390520493777193e-01, 8.650651080976479e-02, 5.942245210433184e-01, -4.662064901266157e-01, 6.800866187330593e-02, 8.732633185547312e-01, -4.830786617878046e-01, 3.820005521117373e-02, 9.940169417092007e-01, -5.045376712354956e-01, 7.120126112016729e-03, 1.779409708250335e-01, -7.757488163218268e-01, 5.827884791218824e-02, 4.005073391747475e-01, -6.369947479572929e-01, 6.761649569671979e-02, 7.533634302760710e-01, -6.535675504807226e-01, 4.601021918794469e-02, 9.508745400630684e-01, -6.595342990537197e-01, 1.723971223901597e-02, 5.934656821653986e-01, -7.906371918772621e-01, 4.568246307970081e-02, 8.773080252379840e-01, -8.055336482199160e-01, 2.484701585804588e-02, 9.795331551151395e-01, -7.535373835857363e-01, 4.313716122504384e-03, 1.571205964662031e-01, -9.564646592216111e-01, 2.535428637556193e-02, 3.914673487680078e-01, -8.902743405153437e-01, 3.753743912008318e-02, 7.595177874407598e-01, -9.033639329394400e-01, 2.457091299936516e-02, 9.385069580562527e-01, -9.372983112686489e-01, 1.007243031836228e-02, 9.896801193034730e-01, -8.636461603424476e-01, 4.961458044877222e-03, 3.750300609078687e-01, -9.938360564492993e-01, 7.598805738926257e-03, 5.836819379362060e-01, -9.619543373713801e-01, 1.776328589308987e-02, 7.176122922535305e-01, -9.918091540820021e-01, 3.747585881528770e-03, 8.571417379883992e-01, -9.847916123916505e-01, 6.289982457940641e-03, 9.860593798999778e-01, -9.885652833157735e-01, 1.697350354991505e-03, 2.208671709154587e-01, -9.233777086225921e-03, 9.688701249182402e-02, 6.137415273447637e-01, -3.041113022131709e-02, 7.643513771616485e-02, 8.786573477860689e-01, -6.708941821053316e-02, 4.373962481400533e-02, 9.928846185998859e-01, -7.276240742442623e-02, 8.791036677341798e-03, 0.000000000000000e+00, 0.000000000000000e+00, 1.459027412974250e-05 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr26 ( ) /******************************************************************************/ /* Purpose: SMR26 returns the SMR rule of degree 26. Discussion: DEGREE: 26 POINTS CARDINALITY: 127 NORM INF MOMS. RESIDUAL: 2.72785e-16 SUM NEGATIVE WEIGHTS: 0 Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR26[3*127], the requested rule. */ { const int degree = 26; int order; double xw[3*127] = { -9.954149432053586e-01, -8.467950645561364e-01, 3.684914976298696e-03, -9.951846316463983e-01, 4.962239287417011e-01, 5.769468055434405e-03, -9.946910061010343e-01, -2.554039039018801e-01, 5.732088976456913e-03, -9.919373759022915e-01, 1.036499274958413e-01, 7.852213957780449e-03, -9.906494815143384e-01, 8.649856906058798e-01, 5.121508648969548e-03, -9.896255204784331e-01, -5.367353086290219e-01, 7.653897829024368e-03, -9.883067004873612e-01, 9.892991458592666e-01, 1.353489163893612e-03, -9.871269120799574e-01, -9.856243972746432e-01, 1.601609521006031e-03, -9.688175505639893e-01, 6.876579842845731e-01, 1.481263227905701e-02, -9.623107235333284e-01, -7.113315046485157e-01, 1.467102553035195e-02, -9.594699050132816e-01, 3.158494085062999e-01, 2.066592783625548e-02, -9.556340329654068e-01, -8.022603946161568e-02, 2.057995991913653e-02, -9.551530447389018e-01, 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6.886649912238812e-03, 9.952509862842531e-01, -2.550697642134186e-01, 6.463763906756026e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr27 ( ) /******************************************************************************/ /* Purpose: SMR27 returns the SMR rule of degree 27. Discussion: DEGREE: 27 ROTATIONALLY INVARIANT: (X, Y), (-Y, X), (-X, -Y), (Y, -X). POINTS CARDINALITY: 132 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR27[3*132], the requested rule. */ { const int degree = 27; int order; double xw[3*132] = { 1.886005021782720e-03, 5.828566738691298e-01, 6.979680609052895e-02, -4.610090446080339e-03, 8.577929872633651e-01, 4.314966506560954e-02, -4.155284083838179e-02, 9.888852930552997e-01, 1.072399831167234e-02, 2.067872320344404e-01, -6.296477862134920e-04, 8.488483237702206e-02, 2.082269167023509e-01, 4.038616639190317e-01, 7.727830308506041e-02, 2.052137161358173e-01, 7.394171390837959e-01, 5.621586102413551e-02, 1.807393256852824e-01, 9.496265565549702e-01, 2.535911050249288e-02, 3.322057475944939e-01, 9.960227386085063e-01, 5.272055511683970e-03, 4.045305824612024e-01, 2.059859741136477e-01, 7.726743635049893e-02, 4.025837874193514e-01, 5.870128510662224e-01, 6.365147357369046e-02, 3.850900136224434e-01, 8.707822203710608e-01, 3.854775533996937e-02, 4.961565809611688e-01, 9.726108752621352e-01, 1.194490281825224e-02, 7.337041416795138e-01, 2.082446806606209e-01, 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-9.293007271718974e-01, 9.170625679716800e-03, 2.247649048332172e-01, -9.365522911144204e-01, 2.753902574626484e-02, 5.146910772747944e-01, -9.913001320610028e-01, 2.062938898431044e-03, 7.177306340863354e-01, -9.892062993684332e-01, 6.213455600316710e-03, 8.526916976863245e-01, -9.825794696006162e-01, 3.542165144107640e-03, 9.559123039771233e-01, -9.899623983816039e-01, 2.789025209960568e-03, 9.931508332329141e-01, -9.454816892922301e-01, 2.641115240694518e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr28 ( ) /******************************************************************************/ /* Purpose: SMR28 returns the SMR rule of degree 28. Discussion: DEGREE: 28 POINTS CARDINALITY: 146 NORM INF MOMS. RESIDUAL: 9.93129e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR28[3*146], the requested rule. */ { const int degree = 28; int order; double xw[3*146] = { 9.948578576510482e-01, 9.951313777393089e-01, 5.350567115949095e-04, 9.942909707974587e-01, 8.766570248036105e-01, 2.791138783606160e-03, 9.936073905803642e-01, 6.435920031349868e-01, 4.975936005062567e-03, 9.955492434128311e-01, 3.009600560431372e-01, 5.028092846274678e-03, 9.953117341967249e-01, -1.155952308989407e-01, 5.404591653487998e-03, 9.945568546430053e-01, -4.954371212342698e-01, 4.995854780477920e-03, 9.941123224157187e-01, -7.686450662183013e-01, 3.412261550099155e-03, 9.973970470314231e-01, -9.477148556565722e-01, 1.256390479179999e-03, 9.688973112117140e-01, 9.519148224233880e-01, 4.713673624203035e-03, 9.613309496208055e-01, 7.757025845655895e-01, 1.166028423822774e-02, 9.659860418437211e-01, 4.681861912724297e-01, 1.519045343561554e-02, 9.698532836823804e-01, 9.733476040026384e-02, 1.657048063461714e-02, 9.676987141452779e-01, -3.026291805415285e-01, 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-9.260008242840237e-01, -1.093506648825247e-01, 2.730772032753251e-02, -9.322554404281427e-01, -4.325018867770137e-01, 2.166809223221262e-02, -9.265407983180678e-01, -6.804488306011519e-01, 1.129035475105887e-02, -8.963660879434896e-01, -8.578428981954751e-01, 1.427796928443569e-02, -9.228669313126245e-01, -9.903049524081384e-01, 2.936908429948614e-03, -9.795076814902403e-01, 9.391999047436536e-01, 3.611764745309799e-03, -9.682018722477242e-01, 7.789774445276537e-01, 9.386909970380832e-03, -9.723852471142062e-01, 4.752904607817341e-01, 1.284048267253543e-02, -9.845164156039091e-01, 4.496231910970377e-02, 1.155742732057478e-02, -9.919138468383984e-01, -5.892431915668860e-01, 5.712369221551983e-03, -9.653607048671382e-01, -7.514078044040550e-01, 9.247990075208610e-03, -9.634968122279486e-01, -9.300634913926976e-01, 6.455717208889276e-03, -9.899993900205354e-01, 9.944294353233614e-01, 6.923818612511791e-04, -9.957478348818433e-01, 8.751512353190156e-01, 1.727894454945418e-03, -9.950143212717456e-01, 6.584454755209226e-01, 3.945702259118487e-03, -9.960292127295011e-01, 3.102415165889851e-01, 4.235815708736188e-03, -9.868438218892031e-01, -2.632466666320823e-01, 1.048005824445523e-02, -9.759292701915939e-01, -4.866196210874411e-01, 4.122542369510386e-03, -9.956803593698287e-01, -8.464174810055828e-01, 2.909040478947865e-03, -9.918608822862665e-01, -9.833582162145671e-01, 1.201280370154266e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr29 ( ) /******************************************************************************/ /* Purpose: SMR29 returns the SMR rule of degree 29. Discussion: DEGREE: 29 SYMMETRY: (X, Y), (-X, -Y) POINTS CARDINALITY: 153 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0 Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR29[3*153], the requested rule. */ { const int degree = 29; int order; double xw[3*153] = { -9.949933127427926e-01, 9.846653902287268e-01, 9.395690393964076e-04, -9.947553807807867e-01, -6.565416135023074e-01, 4.902572703976801e-03, -9.945746533289254e-01, 8.095347588811120e-01, 3.444322300954131e-03, -9.936165836110393e-01, 5.449460871204653e-01, 5.048778280493792e-03, -9.916021206491545e-01, -3.020588266272870e-01, 7.856326443021280e-03, -9.885493346511697e-01, 3.754426235872674e-02, 9.243463352453171e-03, -9.853949921317231e-01, -9.716885948870212e-01, 2.611624046143479e-03, -9.850307952194871e-01, -8.619821821525152e-01, 5.756950705760731e-03, -9.832827588628299e-01, 3.064831307139635e-01, 1.011037458386165e-02, -9.721136174490276e-01, 9.199575449706859e-01, 6.336945711231334e-03, -9.629324251458530e-01, 6.898499209334040e-01, 1.253275279199627e-02, -9.611009120701418e-01, -5.003853392931935e-01, 1.712154659313199e-02, -9.462498240094916e-01, 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5.358828824792165e-02, 4.208552091936032e-01, -9.360365856546493e-01, 1.371343151498627e-02, 4.243714642725596e-01, 7.749351121311114e-01, 3.571480883454545e-02, 4.253276863946355e-01, 1.030467857496181e-01, 5.661499125196553e-02, 4.518891563243193e-01, 9.565726413410444e-01, 1.899592320434612e-02, 4.747713831994940e-01, -6.303175260200578e-01, 3.769491458931831e-02, 5.205683240806913e-01, -9.943357320308036e-01, 5.022863813128614e-03, 5.213722221542452e-01, -8.717053530032300e-01, 2.354037231951190e-02, 5.315715479129545e-01, 6.485017442148879e-01, 2.380537349256804e-02, 5.600226162335806e-01, -7.201761109528224e-02, 4.732804187723799e-02, 5.602688054023620e-01, -3.829382266085970e-01, 5.473125938490646e-02, 5.728456567073842e-01, 5.296576709425311e-01, 3.639826598763665e-02, 5.778077209855854e-01, 8.675249675625395e-01, 2.457244823370517e-02, 5.979003962373047e-01, 2.796891406409555e-01, 4.888297756206456e-02, 6.116292887899047e-01, -7.357056852576344e-01, 3.377020044412275e-02, 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-3.658292544818590e-01, 3.507289015514854e-02, 8.296887505977431e-01, 8.345532619659806e-01, 2.285324032183532e-02, 8.333294144744180e-01, 1.839129116833924e-01, 9.917901422164995e-03, 8.432406244370808e-01, -2.340798283350513e-02, 3.394457013361375e-02, 8.501310025536610e-01, -9.347168973702518e-01, 1.258009305624949e-02, 8.637451657209914e-01, 6.139023089531026e-01, 2.808904163947544e-02, 8.779301509485118e-01, -5.948583749922939e-01, 2.015897400394908e-02, 8.888912692909274e-01, 3.479987971871291e-01, 2.796225817458692e-02, 9.116574969716016e-01, -8.271123341203230e-01, 1.601077862848477e-02, 9.183317763520688e-01, 9.261208167798600e-01, 1.096195080681892e-02, 9.232310041137312e-01, -1.804372056187648e-01, 2.477443727803194e-02, 9.295735875484051e-01, -9.848152682310835e-01, 4.103888501222671e-03, 9.314612798308124e-01, 9.967694998936406e-01, 1.745679805743830e-03, 9.382448848401312e-01, -4.599118226520993e-01, 1.718439080989034e-02, 9.411359576419225e-01, 7.505472392355358e-01, 1.603065396164867e-02, 9.462498240094916e-01, 1.332608625807670e-01, 2.240075719199047e-02, 9.611009120701418e-01, 5.003853392931935e-01, 1.712154659313199e-02, 9.629324251458530e-01, -6.898499209334040e-01, 1.253275279199627e-02, 9.721136174490276e-01, -9.199575449706859e-01, 6.336945711231334e-03, 9.832827588628299e-01, -3.064831307139635e-01, 1.011037458386165e-02, 9.850307952194871e-01, 8.619821821525152e-01, 5.756950705760731e-03, 9.853949921317231e-01, 9.716885948870212e-01, 2.611624046143479e-03, 9.885493346511697e-01, -3.754426235872674e-02, 9.243463352453171e-03, 9.916021206491545e-01, 3.020588266272870e-01, 7.856326443021280e-03, 9.936165836110393e-01, -5.449460871204653e-01, 5.048778280493792e-03, 9.945746533289254e-01, -8.095347588811120e-01, 3.444322300954131e-03, 9.947553807807867e-01, 6.565416135023074e-01, 4.902572703976801e-03, 9.949933127427926e-01, -9.846653902287268e-01, 9.395690393964076e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr29s ( ) /******************************************************************************/ /* Purpose: SMR29S returns the rotatonally invariant SMR rule of degree 29. Discussion: DEGREE: 29 ROTATIONALLY INVARIANT: (X, Y), (-Y, X), (-X, -Y), (Y, -X). POINTS CARDINALITY: 153 NORM INF MOMS. RESIDUAL: 2.70617e-16 SUM NEGATIVE WEIGHTS: 1.46517e-02, Licensing: This code is distributed under the MIT license. Modified: 21 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR29S[3*153], the requested rule. */ { const int degree = 29; int order; double xw[3*153] = { 1.995554510347695e-01, 1.887353707822548e-01, 7.344881047788419e-02, 2.130905764129401e-01, 5.455267403580432e-01, 6.209984035978915e-02, 2.388895084628246e-01, 8.131809651972651e-01, 4.184384427163037e-02, 2.038938767743064e-01, 9.863343822191492e-01, 5.131075694716339e-04, 2.698246240215212e-01, 9.643706705539593e-01, 1.712164807226155e-02, 3.928345180603147e-01, 3.686184349052021e-01, 6.489527207591311e-02, 4.144000559327279e-01, 6.845975817877503e-01, 4.944840913356350e-02, 4.402759350876731e-01, 9.010432195863607e-01, 2.757405514679618e-02, 4.527181508614157e-01, 9.952152161758490e-01, 4.925532618002394e-03, 5.579522261495289e-01, 1.761582819832321e-01, 6.188238970001386e-02, 5.753347655779147e-01, 5.292288412881652e-01, 5.168460836650747e-02, 5.934515082334419e-01, 8.046750403390426e-01, 3.415204165486681e-02, 6.055261390114883e-01, 9.669899640916929e-01, 1.323441713883155e-02, 7.154909664472028e-01, 3.523839334728593e-01, 4.846803145137081e-02, 7.268010683419225e-01, 6.783203685825908e-01, 3.602990566434557e-02, 7.296280734021068e-01, 9.091222948855520e-01, 1.863026896557024e-02, 7.685318705419301e-01, 9.932162015282956e-01, 4.367730318652883e-03, 8.289969889442530e-01, 1.584524092289804e-01, 4.058748981808668e-02, 8.365897953958084e-01, 5.258522155386184e-01, 3.319785897530852e-02, 8.316719666113968e-01, 8.204370730256448e-01, 2.075970864095848e-02, 8.871860429369979e-01, 9.163805447338561e-01, 3.893022115627323e-03, 8.690236026560919e-01, 9.647484296790166e-01, 7.335685071586549e-03, 9.187533470826308e-01, 3.499840453590620e-01, 2.626466692941504e-02, 9.134702524025524e-01, 7.013486396820177e-01, 1.955810454123038e-02, 9.541746623132917e-01, 8.410023019359270e-01, 6.965850473298985e-03, 9.475721149242626e-01, 9.269025845479085e-01, 5.802838252293684e-03, 9.568636719971328e-01, 9.889975414443256e-01, 3.098510717291335e-03, 9.700539003410525e-01, 1.462690094803091e-01, 1.684436628110278e-02, 9.961052572257075e-01, 3.632658507871842e-01, 5.146979760206008e-03, 9.696607933775031e-01, 5.511605113182357e-01, 1.411592369017692e-02, 9.930889935835162e-01, 7.238541535398172e-01, 4.366281743177327e-03, 9.861368659264031e-01, 8.483603521407402e-01, 2.558788266617852e-03, 9.703225207695876e-01, 9.617299191521864e-01, -3.662923776765502e-03, 9.877929897983208e-01, 9.572988789445298e-01, 3.811762688023674e-03, 1.110886537223573e-02, 3.805232776392568e-01, 7.048655460716827e-02, 2.656770164537742e-02, 7.008980905724834e-01, 5.369718262729808e-02, 5.439701833656795e-02, 9.100556589521321e-01, 2.989488274476401e-02, 7.142859722990462e-02, 9.946044556670760e-01, 5.849868982269740e-03, -1.887353707822548e-01, 1.995554510347695e-01, 7.344881047788419e-02, -5.455267403580432e-01, 2.130905764129401e-01, 6.209984035978915e-02, -8.131809651972651e-01, 2.388895084628246e-01, 4.184384427163037e-02, -9.863343822191492e-01, 2.038938767743064e-01, 5.131075694716339e-04, -9.643706705539593e-01, 2.698246240215212e-01, 1.712164807226155e-02, -3.686184349052021e-01, 3.928345180603147e-01, 6.489527207591311e-02, -6.845975817877503e-01, 4.144000559327279e-01, 4.944840913356350e-02, -9.010432195863607e-01, 4.402759350876731e-01, 2.757405514679618e-02, -9.952152161758490e-01, 4.527181508614157e-01, 4.925532618002394e-03, -1.761582819832321e-01, 5.579522261495289e-01, 6.188238970001386e-02, -5.292288412881652e-01, 5.753347655779147e-01, 5.168460836650747e-02, -8.046750403390426e-01, 5.934515082334419e-01, 3.415204165486681e-02, -9.669899640916929e-01, 6.055261390114883e-01, 1.323441713883155e-02, -3.523839334728593e-01, 7.154909664472028e-01, 4.846803145137081e-02, -6.783203685825908e-01, 7.268010683419225e-01, 3.602990566434557e-02, -9.091222948855520e-01, 7.296280734021068e-01, 1.863026896557024e-02, -9.932162015282956e-01, 7.685318705419301e-01, 4.367730318652883e-03, -1.584524092289804e-01, 8.289969889442530e-01, 4.058748981808668e-02, -5.258522155386184e-01, 8.365897953958084e-01, 3.319785897530852e-02, -8.204370730256448e-01, 8.316719666113968e-01, 2.075970864095848e-02, -9.163805447338561e-01, 8.871860429369979e-01, 3.893022115627323e-03, -9.647484296790166e-01, 8.690236026560919e-01, 7.335685071586549e-03, -3.499840453590620e-01, 9.187533470826308e-01, 2.626466692941504e-02, -7.013486396820177e-01, 9.134702524025524e-01, 1.955810454123038e-02, -8.410023019359270e-01, 9.541746623132917e-01, 6.965850473298985e-03, -9.269025845479085e-01, 9.475721149242626e-01, 5.802838252293684e-03, -9.889975414443256e-01, 9.568636719971328e-01, 3.098510717291335e-03, -1.462690094803091e-01, 9.700539003410525e-01, 1.684436628110278e-02, -3.632658507871842e-01, 9.961052572257075e-01, 5.146979760206008e-03, -5.511605113182357e-01, 9.696607933775031e-01, 1.411592369017692e-02, -7.238541535398172e-01, 9.930889935835162e-01, 4.366281743177327e-03, -8.483603521407402e-01, 9.861368659264031e-01, 2.558788266617852e-03, -9.617299191521864e-01, 9.703225207695876e-01, -3.662923776765502e-03, -9.572988789445298e-01, 9.877929897983208e-01, 3.811762688023674e-03, -3.805232776392568e-01, 1.110886537223573e-02, 7.048655460716827e-02, -7.008980905724834e-01, 2.656770164537742e-02, 5.369718262729808e-02, -9.100556589521321e-01, 5.439701833656795e-02, 2.989488274476401e-02, -9.946044556670760e-01, 7.142859722990462e-02, 5.849868982269740e-03, -1.995554510347695e-01, -1.887353707822548e-01, 7.344881047788419e-02, -2.130905764129401e-01, -5.455267403580432e-01, 6.209984035978915e-02, -2.388895084628246e-01, -8.131809651972651e-01, 4.184384427163037e-02, -2.038938767743064e-01, -9.863343822191492e-01, 5.131075694716339e-04, -2.698246240215212e-01, -9.643706705539593e-01, 1.712164807226155e-02, -3.928345180603147e-01, -3.686184349052021e-01, 6.489527207591311e-02, -4.144000559327279e-01, -6.845975817877503e-01, 4.944840913356350e-02, -4.402759350876731e-01, -9.010432195863607e-01, 2.757405514679618e-02, -4.527181508614157e-01, -9.952152161758490e-01, 4.925532618002394e-03, -5.579522261495289e-01, -1.761582819832321e-01, 6.188238970001386e-02, -5.753347655779147e-01, -5.292288412881652e-01, 5.168460836650747e-02, -5.934515082334419e-01, -8.046750403390426e-01, 3.415204165486681e-02, -6.055261390114883e-01, -9.669899640916929e-01, 1.323441713883155e-02, -7.154909664472028e-01, -3.523839334728593e-01, 4.846803145137081e-02, -7.268010683419225e-01, -6.783203685825908e-01, 3.602990566434557e-02, -7.296280734021068e-01, -9.091222948855520e-01, 1.863026896557024e-02, -7.685318705419301e-01, -9.932162015282956e-01, 4.367730318652883e-03, -8.289969889442530e-01, -1.584524092289804e-01, 4.058748981808668e-02, -8.365897953958084e-01, -5.258522155386184e-01, 3.319785897530852e-02, -8.316719666113968e-01, -8.204370730256448e-01, 2.075970864095848e-02, -8.871860429369979e-01, -9.163805447338561e-01, 3.893022115627323e-03, -8.690236026560919e-01, -9.647484296790166e-01, 7.335685071586549e-03, -9.187533470826308e-01, -3.499840453590620e-01, 2.626466692941504e-02, -9.134702524025524e-01, -7.013486396820177e-01, 1.955810454123038e-02, -9.541746623132917e-01, -8.410023019359270e-01, 6.965850473298985e-03, -9.475721149242626e-01, -9.269025845479085e-01, 5.802838252293684e-03, -9.568636719971328e-01, -9.889975414443256e-01, 3.098510717291335e-03, -9.700539003410525e-01, -1.462690094803091e-01, 1.684436628110278e-02, -9.961052572257075e-01, -3.632658507871842e-01, 5.146979760206008e-03, -9.696607933775031e-01, -5.511605113182357e-01, 1.411592369017692e-02, -9.930889935835162e-01, -7.238541535398172e-01, 4.366281743177327e-03, -9.861368659264031e-01, -8.483603521407402e-01, 2.558788266617852e-03, -9.703225207695876e-01, -9.617299191521864e-01, -3.662923776765502e-03, -9.877929897983208e-01, -9.572988789445298e-01, 3.811762688023674e-03, -1.110886537223573e-02, -3.805232776392568e-01, 7.048655460716827e-02, -2.656770164537742e-02, -7.008980905724834e-01, 5.369718262729808e-02, -5.439701833656795e-02, -9.100556589521321e-01, 2.989488274476401e-02, -7.142859722990462e-02, -9.946044556670760e-01, 5.849868982269740e-03, 1.887353707822548e-01, -1.995554510347695e-01, 7.344881047788419e-02, 5.455267403580432e-01, -2.130905764129401e-01, 6.209984035978915e-02, 8.131809651972651e-01, -2.388895084628246e-01, 4.184384427163037e-02, 9.863343822191492e-01, -2.038938767743064e-01, 5.131075694716339e-04, 9.643706705539593e-01, -2.698246240215212e-01, 1.712164807226155e-02, 3.686184349052021e-01, -3.928345180603147e-01, 6.489527207591311e-02, 6.845975817877503e-01, -4.144000559327279e-01, 4.944840913356350e-02, 9.010432195863607e-01, -4.402759350876731e-01, 2.757405514679618e-02, 9.952152161758490e-01, -4.527181508614157e-01, 4.925532618002394e-03, 1.761582819832321e-01, -5.579522261495289e-01, 6.188238970001386e-02, 5.292288412881652e-01, -5.753347655779147e-01, 5.168460836650747e-02, 8.046750403390426e-01, -5.934515082334419e-01, 3.415204165486681e-02, 9.669899640916929e-01, -6.055261390114883e-01, 1.323441713883155e-02, 3.523839334728593e-01, -7.154909664472028e-01, 4.846803145137081e-02, 6.783203685825908e-01, -7.268010683419225e-01, 3.602990566434557e-02, 9.091222948855520e-01, -7.296280734021068e-01, 1.863026896557024e-02, 9.932162015282956e-01, -7.685318705419301e-01, 4.367730318652883e-03, 1.584524092289804e-01, -8.289969889442530e-01, 4.058748981808668e-02, 5.258522155386184e-01, -8.365897953958084e-01, 3.319785897530852e-02, 8.204370730256448e-01, -8.316719666113968e-01, 2.075970864095848e-02, 9.163805447338561e-01, -8.871860429369979e-01, 3.893022115627323e-03, 9.647484296790166e-01, -8.690236026560919e-01, 7.335685071586549e-03, 3.499840453590620e-01, -9.187533470826308e-01, 2.626466692941504e-02, 7.013486396820177e-01, -9.134702524025524e-01, 1.955810454123038e-02, 8.410023019359270e-01, -9.541746623132917e-01, 6.965850473298985e-03, 9.269025845479085e-01, -9.475721149242626e-01, 5.802838252293684e-03, 9.889975414443256e-01, -9.568636719971328e-01, 3.098510717291335e-03, 1.462690094803091e-01, -9.700539003410525e-01, 1.684436628110278e-02, 3.632658507871842e-01, -9.961052572257075e-01, 5.146979760206008e-03, 5.511605113182357e-01, -9.696607933775031e-01, 1.411592369017692e-02, 7.238541535398172e-01, -9.930889935835162e-01, 4.366281743177327e-03, 8.483603521407402e-01, -9.861368659264031e-01, 2.558788266617852e-03, 9.617299191521864e-01, -9.703225207695876e-01, -3.662923776765502e-03, 9.572988789445298e-01, -9.877929897983208e-01, 3.811762688023674e-03, 3.805232776392568e-01, -1.110886537223573e-02, 7.048655460716827e-02, 7.008980905724834e-01, -2.656770164537742e-02, 5.369718262729808e-02, 9.100556589521321e-01, -5.439701833656795e-02, 2.989488274476401e-02, 9.946044556670760e-01, -7.142859722990462e-02, 5.849868982269740e-03, 0.000000000000000e+00, 0.000000000000000e+00, 7.642673546157117e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr30 ( ) /******************************************************************************/ /* Purpose: SMR30 returns the SMR rule of degree 30. Discussion: DEGREE: 30 XIAO-GIMBUTAS ESTIMATE: 166 MOELLER-RASPUTIN LOWER BOUND: 128 POINTS CARDINALITY: 167 NORM INF MOMS. RESIDUAL: 4.44089e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR30[3*167], the requested rule. */ { const int degree = 30; int order; double xw[3*167] = { -9.962695953022117e-01, -4.432378837780777e-01, 3.892630177515625e-03, -9.954114173626548e-01, 2.643526556843364e-01, 4.534201620636273e-03, -9.949269819594637e-01, 5.811458193160586e-01, 3.690610442210943e-03, -9.947791036081476e-01, -8.855392187642179e-02, 4.945989612847814e-03, -9.940513238712086e-01, -9.629012381127487e-01, 1.529980279640927e-03, -9.937009284888235e-01, -7.963627749974038e-01, 3.806704824353705e-03, -9.914129974533056e-01, 9.014542533695618e-01, 3.250642010093910e-03, -9.912570981356938e-01, 9.920951313914597e-01, 7.737085001637529e-04, -9.823632199537923e-01, 7.480690374805677e-01, 7.178107249226476e-03, -9.774418651328238e-01, -6.195633189134538e-01, 1.012484081798108e-02, -9.706544108267556e-01, -2.729784055929617e-01, 1.355763804090731e-02, -9.703903424063889e-01, -9.055390715558742e-01, 3.812394374457089e-03, -9.694574072185448e-01, 4.341328372213230e-01, 1.266804574330651e-02, -9.690407115962437e-01, 9.341819736700314e-02, 1.394979228044907e-02, -9.628802671374306e-01, -9.944496567185225e-01, 1.467879560257003e-03, -9.527399448624597e-01, 9.624994157286473e-01, 5.260700927456429e-03, -9.514132521336506e-01, -8.829098392211954e-01, 4.907118941168890e-03, -9.345046955574380e-01, 8.387639945783492e-01, 1.172001054121250e-02, -9.297146520577916e-01, 6.183698468095432e-01, 1.825632348450637e-02, -9.293314444347445e-01, -7.435030197515320e-01, 1.482615873422928e-02, -9.260149110397602e-01, -4.576849499932223e-01, 2.118348083720268e-02, -9.191878620227047e-01, -8.576798396420104e-02, 2.272620841149477e-02, -9.158350185631037e-01, 2.692275128511369e-01, 2.304963366190331e-02, -9.100695669977138e-01, -9.705559713759557e-01, 3.389088115670371e-03, -8.849735191782050e-01, -9.517932989228655e-01, 5.364563594773305e-03, -8.808666812238437e-01, 9.943730059848226e-01, 2.875519471334203e-03, -8.595716686462112e-01, 9.162278791175735e-01, 1.277710664857529e-02, -8.534233518158184e-01, -8.449957910924649e-01, 1.658096280776431e-02, -8.532333609215695e-01, -2.640000860004303e-01, 2.911278919189067e-02, -8.500329899131518e-01, 4.606540637675913e-01, 2.965001937514069e-02, -8.500143920229317e-01, -6.082069619792305e-01, 2.645748804192720e-02, -8.454025250745086e-01, 7.354238754987793e-01, 2.316800651593019e-02, -8.367229347951246e-01, 8.990264524394226e-02, 3.373621057431971e-02, -8.001347673092866e-01, -9.933648582561806e-01, 3.850587028399502e-03, -7.695116274930011e-01, -4.118889027088976e-01, 3.034893498538439e-02, -7.555576680657612e-01, 9.700425147325877e-01, 1.014999644279535e-02, -7.538647716837412e-01, -9.239685866946878e-01, 1.517356711708827e-02, -7.462990724884444e-01, -7.378827886824465e-01, 2.842931310687195e-02, -7.435929466472898e-01, 2.881630895110476e-01, 4.096030346800239e-02, -7.367231344637399e-01, 8.391743049717729e-01, 2.371595352428025e-02, -7.360783333119556e-01, 5.954890113834164e-01, 3.505479981673820e-02, -7.358357631903044e-01, -9.719765309238697e-02, 4.351226764743989e-02, -6.784458770150876e-01, -5.334164281651258e-01, 2.989403195373217e-02, -6.323142754086236e-01, -9.756719768853687e-01, 1.050599811708168e-02, -6.187413218331049e-01, 9.957562917639060e-01, 3.877141507331312e-03, -6.185869146576074e-01, -8.450016714647949e-01, 2.679921705344681e-02, -6.169842175479411e-01, 1.078804071378743e-01, 5.004167045667424e-02, -6.111438437236484e-01, -2.769425577256663e-01, 4.921646796209538e-02, -6.028539652311193e-01, 7.224797085796002e-01, 3.569438969843478e-02, -6.024568031179428e-01, 9.184105677646670e-01, 2.045602682082065e-02, -6.005929876780286e-01, 4.366655576377152e-01, 4.708714185331756e-02, -5.721476752159514e-01, -6.455779128450227e-01, 3.324435055130232e-02, -4.821587643276321e-01, -6.552912695280060e-02, 5.000579378842215e-02, -4.693508512146036e-01, -9.251514364207605e-01, 2.156482363680663e-02, -4.664112078735353e-01, -4.442366390500550e-01, 5.080565593127764e-02, -4.618665119648193e-01, -9.967579491010828e-01, 3.816748009617544e-03, -4.494143273856875e-01, 8.276470129494764e-01, 3.214308262649934e-02, -4.479160038776199e-01, 9.719485124728899e-01, 1.285989649810473e-02, -4.448715957603111e-01, 5.834707035806619e-01, 4.762439584447587e-02, -4.443814635842437e-01, 2.635713157455042e-01, 5.794249352621746e-02, -4.396198113354752e-01, -7.548969208964118e-01, 3.428224841661166e-02, -3.657940226461420e-01, -2.006533706330268e-01, 3.755546975167398e-02, -3.068381707303870e-01, -5.934274157892239e-01, 4.806217780828576e-02, -2.967415958337351e-01, 9.138477267737024e-01, 2.179374136407983e-02, -2.843623923239886e-01, -9.723235086081026e-01, 1.370404412777051e-02, -2.841773158810993e-01, -8.502214939557524e-01, 3.069486488068529e-02, -2.774502955157072e-01, 7.783800443548269e-02, 6.485367831170855e-02, -2.754822583044234e-01, 7.120414326285063e-01, 4.331711859331076e-02, -2.687871739273696e-01, 4.269851621611526e-01, 5.836353670839110e-02, -2.567436168928061e-01, 9.945360458235195e-01, 5.240836616582418e-03, -2.513499007569119e-01, -3.210828524214602e-01, 4.844638715484986e-02, -1.598590614299937e-01, 8.482252908489760e-01, 1.682030776305622e-02, -1.427799257579916e-01, -7.170460184955041e-01, 3.915904236744100e-02, -1.091853236351570e-01, 9.580261183543186e-01, 1.474658041347025e-02, -1.022992873259765e-01, -1.104785543258961e-01, 6.661932528640521e-02, -1.015416677161510e-01, -9.200676756186152e-01, 2.329668513832093e-02, -9.412889158601399e-02, -4.688846642550997e-01, 5.469210938337690e-02, -8.907000954456333e-02, 2.513213963693643e-01, 6.571594126284511e-02, -8.744029029762249e-02, 5.785381919572421e-01, 5.300724353175888e-02, -7.712410713884485e-02, -9.943194119246959e-01, 5.499112875651152e-03, -6.523473573640569e-02, 7.983102424409952e-01, 2.517854943448727e-02, -1.095721736720509e-02, -7.962228644585386e-01, 2.085587531389086e-02, 2.518303411594268e-02, 9.860853647300887e-01, 6.488533578084030e-03, 6.649367018609328e-02, 8.934359950937619e-01, 2.271313073139822e-02, 7.864093675987820e-02, -2.918181793455348e-01, 6.374671736339137e-02, 8.230594852824862e-02, -6.128716323876445e-01, 5.036688164272618e-02, 8.994595584407042e-02, 6.378488548700582e-02, 6.747055946750249e-02, 9.331414467274024e-02, 4.264112897435890e-01, 5.762076485371190e-02, 1.036274934285688e-01, -9.654170079830869e-01, 1.528097665571535e-02, 1.062379679357248e-01, 7.045294352966426e-01, 4.336208027576573e-02, 1.070563588959354e-01, -8.518818738785342e-01, 2.527278435887043e-02, 1.702241024982263e-01, 9.972150656853269e-01, 2.645999354440326e-03, 1.941498146775931e-01, 9.416958566669819e-01, 1.299271762491835e-02, 2.382851494989787e-01, 2.794121411070278e-01, 3.900619436018987e-02, 2.597216208260207e-01, -4.600772470944473e-01, 5.666664739064187e-02, 2.633608715116675e-01, -7.382703813686664e-01, 4.094242606634930e-02, 2.680422728449567e-01, -1.237008749685800e-01, 6.430917939062639e-02, 2.709535102430495e-01, -9.946277851787007e-01, 4.995330134500549e-03, 2.717768185055764e-01, 8.206386870412736e-01, 3.322490081159339e-02, 2.860865029604455e-01, 5.750046796390000e-01, 5.163715243954464e-02, 2.919629207724195e-01, -9.126523283762582e-01, 2.290547444002006e-02, 3.221994725736586e-01, 9.739873036170995e-01, 9.709787500403095e-03, 3.238916864531801e-01, 1.871152636079638e-01, 3.786953410800285e-02, 4.326883824490022e-01, 9.054252916722543e-01, 2.179426150033381e-02, 4.334112862933560e-01, -6.113886295669552e-01, 4.619739021387417e-02, 4.393894435610020e-01, -3.042985645127746e-01, 5.679993404477694e-02, 4.457707596424073e-01, 4.135856200591693e-01, 5.369039030443149e-02, 4.480222422807119e-01, -9.699096000438506e-01, 1.284524266997323e-02, 4.481245405662154e-01, 7.226095819651619e-01, 3.875267371369672e-02, 4.548106144879556e-01, -8.323081975450645e-01, 3.014220006377240e-02, 4.581780858586828e-01, 3.469977407352812e-02, 5.500861406890217e-02, 4.703860920439117e-01, 9.947418373536940e-01, 3.780014001903630e-03, 5.804007745393400e-01, 9.618079992566787e-01, 1.161053553740031e-02, 5.833354854251170e-01, 5.930231848340044e-01, 3.799033003205411e-02, 5.930856097062387e-01, 2.373877943575770e-01, 5.084589261923814e-02, 5.965641353176369e-01, -4.730726160978409e-01, 4.627028009154758e-02, 6.019409638213139e-01, 8.388951235924025e-01, 2.676118014960469e-02, 6.025337137784932e-01, -1.408542441091471e-01, 4.904466589737274e-02, 6.035685372806677e-01, -9.184860893047954e-01, 1.930940686592521e-02, 6.054544565337695e-01, -9.957236600655665e-01, 3.768877425809975e-03, 6.057207734020840e-01, -7.322512010108688e-01, 3.339567747677206e-02, 6.875670307888612e-01, 4.627278058070344e-01, 2.947065495054275e-02, 7.023177466590736e-01, 9.928205536558483e-01, 3.885728390925337e-03, 7.195170980700635e-01, 7.305873329417671e-01, 2.847807814619499e-02, 7.238446910374424e-01, 5.561690048457465e-02, 4.419832996135477e-02, 7.240210652949475e-01, -3.007014261972007e-01, 3.439575422315153e-02, 7.371020667711963e-01, 9.252595362582127e-01, 1.536785201342859e-02, 7.384195707426535e-01, -8.438380733571678e-01, 2.235510768691872e-02, 7.405499034588905e-01, -6.188990137603514e-01, 3.246295613758222e-02, 7.440604692168796e-01, -9.713408547253161e-01, 9.607479120263517e-03, 7.685900922442980e-01, 3.465057508910690e-01, 2.776338673026480e-02, 7.954329735534109e-01, -4.109341533557886e-01, 1.857369110661608e-02, 8.140461518894649e-01, 5.964637576338830e-01, 2.604275460103966e-02, 8.298209513184971e-01, 8.417188363600208e-01, 1.770911353888088e-02, 8.337255108415906e-01, -1.237521420338796e-01, 3.477363818844623e-02, 8.450338467672839e-01, 9.803989034344278e-01, 6.318914249788773e-03, 8.499053424569023e-01, -7.513598048680518e-01, 2.131459811435620e-02, 8.505494208495452e-01, 2.034903566768354e-01, 2.898559224509574e-02, 8.534375448038033e-01, -9.226288272371441e-01, 1.212457346099878e-02, 8.655964520277111e-01, -9.954157466834472e-01, 2.574302443652325e-03, 8.685850946711341e-01, -5.228140183251604e-01, 2.217355893689261e-02, 8.943292225583946e-01, 4.595261118728072e-01, 2.187541877465294e-02, 9.006171857996548e-01, 7.180830553785196e-01, 1.636671200905273e-02, 9.075612511758971e-01, 9.205481097893897e-01, 7.889096906380533e-03, 9.108243128554621e-01, -3.093255095271237e-01, 2.247790975299426e-02, 9.242976221883992e-01, 4.079387457619278e-02, 2.269368117980306e-02, 9.336780332840566e-01, -8.547607055279791e-01, 1.092484696375260e-02, 9.387265328139205e-01, -6.592057435752858e-01, 1.490210373726517e-02, 9.446037800832677e-01, -9.693489449953010e-01, 4.898536170872587e-03, 9.479408185885666e-01, 9.575588886071763e-01, 2.997578905083853e-03, 9.495353577945340e-01, 3.186288923971684e-01, 1.513823435852647e-02, 9.520042531581182e-01, 9.980612732198122e-01, 1.202725556369299e-03, 9.525533437955715e-01, 8.141508738554407e-01, 8.188829215294804e-03, 9.641212854564828e-01, 5.940993549166920e-01, 1.301642639806175e-02, 9.664957432070073e-01, -1.581005636616589e-01, 1.326365917546207e-02, 9.683352852913805e-01, -4.679098763469015e-01, 1.225127303731581e-02, 9.842518603414726e-01, 1.836896830126595e-01, 8.499315253887113e-03, 9.846619943535420e-01, -7.799717728866817e-01, 5.965044619806711e-03, 9.852422761112879e-01, 8.861974810453618e-01, 3.934308818294981e-03, 9.891880847194036e-01, -9.191530362600531e-01, 3.253203726665571e-03, 9.897009343507575e-01, -9.931297876049714e-01, 7.600893882678305e-04, 9.922603853169076e-01, 9.719335327349238e-01, 1.435167883917694e-03, 9.936305551862550e-01, -1.510608205895069e-02, 4.418790732679170e-03, 9.941000705774204e-01, 4.425041797515226e-01, 5.144918479154803e-03, 9.941101780466588e-01, 7.280757215015392e-01, 3.867855257271212e-03, 9.950172763856067e-01, -6.114888124308911e-01, 3.581501248642710e-03, 9.951171305214461e-01, -3.168085664517462e-01, 4.336888090484613e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr31 ( ) /******************************************************************************/ /* Purpose: SMR31 returns the SMR rule of degree 31. Discussion: DEGREE: 31 POINTS CARDINALITY: 172 NORM INF MOMS. RESIDUAL: 2.07733e-16 SUM NEGATIVE WEIGHTS: 0 Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR31[3*172], the requested rule. */ { const int degree = 31; int order; double xw[3*172] = { -9.955496296346447e-01, -2.757893737712724e-02, 3.952828126515200e-03, -9.952093636515903e-01, 3.078672596656606e-01, 4.465674926565594e-03, -9.949219293696708e-01, -4.293313749761587e-01, 4.734653786951149e-03, -9.947777289433332e-01, 6.082007753656643e-01, 3.687083706716762e-03, -9.947711657807963e-01, -7.291135575065112e-01, 3.567753282746309e-03, -9.898665478412323e-01, 9.927280498683013e-01, 7.900353557535870e-04, -9.896201133119824e-01, 9.148303804608103e-01, 3.248339703605548e-03, -9.894221797194233e-01, -8.963309110540300e-01, 2.680927365865937e-03, -9.891613515312806e-01, -9.745514901909140e-01, 1.655992514072826e-03, -9.847115459075805e-01, 7.760597415217361e-01, 5.791133619187665e-03, -9.822216835749852e-01, -1.858981354343795e-01, 6.958199118327337e-03, -9.688290280461213e-01, 1.346435658626332e-01, 1.394269523344841e-02, -9.681141160941403e-01, 4.618217642366224e-01, 1.249777398451523e-02, -9.680653032337156e-01, -5.892528308394193e-01, 1.229021380297149e-02, -9.621105113004819e-01, -8.293868331873653e-01, 7.471001060877273e-03, -9.565888110363844e-01, -2.955138587991344e-01, 1.254260666301101e-02, -9.469790068617381e-01, -9.968106327216375e-01, 1.362946951896202e-03, -9.464971595212447e-01, 9.668261260754121e-01, 5.057521838548747e-03, -9.404074979033311e-01, 6.594952112812067e-01, 1.377422226180171e-02, -9.386071275818250e-01, -9.447266737672761e-01, 4.408092682116897e-03, -9.340283957363398e-01, 8.488905145274273e-01, 1.096662540259541e-02, -9.198052061978775e-01, -5.136717870605163e-02, 2.325563069027812e-02, -9.125069846160752e-01, 2.965175005985400e-01, 2.317349030282014e-02, -9.123983052278903e-01, -7.247489113905282e-01, 1.632786836774140e-02, -9.068677187647651e-01, -4.445873104662686e-01, 2.188675357029590e-02, -9.055659136341394e-01, -9.175374320754922e-01, 5.491385731273308e-03, -8.774069425333418e-01, 5.321738640857572e-01, 1.982544206254361e-02, -8.706320657782287e-01, 9.948789461128177e-01, 2.722659841902658e-03, -8.559872594373867e-01, 9.178926105523973e-01, 1.239374987476905e-02, -8.487413771127457e-01, 7.442213847925441e-01, 2.151797328509033e-02, -8.475645530398180e-01, -2.209115672323100e-01, 2.832498125828596e-02, -8.449119525088070e-01, -9.790174363372415e-01, 6.490465735794747e-03, -8.388493423999792e-01, -8.442105015209006e-01, 1.741719558530361e-02, -8.320878189683433e-01, 1.145771146767555e-01, 3.439308836971967e-02, -8.290267482829156e-01, -5.972601809233826e-01, 2.702368742037717e-02, -8.034575695173080e-01, 4.094748377144478e-01, 2.509623581221141e-02, -7.760357257316899e-01, -3.548922765484357e-01, 2.268460217050950e-02, -7.509248997072229e-01, 9.692871949519603e-01, 9.979338077119586e-03, -7.387062791288017e-01, 8.375261790603876e-01, 2.279562252991799e-02, -7.382550697652832e-01, 6.100825289389331e-01, 3.294515892662506e-02, -7.363260972595160e-01, -9.253776418251520e-01, 1.584321780127163e-02, -7.298334096260589e-01, -7.360653624762051e-01, 2.903866461556321e-02, -7.251547137883499e-01, -6.996863675606819e-02, 4.370775795972342e-02, -7.113581145093257e-01, 2.702313355967329e-01, 3.714680586412551e-02, -7.071807441641457e-01, -9.944876613499617e-01, 3.480514785349274e-03, -7.039820156247814e-01, -4.654525806221809e-01, 3.069975493264972e-02, -6.184086297782805e-01, 9.955258965189302e-01, 3.881568620917195e-03, -6.077218310997582e-01, 9.148419450019303e-01, 1.991703166278273e-02, -6.034954341682667e-01, 7.237932367497814e-01, 3.389141507458358e-02, -6.013041765209197e-01, -8.425771299300596e-01, 2.713141610820029e-02, -5.995171839175090e-01, 4.575307437971686e-01, 4.553732637405976e-02, -5.987747492273197e-01, -2.555243484270794e-01, 5.002646306167845e-02, -5.927778548664734e-01, -6.036103424582507e-01, 3.911054484449386e-02, -5.901440107668018e-01, 1.060198712260184e-01, 4.821287790488322e-02, -5.784913015945842e-01, -9.666586674693635e-01, 1.173664788888666e-02, -4.588198301439419e-01, 9.693486333009571e-01, 1.313666284266309e-02, -4.567271123348693e-01, 8.253817273249296e-01, 3.056135412459628e-02, -4.537069399424983e-01, -4.303216184624180e-01, 5.246566824474671e-02, -4.528331550744691e-01, -6.765968580901423e-02, 5.225928157201407e-02, -4.472686223717481e-01, -7.308744706376724e-01, 3.864973973872291e-02, -4.434291687664805e-01, 2.865506865407869e-01, 5.621007974099101e-02, -4.406325252658753e-01, -9.951861574277165e-01, 3.942690951645789e-03, -4.393137941615871e-01, 5.923410411408412e-01, 4.592935024129758e-02, -4.286656345744738e-01, -9.110553186427821e-01, 2.189185972353471e-02, -3.378031689425344e-01, -2.004187762207998e-01, 2.823318588167162e-02, -3.044251614615452e-01, 9.086361595553306e-01, 2.309284749670321e-02, -2.928456536563019e-01, -9.712927609897977e-01, 1.017404407973292e-02, -2.912502627343641e-01, -5.842088648173278e-01, 4.974674817513107e-02, -2.811322114845757e-01, 9.943304190452436e-01, 5.253366355617619e-03, -2.766540265223294e-01, 7.185096774489679e-01, 4.096115975792532e-02, -2.748922538693097e-01, 1.033654937792176e-01, 6.398840590311936e-02, -2.658691893989600e-01, -8.278100124889359e-01, 3.306728584127568e-02, -2.638448388909397e-01, 4.383380334499116e-01, 5.665306321736824e-02, -2.509255177911933e-01, -2.932348684939356e-01, 4.608256320492199e-02, -1.873924507634162e-01, -9.941405263712252e-01, 2.154767334708834e-03, -1.623725117112911e-01, -9.393770105705175e-01, 1.234737326017698e-02, -1.321095433217891e-01, 8.321796504932829e-01, 2.707477268062864e-02, -1.248486574319150e-01, 9.596829430320998e-01, 1.572343508516491e-02, -1.160794606213587e-01, -7.051458563086379e-01, 3.871928288060491e-02, -9.862627315703923e-02, -8.426927609780714e-02, 6.684928152189812e-02, -9.324738040737274e-02, -4.443546718135858e-01, 5.655998204740964e-02, -9.000433791884935e-02, 5.901154946888999e-01, 5.092306051816564e-02, -8.193230202578333e-02, 2.677472052197009e-01, 6.437330348690143e-02, -4.921447280582370e-02, -8.948195724899560e-01, 2.115122029930462e-02, -2.766233245972495e-02, -9.904468342292223e-01, 6.140727614369517e-03, -1.010525418227002e-02, 7.724429823094344e-01, 1.672977767715024e-02, 1.010525418227002e-02, -7.724429823094344e-01, 1.672977767715024e-02, 2.766233245972495e-02, 9.904468342292223e-01, 6.140727614369517e-03, 4.921447280582370e-02, 8.948195724899560e-01, 2.115122029930462e-02, 8.193230202578333e-02, -2.677472052197009e-01, 6.437330348690143e-02, 9.000433791884935e-02, -5.901154946888999e-01, 5.092306051816564e-02, 9.324738040737274e-02, 4.443546718135858e-01, 5.655998204740964e-02, 9.862627315703923e-02, 8.426927609780714e-02, 6.684928152189812e-02, 1.160794606213587e-01, 7.051458563086379e-01, 3.871928288060491e-02, 1.248486574319150e-01, -9.596829430320998e-01, 1.572343508516491e-02, 1.321095433217891e-01, -8.321796504932829e-01, 2.707477268062864e-02, 1.623725117112911e-01, 9.393770105705175e-01, 1.234737326017698e-02, 1.873924507634162e-01, 9.941405263712252e-01, 2.154767334708834e-03, 2.509255177911933e-01, 2.932348684939356e-01, 4.608256320492199e-02, 2.638448388909397e-01, -4.383380334499116e-01, 5.665306321736824e-02, 2.658691893989600e-01, 8.278100124889359e-01, 3.306728584127568e-02, 2.748922538693097e-01, -1.033654937792176e-01, 6.398840590311936e-02, 2.766540265223294e-01, -7.185096774489679e-01, 4.096115975792532e-02, 2.811322114845757e-01, -9.943304190452436e-01, 5.253366355617619e-03, 2.912502627343641e-01, 5.842088648173278e-01, 4.974674817513107e-02, 2.928456536563019e-01, 9.712927609897977e-01, 1.017404407973292e-02, 3.044251614615452e-01, -9.086361595553306e-01, 2.309284749670321e-02, 3.378031689425344e-01, 2.004187762207998e-01, 2.823318588167162e-02, 4.286656345744738e-01, 9.110553186427821e-01, 2.189185972353471e-02, 4.393137941615871e-01, -5.923410411408412e-01, 4.592935024129758e-02, 4.406325252658753e-01, 9.951861574277165e-01, 3.942690951645789e-03, 4.434291687664805e-01, -2.865506865407869e-01, 5.621007974099101e-02, 4.472686223717481e-01, 7.308744706376724e-01, 3.864973973872291e-02, 4.528331550744691e-01, 6.765968580901423e-02, 5.225928157201407e-02, 4.537069399424983e-01, 4.303216184624180e-01, 5.246566824474671e-02, 4.567271123348693e-01, -8.253817273249296e-01, 3.056135412459628e-02, 4.588198301439419e-01, -9.693486333009571e-01, 1.313666284266309e-02, 5.784913015945842e-01, 9.666586674693635e-01, 1.173664788888666e-02, 5.901440107668018e-01, -1.060198712260184e-01, 4.821287790488322e-02, 5.927778548664734e-01, 6.036103424582507e-01, 3.911054484449386e-02, 5.987747492273197e-01, 2.555243484270794e-01, 5.002646306167845e-02, 5.995171839175090e-01, -4.575307437971686e-01, 4.553732637405976e-02, 6.013041765209197e-01, 8.425771299300596e-01, 2.713141610820029e-02, 6.034954341682667e-01, -7.237932367497814e-01, 3.389141507458358e-02, 6.077218310997582e-01, -9.148419450019303e-01, 1.991703166278273e-02, 6.184086297782805e-01, -9.955258965189302e-01, 3.881568620917195e-03, 7.039820156247814e-01, 4.654525806221809e-01, 3.069975493264972e-02, 7.071807441641457e-01, 9.944876613499617e-01, 3.480514785349274e-03, 7.113581145093257e-01, -2.702313355967329e-01, 3.714680586412551e-02, 7.251547137883499e-01, 6.996863675606819e-02, 4.370775795972342e-02, 7.298334096260589e-01, 7.360653624762051e-01, 2.903866461556321e-02, 7.363260972595160e-01, 9.253776418251520e-01, 1.584321780127163e-02, 7.382550697652832e-01, -6.100825289389331e-01, 3.294515892662506e-02, 7.387062791288017e-01, -8.375261790603876e-01, 2.279562252991799e-02, 7.509248997072229e-01, -9.692871949519603e-01, 9.979338077119586e-03, 7.760357257316899e-01, 3.548922765484357e-01, 2.268460217050950e-02, 8.034575695173080e-01, -4.094748377144478e-01, 2.509623581221141e-02, 8.290267482829156e-01, 5.972601809233826e-01, 2.702368742037717e-02, 8.320878189683433e-01, -1.145771146767555e-01, 3.439308836971967e-02, 8.388493423999792e-01, 8.442105015209006e-01, 1.741719558530361e-02, 8.449119525088070e-01, 9.790174363372415e-01, 6.490465735794747e-03, 8.475645530398180e-01, 2.209115672323100e-01, 2.832498125828596e-02, 8.487413771127457e-01, -7.442213847925441e-01, 2.151797328509033e-02, 8.559872594373867e-01, -9.178926105523973e-01, 1.239374987476905e-02, 8.706320657782287e-01, -9.948789461128177e-01, 2.722659841902658e-03, 8.774069425333418e-01, -5.321738640857572e-01, 1.982544206254361e-02, 9.055659136341394e-01, 9.175374320754922e-01, 5.491385731273308e-03, 9.068677187647651e-01, 4.445873104662686e-01, 2.188675357029590e-02, 9.123983052278903e-01, 7.247489113905282e-01, 1.632786836774140e-02, 9.125069846160752e-01, -2.965175005985400e-01, 2.317349030282014e-02, 9.198052061978775e-01, 5.136717870605163e-02, 2.325563069027812e-02, 9.340283957363398e-01, -8.488905145274273e-01, 1.096662540259541e-02, 9.386071275818250e-01, 9.447266737672761e-01, 4.408092682116897e-03, 9.404074979033311e-01, -6.594952112812067e-01, 1.377422226180171e-02, 9.464971595212447e-01, -9.668261260754121e-01, 5.057521838548747e-03, 9.469790068617381e-01, 9.968106327216375e-01, 1.362946951896202e-03, 9.565888110363844e-01, 2.955138587991344e-01, 1.254260666301101e-02, 9.621105113004819e-01, 8.293868331873653e-01, 7.471001060877273e-03, 9.680653032337156e-01, 5.892528308394193e-01, 1.229021380297149e-02, 9.681141160941403e-01, -4.618217642366224e-01, 1.249777398451523e-02, 9.688290280461213e-01, -1.346435658626332e-01, 1.394269523344841e-02, 9.822216835749852e-01, 1.858981354343795e-01, 6.958199118327337e-03, 9.847115459075805e-01, -7.760597415217361e-01, 5.791133619187665e-03, 9.891613515312806e-01, 9.745514901909140e-01, 1.655992514072826e-03, 9.894221797194233e-01, 8.963309110540300e-01, 2.680927365865937e-03, 9.896201133119824e-01, -9.148303804608103e-01, 3.248339703605548e-03, 9.898665478412323e-01, -9.927280498683013e-01, 7.900353557535870e-04, 9.947711657807963e-01, 7.291135575065112e-01, 3.567753282746309e-03, 9.947777289433332e-01, -6.082007753656643e-01, 3.687083706716762e-03, 9.949219293696708e-01, 4.293313749761587e-01, 4.734653786951149e-03, 9.952093636515903e-01, -3.078672596656606e-01, 4.465674926565594e-03, 9.955496296346447e-01, 2.757893737712724e-02, 3.952828126515200e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr31s ( ) /******************************************************************************/ /* Purpose: SMR31S returns the rotationally invariant SMR rule of degree 31. Discussion: DEGREE: 31 ROTATIONALLY INVARIANT: (X, Y), (-Y, X), (-X, -Y), (Y, -X). POINTS CARDINALITY: 172 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 2.84496e-03, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR31S[3*172], the requested rule. */ { const int degree = 31; int order; double xw[3*172] = { -3.904806549599655e-02, -4.335393037614890e-02, 1.983961989686983e-02, 1.361882296394248e-01, 5.339885716159022e-01, 5.606290851087755e-02, -6.237898572307624e-02, 6.650680443410015e-01, 4.954926631685492e-02, 1.043081290265589e-01, 7.978188386199707e-01, 3.954352049027991e-02, 8.175663994339752e-02, 9.536337842041479e-01, 1.819407316290299e-02, 1.326909285794375e-01, 9.985967754075993e-01, 2.198741271493517e-03, 3.573200269387108e-01, 3.083812309752531e-02, 6.275382718191864e-02, 1.711251846005631e-01, 2.023226798991783e-01, 6.312752392564865e-02, 3.301558760691319e-01, 3.890928730006515e-01, 5.784814622385932e-02, 2.990255438330413e-01, 6.899278622152941e-01, 4.599430975257399e-02, 2.743945607890017e-01, 8.969874117510032e-01, 2.722794626350406e-02, 2.930343597574347e-01, 9.866371893249537e-01, 7.297804329258834e-03, 4.811149130969350e-01, 5.627097593501477e-01, 4.813376158498228e-02, 4.577756563378687e-01, 8.164852603726748e-01, 3.369675352996792e-02, 4.576603028130140e-01, 9.581255271049165e-01, 1.483779460064700e-02, 5.011717697509331e-01, 9.967514621307949e-01, 1.375821755782986e-03, 5.066615476347207e-01, 2.279797220835331e-01, 5.617345067788202e-02, 6.414891770627443e-01, 4.165197368067110e-01, 4.637789988502160e-02, 6.218437939846863e-01, 7.124331860455828e-01, 3.611694132069662e-02, 6.189705671147399e-01, 9.055771549707593e-01, 2.142589064473523e-02, 6.257695254024579e-01, 9.920525612037021e-01, 4.604781337595676e-03, 7.773558757000041e-01, 2.587982643359955e-01, 3.995025536141079e-02, 7.602620921526432e-01, 5.866806545003377e-01, 3.454986715153591e-02, 8.080239490235996e-01, 7.466133969130114e-01, -7.112408612016088e-04, 7.578022960040215e-01, 8.250210799807152e-01, 2.415241529771807e-02, 7.572699136795021e-01, 9.658518182607879e-01, 1.066322723829567e-02, 8.838106230790593e-01, 9.517499798893064e-02, 3.021344380673574e-02, 8.690870146928319e-01, 4.428796688721532e-01, 2.841376547681653e-02, 8.664188675223524e-01, 7.215367573910633e-01, 2.259238404505909e-02, 8.662357931878873e-01, 9.115792712550360e-01, 1.298465964143997e-02, 8.668639312200342e-01, 9.955281129139882e-01, 2.678582771124035e-03, 9.464084279482877e-01, 2.901035303505874e-01, 1.924589937161483e-02, 9.432116671128417e-01, 5.914789709627410e-01, 1.601261396001407e-02, 9.621603453340629e-01, 7.421113162391834e-01, 2.850658278802631e-03, 9.424551010209468e-01, 8.371341836669535e-01, 1.039571899572164e-02, 9.455364284543073e-01, 9.674856486387888e-01, 5.118117255528941e-03, 9.860981807821748e-01, 1.062982665761813e-01, 8.507743310820945e-03, 9.968354200302058e-01, 2.730105297008364e-01, 1.865603977042027e-03, 9.874604632806666e-01, 4.594240145656220e-01, 7.587442385373269e-03, 9.914773043212965e-01, 6.790808250554563e-01, 3.856894243772012e-03, 9.892581143506071e-01, 7.901123155310413e-01, 2.579633577944600e-03, 9.893015185233809e-01, 9.146983422640488e-01, 3.312018439594244e-03, 9.898894503566366e-01, 9.925022404589151e-01, 7.995136114829527e-04, 4.335393037614890e-02, -3.904806549599655e-02, 1.983961989686983e-02, -5.339885716159022e-01, 1.361882296394248e-01, 5.606290851087755e-02, -6.650680443410015e-01, -6.237898572307624e-02, 4.954926631685492e-02, -7.978188386199707e-01, 1.043081290265589e-01, 3.954352049027991e-02, -9.536337842041479e-01, 8.175663994339752e-02, 1.819407316290299e-02, -9.985967754075993e-01, 1.326909285794375e-01, 2.198741271493517e-03, -3.083812309752531e-02, 3.573200269387108e-01, 6.275382718191864e-02, -2.023226798991783e-01, 1.711251846005631e-01, 6.312752392564865e-02, -3.890928730006515e-01, 3.301558760691319e-01, 5.784814622385932e-02, -6.899278622152941e-01, 2.990255438330413e-01, 4.599430975257399e-02, -8.969874117510032e-01, 2.743945607890017e-01, 2.722794626350406e-02, -9.866371893249537e-01, 2.930343597574347e-01, 7.297804329258834e-03, -5.627097593501477e-01, 4.811149130969350e-01, 4.813376158498228e-02, -8.164852603726748e-01, 4.577756563378687e-01, 3.369675352996792e-02, -9.581255271049165e-01, 4.576603028130140e-01, 1.483779460064700e-02, -9.967514621307949e-01, 5.011717697509331e-01, 1.375821755782986e-03, -2.279797220835331e-01, 5.066615476347207e-01, 5.617345067788202e-02, -4.165197368067110e-01, 6.414891770627443e-01, 4.637789988502160e-02, -7.124331860455828e-01, 6.218437939846863e-01, 3.611694132069662e-02, -9.055771549707593e-01, 6.189705671147399e-01, 2.142589064473523e-02, -9.920525612037021e-01, 6.257695254024579e-01, 4.604781337595676e-03, -2.587982643359955e-01, 7.773558757000041e-01, 3.995025536141079e-02, -5.866806545003377e-01, 7.602620921526432e-01, 3.454986715153591e-02, -7.466133969130114e-01, 8.080239490235996e-01, -7.112408612016088e-04, -8.250210799807152e-01, 7.578022960040215e-01, 2.415241529771807e-02, -9.658518182607879e-01, 7.572699136795021e-01, 1.066322723829567e-02, -9.517499798893064e-02, 8.838106230790593e-01, 3.021344380673574e-02, -4.428796688721532e-01, 8.690870146928319e-01, 2.841376547681653e-02, -7.215367573910633e-01, 8.664188675223524e-01, 2.259238404505909e-02, -9.115792712550360e-01, 8.662357931878873e-01, 1.298465964143997e-02, -9.955281129139882e-01, 8.668639312200342e-01, 2.678582771124035e-03, -2.901035303505874e-01, 9.464084279482877e-01, 1.924589937161483e-02, -5.914789709627410e-01, 9.432116671128417e-01, 1.601261396001407e-02, -7.421113162391834e-01, 9.621603453340629e-01, 2.850658278802631e-03, -8.371341836669535e-01, 9.424551010209468e-01, 1.039571899572164e-02, -9.674856486387888e-01, 9.455364284543073e-01, 5.118117255528941e-03, -1.062982665761813e-01, 9.860981807821748e-01, 8.507743310820945e-03, -2.730105297008364e-01, 9.968354200302058e-01, 1.865603977042027e-03, -4.594240145656220e-01, 9.874604632806666e-01, 7.587442385373269e-03, -6.790808250554563e-01, 9.914773043212965e-01, 3.856894243772012e-03, -7.901123155310413e-01, 9.892581143506071e-01, 2.579633577944600e-03, -9.146983422640488e-01, 9.893015185233809e-01, 3.312018439594244e-03, -9.925022404589151e-01, 9.898894503566366e-01, 7.995136114829527e-04, 3.904806549599655e-02, 4.335393037614890e-02, 1.983961989686983e-02, -1.361882296394248e-01, -5.339885716159022e-01, 5.606290851087755e-02, 6.237898572307624e-02, -6.650680443410015e-01, 4.954926631685492e-02, -1.043081290265589e-01, -7.978188386199707e-01, 3.954352049027991e-02, -8.175663994339752e-02, -9.536337842041479e-01, 1.819407316290299e-02, -1.326909285794375e-01, -9.985967754075993e-01, 2.198741271493517e-03, -3.573200269387108e-01, -3.083812309752531e-02, 6.275382718191864e-02, -1.711251846005631e-01, -2.023226798991783e-01, 6.312752392564865e-02, -3.301558760691319e-01, -3.890928730006515e-01, 5.784814622385932e-02, -2.990255438330413e-01, -6.899278622152941e-01, 4.599430975257399e-02, -2.743945607890017e-01, -8.969874117510032e-01, 2.722794626350406e-02, -2.930343597574347e-01, -9.866371893249537e-01, 7.297804329258834e-03, -4.811149130969350e-01, -5.627097593501477e-01, 4.813376158498228e-02, -4.577756563378687e-01, -8.164852603726748e-01, 3.369675352996792e-02, -4.576603028130140e-01, -9.581255271049165e-01, 1.483779460064700e-02, -5.011717697509331e-01, -9.967514621307949e-01, 1.375821755782986e-03, -5.066615476347207e-01, -2.279797220835331e-01, 5.617345067788202e-02, -6.414891770627443e-01, -4.165197368067110e-01, 4.637789988502160e-02, -6.218437939846863e-01, -7.124331860455828e-01, 3.611694132069662e-02, -6.189705671147399e-01, -9.055771549707593e-01, 2.142589064473523e-02, -6.257695254024579e-01, -9.920525612037021e-01, 4.604781337595676e-03, -7.773558757000041e-01, -2.587982643359955e-01, 3.995025536141079e-02, -7.602620921526432e-01, -5.866806545003377e-01, 3.454986715153591e-02, -8.080239490235996e-01, -7.466133969130114e-01, -7.112408612016088e-04, -7.578022960040215e-01, -8.250210799807152e-01, 2.415241529771807e-02, -7.572699136795021e-01, -9.658518182607879e-01, 1.066322723829567e-02, -8.838106230790593e-01, -9.517499798893064e-02, 3.021344380673574e-02, -8.690870146928319e-01, -4.428796688721532e-01, 2.841376547681653e-02, -8.664188675223524e-01, -7.215367573910633e-01, 2.259238404505909e-02, -8.662357931878873e-01, -9.115792712550360e-01, 1.298465964143997e-02, -8.668639312200342e-01, -9.955281129139882e-01, 2.678582771124035e-03, -9.464084279482877e-01, -2.901035303505874e-01, 1.924589937161483e-02, -9.432116671128417e-01, -5.914789709627410e-01, 1.601261396001407e-02, -9.621603453340629e-01, -7.421113162391834e-01, 2.850658278802631e-03, -9.424551010209468e-01, -8.371341836669535e-01, 1.039571899572164e-02, -9.455364284543073e-01, -9.674856486387888e-01, 5.118117255528941e-03, -9.860981807821748e-01, -1.062982665761813e-01, 8.507743310820945e-03, -9.968354200302058e-01, -2.730105297008364e-01, 1.865603977042027e-03, -9.874604632806666e-01, -4.594240145656220e-01, 7.587442385373269e-03, -9.914773043212965e-01, -6.790808250554563e-01, 3.856894243772012e-03, -9.892581143506071e-01, -7.901123155310413e-01, 2.579633577944600e-03, -9.893015185233809e-01, -9.146983422640488e-01, 3.312018439594244e-03, -9.898894503566366e-01, -9.925022404589151e-01, 7.995136114829527e-04, -4.335393037614890e-02, 3.904806549599655e-02, 1.983961989686983e-02, 5.339885716159022e-01, -1.361882296394248e-01, 5.606290851087755e-02, 6.650680443410015e-01, 6.237898572307624e-02, 4.954926631685492e-02, 7.978188386199707e-01, -1.043081290265589e-01, 3.954352049027991e-02, 9.536337842041479e-01, -8.175663994339752e-02, 1.819407316290299e-02, 9.985967754075993e-01, -1.326909285794375e-01, 2.198741271493517e-03, 3.083812309752531e-02, -3.573200269387108e-01, 6.275382718191864e-02, 2.023226798991783e-01, -1.711251846005631e-01, 6.312752392564865e-02, 3.890928730006515e-01, -3.301558760691319e-01, 5.784814622385932e-02, 6.899278622152941e-01, -2.990255438330413e-01, 4.599430975257399e-02, 8.969874117510032e-01, -2.743945607890017e-01, 2.722794626350406e-02, 9.866371893249537e-01, -2.930343597574347e-01, 7.297804329258834e-03, 5.627097593501477e-01, -4.811149130969350e-01, 4.813376158498228e-02, 8.164852603726748e-01, -4.577756563378687e-01, 3.369675352996792e-02, 9.581255271049165e-01, -4.576603028130140e-01, 1.483779460064700e-02, 9.967514621307949e-01, -5.011717697509331e-01, 1.375821755782986e-03, 2.279797220835331e-01, -5.066615476347207e-01, 5.617345067788202e-02, 4.165197368067110e-01, -6.414891770627443e-01, 4.637789988502160e-02, 7.124331860455828e-01, -6.218437939846863e-01, 3.611694132069662e-02, 9.055771549707593e-01, -6.189705671147399e-01, 2.142589064473523e-02, 9.920525612037021e-01, -6.257695254024579e-01, 4.604781337595676e-03, 2.587982643359955e-01, -7.773558757000041e-01, 3.995025536141079e-02, 5.866806545003377e-01, -7.602620921526432e-01, 3.454986715153591e-02, 7.466133969130114e-01, -8.080239490235996e-01, -7.112408612016088e-04, 8.250210799807152e-01, -7.578022960040215e-01, 2.415241529771807e-02, 9.658518182607879e-01, -7.572699136795021e-01, 1.066322723829567e-02, 9.517499798893064e-02, -8.838106230790593e-01, 3.021344380673574e-02, 4.428796688721532e-01, -8.690870146928319e-01, 2.841376547681653e-02, 7.215367573910633e-01, -8.664188675223524e-01, 2.259238404505909e-02, 9.115792712550360e-01, -8.662357931878873e-01, 1.298465964143997e-02, 9.955281129139882e-01, -8.668639312200342e-01, 2.678582771124035e-03, 2.901035303505874e-01, -9.464084279482877e-01, 1.924589937161483e-02, 5.914789709627410e-01, -9.432116671128417e-01, 1.601261396001407e-02, 7.421113162391834e-01, -9.621603453340629e-01, 2.850658278802631e-03, 8.371341836669535e-01, -9.424551010209468e-01, 1.039571899572164e-02, 9.674856486387888e-01, -9.455364284543073e-01, 5.118117255528941e-03, 1.062982665761813e-01, -9.860981807821748e-01, 8.507743310820945e-03, 2.730105297008364e-01, -9.968354200302058e-01, 1.865603977042027e-03, 4.594240145656220e-01, -9.874604632806666e-01, 7.587442385373269e-03, 6.790808250554563e-01, -9.914773043212965e-01, 3.856894243772012e-03, 7.901123155310413e-01, -9.892581143506071e-01, 2.579633577944600e-03, 9.146983422640488e-01, -9.893015185233809e-01, 3.312018439594244e-03, 9.925022404589151e-01, -9.898894503566366e-01, 7.995136114829527e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr32 ( ) /******************************************************************************/ /* Purpose: SMR32 returns the SMR rule of degree 32. Discussion: DEGREE: 32 POINTS CARDINALITY: 189 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR32[3*189], the requested rule. */ { const int degree = 32; int order; double xw[3*189] = { -9.961443158376487e-01, -6.887572767977012e-01, 3.176265881806332e-03, -9.955150539895966e-01, 6.626971407249244e-01, 3.280265060152129e-03, -9.953673174806560e-01, -3.490320479685254e-02, 4.675890937959332e-03, -9.953301148958117e-01, 8.868398804037936e-01, 2.284058312905218e-03, -9.949536289138113e-01, -3.776956946778952e-01, 4.510771123627496e-03, -9.947581001027488e-01, 3.329076528027972e-01, 4.813891020740236e-03, -9.941343268456251e-01, 9.936112061216035e-01, 5.375112498857630e-04, -9.932824877107631e-01, -9.089949394457987e-01, 2.496612709475389e-03, -9.901718606538452e-01, -9.916407321742867e-01, 7.701138529293439e-04, -9.751008630866578e-01, 5.195777161215037e-01, 9.359036791703322e-03, -9.723977826522521e-01, 1.379087148825411e-01, 1.181986792189326e-02, -9.716123205144245e-01, -8.031016796856173e-01, 7.898808335562373e-03, -9.711842977478065e-01, -5.430726346303393e-01, 1.151176330250413e-02, -9.704055489123288e-01, 7.804584258849506e-01, 8.644818596664990e-03, -9.694199058988444e-01, 9.557067749284750e-01, 4.223280833025587e-03, -9.683411874625077e-01, -2.103323587190609e-01, 1.388419379439329e-02, -9.567808007888505e-01, -9.626987460709516e-01, 4.415431329419894e-03, -9.437425354265615e-01, 4.414088771841401e-01, 6.832573539725157e-03, -9.402218788200302e-01, 2.637892865469118e-01, 1.016860955591373e-02, -9.217551019965182e-01, 8.766772417195136e-01, 1.065665181655526e-02, -9.201345036606068e-01, -6.784459223365346e-01, 1.669398948968993e-02, -9.187137927257295e-01, -8.800989100992515e-01, 9.591064623401077e-03, -9.181947091905479e-01, -3.841738613909663e-01, 2.059591744256289e-02, -9.181637384564565e-01, 6.508430144443661e-01, 1.797283563699723e-02, -9.180866951945609e-01, -3.534323022121689e-02, 2.220774903033375e-02, -9.162233942094903e-01, 9.905273384808887e-01, 2.842827696268136e-03, -8.942375422053964e-01, -9.940021310654177e-01, 2.526325882162393e-03, -8.925004548103099e-01, 3.675577608041817e-01, 1.740206879654378e-02, -8.567376342667298e-01, 1.327464071241374e-01, 2.320747945421362e-02, -8.527049285163805e-01, -5.173331471639295e-01, 1.823135215166509e-02, -8.500346292659974e-01, 9.424744256560323e-01, 9.628825949603207e-03, -8.495936622225249e-01, -9.366589616617220e-01, 8.808214017805906e-03, -8.420958026145372e-01, 7.746496697546652e-01, 2.038980506805041e-02, -8.405229008591695e-01, -2.146465103993396e-01, 3.115686395982564e-02, -8.400130747502382e-01, -7.833828505922565e-01, 1.867972088056950e-02, -8.251520015911556e-01, 5.249712695416556e-01, 2.828908020393419e-02, -7.960746843247288e-01, -5.942990648607921e-01, 1.818720327698928e-02, -7.884200501943553e-01, 2.431068089032147e-01, 2.161873511289656e-02, -7.832198580393690e-01, 9.882893671904269e-01, 4.201313239030647e-03, -7.691956377515223e-01, -9.767479568418154e-01, 6.899965676153335e-03, -7.509072411287868e-01, -2.719575746898528e-02, 3.742043760045523e-02, -7.427461156257692e-01, 8.707027609907575e-01, 1.919642121969817e-02, -7.393846173049607e-01, -8.675695987661742e-01, 1.783672443816834e-02, -7.351194183949288e-01, -3.774494681673472e-01, 3.738148314393595e-02, -7.213398259926092e-01, 6.655990116514867e-01, 3.145176228542881e-02, -7.120543004828920e-01, 3.695232160325304e-01, 3.307881993920563e-02, -7.013785314229773e-01, -6.985386189309318e-01, 2.798941131398564e-02, -6.889047528617022e-01, 9.602183527522339e-01, 7.162747081491355e-03, -6.522280372437145e-01, -9.963873207253509e-01, 2.915037219722206e-03, -6.294141334037263e-01, -9.321941144432188e-01, 1.469357873203910e-02, -6.289236482231599e-01, 1.335531135944628e-01, 4.333999853931410e-02, -6.250143551668971e-01, -1.909837821647839e-01, 4.450992385162815e-02, -6.162479185474388e-01, 9.957592249563421e-01, 2.859071377057288e-03, -6.099556060400831e-01, -5.265301674660628e-01, 3.737333144755796e-02, -6.035692437775674e-01, 5.068951570189404e-01, 3.542431096817152e-02, -5.957398931434491e-01, 7.867544741614058e-01, 2.991054273162196e-02, -5.880720673778793e-01, 9.278338224868491e-01, 1.311047438579956e-02, -5.642943955874409e-01, -8.045488845290898e-01, 2.887379909514902e-02, -5.341453289511777e-01, -6.580308812592266e-01, 8.231881868308998e-03, -5.037785265285627e-01, 2.911261252936324e-01, 4.463898222861209e-02, -5.035520020780221e-01, 6.164678558380017e-01, 2.375628980516375e-02, -4.988776272469410e-01, -9.742698071302808e-01, 9.972534732384969e-03, -4.880950860424961e-01, -3.570330104924047e-01, 4.487865776129120e-02, -4.782873763449944e-01, -3.311071102284526e-02, 4.988651815705654e-02, -4.611263669660518e-01, 9.776538472171752e-01, 9.547170864365405e-03, -4.460940172737087e-01, -5.617163266009875e-01, 1.057733340125433e-02, -4.344215560289689e-01, 8.756198694104429e-01, 2.433238295221333e-02, -4.303464547840762e-01, -6.717475726140401e-01, 3.186301410699670e-02, -4.140771909387703e-01, -8.929909921505204e-01, 2.416865558364916e-02, -4.106621334605041e-01, 6.984143047391834e-01, 2.962064813188227e-02, -3.860009662973919e-01, 4.525655697932027e-01, 3.233946990986691e-02, -3.484793436284917e-01, -9.955996498738462e-01, 3.793172976457341e-03, -3.391951567840935e-01, 1.226820429084983e-01, 4.753908343744886e-02, -3.357809356696743e-01, -2.224706747610833e-01, 4.615497828125666e-02, -3.250278129966869e-01, 3.756115481576739e-01, 1.811254509132181e-02, -3.056089159859502e-01, -4.920724555612009e-01, 4.554710841070758e-02, -2.961214860982124e-01, 9.974527284699805e-01, 3.060823576685151e-03, -2.847932366229927e-01, -7.838348732823106e-01, 3.404104052520678e-02, -2.713478176538165e-01, 9.439481248438524e-01, 1.788329467008652e-02, -2.623891895744691e-01, 7.934963660422401e-01, 3.215348099890328e-02, -2.474541407740861e-01, -9.552431364916144e-01, 1.619837106218808e-02, -2.350008942691768e-01, 5.720771765501166e-01, 4.318782866021313e-02, -2.029577973797018e-01, -8.943368639764357e-02, 4.553097782784060e-02, -1.990670168540228e-01, 2.443876445322615e-01, 4.110363868615158e-02, -1.549456091317803e-01, -6.415558836799513e-01, 4.267610300710221e-02, -1.347331997024815e-01, -3.592068358004468e-01, 5.074071162472863e-02, -1.193212615110174e-01, -8.772608465003254e-01, 2.680897358145306e-02, -9.608366181884005e-02, 8.811478309456120e-01, 2.657134265114752e-02, -8.724674317423464e-02, 9.829741829813902e-01, 9.730625947232925e-03, -8.357262181047176e-02, 4.013140117181199e-01, 4.841144500118764e-02, -7.663657189401872e-02, -9.904382181654797e-01, 6.943274642780526e-03, -7.506542665125431e-02, 6.928415737526518e-01, 4.014342124479778e-02, -6.814560175398696e-02, 4.856261470453426e-02, 4.844065720660989e-02, 1.062473558727759e-02, -7.650288124962303e-01, 3.558799833293477e-02, 1.660017546118934e-02, -2.180239260173989e-01, 4.831607231731554e-02, 2.272453179480726e-02, -5.268496064565723e-01, 4.537055458358597e-02, 4.931212356041427e-02, -9.458603059209817e-01, 1.758790356906201e-02, 7.613140727060182e-02, 5.418798866250614e-01, 4.818153253307554e-02, 7.853294280823510e-02, 2.224042128166391e-01, 5.465464269071678e-02, 8.372125557287705e-02, 9.435131549853616e-01, 1.786409990626899e-02, 9.337887407842871e-02, 8.001140509910943e-01, 3.410288806782028e-02, 1.089074910813305e-01, -3.010076469468967e-02, 2.723309734230703e-02, 1.411908890075358e-01, 9.960717448990448e-01, 3.569321191002430e-03, 1.751413474655890e-01, -3.997261685283053e-01, 4.541750739879684e-02, 1.756054472282782e-01, -8.658339890881421e-01, 2.662577185951133e-02, 1.864598899283486e-01, -6.697382946526751e-01, 3.827435555232931e-02, 2.015902009534723e-01, -9.883931352101503e-01, 7.887679303175682e-03, 2.044045426953899e-01, -1.421919981535460e-01, 3.558139723631710e-02, 2.378932317489774e-01, 3.802228852670115e-01, 5.139141457782884e-02, 2.453419557836962e-01, 6.743539422739147e-01, 4.230440026330508e-02, 2.673924056388006e-01, 8.857386065554180e-01, 2.512168130932855e-02, 2.794321878522678e-01, 8.071672703911634e-02, 5.195889525876823e-02, 2.852750321228393e-01, 9.766942081801527e-01, 8.542384805675192e-03, 3.366919097882969e-01, -5.582614071904248e-01, 3.957781174651530e-02, 3.382140442187983e-01, -9.401781089769999e-01, 1.829722847974778e-02, 3.432642241169637e-01, -7.908855951561965e-01, 2.972251194098460e-02, 3.455152732870559e-01, -2.966429396872311e-01, 4.443118022793111e-02, 3.916837958172765e-01, 5.230067770696570e-01, 4.184441753133104e-02, 4.120040722413747e-01, 7.890179305884122e-01, 3.275969268754896e-02, 4.238156081215504e-01, 9.950429590804639e-01, 2.471843473128421e-03, 4.287156692418698e-01, 2.431531997648373e-01, 5.098740815593964e-02, 4.406392416503330e-01, -9.421351055228980e-02, 4.952434585624959e-02, 4.435754669288457e-01, 9.470031795306957e-01, 1.407985409716778e-02, 4.558595671892888e-01, -9.896709958176498e-01, 6.495456224860164e-03, 4.873417746475680e-01, -6.968841503239952e-01, 3.193924196689053e-02, 4.970448498999819e-01, -4.587836752559167e-01, 4.009098722323794e-02, 5.026246934476013e-01, -8.799728934510729e-01, 2.251236322931963e-02, 5.142480616051717e-01, 6.332295493966821e-01, 2.628145641746948e-02, 5.635007809294439e-01, 8.830661523985979e-01, 2.138158866985076e-02, 5.760177960925376e-01, 4.034390879812669e-01, 4.461725888680534e-02, 5.786951907000567e-01, 7.200913403128530e-02, 4.714224835011449e-02, 5.850240866763193e-01, 9.892820474952123e-01, 5.400839750748531e-03, 5.851379143554923e-01, -2.686399203625244e-01, 4.255109067146633e-02, 6.039149631797995e-01, -9.595657522395640e-01, 1.243029383620673e-02, 6.085940759319246e-01, 7.188102110043890e-01, 2.414801307071736e-02, 6.367455861073830e-01, -6.091087379827170e-01, 3.181043163725066e-02, 6.389771216716220e-01, -8.001263004879462e-01, 2.306016831136437e-02, 6.923848443111713e-01, 9.536668248480320e-01, 1.203359878310568e-02, 7.027796911591175e-01, -9.965270954213804e-01, 3.064467191573918e-03, 7.057525331398022e-01, -1.091230879464752e-01, 3.965634211274827e-02, 7.065644544049037e-01, 5.577016427729737e-01, 3.480317597803997e-02, 7.069599688705653e-01, 2.377081331437708e-01, 4.015277537380656e-02, 7.089592747680905e-01, -4.373982847931744e-01, 3.119546438021227e-02, 7.146942471573201e-01, 8.178315170110126e-01, 2.161251683469453e-02, 7.307538954522923e-01, -9.007096403390665e-01, 1.565402681403319e-02, 7.704504545341444e-01, -7.396567334921220e-01, 2.067477223680419e-02, 7.897818448390646e-01, -3.232780509941606e-01, 1.880267861755378e-02, 7.934484404832058e-01, 9.926504167490636e-01, 3.776050573673320e-03, 8.026728771155266e-01, -9.655537841075670e-01, 7.156659310810854e-03, 8.087651311032129e-01, -5.934295716161285e-01, 2.201032582341064e-02, 8.095585492822206e-01, 5.131014033000776e-02, 3.161252035941158e-02, 8.113041827662109e-01, 4.018532940895451e-01, 2.993617375756685e-02, 8.140746090133947e-01, 6.999506065348030e-01, 2.452891312211182e-02, 8.175220739845873e-01, 9.045678533269281e-01, 1.408878624403073e-02, 8.442244744987026e-01, -2.330660680257271e-01, 2.137248847571907e-02, 8.577022609864510e-01, -8.347496637512480e-01, 1.302733239156259e-02, 8.740997306450817e-01, -9.016947723266606e-01, 5.700088397798577e-03, 8.760720409171439e-01, 1.798424501465092e-01, 1.242534941969138e-02, 8.776595540145247e-01, -9.871736549700062e-01, 3.023298480841919e-03, 8.876770756008736e-01, -4.737591614279884e-01, 2.114261952758863e-02, 8.893257290546359e-01, 5.497239097424972e-01, 1.883916296392102e-02, 8.966119498774410e-01, 9.684306041089270e-01, 6.514509999621678e-03, 9.005134923798872e-01, 8.160726922491325e-01, 1.492277961359557e-02, 9.028454350969399e-01, 2.628603055474125e-01, 1.525753840078711e-02, 9.099332057150961e-01, -9.463222058681031e-02, 2.114989721521207e-02, 9.116851421787586e-01, -6.958768581382581e-01, 1.656213645188111e-02, 9.380791852794695e-01, -9.390761885813872e-01, 6.505278755513709e-03, 9.395276649285043e-01, 6.496946928511224e-01, 9.668304772455727e-03, 9.473470605468867e-01, -3.346050342634356e-01, 1.585630013135186e-02, 9.491070785582369e-01, -9.969373733490615e-01, 9.281179028714037e-04, 9.509854262005244e-01, 3.980404983361944e-01, 1.284569365696251e-02, 9.538750327616107e-01, 4.953599709321822e-02, 1.055872402608265e-02, 9.578780270631901e-01, 9.103067048312464e-01, 6.968260865746103e-03, 9.592991308680465e-01, 9.968731654931796e-01, 1.210463692085719e-03, 9.601302552519683e-01, -8.180789200488630e-01, 8.683841565713000e-03, 9.694392229219877e-01, -5.715033820344755e-01, 1.159851379681130e-02, 9.703910299123225e-01, 1.632899951759109e-01, 7.363467408097404e-03, 9.719159673797005e-01, 7.411274370572497e-01, 7.318477013269155e-03, 9.829103770479058e-01, 5.055355492737967e-01, 6.282928001932651e-03, 9.833554207795926e-01, -1.837381133019218e-01, 8.880000815180237e-03, 9.874031706544697e-01, -9.781008326752658e-01, 1.660786069380725e-03, 9.915132744979969e-01, -8.960126036794243e-01, 2.711324914646742e-03, 9.917024709452258e-01, 9.678533470873883e-01, 1.688735729327407e-03, 9.937579585467250e-01, -7.180615094407097e-01, 3.871787826906220e-03, 9.940901871599539e-01, 8.430528621092570e-01, 2.741339818214634e-03, 9.941340765584503e-01, 2.870098965170506e-01, 4.589322892855804e-03, 9.949661609261387e-01, 6.084106757497423e-03, 3.583734039365401e-03, 9.960704002907521e-01, -4.220415957631232e-01, 4.045718926806529e-03, 9.968040423856127e-01, 6.246991365652024e-01, 2.405030444120247e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr33 ( ) /******************************************************************************/ /* Purpose: SMR33 returns the SMR rule of degree 33. Discussion: DEGREE: 33 ROTATIONALLY INVARIANT: (X, Y), (-Y, X), (-X, -Y), (Y, -X). POINTS CARDINALITY: 197 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR33[3*197], the requested rule. */ { const int degree = 33; int order; double xw[3*197] = { 1.485098538731679e-01, 1.694663949097438e-01, 4.808148867803772e-02, 1.172505996395537e-01, 5.208793654128182e-01, 4.731056174816773e-02, 1.066031447166868e-01, 7.778079973351806e-01, 3.516295123254670e-02, 1.132061108674796e-01, 9.316643176349618e-01, 1.951472105421867e-02, 1.073615219898776e-01, 9.959676005278353e-01, 3.976501444162358e-03, 3.462608583072413e-01, 2.127438914067375e-01, 2.293908286621284e-02, 2.637306543835957e-01, 3.722947344432351e-01, 4.432068637015522e-02, 2.693795841838618e-01, 6.682002420169654e-01, 3.897921421448901e-02, 2.875348204916399e-01, 8.670452225998134e-01, 2.660382508243975e-02, 2.893753551193369e-01, 9.757288452382227e-01, 1.105346348813268e-02, 5.338546425019925e-01, 2.183801117113716e-01, 3.966360973156654e-02, 4.549680885072975e-01, 3.744677210475233e-01, 2.597222014353591e-02, 4.090144024608169e-01, 5.458954883248048e-01, 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0.000000000000000e+00, 0.000000000000000e+00, 5.152213536694609e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr34 ( ) /******************************************************************************/ /* Purpose: SMR34 returns the SMR rule of degree 34. Discussion: DEGREE: 34 POINTS CARDINALITY: 211 NORM INF MOMS. RESIDUAL: 1.28603e-15 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR34[3*211], the requested rule. */ { const int degree = 34; int order; double xw[3*211] = { 9.957894049234940e-01, 9.315910675787362e-01, 1.269921938742754e-03, 9.966819226158441e-01, 1.990628617847345e-01, 2.824785894655117e-03, 9.964232568659287e-01, -8.647711926777550e-02, 2.645466616758639e-03, 9.956972863904161e-01, -3.319481955508262e-01, 2.579394947779080e-03, 9.970019011095760e-01, -9.813822644968530e-01, 5.121822735027154e-04, 9.927566324556228e-01, 9.934870866426487e-01, 4.095595153407374e-04, 9.703323011961618e-01, 9.728958623239660e-01, 2.212600752526410e-03, 9.900242513895626e-01, 8.165046961752659e-01, 3.489632874382814e-03, 9.902525852068462e-01, 6.514930247782849e-01, 4.815598056643366e-03, 9.917712097306959e-01, 4.326945836209130e-01, 4.941854039948059e-03, 9.773024985383031e-01, 5.683940605635036e-02, 8.460270092591574e-03, 9.765396060948102e-01, -2.098648339912204e-01, 7.323452637054356e-03, 9.880323477406699e-01, -5.045486513331958e-01, 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9.749008815310916e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr35 ( ) /******************************************************************************/ /* Purpose: SMR35 returns the SMR rule of degree 35. Discussion: DEGREE: 35 SYMMETRY: (X, Y), (-X, -Y) POINTS CARDINALITY: 220 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0 Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR35[3*220], the requested rule. */ { const int degree = 35; int order; double xw[3*220] = { -9.963054076831271e-01, 5.895227652492729e-01, 3.040448117975760e-03, -9.959258876623405e-01, 8.226232166208993e-01, 1.978505799274471e-03, -9.954067560360648e-01, -9.074145716693054e-01, 1.810898452206362e-03, -9.952722138287154e-01, 2.785041956973013e-01, 4.045664611880734e-03, -9.949775104476455e-01, -7.198973208744759e-01, 2.841977406946536e-03, -9.943750550352027e-01, -3.487179202856331e-02, 4.548201806973092e-03, -9.943677037885296e-01, 9.710410041880194e-01, 1.118097006661641e-03, -9.937038655576630e-01, -9.931050540649843e-01, 4.830386117047890e-04, -9.930202681222473e-01, -3.214092657950156e-01, 4.905950396961080e-03, -9.861727841676639e-01, -5.400704381496431e-01, 6.797444141494801e-03, -9.810071147719678e-01, 8.985653300431585e-01, 3.181435032108599e-03, -9.740062631571137e-01, 7.162386303873299e-01, 7.887232214456035e-03, -9.739587062789258e-01, 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6.797444141494801e-03, 9.930202681222473e-01, 3.214092657950156e-01, 4.905950396961080e-03, 9.937038655576630e-01, 9.931050540649843e-01, 4.830386117047890e-04, 9.943677037885296e-01, -9.710410041880194e-01, 1.118097006661641e-03, 9.943750550352027e-01, 3.487179202856331e-02, 4.548201806973092e-03, 9.949775104476455e-01, 7.198973208744759e-01, 2.841977406946536e-03, 9.952722138287154e-01, -2.785041956973013e-01, 4.045664611880734e-03, 9.954067560360648e-01, 9.074145716693054e-01, 1.810898452206362e-03, 9.959258876623405e-01, -8.226232166208993e-01, 1.978505799274471e-03, 9.963054076831271e-01, -5.895227652492729e-01, 3.040448117975760e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr35s ( ) /******************************************************************************/ /* Purpose: SMR35S returns the rotationally invariant SMR rule of degree 35. Discussion: DEGREE: 35 ROTATIONALLY INVARIANT: (X, Y), (-Y, X), (-X, -Y), (Y, -X). POINTS CARDINALITY: 220 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 6.54962e-03, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR35S[3*220], the requested rule. */ { const int degree = 35; int order; double xw[3*220] = { 4.889904361809121e-03, 1.619503174532069e-01, 5.226179892467336e-02, 1.430680275835659e-02, 4.693240762340044e-01, 4.679625079080703e-02, 1.814469250768224e-01, 6.009339848917693e-01, 4.158929555209646e-02, 3.513535787046233e-02, 9.092730028762068e-01, 2.186232237110550e-02, 7.640565697368758e-02, 9.941574902382558e-01, 4.878857836280798e-03, 1.713276929018882e-01, 3.150081208611397e-01, 4.954187274939539e-02, 1.878090789211163e-01, 8.240874417949644e-01, 2.885775015170009e-02, 3.641585032828187e-01, 8.926397561302795e-01, 1.380108158872173e-02, 3.369795025432279e-01, 8.976793330525874e-01, 7.578776426504399e-03, 2.085987555621641e-01, 9.589188503432696e-01, 1.412626006558690e-02, 3.243221451814440e-01, 1.522018736597065e-01, 4.954596092265034e-02, 3.340889484224165e-01, 4.585908563274512e-01, 4.430990763422363e-02, 3.416707320282502e-01, 7.180689168234468e-01, 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4.286571189420782e-03, 7.973265219048701e-01, -9.238348699268808e-01, 1.108551064645477e-02, 9.731930158842385e-01, -9.785171113227352e-01, 1.481797407435137e-03, 2.646515091977871e-01, -9.963425748043122e-01, 3.893391747905868e-03, 4.106568212342961e-01, -9.988925112672711e-01, -1.617960722779276e-04, 5.751170429423320e-01, -9.959215572099368e-01, 3.902216245624410e-03, 6.867546611123180e-01, -9.640572980293431e-01, 5.691309854696837e-03, 8.085573402233180e-01, -9.945462995891214e-01, 2.563895647702530e-03, 8.888800101355042e-01, -9.670114428073955e-01, 5.609754500116319e-03, 9.383693663697456e-01, -9.958210056699377e-01, 1.047624717542818e-03, 9.949415630191232e-01, -9.943379392579890e-01, 2.607250337397194e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr36 ( ) /******************************************************************************/ /* Purpose: SMR36 returns the SMR rule of degree 36. Discussion: DEGREE: 36 POINTS CARDINALITY: 238 NORM INF MOMS. RESIDUAL: 4.27609e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR36[3*238], the requested rule. */ { const int degree = 36; int order; double xw[3*238] = { -9.978817227614055e-01, -8.890076517204608e-04, 2.330741632674278e-03, -9.973711646404451e-01, -9.532092615783677e-01, 9.627969193226537e-04, -9.961692635944124e-01, 9.688495936611794e-01, 8.898973486835064e-04, -9.959497922009879e-01, -3.098803144458146e-01, 2.832831652522147e-03, -9.959465092993447e-01, -5.592627350774092e-01, 2.519247595249725e-03, -9.954980310007229e-01, 6.046856101951658e-01, 2.958064214592169e-03, -9.946662450102303e-01, 3.416260049342602e-01, 3.892327512506448e-03, -9.934973736752050e-01, -7.316617433785163e-01, 2.288706827417947e-03, -9.917053480435074e-01, 8.702611598150070e-01, 1.958941400105963e-03, -9.916694420714898e-01, 7.809558409845563e-01, 2.395428553360631e-03, -9.895910809889403e-01, -8.396046316830745e-01, 2.802701148321813e-03, -9.831331889300642e-01, 1.393223594054985e-01, 6.694472389602429e-03, -9.807631633241672e-01, 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r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr37 ( ) /******************************************************************************/ /* Purpose: SMR37 returns the SMR rule of degree 37. Discussion: DEGREE: 37 ROTATIONALLY INVARIANT: (X, Y), (-Y, X), (-X, -Y), (Y, -X). POINTS CARDINALITY: 245 NORM INF MOMS. RESIDUAL: 1.97628e-15 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR37[3*245], the requested rule. */ { const int degree = 37; int order; double xw[3*245] = { 1.688689515698905e-01, 1.899855310442196e-01, 1.879076889689011e-02, 1.664922672701527e-01, 3.429140133623845e-01, 3.449917667815708e-02, 1.789345703476781e-01, 5.923725055750423e-01, 3.712580251685875e-02, 1.912828097143278e-01, 8.047865391631479e-01, 2.804589806906416e-02, 1.909000554810340e-01, 9.526563151843329e-01, 1.317147780067464e-02, 1.197461382570857e-01, 9.934656721499447e-01, 3.246786042836229e-03, 3.142942982925143e-01, 1.085212763112808e-01, 4.073944662145704e-02, 3.136430658350363e-01, 3.168853969514266e-01, 1.888769495196758e-02, 3.257592735310023e-01, 4.744364447040339e-01, 3.338316795412049e-02, 3.370767611845376e-01, 7.018261748212855e-01, 3.160597567460757e-02, 3.228638369287264e-01, 8.961218476259737e-01, 1.882298564453669e-02, 3.240794429041066e-01, 9.907076643667728e-01, 5.242591968441962e-03, 4.573043899331568e-01, 2.595246348757670e-01, 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1.723425646848686e-01, -3.032916166693916e-02, 3.655713458299797e-02, 4.663686154054668e-01, -2.217294601980654e-02, 4.027271183981319e-02, 7.102048981913600e-01, -2.814942665710626e-02, 3.373438958414005e-02, 8.900063168860698e-01, -4.438627067827642e-02, 2.195827523559744e-02, 9.762502292316360e-01, -6.744822333126831e-03, 5.490273279461521e-03, 9.956312481563887e-01, 4.982446329191517e-02, 1.436535699659639e-03, 0.000000000000000e+00, 0.000000000000000e+00, 2.733962437974891e-02 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr38 ( ) /******************************************************************************/ /* Purpose: SMR38 returns the SMR rule of degree 38. Discussion: DEGREE: 38 POINTS CARDINALITY: 265 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR38[3*265], the requested rule. */ { const int degree = 38; int order; double xw[3*265] = { -9.986728061698944e-01, 7.022053437776367e-01, 1.480107210669759e-03, -9.982717636746820e-01, -9.054654092610701e-01, 1.075250999194606e-03, -9.967818426696908e-01, 2.010254208991262e-01, 2.641794509284354e-03, -9.961752169855568e-01, -4.510641362412657e-01, 2.262884166375447e-03, -9.956850333584866e-01, -2.104763459288764e-01, 2.878520269955461e-03, -9.954228554916262e-01, -6.464904618069326e-01, 2.018600540825626e-03, -9.952874532053829e-01, 4.651922890152825e-01, 2.686536492992849e-03, -9.948147421400011e-01, 8.983435291372217e-01, 1.575355209064770e-03, -9.944162554551003e-01, 9.804886839517453e-01, 7.123537841476463e-04, -9.941508164951945e-01, -9.924193701187594e-01, 4.124555173483122e-04, -9.895909168262721e-01, -7.760059158658976e-01, 3.335123807315742e-03, -9.884130043110948e-01, -6.828120541211876e-03, 5.966363865455931e-03, -9.825019022108045e-01, 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/******************************************************************************/ double *smr39 ( ) /******************************************************************************/ /* Purpose: SMR39 returns the SMR rule of degree 39. Discussion: DEGREE: 39 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 274 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR39[3*274], the requested rule. */ { const int degree = 39; int order; double xw[3*274] = { -9.983507138390728e-01, 2.846923651716276e-02, 1.946248040917511e-03, -9.974803510397799e-01, 5.723617700464594e-01, 1.730177315995435e-03, -9.970436400762405e-01, 8.380584517844631e-01, 1.474635255954523e-03, -9.966780331371223e-01, 9.694002319896394e-01, 7.243867200693780e-04, -9.966744182696170e-01, -5.085108565093430e-01, 1.980803681920541e-03, -9.958948247928820e-01, 3.162381906315679e-01, 2.750378228739487e-03, -9.957989442071229e-01, -2.946071658312556e-01, 2.310642786475657e-03, -9.951431493831727e-01, -9.317895250779106e-01, 1.311747532188187e-03, -9.947425612275775e-01, -6.937725695763057e-01, 2.327244406193416e-03, -9.946606993258253e-01, -9.946657556011911e-01, 3.187304244494703e-04, -9.891970457464104e-01, -8.249338141061554e-01, 3.217331206511223e-03, -9.882776096714541e-01, 6.996008607192892e-01, 3.762641387033618e-03, -9.863969994171303e-01, 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2.946071658312556e-01, 2.310642786475657e-03, 9.958948247928820e-01, -3.162381906315679e-01, 2.750378228739487e-03, 9.966744182696170e-01, 5.085108565093430e-01, 1.980803681920541e-03, 9.966780331371223e-01, -9.694002319896394e-01, 7.243867200693780e-04, 9.970436400762405e-01, -8.380584517844631e-01, 1.474635255954523e-03, 9.974803510397799e-01, -5.723617700464594e-01, 1.730177315995435e-03, 9.983507138390728e-01, -2.846923651716276e-02, 1.946248040917511e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr40 ( ) /******************************************************************************/ /* Purpose: SMR40 returns the SMR rule of degree 40. Discussion: DEGREE: 40 POINTS CARDINALITY: 296 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR40[3*296], the requested rule. */ { const int degree = 40; int order; double xw[3*296] = { -9.980037227196614e-01, 1.255939443038006e-01, 2.053102393463664e-03, -9.975374042816840e-01, 5.936108405943805e-01, 1.276187489810420e-03, -9.972861466402049e-01, -6.627439420820114e-01, 1.580298156253981e-03, -9.968592673741854e-01, 8.736423351683437e-01, 1.202945592410809e-03, -9.966589451189311e-01, -4.369849380251707e-01, 2.115586380569520e-03, -9.965219690837533e-01, -1.944638806631471e-01, 2.429919318140257e-03, -9.964358393204891e-01, 4.092031891829934e-01, 2.182777026867640e-03, -9.963941450446951e-01, 9.775574537230374e-01, 5.781357112514488e-04, -9.955864247426816e-01, -9.784933334191470e-01, 6.423485659734820e-04, -9.955242411687316e-01, -8.201261852647251e-01, 1.452148584658119e-03, -9.913130830294992e-01, 7.451950336474148e-01, 3.099740554076827e-03, -9.900337734476057e-01, -9.112486663047842e-01, 2.026525375041507e-03, -9.844939696556546e-01, 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1.444993876112607e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr41 ( ) /******************************************************************************/ /* Purpose: SMR41 returns the SMR rule of degree 41. Discussion: DEGREE: 41 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 303 NORM INF MOMS. RESIDUAL: 1.77636e-15 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR41[3*303], the requested rule. */ { const int degree = 41; int order; double xw[3*303] = { -9.979788599279722e-01, -8.620321171404657e-01, 9.843080418879427e-04, -9.972233023651127e-01, -1.689643304774283e-01, 2.334603070096216e-03, -9.968865665492763e-01, 1.264946845449128e-01, 2.336585497171821e-03, -9.967543799903597e-01, 3.758322880625192e-01, 1.965952058779777e-03, -9.966030911672050e-01, -4.414700559329136e-01, 2.103309187930437e-03, -9.965848495106024e-01, -6.842725498070605e-01, 1.783337849517576e-03, -9.961989059670616e-01, 9.525971037206539e-01, 6.660472520224310e-04, -9.955131940127476e-01, 8.558413309721239e-01, 1.466862509992409e-03, -9.955121819465780e-01, -9.789979211029449e-01, 6.407210056305082e-04, -9.954861320163209e-01, 5.681416527913361e-01, 2.181094091195068e-03, -9.929582393871068e-01, -5.370347553708347e-01, 6.365700160628349e-04, -9.918341858607898e-01, 7.250215468408083e-01, 2.925047225366992e-03, -9.897982334464747e-01, 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1.466862509992409e-03, 9.961989059670616e-01, -9.525971037206539e-01, 6.660472520224310e-04, 9.965848495106024e-01, 6.842725498070605e-01, 1.783337849517576e-03, 9.966030911672050e-01, 4.414700559329136e-01, 2.103309187930437e-03, 9.967543799903597e-01, -3.758322880625192e-01, 1.965952058779777e-03, 9.968865665492763e-01, -1.264946845449128e-01, 2.336585497171821e-03, 9.972233023651127e-01, 1.689643304774283e-01, 2.334603070096216e-03, 9.979788599279722e-01, 8.620321171404657e-01, 9.843080418879427e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr42 ( ) /******************************************************************************/ /* Purpose: SMR42 returns the SMR rule of degree 42. Discussion: DEGREE: 42 POINTS CARDINALITY: 326 NORM INF MOMS. RESIDUAL: 8.67145e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR42[3*326], the requested rule. */ { const int degree = 42; int order; double xw[3*326] = { -9.983895853811665e-01, 2.969737627802563e-01, 1.238150356046483e-03, -9.977615781174913e-01, -7.709519815994472e-01, 1.081616267165998e-03, -9.975604363093720e-01, 5.337316075161648e-01, 1.249737376093271e-03, -9.975457919908404e-01, 9.770830229535153e-01, 4.470299430512570e-04, -9.969809246965443e-01, -2.451005993701799e-01, 2.074243379361551e-03, -9.968120260128331e-01, 8.514136675024221e-01, 1.167341137105932e-03, -9.966778899972677e-01, -9.176221669397234e-01, 8.041981836080290e-04, -9.964245708711565e-01, -4.889240550109979e-01, 2.074108808482804e-03, -9.960265424615615e-01, -3.206262163873343e-03, 2.541774019152315e-03, -9.957193536867802e-01, 6.973486149261381e-01, 1.833470374408153e-03, -9.954906184608173e-01, -9.834651303414434e-01, 4.837448015994170e-04, -9.923170394980639e-01, -6.488504511621852e-01, 2.300838549325460e-03, -9.894941295318366e-01, 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9.977589931512456e-01, -2.851699647482527e-01, 1.679574685965640e-03, 9.982095733525613e-01, 7.942486273250328e-01, 1.122778646693857e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr43 ( ) /******************************************************************************/ /* Purpose: SMR43 returns the SMR rule of degree 43. Discussion: DEGREE: 43 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 331 NORM INF MOMS. RESIDUAL: 6.57894e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR43[3*331], the requested rule. */ { const int degree = 43; int order; double xw[3*331] = { -9.991322806157538e-01, -7.967757949667039e-01, 8.678616486509684e-04, -9.982000639809739e-01, 3.046004953209935e-01, 1.439841751685133e-03, -9.980850962629690e-01, 9.781664509229353e-01, 3.796649877334044e-04, -9.980230789357751e-01, -2.518043976244231e-01, 1.843703555135520e-03, -9.977169911523626e-01, 5.459548078849698e-01, 1.239809456010988e-03, -9.960955981451255e-01, 1.323322402607845e-02, 2.563511060458582e-03, -9.949822864151898e-01, -9.238195120237386e-01, 9.266737007306744e-04, -9.949201671743397e-01, 7.003454609915903e-01, 2.006603012946033e-03, -9.948772524482939e-01, 8.407859395090261e-01, 1.605442474502514e-03, -9.948541523937408e-01, -9.833293028860037e-01, 5.491048194068317e-04, -9.947061632621746e-01, -6.355790382588226e-01, 1.778286578522554e-03, -9.945229623313124e-01, -4.853023018396496e-01, 2.560515852421350e-03, -9.892614872423598e-01, 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9.949822864151898e-01, 9.238195120237386e-01, 9.266737007306744e-04, 9.960955981451255e-01, -1.323322402607845e-02, 2.563511060458582e-03, 9.977169911523626e-01, -5.459548078849698e-01, 1.239809456010988e-03, 9.980230789357751e-01, 2.518043976244231e-01, 1.843703555135520e-03, 9.980850962629690e-01, -9.781664509229353e-01, 3.796649877334044e-04, 9.982000639809739e-01, -3.046004953209935e-01, 1.439841751685133e-03, 9.991322806157538e-01, 7.967757949667039e-01, 8.678616486509684e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr44 ( ) /******************************************************************************/ /* Purpose: SMR44 returns the SMR rule of degree 44. Discussion: DEGREE: 44 POINTS CARDINALITY: 353 NORM INF MOMS. RESIDUAL: 1.39699e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR44[3*353], the requested rule. */ { const int degree = 44; int order; double xw[3*353] = { -9.984320392366739e-01, -8.069907738631789e-01, 9.129188297919674e-04, -9.983184531837725e-01, 9.305063618730227e-01, 5.320673982755963e-04, -9.978244272094350e-01, 2.010518994335128e-01, 1.598745058561678e-03, -9.974547076089806e-01, 4.442097888110615e-01, 1.587226283438206e-03, -9.971550888623897e-01, -5.768131021729579e-02, 1.901977649434094e-03, -9.967647638775373e-01, 8.058203134817903e-01, 1.263686704072662e-03, -9.963702759402693e-01, 9.957226369845009e-01, 1.818612016165590e-04, -9.963181710633270e-01, -6.304189998071564e-01, 1.671127161151113e-03, -9.961268807224163e-01, 6.456711899254686e-01, 1.750825399677425e-03, -9.957522090105495e-01, -9.275216968816482e-01, 8.507815083388259e-04, -9.951693924758787e-01, -2.761138658566978e-01, 2.436900445069140e-03, -9.950054666513565e-01, -9.849707705106929e-01, 4.757246968341718e-04, -9.946541882949513e-01, 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8.953088235569292e-01, 3.456123608241670e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr45 ( ) /******************************************************************************/ /* Purpose: SMR45 returns the SMR rule of degree 45. Discussion: DEGREE: 45 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 359 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR45[3*359], the requested rule. */ { const int degree = 45; int order; double xw[3*359] = { -9.987809239831112e-01, 9.748368623655366e-01, 2.993356033786765e-04, -9.985808631696838e-01, -8.990756328646153e-01, 5.055415778287732e-04, -9.985157152594889e-01, -9.971971012987678e-01, 9.821637011466654e-05, -9.976075303722897e-01, 2.495954379536609e-01, 1.726273349986223e-03, -9.975896723770977e-01, -7.481282641542352e-01, 1.186765908765057e-03, -9.970359077153589e-01, -4.973042782590862e-03, 1.976200444087838e-03, -9.970034303329453e-01, 4.805125358625656e-01, 1.717119689979895e-03, -9.961579504388610e-01, -2.339319461038794e-01, 2.197631276825098e-03, -9.959759411587524e-01, 8.966138075526434e-01, 9.460145783264427e-04, -9.955607724070924e-01, 6.629654417179892e-01, 1.734396121578630e-03, -9.944882643568679e-01, -5.822713014974841e-01, 2.130704201950450e-03, -9.944839549832385e-01, -4.305503528241359e-01, 2.332655357182134e-03, -9.943103084508153e-01, 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-4.805125358625656e-01, 1.717119689979895e-03, 9.970359077153589e-01, 4.973042782590862e-03, 1.976200444087838e-03, 9.975896723770977e-01, 7.481282641542352e-01, 1.186765908765057e-03, 9.976075303722897e-01, -2.495954379536609e-01, 1.726273349986223e-03, 9.985157152594889e-01, 9.971971012987678e-01, 9.821637011466654e-05, 9.985808631696838e-01, 8.990756328646153e-01, 5.055415778287732e-04, 9.987809239831112e-01, -9.748368623655366e-01, 2.993356033786765e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr46 ( ) /******************************************************************************/ /* Purpose: SMR46 returns the SMR rule of degree 46. Discussion: DEGREE: 46 POINTS CARDINALITY: 387 NORM INF MOMS. RESIDUAL: 2.03396e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR46[3*387], the requested rule. */ { const int degree = 46; int order; double xw[3*387] = { -9.983111423300403e-01, 5.288463379177194e-01, 1.077877162071936e-03, -9.980165536240384e-01, 9.771577113964515e-01, 3.358785950069629e-04, -9.980057366593107e-01, 8.627358726346877e-01, 6.003479857328445e-04, -9.979683379344829e-01, -6.942209080688336e-01, 1.101994373218053e-03, -9.978846063972665e-01, 7.248634355141237e-01, 9.622706165518573e-04, -9.976343783636701e-01, -4.901932224780105e-01, 1.459301786243793e-03, -9.975456640840761e-01, 3.145719096243388e-01, 1.435508298339834e-03, -9.974305941584383e-01, -8.537788458635189e-01, 8.063821753970214e-04, -9.963467450618418e-01, -9.821437303584432e-01, 4.013000811661125e-04, -9.963009121112316e-01, -2.704146437985034e-01, 2.094359395819190e-03, -9.962806106099757e-01, 1.123075934443736e-01, 1.888774287809683e-03, -9.940902248165704e-01, -9.316901409499205e-01, 8.495707747103506e-04, -9.938933215260991e-01, 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1.645309636276182e-03, 9.973259273035801e-01, -8.408771000383591e-01, 8.974177017096658e-04, 9.974702963631498e-01, -4.748457085080076e-01, 1.419898311116791e-03, 9.979804912008889e-01, 2.058903880121162e-01, 1.332855555539681e-03, 9.980187128543290e-01, 8.423741210647242e-01, 8.934394663320126e-04, 9.983634077785508e-01, -9.445962296620326e-01, 4.424225813983324e-04, 9.983713812941385e-01, -6.766009567160710e-01, 9.089763434524134e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr47 ( ) /******************************************************************************/ /* Purpose: SMR47 returns the SMR rule of degree 47. Discussion: DEGREE: 47 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 396 NORM INF MOMS. RESIDUAL: 1.56234e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR47[3*396], the requested rule. */ { const int degree = 47; int order; double xw[3*396] = { -9.988860641396040e-01, -2.373189065242698e-01, 1.059226498023405e-03, -9.984308298705356e-01, 6.810115040142974e-01, 8.742573242503645e-04, -9.982849649115821e-01, 9.436218579698744e-01, 4.239310404841323e-04, -9.975891711682796e-01, -9.308337785182597e-01, 5.005025838481359e-04, -9.975758902464168e-01, -8.292104265892648e-01, 8.758182348185454e-04, -9.975527094081703e-01, -6.722468138431469e-01, 1.228393951475139e-03, -9.975229416008712e-01, 8.411146681674510e-01, 8.259054712338016e-04, -9.974354703237893e-01, -9.972347569712511e-01, 1.178428627156954e-04, -9.973323543291763e-01, 4.909591129622543e-01, 1.363609590112511e-03, -9.971337685886393e-01, 7.632359805747738e-02, 1.695014196901724e-03, -9.969508496154019e-01, 9.951320518801803e-01, 1.406624476607038e-04, -9.967221454497780e-01, 2.967880150220837e-01, 1.788242723078713e-03, -9.958488160214992e-01, 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9.988860641396040e-01, 2.373189065242698e-01, 1.059226498023405e-03 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr48 ( ) /******************************************************************************/ /* Purpose: SMR48 returns the SMR rule of degree 48. Discussion: DEGREE: 48 POINTS CARDINALITY: 417 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR48[3*417], the requested rule. */ { const int degree = 48; int order; double xw[3*417] = { -9.990552580516567e-01, -7.479254608809505e-01, 4.845844320013029e-04, -9.988760883998153e-01, 2.568938088751835e-01, 8.877453242046397e-04, -9.986091048294213e-01, 1.475403353286808e-02, 8.040816649406732e-04, -9.985122425173576e-01, -5.059783159128129e-01, 1.125983578847775e-03, -9.984530690329705e-01, -9.196817917920462e-01, 3.211292464713594e-04, -9.982842816947866e-01, 4.893019742012896e-01, 8.282364851453019e-04, -9.982520488850405e-01, 9.821235324863609e-01, 2.422927481790814e-04, -9.973280389800840e-01, 6.756520485620866e-01, 8.237504025536671e-04, -9.967449468871785e-01, 7.716874829711365e-01, 7.452334436535652e-04, -9.964365335122727e-01, 8.681515615582738e-01, 8.902528711682726e-04, -9.962098577937705e-01, -2.857266397360828e-01, 1.937344458762579e-03, -9.957795050622651e-01, -9.876433080318017e-01, 3.429533984696887e-04, -9.946778211699915e-01, 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9.972221719684324e-01, 9.836266811867312e-01, 3.070667564929675e-04, 9.973502464780108e-01, 5.703835859132225e-01, 1.316377370815025e-03, 9.977903571993221e-01, -2.217334569601657e-01, 1.243787387237352e-03, 9.981467866635252e-01, -7.852760517081546e-01, 5.930241241924707e-04, 9.983950085519884e-01, -6.322084663179760e-01, 8.416135676187133e-04, 9.984871758203762e-01, -9.838507269098428e-01, 1.976555370458795e-04, 9.985986690813101e-01, -4.178930731294047e-01, 8.254856487030795e-04, 9.989532297601191e-01, 3.236183825764132e-02, 9.014418551948622e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr49 ( ) /******************************************************************************/ /* Purpose: SMR49 returns the SMR rule of degree 49. Discussion: DEGREE: 49 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 427 NORM INF MOMS. RESIDUAL: 4.24357e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR49[3*427], the requested rule. */ { const int degree = 49; int order; double xw[3*427] = { -9.987179482199477e-01, -6.341055456784180e-02, 9.427263494251746e-04, -9.985820051950299e-01, 4.311577677096960e-01, 8.506787489756313e-04, -9.985812215892117e-01, 2.098862069679553e-01, 9.932558396288942e-04, -9.984853824993025e-01, -7.518363345257131e-01, 6.752050933944545e-04, -9.984416520882323e-01, -8.814083896367437e-01, 4.548127997472904e-04, -9.983293539476047e-01, 9.835120159252404e-01, 2.181997023647311e-04, -9.982115219214424e-01, 6.305047404346029e-01, 8.136830618787968e-04, -9.978435386703776e-01, 7.692080150297662e-01, 6.598609318533428e-04, -9.977221586114358e-01, -6.019952067889782e-01, 5.964708123101566e-04, -9.966209533584591e-01, -3.237318892379179e-01, 1.736701233968157e-03, -9.963148656482692e-01, -9.861307505276973e-01, 3.364793817721787e-04, -9.959458468526172e-01, 8.639404647925895e-01, 9.127209022448868e-04, -9.955419051779023e-01, 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7.518363345257131e-01, 6.752050933944545e-04, 9.985812215892117e-01, -2.098862069679553e-01, 9.932558396288942e-04, 9.985820051950299e-01, -4.311577677096960e-01, 8.506787489756313e-04, 9.987179482199477e-01, 6.341055456784180e-02, 9.427263494251746e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr50 ( ) /******************************************************************************/ /* Purpose: SMR50 returns the SMR rule of degree 50. Discussion: DEGREE: 50 POINTS CARDINALITY: 454 NORM INF MOMS. RESIDUAL: 1.97162e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR50[3*454], the requested rule. */ { const int degree = 50; int order; double xw[3*454] = { -9.985305923732772e-01, 9.480346609653949e-01, 3.637016044890175e-04, -9.983266574377337e-01, -7.135200571279965e-01, 8.681375001274842e-04, -9.982410599710965e-01, -5.061487385182745e-02, 1.210422814431021e-03, -9.981979630190073e-01, 1.943963798210550e-01, 1.209667416017868e-03, -9.981164101550407e-01, 4.217828123427281e-01, 1.100221981712261e-03, -9.980251256813344e-01, 8.409407151717826e-01, 6.790256958329296e-04, -9.978977416561483e-01, -5.175396836215825e-01, 1.209910288842294e-03, -9.977749235731450e-01, -8.787383894116004e-01, 6.804808145574241e-04, -9.974790513531082e-01, -2.987478462479398e-01, 1.485043396561256e-03, -9.973044005791738e-01, 9.966139708183203e-01, 1.024593234561320e-04, -9.965354581345114e-01, -9.625283354808352e-01, 5.086867261717252e-04, -9.959776528397177e-01, 5.865612330340912e-01, 1.475073552829635e-03, -9.958674494848857e-01, 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9.490722861303337e-01, 9.334114310415029e-04, 9.939722207265341e-01, 8.378634329981455e-01, 1.348862490324860e-03, 9.941405572471773e-01, 5.032506127587780e-01, 1.321839206521580e-03, 9.943806852246193e-01, -6.860004265918137e-01, 7.140588378827780e-04, 9.957633794231534e-01, -8.360114490592742e-01, 1.041707929268540e-03, 9.966883143052717e-01, 7.278496613238243e-01, 1.191792248759857e-03, 9.968035955361948e-01, -5.651523666439144e-01, 1.374867996046609e-03, 9.968605145969733e-01, -7.336757652882711e-01, 7.990342391998070e-04, 9.969384946190634e-01, 9.868433854178995e-01, 2.740236217096047e-04, 9.975593611800366e-01, 3.360863758987011e-01, 1.454585250009066e-03, 9.978110877289340e-01, -1.465750773024727e-01, 1.377030312298262e-03, 9.979325714446113e-01, -9.249224066288942e-01, 4.777113521207585e-04, 9.979765814666188e-01, -3.775646733998311e-01, 1.202710471482889e-03, 9.980898127659592e-01, -9.874218489978037e-01, 1.849225119026519e-04, 9.981040178114964e-01, 8.280125816480956e-02, 1.248796781507533e-03, 9.982820382418868e-01, 5.769543802463517e-01, 7.519588496094478e-04, 9.990895206748026e-01, 9.085198717249964e-01, 3.348164725496207e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr51 ( ) /******************************************************************************/ /* Purpose: SMR51 returns the SMR rule of degree 51. Discussion: DEGREE: 51 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 462 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR51[3*462], the requested rule. */ { const int degree = 51; int order; double xw[3*462] = { -9.984626432301655e-01, 7.731810731373280e-01, 2.930667077040843e-04, -9.982950612817629e-01, -8.809790422529455e-01, 5.868976563494200e-04, -9.982657143502113e-01, -5.472808378415216e-01, 8.825961580004602e-04, -9.982465196344393e-01, -5.720585061776626e-02, 1.187360397727577e-03, -9.980752383111224e-01, 9.895590629954322e-01, 1.605142582803498e-04, -9.980563050335864e-01, 4.026468127795991e-01, 1.133746118823456e-03, -9.980560915023025e-01, 9.316458011937399e-01, 4.410107005989047e-04, -9.979331148437312e-01, -7.218491788727829e-01, 9.165568347185002e-04, -9.978969853829365e-01, 1.744352247684945e-01, 1.316676065658554e-03, -9.976266468967849e-01, -3.120694338997498e-01, 1.417265243059842e-03, -9.967076640629561e-01, 5.825966069019469e-01, 1.351054193096153e-03, -9.966756982844908e-01, -9.646209210318417e-01, 4.673464325522125e-04, -9.958487588813290e-01, 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-5.825966069019469e-01, 1.351054193096153e-03, 9.976266468967849e-01, 3.120694338997498e-01, 1.417265243059842e-03, 9.978969853829365e-01, -1.744352247684945e-01, 1.316676065658554e-03, 9.979331148437312e-01, 7.218491788727829e-01, 9.165568347185002e-04, 9.980560915023025e-01, -9.316458011937399e-01, 4.410107005989047e-04, 9.980563050335864e-01, -4.026468127795991e-01, 1.133746118823456e-03, 9.980752383111224e-01, -9.895590629954322e-01, 1.605142582803498e-04, 9.982465196344393e-01, 5.720585061776626e-02, 1.187360397727577e-03, 9.982657143502113e-01, 5.472808378415216e-01, 8.825961580004602e-04, 9.982950612817629e-01, 8.809790422529455e-01, 5.868976563494200e-04, 9.984626432301655e-01, -7.731810731373280e-01, 2.930667077040843e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr52 ( ) /******************************************************************************/ /* Purpose: SMR52 returns the SMR rule of degree 52. Discussion: DEGREE: 52 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 493 NORM INF MOMS. RESIDUAL: 1.77636e-15 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR52[3*493], the requested rule. */ { const int degree = 52; int order; double xw[3*493] = { -9.988127014409341e-01, -9.436524211624828e-01, 2.625526521558170e-04, -9.987672107909694e-01, 6.030206237885709e-01, 5.695969962358445e-04, -9.985482219550218e-01, -5.039975022757507e-02, 7.711038314291110e-04, -9.984716084730251e-01, 1.648465544907188e-01, 9.876559621004946e-04, -9.983247427277899e-01, 9.851489095644117e-01, 1.847474866940015e-04, -9.981050848344004e-01, -9.897892710317165e-01, 1.455770846634000e-04, -9.980702320219238e-01, -8.710220064295600e-01, 3.606265853945270e-04, -9.980157290478607e-01, 3.823621895788871e-01, 1.084084984803506e-03, -9.979067456855214e-01, -3.075229364789735e-01, 1.191752453605670e-03, -9.976530827260541e-01, -4.990560694030991e-01, 1.016394895068514e-03, -9.976275687951667e-01, 7.452570753224620e-01, 7.962338632193404e-04, -9.975360338995326e-01, 9.261655839467660e-01, 4.882884021026501e-04, -9.975285341692078e-01, 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r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr53 ( ) /******************************************************************************/ /* Purpose: SMR53 returns the SMR rule of degree 53. Discussion: DEGREE: 53 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 498 NORM INF MOMS. RESIDUAL: 8.88178e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR53[3*498], the requested rule. */ { const int degree = 53; int order; double xw[3*498] = { -9.985928023604709e-01, 9.844383621232889e-01, 1.770821645368481e-04, -9.985845745857554e-01, -8.445726670375774e-01, 5.544684199320334e-04, -9.983624118051736e-01, 7.608024429037672e-01, 5.224452130672093e-04, -9.981572755902530e-01, 3.878455559966326e-01, 9.745188221536845e-04, -9.979563603675163e-01, 1.627643560997576e-01, 1.227147024948174e-03, -9.979346610008943e-01, -5.127640690222413e-01, 1.033714442116006e-03, -9.979107729485314e-01, 6.529159127971879e-01, 3.633867657362896e-04, -9.978751297025205e-01, -6.879180639452499e-01, 9.450952193083518e-04, -9.974464559033922e-01, -2.966888554053162e-01, 1.397441303354461e-03, -9.973454309733404e-01, -9.423377622335455e-01, 4.836806630189215e-04, -9.972986691166712e-01, -2.707798066058207e-02, 8.893716616277809e-04, -9.965532592866511e-01, 9.290044070830288e-01, 6.034648824689160e-04, -9.964624830190837e-01, 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1.361348704496822e-03, 9.914304305563754e-01, 4.291601514462605e-01, 1.270285302858647e-03, 9.933660348449341e-01, -3.161717536031352e-01, 8.387713122106447e-04, 9.953468235808713e-01, -8.508120228627952e-01, 1.028310207063733e-03, 9.954005028570233e-01, 1.203116337354302e-01, 1.443569885953676e-03, 9.964167588339662e-01, 9.892036664406441e-01, 2.516135591721077e-04, 9.964624830190837e-01, -5.570787045642487e-01, 1.303165763686854e-03, 9.965532592866511e-01, -9.290044070830288e-01, 6.034648824689160e-04, 9.972986691166712e-01, 2.707798066058207e-02, 8.893716616277809e-04, 9.973454309733404e-01, 9.423377622335455e-01, 4.836806630189215e-04, 9.974464559033922e-01, 2.966888554053162e-01, 1.397441303354461e-03, 9.978751297025205e-01, 6.879180639452499e-01, 9.450952193083518e-04, 9.979107729485314e-01, -6.529159127971879e-01, 3.633867657362896e-04, 9.979346610008943e-01, 5.127640690222413e-01, 1.033714442116006e-03, 9.979563603675163e-01, -1.627643560997576e-01, 1.227147024948174e-03, 9.981572755902530e-01, -3.878455559966326e-01, 9.745188221536845e-04, 9.983624118051736e-01, -7.608024429037672e-01, 5.224452130672093e-04, 9.985845745857554e-01, 8.445726670375774e-01, 5.544684199320334e-04, 9.985928023604709e-01, -9.844383621232889e-01, 1.770821645368481e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr54 ( ) /******************************************************************************/ /* Purpose: SMR54 returns the SMR rule of degree 54. Discussion: DEGREE: 54 POINTS CARDINALITY: 530 NORM INF MOMS. RESIDUAL: 5.39824e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR54[3*530], the requested rule. */ { const int degree = 54; int order; double xw[3*530] = { -9.989817717908982e-01, 5.919272627455578e-01, 4.192164731453299e-04, -9.987404572465737e-01, 1.308084790329531e-01, 1.021852171532156e-03, -9.986234267784356e-01, -4.263738548438438e-01, 8.092096026376365e-04, -9.981931054807527e-01, 3.646305906433953e-01, 9.508207239229283e-04, -9.981856973324568e-01, 7.272619710460589e-01, 6.106524315481551e-04, -9.980709749587124e-01, -6.001863748613854e-01, 8.126915872357096e-04, -9.980413923840483e-01, -8.914269607610811e-01, 4.682310231840477e-04, -9.979705384624564e-01, -7.412992362417765e-01, 7.151087866573973e-04, -9.978690979638140e-01, 9.663123578468790e-01, 3.202153175837222e-04, -9.977258243819628e-01, -9.652827050987313e-01, 3.283439367940058e-04, -9.971716111831748e-01, -9.974268956785031e-01, 8.241300038356087e-05, -9.970352418855494e-01, 9.974255320425583e-01, 8.212998926647490e-05, -9.969224314551484e-01, 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-7.553329869388302e-01, 7.631344418442518e-04, 9.981736018665053e-01, -4.034601085328000e-01, 1.016793649077959e-03, 9.983952046173069e-01, -1.854951369598807e-01, 1.067126311051324e-03, 9.985153359170394e-01, 5.693738362511860e-01, 6.823019963489656e-04, 9.985841032168380e-01, 8.747010290291490e-01, 4.298394507636698e-04, 9.986616936036532e-01, 7.274743018665757e-01, 5.666609989068298e-04, 9.986723383294596e-01, -6.114560344532366e-01, 5.878791899273640e-04, 9.988762864534128e-01, 3.869508160326630e-01, 6.702204214070438e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *smr55 ( ) /******************************************************************************/ /* Purpose: SMR55 returns the SMR rule of degree 55. Discussion: DEGREE: 55 SYMMETRY: (X, Y), (-X, -Y). POINTS CARDINALITY: 536 NORM INF MOMS. RESIDUAL: 2.29858e-16 SUM NEGATIVE WEIGHTS: 0.00000e+00, Licensing: This code is distributed under the MIT license. Modified: 22 February 2018 Author: Original MATLAB version by Mattia Festa, Alvise Sommariva, C version by John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, double *SMR55[3*536], the requested rule. */ { const int degree = 55; int order; double xw[3*536] = { -9.988024883360008e-01, -4.104312670989267e-01, 7.247738364520382e-04, -9.986637711500016e-01, 5.835721439184322e-01, 6.485919247428450e-04, -9.985517638075939e-01, 1.568937801098068e-01, 1.047965997616771e-03, -9.983624199159088e-01, -7.311569400195331e-01, 6.234634259867621e-04, -9.982698924284638e-01, -5.843339960398285e-01, 7.392815641484668e-04, -9.982434766908749e-01, 3.819333028903024e-01, 9.500630778549490e-04, -9.982097416579526e-01, -8.832442641091779e-01, 4.604405106623550e-04, -9.978286522784937e-01, -9.628763075615144e-01, 3.344694660648939e-04, -9.976963561372271e-01, 9.678687355428703e-01, 3.138082077303478e-04, -9.976125620794032e-01, 7.261201764578525e-01, 8.279866159627213e-04, -9.973396392196245e-01, -9.972222237164891e-01, 8.333752929282447e-05, -9.972455951936153e-01, -7.187156081293171e-02, 1.410032513903549e-03, -9.971099785500591e-01, 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2.479538638037798e-01, 1.491315677106339e-03, 9.971099785500591e-01, -9.975224120853297e-01, 7.798442798007867e-05, 9.972455951936153e-01, 7.187156081293171e-02, 1.410032513903549e-03, 9.973396392196245e-01, 9.972222237164891e-01, 8.333752929282447e-05, 9.976125620794032e-01, -7.261201764578525e-01, 8.279866159627213e-04, 9.976963561372271e-01, -9.678687355428703e-01, 3.138082077303478e-04, 9.978286522784937e-01, 9.628763075615144e-01, 3.344694660648939e-04, 9.982097416579526e-01, 8.832442641091779e-01, 4.604405106623550e-04, 9.982434766908749e-01, -3.819333028903024e-01, 9.500630778549490e-04, 9.982698924284638e-01, 5.843339960398285e-01, 7.392815641484668e-04, 9.983624199159088e-01, 7.311569400195331e-01, 6.234634259867621e-04, 9.985517638075939e-01, -1.568937801098068e-01, 1.047965997616771e-03, 9.986637711500016e-01, -5.835721439184322e-01, 6.485919247428450e-04, 9.988024883360008e-01, 4.104312670989267e-01, 7.247738364520382e-04 }; double *xw_copy; order = square_minimal_rule_order ( degree ); xw_copy = r8mat_copy_new ( 3, order, xw ); return xw_copy; } /******************************************************************************/ double *square_minimal_rule ( int degree ) /******************************************************************************/ /* Purpose: SQUARE_MINIMAL_RULE returns a minimal rule for the square. Licensing: This code is distributed under the GNU GPL license. Modified: 20 February 2018 Author: John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Input, int DEGREE, the degree, between 0 and 55. Output, double *SQUARE_MINIMAL_RULE[3*O], the rule. */ { double *xw; if ( degree == 0 ) { xw = smr00 ( ); } else if ( degree == 1 ) { xw = smr01 ( ); } else if ( degree == 2 ) { xw = smr02 ( ); } else if ( degree == 3 ) { xw = smr03 ( ); } else if ( degree == 4 ) { xw = smr04 ( ); } else if ( degree == 5 ) { xw = smr05 ( ); } else if ( degree == 6 ) { xw = smr06 ( ); } else if ( degree == 7 ) { xw = smr07 ( ); } else if ( degree == 8 ) { xw = smr08 ( ); } else if ( degree == 9 ) { xw = smr09 ( ); } else if ( degree == 10 ) { xw = smr10 ( ); } else if ( degree == 11 ) { xw = smr11 ( ); } else if ( degree == 12 ) { xw = smr12 ( ); } else if ( degree == 13 ) { xw = smr13 ( ); } else if ( degree == 14 ) { xw = smr14 ( ); } else if ( degree == 15 ) { xw = smr15 ( ); } else if ( degree == 16 ) { xw = smr16 ( ); } else if ( degree == 17 ) { xw = smr17 ( ); } else if ( degree == 18 ) { xw = smr18 ( ); } else if ( degree == 19 ) { xw = smr19 ( ); } else if ( degree == 20 ) { xw = smr20 ( ); } else if ( degree == 21 ) { xw = smr21 ( ); } else if ( degree == 22 ) { xw = smr22 ( ); } else if ( degree == 23 ) { xw = smr23 ( ); } else if ( degree == 24 ) { xw = smr24 ( ); } else if ( degree == 25 ) { xw = smr25 ( ); } else if ( degree == 26 ) { xw = smr26 ( ); } else if ( degree == 27 ) { xw = smr27 ( ); } else if ( degree == 28 ) { xw = smr28 ( ); } else if ( degree == 29 ) { xw = smr29 ( ); } else if ( degree == 30 ) { xw = smr30 ( ); } else if ( degree == 31 ) { xw = smr31 ( ); } else if ( degree == 32 ) { xw = smr32 ( ); } else if ( degree == 33 ) { xw = smr33 ( ); } else if ( degree == 34 ) { xw = smr34 ( ); } else if ( degree == 35 ) { xw = smr35 ( ); } else if ( degree == 36 ) { xw = smr36 ( ); } else if ( degree == 37 ) { xw = smr37 ( ); } else if ( degree == 38 ) { xw = smr38 ( ); } else if ( degree == 39 ) { xw = smr39 ( ); } else if ( degree == 40 ) { xw = smr40 ( ); } else if ( degree == 41 ) { xw = smr41 ( ); } else if ( degree == 42 ) { xw = smr42 ( ); } else if ( degree == 43 ) { xw = smr43 ( ); } else if ( degree == 44 ) { xw = smr44 ( ); } else if ( degree == 45 ) { xw = smr45 ( ); } else if ( degree == 46 ) { xw = smr46 ( ); } else if ( degree == 47 ) { xw = smr47 ( ); } else if ( degree == 48 ) { xw = smr48 ( ); } else if ( degree == 49 ) { xw = smr49 ( ); } else if ( degree == 50 ) { xw = smr50 ( ); } else if ( degree == 51 ) { xw = smr51 ( ); } else if ( degree == 52 ) { xw = smr52 ( ); } else if ( degree == 53 ) { xw = smr53 ( ); } else if ( degree == 54 ) { xw = smr54 ( ); } else if ( degree == 55 ) { xw = smr55 ( ); } else { xw = NULL; printf ( "\n" ); printf ( "SQUARE_MINIMAL_RULE - Fatal error!\n" ); printf ( " 0 <= DEGREE <= 55 is required.\n" ); exit ( 1 ); } return xw; } /******************************************************************************/ int square_minimal_rule_degree_max ( ) /******************************************************************************/ /* Purpose: SQUARE_MINIMAL_RULE_DEGREE_MAX returns the maximum rule degree. Licensing: This code is distributed under the GNU GPL license. Modified: 20 February 2018 Author: John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Output, integer DEGREE_MAX, the maximum degree of the minimal rules for the square. */ { const int degree_max = 55; return degree_max; } /******************************************************************************/ double square_minimal_rule_error_max ( int degree ) /******************************************************************************/ /* Purpose: SQUARE_MINIMAL_RULE_ERROR_MAX returns the maximum error. Discussion: The rule of given DEGREE should theoretically have zero error for all monomials of degrees 0 <= D <= DEGREE. This function checks every such monomial and reports the maximum error. Licensing: This code is distributed under the GNU GPL license. Modified: 22 February 2018 Author: John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Input, int DEGREE, the desired total polynomial degree exactness of the quadrature rule. Output, double SQUARE_MINIMAL_RULE_ERROR_MAX, the maximum error observed when using the rule to compute the integrals of all monomials of degree between 0 and DEGREE. */ { int d; int e[2]; double err; double error_max; double exact; int i; int j; int k; int order; double s; double *xyw; order = square_minimal_rule_order ( degree ); xyw = square_minimal_rule ( degree ); error_max = 0.0; for ( d = 0; d <= degree; d++ ) { for ( i = 0; i <= d; i++ ) { j = d - i; e[0] = i; e[1] = j; exact = squaresym_monomial_integral ( e ); s = 0.0; for ( k = 0; k < order; k++ ) { s = s + xyw[2+k*3] * pow ( xyw[0+k*3], i ) * pow ( xyw[1+k*3], j ); } err = fabs ( exact - s ); if ( error_max < err ) { error_max = err; } } } return error_max; } /******************************************************************************/ int square_minimal_rule_order ( int degree ) /******************************************************************************/ /* Purpose: SQUARE_MINIMAL_RULE_ORDER returns the order of a minimal square rule. Licensing: This code is distributed under the GNU GPL license. Modified: 20 February 2018 Author: John Burkardt. Reference: Mattia Festa, Alvise Sommariva, Computing almost minimal formulas on the square, Journal of Computational and Applied Mathematics, Volume 17, Number 236, November 2012, pages 4296-4302. Parameters: Input, int DEGREE, the degree of the rule, between 0 and 55. Output, int SQUARE_MINIMAL_RULE_ORDER, the order of the rule. */ { int degree_max; int order; int order_list[56] = { 1, 1, 3, 4, 6, 7, 10, 12, 16, 17, 22, 24, 31, 33, 40, 43, 52, 54, 64, 67, 78, 81, 93, 96, 109, 113, 127, 132, 146, 153, 167, 172, 189, 197, 211, 220, 238, 245, 265, 274, 296, 303, 326, 331, 353, 359, 387, 396, 417, 427, 454, 462, 493, 498, 530, 536 }; degree_max = square_minimal_rule_degree_max ( ); if ( degree < 0 ) { order = 0; } else if ( degree_max < degree ) { order = 0; } else { order = order_list[degree]; } return order; } /******************************************************************************/ double squaresym_area ( ) /******************************************************************************/ /* Purpose: SQUARESYM_AREA: area of the symmetric unit square in 2D. Licensing: This code is distributed under the MIT license. Modified: 20 February 2018 Author: John Burkardt Parameters: Output, double SQUARE01_AREA, the area. */ { double area; area = 4.0; return area; } /******************************************************************************/ double squaresym_monomial_integral ( int e[2] ) /******************************************************************************/ /* Purpose: SQUARESYM_MONOMIAL_INTEGRAL: integrals over the symmetric unit square in 2D. Discussion: The integration region is -1 <= X <= 1, -1 <= Y <= 1. The monomial is F(X,Y) = X^E(1) * Y^E(2). Licensing: This code is distributed under the MIT license. Modified: 20 February 2018 Author: John Burkardt Reference: Philip Davis, Philip Rabinowitz, Methods of Numerical Integration, Second Edition, Academic Press, 1984, page 263. Parameters: Input, int E[2], the exponents. Each exponent must be nonnegative. Output, double SQUARESYM_MONOMIAL_INTEGRAL, the integral. */ { int i; double integral; const int m = 2; for ( i = 0; i < m; i++ ) { if ( e[i] < 0 ) { fprintf ( stderr, "\n" ); fprintf ( stderr, "SQUARESYM_MONOMIAL_INTEGRAL - Fatal error!\n" ); fprintf ( stderr, " All exponents must be nonnegative.\n" ); fprintf ( stderr, " E[%d] = %d\n", i, e[i] ); exit ( 1 ); } } if ( ( e[0] % 2 ) == 1 || ( e[1] % 2 ) == 1 ) { integral = 0.0; } else { integral = 4.0 / ( double ) ( e[0] + 1 ) / ( double ) ( e[1] + 1 ); } return integral; } /******************************************************************************/ void timestamp ( ) /******************************************************************************/ /* Purpose: TIMESTAMP prints the current YMDHMS date as a time stamp. Example: 31 May 2001 09:45:54 AM Licensing: This code is distributed under the MIT license. Modified: 24 September 2003 Author: John Burkardt Parameters: None */ { # define TIME_SIZE 40 static char time_buffer[TIME_SIZE]; const struct tm *tm; time_t now; now = time ( NULL ); tm = localtime ( &now ); strftime ( time_buffer, TIME_SIZE, "%d %B %Y %I:%M:%S %p", tm ); fprintf ( stdout, "%s\n", time_buffer ); return; # undef TIME_SIZE }