07-Jan-2022 20:32:28 gen_laguerre_exactness_test(): MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Test gen_laguerre_exactness(). 07-Jan-2022 20:32:28 GEN_LAGUERRE_EXACTNESS MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Investigate the polynomial exactness of a generalized Gauss-Laguerre quadrature rule by integrating exponentially weighted monomials up to a given degree over the [0,+oo) interval. The rule may be defined on another interval, [A,+oo) in which case it is adjusted to the [0,+oo) interval. GEN_LAGUERRE_EXACTNESS: User input: Quadrature rule X file = "gen_lag_o8_a0.5_x.txt". Quadrature rule W file = "gen_lag_o8_a0.5_w.txt". Quadrature rule R file = "gen_lag_o8_a0.5_r.txt". Maximum degree to check = 18 Weighting function exponent ALPHA = 0.500000 OPTION = 0, integrate x^alpha*exp(-x)*f(x). Spatial dimension = 1 Number of points = 8 The quadrature rule to be tested is a generalized Gauss-Laguerre rule ORDER = 8 A = 0.000000 ALPHA = 0.500000 OPTION = 0, standard rule: Integral ( A <= x < oo ) x^alpha exp(-x) f(x) dx is to be approximated by sum ( 1 <= I <= ORDER ) w(i) * f(x(i)). Weights W: w(1) = 0.2271393619524718 w(2) = 0.3935945428036146 w(3) = 0.2129089708672283 w(4) = 0.0478774832031382 w(5) = 0.0045425174747626 w(6) = 0.0001624046001853 w(7) = 0.0000016423774138 w(8) = 0.0000000021739431 Abscissas X: x(1) = 0.2826336481165992 x(2) = 1.1398738015816141 x(3) = 2.6015248434060290 x(4) = 4.7241145375277904 x(5) = 7.6052562992316144 x(6) = 11.4171820765458296 x(7) = 16.4994107976558197 x(8) = 23.7300039959347089 Region R: r(1) = 0.000000e+00 r(2) = 1.000000e+30 A generalized Gauss-Laguerre rule would be able to exactly integrate monomials up to and including degree = 15 Error Degree 0.0000000000000001 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000003 3 0.0000000000000001 4 0.0000000000000004 5 0.0000000000000004 6 0.0000000000000004 7 0.0000000000000005 8 0.0000000000000006 9 0.0000000000000008 10 0.0000000000000017 11 0.0000000000000018 12 0.0000000000000035 13 0.0000000000000053 14 0.0000000000000045 15 0.0000561671454582 16 0.0004926661044402 17 0.0022799523824517 18 GEN_LAGUERRE_EXACTNESS: Normal end of execution. 07-Jan-2022 20:32:28 07-Jan-2022 20:32:28 GEN_LAGUERRE_EXACTNESS MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2 Investigate the polynomial exactness of a generalized Gauss-Laguerre quadrature rule by integrating exponentially weighted monomials up to a given degree over the [0,+oo) interval. The rule may be defined on another interval, [A,+oo) in which case it is adjusted to the [0,+oo) interval. GEN_LAGUERRE_EXACTNESS: User input: Quadrature rule X file = "gen_lag_o8_a0.5_modified_x.txt". Quadrature rule W file = "gen_lag_o8_a0.5_modified_w.txt". Quadrature rule R file = "gen_lag_o8_a0.5_modified_r.txt". Maximum degree to check = 18 Weighting function exponent ALPHA = 0.500000 OPTION = 1, integrate f(x). Spatial dimension = 1 Number of points = 8 The quadrature rule to be tested is a generalized Gauss-Laguerre rule ORDER = 8 A = 0.000000 ALPHA = 0.500000 OPTION = 1, modified rule: Integral ( A <= x < oo ) f(x) dx is to be approximated by sum ( 1 <= I <= ORDER ) w(i) * f(x(i)). Weights W: w(1) = 0.5667959040373108 w(2) = 1.1525548015354481 w(3) = 1.7799502176328139 w(4) = 2.4810069381384330 w(5) = 3.3087238631029070 w(6) = 4.3675515321753773 w(7) = 5.9202740429118927 w(8) = 9.0242073059178782 Abscissas X: x(1) = 0.2826336481165992 x(2) = 1.1398738015816141 x(3) = 2.6015248434060290 x(4) = 4.7241145375277904 x(5) = 7.6052562992316144 x(6) = 11.4171820765458296 x(7) = 16.4994107976558197 x(8) = 23.7300039959347089 Region R: r(1) = 0.000000e+00 r(2) = 1.000000e+30 A generalized Gauss-Laguerre rule would be able to exactly integrate monomials up to and including degree = 15 Error Degree 0.0000000000000003 0 0.0000000000000000 1 0.0000000000000001 2 0.0000000000000002 3 0.0000000000000001 4 0.0000000000000000 5 0.0000000000000000 6 0.0000000000000001 7 0.0000000000000002 8 0.0000000000000002 9 0.0000000000000005 10 0.0000000000000030 11 0.0000000000000000 12 0.0000000000000053 13 0.0000000000000031 14 0.0000000000000024 15 0.0000561671454600 16 0.0004926661044420 17 0.0022799523824534 18 GEN_LAGUERRE_EXACTNESS: Normal end of execution. 07-Jan-2022 20:32:28 gen_laguerre_exactness_test(): Normal end of execution. 07-Jan-2022 20:32:28