04 April 2024 08:28:25 PM test_matrix_test(): C version test_matrix() provides many test matrices with known properties. bvec_next_grlex_test(): bvec_next_grlex() computes binary vectors in GRLEX order. 0: 0000 1: 0001 2: 0010 3: 0100 4: 1000 5: 0011 6: 0101 7: 0110 8: 1001 9: 1010 10: 1100 11: 0111 12: 1011 13: 1101 14: 1110 15: 1111 16: 0000 legendre_zeros_test(): legendre_zeros() computes the zeros of the N-th Legendre polynomial. Legendre zeros 0: 0.000000 Legendre zeros 0: -0.577350 1: 0.577350 Legendre zeros 0: -0.774597 1: 0.000000 2: 0.774597 Legendre zeros 0: -0.861136 1: -0.339981 2: 0.339981 3: 0.861136 Legendre zeros 0: -0.906180 1: -0.538469 2: 0.000000 3: 0.538469 4: 0.906180 Legendre zeros 0: -0.932470 1: -0.661209 2: -0.238619 3: 0.238619 4: 0.661209 5: 0.932470 Legendre zeros 0: -0.949108 1: -0.741531 2: -0.405845 3: 0.000000 4: 0.405845 5: 0.741531 6: 0.949108 mertens_test(): mertens() computes the Mertens function. N Exact MERTENS(N) 1 1 1 2 0 0 3 -1 -1 4 -1 -1 5 -2 -2 6 -1 -1 7 -2 -2 8 -2 -2 9 -2 -2 10 -1 -1 11 -2 -2 12 -2 -2 100 1 1 1000 2 2 10000 -23 -23 moebius_test() moebius() computes the Moebius function. N Exact MOEBIUS(N) 1 1 1 2 -1 -1 3 -1 -1 4 0 0 5 -1 -1 6 1 1 7 -1 -1 8 0 0 9 0 0 10 1 1 11 -1 -1 12 0 0 13 -1 -1 14 1 1 15 1 1 16 0 0 17 -1 -1 18 0 0 19 -1 -1 20 0 0 r8mat_is_eigen_left_test(): r8mat_is_eigen_left() tests the error in the left eigensystem A' * X - X * LAMBDA = 0 Matrix A: Col: 1 2 3 4 Row 1 0.136719 0.605469 0.253906 0.003906 2 0.058594 0.527344 0.394531 0.019531 3 0.019531 0.394531 0.527344 0.058594 4 0.003906 0.253906 0.605469 0.136719 Eigenmatrix X: Col: 1 2 3 4 Row 1 1.000000 1.000000 1.000000 1.000000 2 11.000000 3.000000 -1.000000 -3.000000 3 11.000000 -3.000000 -1.000000 3.000000 4 1.000000 -1.000000 1.000000 -1.000000 Eigenvalues LAM: 0: 1.000000 1: 0.250000 2: 0.062500 3: 0.015625 Frobenius norm of A'*X-X*LAMBDA is 9.40908 r8mat_is_llt_test(): r8mat_is_llt() tests the error in a lower triangular Cholesky factorization A = L * L' by looking at A-L*L' Matrix A: Col: 1 2 3 4 Row 1 2.000000 1.000000 0.000000 0.000000 2 1.000000 2.000000 1.000000 0.000000 3 0.000000 1.000000 2.000000 1.000000 4 0.000000 0.000000 1.000000 2.000000 Cholesky factor L: Col: 1 2 3 4 Row 1 1.414214 0.000000 0.000000 0.000000 2 0.707107 1.224745 0.000000 0.000000 3 0.000000 0.816497 1.154701 0.000000 4 0.000000 0.000000 0.866025 1.118034 Frobenius norm of A-L*L' is 2.18689e-15 r8mat_is_null_left_test(): r8mat_is_null_left() tests whether the M vector X is a left null vector of A, that is, x'*A=0. Matrix A: Col: 1 2 3 Row 1 1.000000 2.000000 3.000000 2 4.000000 5.000000 6.000000 3 7.000000 8.000000 9.000000 Vector X: 0: 1.000000 1: -2.000000 2: 1.000000 Frobenius norm of X'*A is 0 r8mat_is_null_right_test(): r8mat_is_null_right() tests whether the N vector X is a right null vector of A, that is, A*x=0. Matrix A: Col: 1 2 3 Row 1 1.000000 2.000000 3.000000 2 4.000000 5.000000 6.000000 3 7.000000 8.000000 9.000000 Vector X: 0: 1.000000 1: -2.000000 2: 1.000000 Frobenius norm of A*x is 0 r8mat_is_solution_test(): r8mat_is_solution() tests whether X is the solution of A*X=B by computing the Frobenius norm of the residual. A is 3 by 10 X is 10 by 9 B is 3 by 9 Frobenius error in A*X-B is 2.36629e-14 r8mat_norm_fro_test(): r8mat_norm_fro() computes the Frobenius norm of a matrix. Matrix A: Col: 1 2 3 4 Row 1 1.000000 2.000000 3.000000 4.000000 2 5.000000 6.000000 7.000000 8.000000 3 9.000000 10.000000 11.000000 12.000000 4 13.000000 14.000000 15.000000 16.000000 5 17.000000 18.000000 19.000000 20.000000 Expected Frobenius norm = 53.5724 Computed Frobenius norm = 53.5724 test_condition(): Compare reported and computed condition numbers of a test matrix. Title N COND COND Reported Computed aegerter 5 24 24 antisummation 5 80 80 BAB 5 8.46751 8.46751 BAUER 6 8.52877e+06 8.52877e+06 BIS 5 42.9756 42.9756 BIW 5 59.9171 59.9171 BODEWIG 4 10.4366 10.4366 BOOTHROYD 5 1.002e+06 1.002e+06 COMBIN 5 11.9644 11.9644 COMPANION 5 66.4264 66.4264 CONEX1 4 68.0622 68.0622 CONEX2 3 17.7034 17.7034 CONEX1 4 68.0622 68.0622 CONEX2 3 17.7034 17.7034 CONEX3 5 80 80 CONEX4 4 4488 4488 DAUB2 4 2 2 DAUB4 8 2.79904 2.79904 DAUB6 12 3.44146 3.44146 DAUB8 16 3.47989 3.47989 DAUB10 20 4.00375 4.00375 DAUB12 24 4.80309 4.80309 DIAGONAL 5 7.39629 7.39629 DIF2 5 18 18 DOWNSHIFT 5 1 1 EXCHANGE 5 1 1 fibonacci2 5 15 15 GFPP 5 12.2633 12.2633 GIVENS 5 50 50 HANKEL_N 5 5.8368 5.8368 HARMAN 8 77.069 77.069 HARTLEY 5 5 5 IDENTITY 5 1 1 ILL3 3 216775 216775 JORDAN 5 2.08956 2.08956 KERSHAW 4 49 49 LIETZKE 5 38 38 MAXIJ 5 100 100 MINIJ 5 60 60 orthogonal_symmetric 5 4.39765 4.39765 OTO 5 18 18 PASCAL1 5 100 100 PASCAL3 5 14333.5 14333.5 PEI 5 4.90227 4.90227 RODMAN 5 -1 5.859 RUTIS1 4 15 15 RUTIS2 4 11.44 11.44 RUTIS3 4 6 6 RUTIS5 4 62608 62608 summation 5 10 10 SWEET1 6 16.9669 16.9669 SWEET2 6 49.2227 49.2227 SWEET3 6 24.7785 24.7785 SWEET4 13 51.1709 51.1709 TRI_UPPER 5 2599.9 2599.9 UPSHIFT 5 1 1 WILK03 3 2.6e+10 2.6e+10 WILK04 4 2.45892e+16 2.85325e+16 WILK05 5 7.93703e+06 7.93703e+06 WILSON 4 4488 4488 test_determinant(): Compute the determinants of an example of each test matrix; compare with the determinant routine, if available. Print the matrix Frobenius norm for an estimate of magnitude. Title N Determ Determ ||A|| A123 3 0 6.66134e-16 16.881943 AEGERTER 5 -25 -25 9.433981 ANTICIRCULANT 3 -235.484 -235.484 10.900784 ANTICIRCULANT 4 1407.78 1407.78 12.647476 ANTICIRCULANT 5 7148.67 7148.67 14.266561 ANTIHADAMARD 5 1 1 3.316625 antisummation 5 1 1 3.872983 antisymmetric_random 5 0 2.873692 antisymmetric_random 6 0.097353 3.334512 BAB 5 -1980.11 -1980.11 14.360526 BAUER 6 1 1 185.854782 bernstein_matrix 5 96 96 25.278449 BIMARKOV_RANDOM 5 -8.62803e-05 1.387933 BIS 5 -177.02 -177.02 11.087580 BIW 5 0.0547223 0.0547223 2.360508 BODEWIG 4 568 568 12.727922 BOOTHROYD 5 1 1 886.710212 BORDERBAND 5 -0.328125 -0.328125 2.766993 CARRY 5 1.65382e-08 1.65382e-08 1.413914 CAUCHY 5 38.7671 38.7671 682.272599 CHEBY_DIFF1 5 -2.13163e-14 13.472194 CHEBY_DIFF1 5 -2.13163e-14 13.472194 CHEBY_T 5 64 64 12.688578 CHEBY_U 5 1024 1024 22.427661 CHEBY_VAN1 5 18 4.301163 CHEBY_VAN2 2 -2 -2 2.000000 CHEBY_VAN2 3 -1.41421 -1.41421 2.000000 CHEBY_VAN2 4 1 1 2.081666 CHEBY_VAN2 5 0.707107 0.707107 2.179449 CHEBY_VAN2 6 -0.5 -0.5 2.280351 CHEBY_VAN2 7 -0.353553 -0.353553 2.380476 CHEBY_VAN2 8 0.25 0.25 2.478479 CHEBY_VAN2 9 0.176777 0.176777 2.573908 CHEBY_VAN2 10 -0.125 -0.125 2.666667 CHEBY_VAN3 5 13.9754 13.9754 3.872983 CHOW 5 -70.5488 -70.5488 202.500592 CIRCULANT 5 7148.67 7148.67 14.266561 CIRCULANT2 3 18 18 6.480741 CIRCULANT2 4 -160 -160 10.954451 CIRCULANT2 5 1875 1875 16.583124 CLEMENT1 5 0 0 6.324555 CLEMENT1 6 -225 -225 8.366600 CLEMENT2 5 0 0 8.979002 CLEMENT2 6 -178.154 -178.154 10.160037 COMBIN 5 1257.33 1257.33 20.777819 COMPANION 5 -2.81582 -2.81582 6.686326 COMPLEX_I 2 1 1 1.414214 CONEX1 4 -2.81582 -2.81582 8.129955 CONEX2 3 -0.355137 -0.355137 2.648756 CONEX3 5 -1 -1 3.872983 CONEX4 4 -1 -1 30.545049 CONFERENCE 6 -125 -125 5.477226 CREATION 5 0 0 5.477226 DAUB2 4 1 1 2.000000 DAUB4 8 -1 -1 2.828427 DAUB6 12 1 1 3.464102 DAUB8 16 -1 -1 4.000000 DAUB10 20 1 1 4.472136 DAUB12 24 -1 -1 4.898979 DIAGONAL 5 22.1228 22.1228 6.380200 DIF1 5 0 0 2.828427 DIF1 6 1 1 3.162278 DIF1CYCLIC 5 0 0 3.162278 DIF2 5 6 6 5.291503 DIF2CYCLIC 5 0 0 5.477226 DORR 5 -6.33817e+10 -6.33817e+10 533.002742 DOWNSHIFT 5 1 1 2.236068 EBERLEIN 5 0 -1.02318e-12 18.100196 EULERIAN 5 1 1 77.298124 EXCHANGE 5 1 1 2.236068 FIBONACCI1 5 0 -0 95.352724 fibonacci2 5 -1 -1 3.000000 FIBONACCI3 5 8 8 3.605551 FIEDLER 7 1332.21 1332.21 30.134997 FORSYTHE 5 1975.68 1975.68 10.772265 FORSYTHE 5 1975.68 1975.68 10.772265 FOURIER_COSINE 5 1 1 2.236068 FOURIER_SINE 5 1 1 2.236068 FRANK 5 1 1 11.618950 GFPP 5 212.007 212.007 9.396183 GIVENS 5 16 16 20.615528 GK323 5 32 32 10.000000 GK324 5 11.953 11.953 11.457701 GRCAR 5 8 3.605551 HADAMARD 5 0 4.000000 HANKEL 5 -2823.88 15.212611 HANKEL_N 5 3125 3125 15.000000 HANOWA 6 1803.1 1803.1 8.693271 HARMAN 8 0.000954779 0.000954779 5.053593 HARTLEY 5 55.9017 55.9017 5.000000 HARTLEY 6 -216 -216 6.000000 HARTLEY 7 -907.493 -907.493 7.000000 HARTLEY 8 -4096 -4096 8.000000 HELMERT 5 1 1 2.236068 HELMERT2 5 1 2.236068 HERMITE 5 1024 1024 54.194096 HERNDON 5 -0.04 -0.04 1.771327 HILBERT 5 3.7493e-12 3.7493e-12 1.580906 HOUSEHOLDER 5 -1 -1 2.236068 idempotent_random 5 0 0 1.000000 IDENTITY 5 1 1 2.236068 IJFACT1 5 7.16636e+09 7.16636e+09 3665587.891976 IJFACT2 5 1.4948e-21 1.4948e-21 0.557720 ILL3 3 6 6 817.763413 INTEGRATION 5 1 1 4.035785 involutory 5 -1 -1 1942.461253 involutory_RANDOM 5 -1 -1 2.236068 JACOBI 5 0 0 1.490712 JACOBI 6 -0.021645 -0.021645 1.651446 JORDAN 5 -177.02 -177.02 6.606370 KAHAN 5 -3.78564e-08 -3.78564e-08 0.715639 KERSHAW 4 1 1 8.246211 KERSHAWTRI 5 553.995 553.995 8.738446 KMS 5 2304.83 2304.83 101.704415 LAGUERRE 5 0.00347222 0.00347222 6.853755 legendre_matrix 5 16.4062 16.4062 6.807624 LEHMER 5 0.065625 0.065625 3.280413 LESLIE 4 0.605244 0.605244 1.784145 LESP 5 -42300 -42300 22.348683 LIETZKE 5 48 48 18.027756 LIGHTS_OUT -----Not ready---- LINE_ADJ 5 0 0 2.828427 LINE_ADJ 6 -1 -1 3.162278 LINE_LOOP_ADJ 5 0 -0 3.605551 LOEWNER 5 -29.0825 20.522700 LOTKIN 5 1.87465e-11 1.87465e-11 2.456757 MARKOV_RANDOM 5 0.00488558 1.335842 MAXIJ 5 5 5 19.874607 MILNES 5 11.953 11.953 11.457701 MINIJ 5 1 1 12.449900 MOLER1 5 1 1 61.884953 MOLER2 5 0 1.02538e-07 101035.360741 MOLER3 5 1 1 8.660254 MOLER4 4 1 1 2.828427 NEUMANN 25 0 0.00124631 23.237900 ONE 5 0 0 5.000000 ORTEGA 5 -16.5253 -16.5253 244.267730 orthogonal_random 5 1 1 2.236068 orthogonal_symmetric 5 1 1 2.236068 OTO 5 6 6 5.291503 PARTER 5 131.917 131.917 6.340772 PASCAL1 5 1 1 9.949874 PASCAL2 5 1 1 92.460803 PASCAL3 5 1 1 124.742251 SPD_RANDOM 5 0.0404187 0.0404187 1.462299 PEI 5 137.311 137.311 6.040361 PERMUTATION_RANDOM 5 1 1 2.236068 PLU 5 1.93261e+07 1.93261e+07 139.254067 POISSON 25 3.25655e+13 3.25655e+13 21.908902 PROLATE 5 -5651.77 12.598427 RECTANGLE_ADJ -----Not ready----- REDHEFFER 5 -2 -2 3.741657 REF_RANDOM 5 0 0 2.635602 REF_RANDOM 5 1 1 2.818935 RIEMANN 5 -66 6.928203 RING_ADJ 1 1 1 1.000000 RING_ADJ 2 -1 -1 1.414214 RING_ADJ 3 2 2 2.449490 RING_ADJ 4 0 0 2.828427 RING_ADJ 5 2 2 3.162278 RING_ADJ 6 -4 -4 3.464102 RING_ADJ 7 2 2 3.741657 RING_ADJ 8 0 0 4.000000 RIS 5 4.12239 4.12239 3.170386 RODMAN 5 -2175.88 -2175.88 12.789703 ROSSER1 8 0 -9480.58 2482.257037 ROUTH 5 7.85813 7.85813 5.154913 RUTIS1 4 -375 -375 16.613248 RUTIS2 4 100 100 11.401754 RUTIS3 4 624 624 14.142136 RUTIS4 4 125 125 50.714889 RUTIS5 4 1 1 23.769729 SCHUR_BLOCK 5 589.771 589.771 8.399778 SKEW_CIRCULANT 5 -10310.4 -10310.4 14.266561 SPLINE 5 -2566.72 -2566.72 20.824413 STIRLING 5 1 1 67.919069 STRIPE 5 2112 14.832397 summation 5 1 1 3.872983 SWEET1 6 -2.04682e+07 -2.04682e+07 70.199715 SWEET2 6 9562.52 9562.52 30.143287 SWEET3 6 -5.40561e+07 -5.40561e+07 73.423430 SWEET4 13 -6.46348e+16 -6.46348e+16 119.703919 SYLVESTER 5 -222.565 12.499506 SYLVESTER_KAC 5 0 0 7.745967 SYLVESTER_KAC 6 -225 -225 10.488088 symmetric_random 5 22.1228 22.1228 6.380200 TOEPLITZ 5 -2823.88 15.212611 TOEPLITZ_5DIAG 5 -747.438 12.846764 TOEPLITZ_5S 25 -1.51735e+17 40.398137 TOEPLITZ_SPD 5 0.0849362 3.415735 TOURNAMENT_RANDOM 5 0 0 4.472136 TRANSITION_RANDOM 5 0.00486764 1.323312 TRENCH 5 -37.7411 7.030318 TRI_UPPER 5 1 1 9.180864 TRIS 5 6683.42 6683.42 13.388753 TRIV 5 -700.369 -700.369 11.120402 TRIW 5 1 1 9.396291 UPSHIFT 5 1 1 2.236068 VAND1 5 133985 133985 466.164260 VAND2 5 133985 133985 466.164260 WATHEN -----Not ready----- WILK03 3 9e-21 9e-21 1.392839 WILK04 4 4.42923e-17 4.42923e-17 1.895450 WILK05 5 3.7995e-15 3.79947e-15 1.514850 WILK12 12 1 1 53.591044 WILK20 20 1.4763e+25 100.388888 WILK21 21 -4.15825e+12 -4.15825e+12 28.460499 WILSON 4 1 1 30.545049 ZERO 5 0 -0 0.000000 ZIELKE 5 469.417 13.695338 test_eigen_left(): Compute the Frobenius norm of the eigenvalue error: X * A - LAMBDA * X given K left eigenvectors X and eigenvalues LAMBDA. Title N K ||A|| ||X*A-LAMBDA*X|| A123 3 3 16.8819 1.23246e-14 CARRY 5 5 1.41391 3.57943e-15 CHOW 5 5 202.501 1.16147e-12 DIAGONAL 5 5 6.3802 0 ROSSER1 8 8 2482.26 2.61994e-11 symmetric_random 5 5 6.3802 2.57279e-15 test_eigen_right() Compute the Frobenius norm of the eigenvalue error: A * X - X * LAMBDA given K right eigenvectors X and eigenvalues LAMBDA. Title N K ||A|| ||(A*X-X*Lambda|| A123 3 3 16.8819 1.33427e-14 BAB 5 5 14.3605 4.36701e-15 BODEWIG 4 4 12.7279 9.17346e-15 CARRY 5 5 1.41391 1.17642e-15 CHOW 5 5 202.501 9.44267e-13 COMBIN 5 5 20.7778 7.10543e-15 DIF2 5 5 5.2915 1.07099e-15 EXCHANGE 5 5 2.23607 0 fibonacci2 5 5 3 1.46869e-16 idempotent_random 5 5 1.73205 7.2129e-16 IDENTITY 5 5 2.23607 0 ILL3 3 3 817.763 1.62356e-11 KERSHAW 4 4 8.24621 4.80549e-15 KMS 5 5 2.32288 3.2055e-08 LINE_ADJ 5 5 2.82843 8.99223e-16 LINE_LOOP_ADJ 5 5 3.60555 9.99459e-16 ONE 5 5 5 0 ORTEGA 5 5 244.268 3.45197e-13 OTO 5 5 5.2915 1.07099e-15 SPD_RANDOM 5 5 1.4623 5.18134e-16 PEI 5 5 6.04036 0 RODMAN 5 5 12.7897 0 ROSSER1 8 8 2482.26 2.61994e-11 RUTIS1 4 4 16.6132 0 RUTIS2 4 4 11.4018 0 RUTIS5 4 4 23.7697 1.46286e-14 SYLVESTER_KAC 5 5 7.74597 0 symmetric_random 5 5 6.3802 2.49712e-15 tribonacci2 5 5 3.31662 4.2629e-15 WILK12 12 12 53.591 1.01528e-07 WILSON 4 4 30.545 2.48731e-14 ZERO 5 5 0 0 test_inverse(): A = a test matrix of order N; B = inverse as computed by a routine. C = inverse as computed by R8MAT_INVERSE. ||I-AB|| = Frobenius norm of I-A*B. ||I-AC|| = Frobenius norm of I-A*C. ||I-AB|| = Frobenius norm of I-A*B. Title N ||A|| ||C|| ||I-AC|| ||I-AB|| AEGERTER 5 9.43398 1.77133 5.72869e-16 7.1089e-16 antisummation 5 3.87298 10.8167 0 0 BAB 5 14.3605 0.720652 1.06903e-15 9.51853e-16 BAUER 6 185.855 21078.4 1.06558e-10 0 bernstein_matrix 5 25.2784 3.19613 0 0 BIS 5 11.0876 3.89011 8.88178e-16 1.98603e-15 BIW 5 2.36051 26.4794 3.76822e-15 2.03507e-15 BODEWIG 4 12.7279 0.683062 7.1575e-16 7.08784e-16 BOOTHROYD 5 886.71 886.71 1.92659e-11 0 BORDERBAND 5 2.76699 6.80053 0 0 CARRY 5 1.41391 3126.49 3.06614e-13 1.06315e-13 CAUCHY 5 682.273 60.9954 3.04637e-13 3.03487e-14 CHEBY_T 5 12.6886 1.87916 0 0 CHEBY_U 5 22.4277 1.22793 0 0 CHEBY_VAN2 5 2.17945 2.5 4.62778e-16 5.91396e-16 CHEBY_VAN3 5 3.87298 1.34164 9.73821e-16 6.57403e-16 CHOW 5 202.501 269.085 3.65461e-13 2.58333e-13 CIRCULANT 5 14.2666 0.405717 1.01758e-15 7.2061e-16 CIRCULANT2 5 16.5831 0.635959 9.22566e-16 1.49288e-15 CLEMENT1 6 8.3666 1.51731 6.47646e-16 0 CLEMENT2 6 10.16 2.66667 4.98728e-16 1.98603e-15 COMBIN 5 20.7778 0.712031 1.00228e-15 1.10466e-15 COMPANION 5 6.68633 2.87405 4.96507e-16 1.11022e-16 COMPLEX_I 2 1.41421 1.41421 0 0 CONEX1 4 8.12995 6.40059 0 0 CONEX2 3 2.64876 4.27223 2.22045e-16 2.22045e-16 CONEX3 5 3.87298 10.8167 0 0 CONFERENCE 6 5.47723 1.09545 7.75668e-16 0 DAUB2 4 2 2 0 8.88178e-16 DAUB4 8 2.82843 2.82843 2.69133e-16 2.26448e-15 DAUB6 12 3.4641 3.4641 1.01953e-15 1.57726e-15 DAUB8 16 4 4 1.70404e-15 4.70844e-15 DAUB10 20 4.47214 4.47214 1.68801e-15 8.75081e-15 DAUB12 24 4.89898 4.89898 2.19403e-15 1.95459e-14 DIAGONAL 5 6.3802 2.07065 0 0 DIF1 6 3.16228 3.4641 0 0 DIF2 5 5.2915 3.91933 1.04148e-15 6.86635e-16 DORR 5 533.003 0.0382712 1.64016e-15 1.60599e-15 DOWNSHIFT 5 2.23607 2.23607 0 0 EULERIAN 5 77.2981 784.774 2.54716e-13 0 EXCHANGE 5 2.23607 2.23607 0 0 fibonacci2 5 3 3.4641 0 0 FIBONACCI3 5 3.60555 1.58114 1.57009e-16 0 FIEDLER 7 30.135 3.26387 6.99264e-15 4.44956e-15 FORSYTHE 5 10.7723 0.520231 2.3017e-16 6.07649e-17 FOURIER_COSINE 5 2.23607 2.23607 1.15085e-15 9.60078e-16 FOURIER_SINE 5 2.23607 2.23607 7.46826e-16 1.75717e-15 FRANK 5 11.619 59.439 5.19793e-14 0 GFPP 5 9.39618 1.02503 1.58393e-16 1.53657e-14 GIVENS 5 20.6155 2.73861 0 0 GK323 5 10 2.30489 0 0 GK324 5 11.4577 5.59148 1.80048e-15 1.79018e-15 HANKEL_N 5 15 0.550372 6.9069e-16 0 HANOWA 6 8.69327 0.714 5.66105e-16 6.47366e-16 HARMAN 8 5.05359 14.6831 4.64392e-15 1.01653e-14 HARTLEY 5 5 1 9.67571e-16 2.75896e-15 HELMERT 5 2.23607 2.23607 4.85723e-16 7.58596e-16 HELMERT2 5 2.23607 2.23607 6.4278e-16 5.36183e-16 HERMITE 5 54.1941 1.80818 0 0 HERNDON 5 1.77133 9.43398 1.113e-15 7.1089e-16 HILBERT 5 1.58091 304160 8.65244e-12 7.27596e-12 HOUSEHOLDER 5 2.23607 2.23607 1.11383e-15 1.00796e-15 IDENTITY 5 2.23607 2.23607 0 0 ILL3 3 817.763 337.323 1.58731e-11 0 INTEGRATION 5 4.03579 7.46725 0 1.01754e-15 involutory 5 1942.46 1942.46 5.2446e-11 7.27596e-12 JACOBI 6 1.65145 6.48074 7.36439e-16 0 JORDAN 5 6.60637 0.837134 2.22045e-16 2.22045e-16 KAHAN 5 0.715639 433.301 3.14325e-16 3.71199e-16 KERSHAW 4 8.24621 8.24621 3.55271e-15 0 KERSHAWTRI 5 8.73845 0.689986 3.61558e-16 4.82153e-16 KMS 5 101.704 2.51888 4.78765e-15 1.94337e-14 LAGUERRE 5 6.85376 202.65 0 0 legendre_matrix 5 6.80762 1.86931 4.96507e-16 2.22045e-16 LEHMER 5 3.28041 7.7202 1.72652e-15 1.41744e-15 LESP 5 22.3487 0.320749 4.24778e-16 7.47998e-16 LIETZKE 5 18.0278 2.3863 3.3605e-15 6.95553e-16 LINE_ADJ 6 3.16228 3.4641 0 0 LOTKIN 5 2.45676 242794 1.70489e-11 0 MAXIJ 5 19.8746 4.65188 2.7555e-15 0 MILNES 5 11.4577 5.59148 1.80048e-15 1.79018e-15 MINIJ 5 12.4499 5 0 0 MOLER1 5 61.885 28220.2 3.95437e-11 4.14105e-11 MOLER3 5 8.66025 115.659 0 0 ORTEGA 5 244.268 91.1607 1.11923e-12 4.86154e-12 orthogonal_symmetric 5 2.23607 2.23607 1.23481e-15 2.20932e-15 OTO 5 5.2915 3.91933 1.04148e-15 6.86635e-16 PARTER 5 6.34077 0.943311 7.34737e-16 5.55112e-17 PASCAL1 5 9.94987 9.94987 0 0 PASCAL2 5 92.4608 92.4608 0 0 PASCAL3 5 124.742 124.742 2.73039e-13 5.72858e-14 SPD_RANDOM 5 1.4623 5.69873 7.22896e-16 3.77912e-15 PEI 5 6.04036 0.845046 1.32995e-15 2.00148e-16 PERMUTATION_RANDOM 5 2.23607 2.23607 0 0 PLU 5 139.254 0.121172 1.58684e-15 1.23361e-15 RIS 5 3.17039 1.88662 7.08478e-16 8.37173e-17 RODMAN 5 12.7897 0.533114 5.48831e-16 8.00593e-16 RUTIS1 4 16.6132 1.04137 1.74485e-15 1.05471e-15 RUTIS2 4 11.4018 1.14018 5.81544e-16 6.83824e-16 RUTIS3 4 14.1421 0.577813 7.5758e-16 5.92697e-16 RUTIS4 5 59.127 51.9905 2.48423e-13 3.36548e-13 RUTIS5 4 23.7697 1871.74 4.03548e-12 0 SCHUR_BLOCK 5 8.39978 0.652412 7.85046e-17 6.32925e-16 SPLINE 5 20.8244 0.966266 5.9712e-16 1.39536e-15 STIRLING 5 67.9191 32.45 2.85044e-14 0 summation 5 3.87298 3 0 0 SWEET1 6 70.1997 0.263219 1.4354e-15 1.08864e-13 SWEET2 6 30.1433 1.39919 3.85043e-15 3.48958e-14 SWEET3 6 73.4234 0.337964 1.30118e-15 1.43485e-13 SWEET4 13 119.704 0.384221 4.39293e-15 2.5206e-13 SYLVESTER_KAC 6 10.4881 2.54209 0 0 symmetric_random 5 6.3802 2.07065 1.47914e-15 4.81381e-15 TRI_UPPER 5 9.18086 167.99 0 6.02916e-14 TRIS 5 13.3888 0.395936 4.83138e-16 7.74923e-16 TRIV 5 11.1204 1.08925 1.83284e-15 9.44401e-16 TRIW 5 9.39629 455.452 0 0 UPSHIFT 5 2.23607 2.23607 0 0 VAND1 5 466.164 1.3093 4.36687e-15 3.83861e-15 VAND2 5 466.164 1.3093 6.5335e-14 6.41494e-15 WILK03 3 1.39284 1.78816e+10 0 0 WILK04 4 1.89545 1.15398e+16 9.64303e-05 10.0957 WILK05 5 1.51485 3.0639e+06 2.98752e-10 1.02452e-09 WILK21 21 28.4605 4.30755 1.49579e-15 3.82082e-15 WILSON 4 30.545 98.5292 1.70382e-13 0 test_llt(): A = a test matrix of order M by M L is an M by N lower triangular Cholesky factor. ||A|| = Frobenius norm of A. ||A-LLT|| = Frobenius norm of A-L*L'. Title M N ||A|| ||A-LLT|| DIF2 5 5 5.2915 8.88178e-16 GIVENS 5 5 20.6155 4.23634e-15 KERSHAW 4 4 8.24621 2.57035e-15 LEHMER 5 5 3.28041 2.07704e-16 MINIJ 5 5 12.4499 0 MOLER1 5 5 61.885 6.15348e-15 MOLER3 5 5 8.66025 0 OTO 5 5 5.2915 7.36439e-16 PASCAL2 5 5 92.4608 0 WILSON 4 4 30.545 5.25453e-15 test_null_left(): A = a test matrix of order M by N x = an M vector, candidate for a left null vector. ||A|| = Frobenius norm of A. ||x|| = L2 norm of x. ||A'*x||/||x|| = L2 norm of A'*x over L2 norm of x. Title M N ||A|| ||x|| ||A'*x||/||x|| A123 3 3 16.8819 2.44949 0 CHEBY_DIFF1 5 5 13.4722 3.74166 4.45079e-16 CREATION 5 5 5.47723 1 0 DIF1 5 5 2.82843 1.73205 0 DIF1CYCLIC 5 5 3.16228 2.23607 0 DIF2CYCLIC 5 5 5.47723 2.23607 0 EBERLEIN 5 5 18.1002 2.23607 5.61733e-16 FIBONACCI1 5 5 95.3527 1.73205 0 LAUCHLI 6 5 6.68163 3.59567 0 LINE_ADJ 7 7 3.4641 2 0 MOLER2 5 5 101035 263.82 0 ONE 5 5 5 1.41421 0 RING_ADJ 12 12 4.89898 3.4641 0 ROSSER1 8 8 2482.26 22.3607 0 ZERO 5 5 0 2.23607 0 test_null_right(): A = a test matrix of order M by N x = an N vector, candidate for a right null vector. ||A|| = Frobenius norm of A. ||x|| = L2 norm of x. ||A*x||/||x|| = L2 norm of A*x over L2 norm of x. Title M N ||A|| ||x|| ||A*x||/||x|| A123 3 3 16.8819 2.44949 0 ARCHIMEDES 7 8 93.397 1.87697e+07 0 CHEBY_DIFF1 5 5 13.4722 2.23607 6.49741e-16 CREATION 5 5 5.47723 1 0 DIF1 5 5 2.82843 1.73205 0 DIF1CYCLIC 5 5 3.16228 2.23607 0 DIF2CYCLIC 5 5 5.47723 2.23607 0 FIBONACCI1 5 5 95.3527 1.73205 0 HAMMING 5 31 8.94427 2.44949 0 LINE_ADJ 7 7 3.4641 2 0 MOLER2 5 5 101035 1016.3 0 NEUMANN 25 25 23.2379 5 0 ONE 5 5 5 1.41421 0 RING_ADJ 12 12 4.89898 3.4641 0 ROSSER1 8 8 2482.26 22.3607 0 ZERO 5 5 0 2.23607 0 test_plu(): A = a test matrix of order M by N P, L, U are the PLU factors. ||A|| = Frobenius norm of A. ||A-PLU|| = Frobenius norm of A-P*L*U. Title M N ||A|| ||A-PLU|| A123 3 3 16.8819 6.8798e-15 BODEWIG 4 4 12.7279 4.1243e-15 BORDERBAND 5 5 2.76699 0 DIF2 5 5 5.2915 0 GFPP 5 5 9.39618 2.92964e-14 GIVENS 5 5 20.6155 0 KMS 5 5 101.704 2.60787e-13 LEHMER 5 5 3.28041 1.11022e-16 MAXIJ 5 5 19.8746 0 MINIJ 5 5 12.4499 0 MOLER1 5 5 61.885 6.15348e-15 MOLER3 5 5 8.66025 0 OTO 5 5 5.2915 0 PASCAL2 5 5 92.4608 0 PLU 5 5 139.254 0 VAND2 4 4 107.076 2.05727e-14 WILSON 4 4 30.545 7.32411e-15 test_solution(): Compute the Frobenius norm of the solution error: A * X - B given MxN matrix A, NxK solution X, MxK right hand side B. Title M N K ||A|| ||A*X-B|| A123 3 3 1 16.8819 0 BODEWIG 4 4 1 12.7279 0 DIF2 10 10 2 7.61577 0 FRANK 10 10 2 38.6652 0 POISSON 20 20 1 19.5448 0 WILK03 3 3 1 1.39284 6.7435e-07 WILK04 4 4 1 1.89545 1.29463e+16 WILSON 4 4 1 30.545 0 test_type() Demonstrate functions which test the type of a matrix. Title M N ||A|| ||Transition Error|| BODEWIG 4 4 12.7279 inf SNAKES 101 101 5.92077 9.80522e-16 TRANSITION_RANDOM 5 5 1.32331 0 test_matrix_test(): Normal end of execution. 04 April 2024 08:28:25 PM