07-Jan-2022 18:00:16 CHEBYSHEV_POLYNOMIAL_TEST: MATLAB/Octave version 9.8.0.1380330 (R2020a) Update 2. Test CHEBYSHEV_POLYNOMIAL. CHEBYSHEV_POLYNOMIAL_TEST01: T_PROJECT_COEFFICIENTS_DATA estimates the Chebyshev polynomial coefficients for a function given as data (x,fx). Here, we use fx = f(x) = x^2 for the data. Since T(0,x) = 1 and T(2,x) = 2*x^2 - 1, the correct expansion is f(x) = 1/2 T(0,x) + 0 T(1,x) + 1/2 T(2,x) + 0 * all other polys, if Chebyshev polynomials are based in [-1,+1]. Chebyshev polynomials will be based in [0,1] Data ( X, D ): 1: 0.814724 0.663775 2: 0.905792 0.820459 3: 0.126987 0.0161257 4: 0.913376 0.834255 5: 0.632359 0.399878 6: 0.0975404 0.00951413 7: 0.278498 0.0775613 8: 0.546882 0.299079 9: 0.957507 0.916819 10: 0.964889 0.93101 11: 0.157613 0.0248419 12: 0.970593 0.94205 13: 0.957167 0.916169 14: 0.485376 0.23559 15: 0.80028 0.640449 16: 0.141886 0.0201317 17: 0.421761 0.177883 18: 0.915736 0.838572 19: 0.792207 0.627592 20: 0.959492 0.920626 Coefficients of Chebyshev expansion of degree 4. 1: 0.375 2: 0.5 3: 0.125 4: -9.36384e-17 I X(I) Data(I) Chebyshev(X(I)) 1 0.814724 0.663775 0.663775 2 0.905792 0.820459 0.820459 3 0.126987 0.0161257 0.0161257 4 0.913376 0.834255 0.834255 5 0.632359 0.399878 0.399878 6 0.0975404 0.00951413 0.00951413 7 0.278498 0.0775613 0.0775613 8 0.546882 0.299079 0.299079 9 0.957507 0.916819 0.916819 10 0.964889 0.93101 0.93101 11 0.157613 0.0248419 0.0248419 12 0.970593 0.94205 0.94205 13 0.957167 0.916169 0.916169 14 0.485376 0.23559 0.23559 15 0.80028 0.640449 0.640449 16 0.141886 0.0201317 0.0201317 17 0.421761 0.177883 0.177883 18 0.915736 0.838572 0.838572 19 0.792207 0.627592 0.627592 20 0.959492 0.920626 0.920626 T_MASS_MATRIX_TEST: T_MASS_MATRIX computes the mass matrix for the Chebyshev T polynomials T(i,x). A(I,J) = integral ( -1 <=x <= +1 ) T(i,x) T(j,x) / sqrt ( 1 - x^2 ) dx 0 if i is not equal to j; pi if i = j = 0; pi/2 if i = j =/= 0. T mass matrix: Col: 1 2 3 4 Row 1 : 3.14159 7.77156e-16 -5.55112e-16 -2.22045e-16 2 : 7.77156e-16 1.5708 1.11022e-16 3.33067e-16 3 :-5.55112e-16 3.33067e-16 1.5708 6.10623e-16 4 :-2.22045e-16 3.33067e-16 6.38378e-16 1.5708 T_MOMENT_TEST: T_MOMENT returns the value of integral ( -1 <=x <= +1 ) x^e / sqrt ( 1 - x^2 ) dx E Integral 0 3.14159 1 0 2 1.5708 3 0 4 1.1781 5 0 6 0.981748 7 0 8 0.859029 9 0 10 0.773126 T_POLYNOMIAL_TEST: T_POLYNOMIAL evaluates the Chebyshev polynomial T(n,x). Tabulated Computed N X T(n,x) T(n,x) Error 0 0.8 1 1 0 1 0.8 0.8 0.8 0 2 0.8 0.28 0.2800000000000002 -2.2e-16 3 0.8 -0.352 -0.3519999999999996 -3.3e-16 4 0.8 -0.8431999999999999 -0.8431999999999997 -2.2e-16 5 0.8 -0.99712 -0.99712 0 6 0.8 -0.752192 -0.7521920000000004 4.4e-16 7 0.8 -0.2063872 -0.2063872000000007 6.7e-16 8 0.8 0.42197248 0.4219724799999993 6.7e-16 9 0.8 0.881543168 0.8815431679999997 3.3e-16 10 0.8 0.9884965888 0.9884965888000001 -1.1e-16 11 0.8 0.7000513741 0.7000513740800007 2e-11 12 0.8 0.1315856097 0.131585609728001 -2.8e-11 T_POLYNOMIAL_AB_TEST: T_POLYNOMIAL_AB evaluates Chebyshev polynomials TAB(n,x) shifted from [-1,+1] to the domain [A,B]. Here, we will use the new domain [0,1] and the desired maximum polynomial degree will be N = 5. Tables of T values: Col: 1 2 3 4 5 Row 1 : 1 -1 1 -1 1 2 : 1 -0.8 0.28 0.352 -0.8432 3 : 1 -0.6 -0.28 0.936 -0.8432 4 : 1 -0.4 -0.68 0.944 -0.0752 5 : 1 -0.2 -0.92 0.568 0.6928 6 : 1 0 -1 -0 1 7 : 1 0.2 -0.92 -0.568 0.6928 8 : 1 0.4 -0.68 -0.944 -0.0752 9 : 1 0.6 -0.28 -0.936 -0.8432 10 : 1 0.8 0.28 -0.352 -0.8432 11 : 1 1 1 1 1 Col: 6 Row 1 : -1 2 : 0.99712 3 : 0.07584 4 : -0.88384 5 : -0.84512 6 : 0 7 : 0.84512 8 : 0.88384 9 : -0.07584 10 : -0.99712 11 : 1 T_POLYNOMIAL_AB_VALUE_TEST: T_POLYNOMIAL_AB_VALUE evaluates the shifted Chebyshev polynomial TAB(n,x). Here, we will use the new domain [0,1]. Tabulated Computed N X01 T01(n,x) T01(n,x) Error -1 0.850000 0.0000000000000000e+00 0.0000000000000000e+00 0 0 0.850000 1.0000000000000000e+00 1.0000000000000000e+00 0 1 0.850000 6.9999999999999996e-01 6.9999999999999996e-01 0 2 0.850000 -2.0000000000000000e-02 -2.0000000000000129e-02 1.3e-16 3 0.850000 -7.2799999999999998e-01 -7.2800000000000009e-01 1.1e-16 4 0.850000 -9.9919999999999998e-01 -9.9919999999999998e-01 0 5 0.850000 -6.7088000000000003e-01 -6.7087999999999981e-01 -2.2e-16 6 0.850000 5.9968000000000000e-02 5.9968000000000354e-02 -3.5e-16 7 0.850000 7.5483520000000004e-01 7.5483520000000026e-01 -2.2e-16 8 0.850000 9.9680128000000001e-01 9.9680128000000001e-01 0 9 0.850000 6.4068659200000000e-01 6.4068659199999978e-01 2.2e-16 10 0.850000 -9.9840051200000002e-02 -9.9840051200000390e-02 3.9e-16 11 0.850000 -7.8046266367999995e-01 -7.8046266368000028e-01 3.3e-16 12 0.850000 -9.9280767795199998e-01 -9.9280767795199987e-01 -1.1e-16 7 0.000000 -1.0000000000000000e+00 -1.0000000000000000e+00 0 7 0.100000 2.0638719999999999e-01 2.0638720000000066e-01 -6.7e-16 7 0.200000 -9.7847039999999996e-01 -9.7847039999999996e-01 0 7 0.300000 2.5802239999999999e-01 2.5802239999999987e-01 1.1e-16 7 0.400000 9.8702080000000003e-01 9.8702080000000003e-01 0 7 0.500000 0.0000000000000000e+00 -0.0000000000000000e+00 0 7 0.600000 -9.8702080000000003e-01 -9.8702080000000003e-01 0 7 0.700000 -2.5802239999999999e-01 -2.5802240000000076e-01 7.8e-16 7 0.800000 9.7847039999999996e-01 9.7847040000000041e-01 -4.4e-16 7 0.900000 -2.0638719999999999e-01 -2.0638720000000066e-01 6.7e-16 7 1.000000 1.0000000000000000e+00 1.0000000000000000e+00 0 t_polynomial_coefficient_table_test t_polynomial_coefficient_table determines a table of ans = 56 coefficients for T(0:n,x). +1.000000 +1.000000 * x +2.000000 * x^2 -1.000000 +4.000000 * x^3 -3.000000 * x +8.000000 * x^4 -8.000000 * x^2 +1.000000 +16.000000 * x^5 -20.000000 * x^3 +5.000000 * x t_polynomial_coefficients_test t_polynomial_coefficients determines the coefficients of T(d,x). Coefficients of T5(x) = +16.000000 * x^5 -20.000000 * x^3 +5.000000 * x T_POLYNOMIAL_VALUE_TEST: T_POLYNOMIAL_VALUE evaluates the Chebyshev polynomial T(n,x). Tabulated Computed N X T(n,x) T(n,x) Error 0 0.800000 1.0000000000000000e+00 1.0000000000000000e+00 0 1 0.800000 8.0000000000000004e-01 8.0000000000000004e-01 0 2 0.800000 2.8000000000000003e-01 2.8000000000000025e-01 -2.2e-16 3 0.800000 -3.5199999999999998e-01 -3.5199999999999965e-01 -3.3e-16 4 0.800000 -8.4319999999999995e-01 -8.4319999999999973e-01 -2.2e-16 5 0.800000 -9.9712000000000001e-01 -9.9712000000000001e-01 0 6 0.800000 -7.5219199999999997e-01 -7.5219200000000042e-01 4.4e-16 7 0.800000 -2.0638719999999999e-01 -2.0638720000000066e-01 6.7e-16 8 0.800000 4.2197247999999998e-01 4.2197247999999932e-01 6.7e-16 9 0.800000 8.8154316799999999e-01 8.8154316799999966e-01 3.3e-16 10 0.800000 9.8849658880000002e-01 9.8849658880000013e-01 -1.1e-16 11 0.800000 7.0005137409999996e-01 7.0005137408000073e-01 2e-11 12 0.800000 1.3158560969999999e-01 1.3158560972800104e-01 -2.8e-11 T_POLYNOMIAL_ZEROS_TEST: T_POLYNOMIAL_ZEROS computes the zeros of T(n,x); N X T(n,x) 1 0.0000 6.12323e-17 1 0.7071 2.22045e-16 2 -0.7071 -2.22045e-16 1 0.8660 3.33067e-16 2 0.0000 -1.83697e-16 3 -0.8660 -3.33067e-16 1 0.9239 -2.22045e-16 2 0.3827 -2.22045e-16 3 -0.3827 1.11022e-16 4 -0.9239 -2.22045e-16 1 0.9511 -4.44089e-16 2 0.5878 0 3 0.0000 3.06162e-16 4 -0.5878 -7.77156e-16 5 -0.9511 4.44089e-16 T_QUADRATURE_RULE_TEST: T_QUADRATURE_RULE computes the quadrature rule associated with T(n,x); X W 1: -0.974928 0.448799 2: -0.781831 0.448799 3: -0.433884 0.448799 4: 2.19909e-16 0.448799 5: 0.433884 0.448799 6: 0.781831 0.448799 7: 0.974928 0.448799 Use the quadrature rule to estimate: Q = Integral ( -1 <= X <= +1 ) X^E / sqrt ( 1-x^2) dx E Q_Estimate Q_Exact 0 3.14159 3.14159 1 -1.11022e-16 0 2 1.5708 1.5708 3 -1.11022e-16 0 4 1.1781 1.1781 5 -1.11022e-16 0 6 0.981748 0.981748 7 5.55112e-17 0 8 0.859029 0.859029 9 1.66533e-16 0 10 0.773126 0.773126 11 2.22045e-16 0 12 0.708699 0.708699 13 3.33067e-16 0 CHEBYSHEV_POLYNOMIAL_TEST07: T_PROJECT_COEFFICIENTS computes the Chebyshev coefficients of a function defined over [-1,+1]. T_PROJECT_COEFFICIENTS_AB works in [A,B]. Chebyshev coefficients for exp(x) in [-1,+1] 1: 1.26607 2: 1.13031 3: 0.27145 4: 0.0437939 Chebyshev coefficients for exp(x) in [-1,+1] 1: 1.26607 2: 1.13032 3: 0.271495 4: 0.0443368 5: 0.00547404 6: 0.000539728 Chebyshev coefficients for sin(x) in [-1,+1] 1: 1.85037e-17 2: 0.880101 3: 1.11022e-16 4: -0.0391267 5: 2.77556e-16 6: 0.00050252 Chebyshev coefficients for sin(x) in [-1,+1] 1: 1.85037e-17 2: 0.880101 3: 1.11022e-16 4: -0.0391267 5: 3.14563e-16 6: 0.00050252 Chebyshev coefficients for sqrt(x) in [0,+1] 1: 0.638441 2: 0.420681 3: -0.0808705 4: 0.0318506 5: -0.01484 6: 0.00614694 CHEBYSHEV_POLYNOMIAL_TEST08: T_PROJECT_COEFFICIENTS_DATA computes the Chebyshev coefficients of a function defined by data. We are looking for an approximation that is good in [-1,+1]. Begin by using equally spaced points in [-1,+1]. Chebyshev coefficients for exp(x) on [-1,+1] 1: 1.26667 2: 1.13052 3: 0.271785 4: 0.0443939 Chebyshev coefficients for exp(x) on [-1,+1] 1: 1.26608 2: 1.13032 3: 0.271515 4: 0.0443381 5: 0.00547599 6: 0.000542394 Chebyshev coefficients for sin(x) on [-1,+1] 1: -4.16178e-18 2: 0.880099 3: 9.24137e-18 4: -0.0391279 5: 6.38809e-17 6: 0.000500014 Now sample equally spaced points in [0,+1]. The approximation still applies to the interval [-1,+1]. Chebyshev coefficients for sin(x) on [0,+1] 1: 0.449926 2: 0.425221 3: -0.0293448 4: -0.00449978 5: 0.00015411 6: 1.41389e-05 Chebyshev coefficients for sqrt(x) on [0,+1] 1: 0.627654 2: 0.439083 3: -0.0996074 4: 0.0460427 5: -0.0260146 6: 0.014499 Now random points in [-1,+1]. Chebyshev coefficients for sin(x) on [-1,+1] 1: 1.35157e-06 2: 0.880107 3: 2.76201e-06 4: -0.0391212 5: 1.23493e-06 6: 0.000504624 CHEBYSHEV_POLYNOMIAL_TEST09: T_PROJECT_COEFFICIENTS computes the Chebyshev interpolant C(F)(N,X) of a function F(X) defined over [-1,+1]. T_PROJECT_VALUE evaluates that projection. Compute projections of order N to exp(x) over [-1,+1], N Max||F(X)-C(F)(N,X)|| 0 1.718 1 0.3722 2 0.05647 3 0.006657 4 0.0006397 5 5.18e-05 6 3.62e-06 7 2.224e-07 8 1.219e-08 9 6.027e-10 10 2.714e-11 CHEBYSHEV_POLYNOMIAL_TEST10: T_PROJECT_COEFFICIENTS_AB computes the Chebyshev interpolant C(F)(N,X) of a function F(X) defined over [A,B]. T_PROJECT_VALUE_AB evaluates that projection. Compute projections of order N to exp(x) over [0.000000, 1.500000], N Max||F(X)-C(F)(N,X)|| 0 2.365 1 0.3967 2 0.04629 3 0.004159 4 0.0003031 5 1.855e-05 6 9.786e-07 7 4.532e-08 8 1.87e-09 9 6.956e-11 10 2.354e-12 TT_PRODUCT_TEST: TT_PRODUCT(I,J;X) = T(I,X) * T(J,X) I J X TI TJ TI*TJ TT_PRODUCT 0 0 0.412092 1 1 1 1 0 3 -0.907657 1 -0.268092 -0.268092 -0.268092 2 4 0.389657 -0.696334 -0.0302366 0.0210548 0.0210548 3 1 -0.931108 -0.435616 -0.931108 0.405606 0.405606 5 0 0.531034 0.335837 1 0.335837 0.335837 3 2 -0.0204712 0.0613793 -0.999162 -0.0613279 -0.0613279 5 0 0.41873 0.831256 1 0.831256 0.831256 4 -1 0.359405 0.100106 0 0 0 3 4 -0.762005 0.516179 -0.947963 -0.489318 -0.489318 4 0 -0.319229 0.267825 1 0.267825 0.267825 TT_PRODUCT_INTEGRAL_TEST: TT_PRODUCT_INTEGRAL computes the product integral of a pair of Chebyshev T polynomials T(i,x) and T(j,x). A(I,J) = integral ( -1 <=x <= +1 ) T(i,x) T(j,x) / sqrt ( 1 - x^2 ) dx 0 if i is not equal to j; pi if i = j = 0; pi/2 if i = j =/= 0. T(i,x)*T(j,x) integral matrix: Col: 1 2 3 4 5 Row 1 : 3.14159 0 0 0 0 2 : 0 1.5708 0 0 0 3 : 0 0 1.5708 0 0 4 : 0 0 0 1.5708 0 5 : 0 0 0 0 1.5708 TTT_PRODUCT_INTEGRAL_TEST: TTT_PRODUCT_INTEGRAL computes the triple integral Tijk = integral ( -1 <= x <= 1 ) T(i,x) T(j,x) T(k,x) / sqrt ( 1-x^2) dx I J K Tijk Tijk computed exact 5 1 2 0 -5.55112e-15 5 3 4 0 -7.27196e-15 4 1 0 0 -4.52416e-15 3 3 1 0 -2.7478e-15 6 1 4 0 -3.55271e-15 3 1 1 0 -2.7478e-15 5 2 1 0 -5.55112e-15 6 2 2 0 -6.80012e-15 6 1 3 0 -6.4948e-15 5 2 2 0 -3.91354e-15 2 1 2 0 -1.77636e-15 5 3 0 0 -4.63518e-15 4 2 0 0 -4.60743e-15 3 1 3 0 -2.7478e-15 3 2 0 0 -2.498e-15 5 1 3 0 -3.66374e-15 5 3 2 0.785398 0.785398 2 1 4 0 -3.91354e-15 2 3 2 0 -3.08087e-15 6 1 2 0 -5.30131e-15 TU_PRODUCT_TEST: TU_PRODUCT(I,J;X) = T(I,X) * U(J,X) I J X TI UJ TI*UJ TU_PRODUCT 6 -1 -0.786694 -0.659556 0 -0 0 5 4 0.549821 0.228799 -1.16544 -0.266652 -0.266652 2 0 -0.831128 0.381549 1 0.381549 0.381549 3 4 0.600137 -0.935819 -1.24648 1.16648 1.16648 1 -1 -0.636306 -0.636306 0 -0 0 6 2 -0.727863 -0.177861 1.11914 -0.199051 -0.199051 1 4 0.0997204 0.0997204 0.882252 0.0879786 0.882252 2 2 0.24411 -0.88082 -0.761641 0.670869 0.670869 0 0 -0.196384 1 1 1 0.5 1 0 -0.753362 -0.753362 1 -0.753362 1 U_MASS_MATRIX_TEST: U_MASS_MATRIX computes the mass matrix for the Chebyshev U polynomials U(i,x). A(I,J) = integral ( -1 <=x <= +1 ) U(i,x) U(j,x) * sqrt ( 1 - x^2 ) dx 0 if i is not equal to j; pi/2 if i = j. U mass matrix: Col: 1 2 3 4 Row 1 : 1.5708 6.66134e-16 5.55112e-17 2.77556e-16 2 : 6.66134e-16 1.5708 1.11022e-15 1.77636e-15 3 : 5.55112e-17 9.99201e-16 1.5708 2.33147e-15 4 : 2.77556e-16 1.77636e-15 2.27596e-15 1.5708 U_MOMENT_TEST: U_MOMENT returns the value of integral ( -1 <=x <= +1 ) x^e * sqrt ( 1 - x^2 ) dx E Integral 0 1.5708 1 0 2 0.392699 3 0 4 0.19635 5 0 6 0.122718 7 0 8 0.0859029 9 0 10 0.0644272 U_POLYNOMIAL_TEST: U_POLYNOMIAL evaluates the Chebyshev polynomial U(n,x). Tabulated Computed N X U(n,x) U(n,x) Error 0 0.8 1 1 0 1 0.8 1.6 1.6 0 2 0.8 1.56 1.56 -4.4e-16 3 0.8 0.896 0.8960000000000008 -7.8e-16 4 0.8 -0.1264 -0.1263999999999992 -8.3e-16 5 0.8 -1.09824 -1.098239999999999 -6.7e-16 6 0.8 -1.630784 -1.630784 0 7 0.8 -1.5110144 -1.511014400000001 6.7e-16 8 0.8 -0.78683904 -0.7868390400000014 1.3e-15 9 0.8 0.252071936 0.2520719359999986 1.4e-15 10 0.8 1.1901541376 1.190154137599999 8.9e-16 11 0.8 1.65217468416 1.65217468416 0 12 0.8 1.453325357056 1.453325357056001 -1.1e-15 U_POLYNOMIAL_AB_TEST: U_POLYNOMIAL_AB evaluates Chebyshev polynomials UAB(n,x) shifted from [-1,+1] to the domain [A,B]. Here, we will use the new domain [0,1] and the desired maximum polynomial degree will be N = 5. Tables of U values: Col: 1 2 3 4 5 Row 1 : 1 -2 3 -4 5 2 : 1 -1.6 1.56 -0.896 -0.1264 3 : 1 -1.2 0.44 0.672 -1.2464 4 : 1 -0.8 -0.36 1.088 -0.5104 5 : 1 -0.4 -0.84 0.736 0.5456 6 : 1 0 -1 -0 1 7 : 1 0.4 -0.84 -0.736 0.5456 8 : 1 0.8 -0.36 -1.088 -0.5104 9 : 1 1.2 0.44 -0.672 -1.2464 10 : 1 1.6 1.56 0.896 -0.1264 11 : 1 2 3 4 5 Col: 6 Row 1 : -6 2 : 1.09824 3 : 0.82368 4 : -0.67968 5 : -0.95424 6 : 0 7 : 0.95424 8 : 0.67968 9 : -0.82368 10 : -1.09824 11 : 6 U_POLYNOMIAL_AB_VALUE_TEST: U_POLYNOMIAL_AB_VALUE evaluates the shifted Chebyshev polynomial UAB(n,x). Here, we will use the new domain [0,1]. Tabulated Computed N X01 U01(n,x) U01(n,x) Error -1 0.850000 0.0000000000000000e+00 0.0000000000000000e+00 0 0 0.850000 1.0000000000000000e+00 1.0000000000000000e+00 0 1 0.850000 1.3999999999999999e+00 1.3999999999999999e+00 0 2 0.850000 9.5999999999999996e-01 9.5999999999999974e-01 2.2e-16 3 0.850000 -5.6000000000000001e-02 -5.6000000000000272e-02 2.7e-16 4 0.850000 -1.0384000000000000e+00 -1.0384000000000002e+00 2.2e-16 5 0.850000 -1.3977599999999999e+00 -1.3977599999999999e+00 0 6 0.850000 -9.1846399999999995e-01 -9.1846399999999950e-01 -4.4e-16 7 0.850000 1.1191039999999999e-01 1.1191040000000063e-01 -6.4e-16 8 0.850000 1.0751385600000001e+00 1.0751385600000003e+00 -2.2e-16 9 0.850000 1.3932835840000000e+00 1.3932835839999997e+00 2.2e-16 10 0.850000 8.7545845759999996e-01 8.7545845759999930e-01 6.7e-16 11 0.850000 -1.6764174336000001e-01 -1.6764174336000082e-01 8e-16 12 0.850000 -1.1101568983040000e+00 -1.1101568983040004e+00 4.4e-16 7 0.000000 -8.0000000000000000e+00 -8.0000000000000000e+00 0 7 0.100000 1.5110144000000001e+00 1.5110144000000008e+00 -6.7e-16 7 0.200000 -1.1332608000000000e+00 -1.1332608000000002e+00 2.2e-16 7 0.300000 -1.6363520000000001e-01 -1.6363520000000020e-01 1.9e-16 7 0.400000 1.0198016000000001e+00 1.0198016000000001e+00 0 7 0.500000 0.0000000000000000e+00 -0.0000000000000000e+00 0 7 0.600000 -1.0198016000000001e+00 -1.0198016000000001e+00 0 7 0.700000 1.6363520000000001e-01 1.6363519999999931e-01 6.9e-16 7 0.800000 1.1332608000000000e+00 1.1332607999999995e+00 4.4e-16 7 0.900000 -1.5110144000000001e+00 -1.5110144000000008e+00 6.7e-16 7 1.000000 8.0000000000000000e+00 8.0000000000000000e+00 0 U_POLYNOMIAL_COEFFICIENTS_TEST U_POLYNOMIAL_COEFFICIENTS determines the polynomial coefficients for U(n,x). +1.000000 +2.000000 * x +4.000000 * x^2 -1.000000 +8.000000 * x^3 -4.000000 * x +16.000000 * x^4 -12.000000 * x^2 +1.000000 +32.000000 * x^5 -32.000000 * x^3 +6.000000 * x U_POLYNOMIAL_VALUE_TEST: U_POLYNOMIAL_VALUE evaluates the Chebyshev polynomial U(n,x). Tabulated Computed N X U(n,x) U(n,x) Error 0 0.800000 1.0000000000000000e+00 1.0000000000000000e+00 0 1 0.800000 1.6000000000000001e+00 1.6000000000000001e+00 0 2 0.800000 1.5600000000000001e+00 1.5600000000000005e+00 -4.4e-16 3 0.800000 8.9600000000000002e-01 8.9600000000000080e-01 -7.8e-16 4 0.800000 -1.2640000000000001e-01 -1.2639999999999918e-01 -8.3e-16 5 0.800000 -1.0982400000000001e+00 -1.0982399999999994e+00 -6.7e-16 6 0.800000 -1.6307840000000000e+00 -1.6307840000000000e+00 0 7 0.800000 -1.5110144000000001e+00 -1.5110144000000008e+00 6.7e-16 8 0.800000 -7.8683904000000005e-01 -7.8683904000000138e-01 1.3e-15 9 0.800000 2.5207193600000000e-01 2.5207193599999855e-01 1.4e-15 10 0.800000 1.1901541376000000e+00 1.1901541375999991e+00 8.9e-16 11 0.800000 1.6521746841600000e+00 1.6521746841600000e+00 0 12 0.800000 1.4533253570560001e+00 1.4533253570560012e+00 -1.1e-15 U_POLYNOMIAL_ZEROS_TEST: U_POLYNOMIAL_ZEROS computes the zeros of U(n,x); N X U(n,x) 1 0.0000 1.22465e-16 1 0.5000 4.44089e-16 2 -0.5000 -8.88178e-16 1 0.7071 6.66134e-16 2 0.0000 -2.44929e-16 3 -0.7071 6.66134e-16 1 0.8090 0 2 0.3090 -1.11022e-16 3 -0.3090 5.55112e-16 4 -0.8090 -8.88178e-16 1 0.8660 1.33227e-15 2 0.5000 -8.88178e-16 3 0.0000 3.67394e-16 4 -0.5000 -1.77636e-15 5 -0.8660 -1.33227e-15 U_QUADRATURE_RULE_TEST: U_QUADRATURE_RULE computes the quadrature rule associated with U(n,x); X W 1: -0.92388 0.0575094 2: -0.707107 0.19635 3: -0.382683 0.33519 4: 7.90193e-18 0.392699 5: 0.382683 0.33519 6: 0.707107 0.19635 7: 0.92388 0.0575094 Use the quadrature rule to estimate: Q = Integral ( -1 <= X <= +1 ) X^E * sqrt ( 1-x^2) dx E Q_Estimate Q_Exact 0 1.5708 1.5708 1 2.77556e-17 0 2 0.392699 0.392699 3 -5.55112e-17 0 4 0.19635 0.19635 5 -1.249e-16 0 6 0.122718 0.122718 7 -1.59595e-16 0 8 0.0859029 0.0859029 9 -1.73472e-16 0 10 0.0644272 0.0644272 11 -1.83881e-16 0 12 0.0506214 0.0506214 13 -1.76942e-16 0 UU_PRODUCT_TEST: UU_PRODUCT(I,J;X) = U(I,X) * U(J,X) I J X UI UJ UI*UJ UU_PRODUCT 0 4 -0.165466 1 0.683446 0.683446 0.683446 3 1 0.889574 2.07337 1.77915 3.68883 3.68883 6 1 -0.324561 0.715248 -0.649122 -0.464284 -0.464284 5 1 -0.777594 1.28266 -1.55519 -1.99478 -1.99478 2 -1 -0.516617 0.0675743 0 0 0 6 4 -0.736053 -1.30161 -0.80498 1.04777 1.04777 0 0 0.150417 1 1 1 1 5 -1 -0.293683 -1.02145 0 -0 0 1 2 -0.913952 -1.8279 2.34124 -4.27956 -4.27956 4 1 0.463445 -0.839278 0.92689 -0.777918 -0.777918 UU_PRODUCT_INTEGRAL_TEST: UU_PRODUCT_INTEGRAL computes the product integral of a pair of Chebyshev U polynomials U(i,x) and U(j,x). A(I,J) = integral ( -1 <=x <= +1 ) U(i,x) U(j,x) sqrt ( 1 - x^2 ) dx 0 if i is not equal to j; pi/2 if i = j U(i,x)*U(j,x) integral matrix: Col: 1 2 3 4 5 Row 1 : 1.5708 0 0 0 0 2 : 0 1.5708 0 0 0 3 : 0 0 1.5708 0 0 4 : 0 0 0 1.5708 0 5 : 0 0 0 0 1.5708 V_MASS_MATRIX_TEST: V_MASS_MATRIX computes the mass matrix for the Chebyshev polynomials V(i,x). A(I,J) = integral ( -1 <=x <= +1 ) V(i,x) V(j,x) sqrt(1+x)/sqrt(1-x) dx 0 if i is not equal to j; pi if i = j. V mass matrix: Col: 1 2 3 4 Row 1 : 3.14159 7.77156e-16 -5.55112e-16 -2.22045e-16 2 : 7.77156e-16 1.5708 1.11022e-16 3.33067e-16 3 :-5.55112e-16 3.33067e-16 1.5708 6.10623e-16 4 :-2.22045e-16 3.33067e-16 6.38378e-16 1.5708 V_MOMENT_TEST: V_MOMENT returns the value of integral ( -1 <=x <= +1 ) x^e * sqrt ( 1 + x ) / sqrt ( 1 - x ) dx E Integral 0 3.14159 1 1.5708 2 1.5708 3 1.1781 4 1.1781 5 0.981748 6 0.981748 7 0.859029 8 0.859029 9 0.773126 10 0.773126 V_POLYNOMIAL_TEST: V_POLYNOMIAL evaluates the Chebyshev polynomial V(n,x). Tabulated Computed N X V(n,x) V(n,x) Error 0 0.8 1 1 0 1 0.8 0.6 0.6000000000000001 -1.1e-16 2 0.8 -0.04 -0.03999999999999981 -1.9e-16 3 0.8 -0.664 -0.6639999999999998 -2.2e-16 4 0.8 -1.0224 -1.0224 0 5 0.8 -0.97184 -0.9718400000000001 1.1e-16 6 0.8 -0.532544 -0.5325440000000004 3.3e-16 7 0.8 0.1197696 0.1197695999999996 4.2e-16 8 0.8 0.72417536 0.7241753599999997 3.3e-16 9 0.8 1.038910976 1.038910976 0 10 0.8 0.9380822016 0.9380822016000002 -2.2e-16 11 0.8 0.46202054656 0.4620205465600005 -4.4e-16 12 0.8 -0.198849327104 -0.1988493271039994 -6.1e-16 V_POLYNOMIAL_AB_TEST: V_POLYNOMIAL_AB evaluates Chebyshev polynomials VAB(n,x) shifted from [-1,+1] to the domain [A,B]. Here, we will use the new domain [0,1] and the desired maximum polynomial degree will be N = 5. Tables of T values: Col: 1 2 3 4 5 Row 1 : 1 -3 5 -7 9 2 : 1 -2.6 3.16 -2.456 0.7696 3 : 1 -2.2 1.64 0.232 -1.9184 4 : 1 -1.8 0.44 1.448 -1.5984 5 : 1 -1.4 -0.44 1.576 -0.1904 6 : 1 -1 -1 1 1 7 : 1 -0.6 -1.24 0.104 1.2816 8 : 1 -0.2 -1.16 -0.728 0.5776 9 : 1 0.2 -0.76 -1.112 -0.5744 10 : 1 0.6 -0.04 -0.664 -1.0224 11 : 1 1 1 1 1 Col: 6 Row 1 : -11 2 : 1.22464 3 : 2.07008 4 : -0.16928 5 : -1.49984 6 : -1 7 : 0.40864 8 : 1.19008 9 : 0.42272 10 : -0.97184 11 : 1 V_POLYNOMIAL_AB_VALUE_TEST: V_POLYNOMIAL_AB_VALUE evaluates the shifted Chebyshev polynomial VAB(n,x). Here, we will use the new domain [0,1]. Tabulated Computed N X01 V01(n,x) V01(n,x) Error -1 0.850000 0.0000000000000000e+00 0.0000000000000000e+00 0 0 0.850000 1.0000000000000000e+00 1.0000000000000000e+00 0 1 0.850000 4.0000000000000002e-01 3.9999999999999991e-01 1.1e-16 2 0.850000 -4.4000000000000000e-01 -4.4000000000000017e-01 1.7e-16 3 0.850000 -1.0160000000000000e+00 -1.0160000000000000e+00 0 4 0.850000 -9.8240000000000005e-01 -9.8239999999999972e-01 -3.3e-16 5 0.850000 -3.5936000000000001e-01 -3.5935999999999946e-01 -5.6e-16 6 0.850000 4.7929600000000000e-01 4.7929600000000050e-01 -5e-16 7 0.850000 1.0303743999999999e+00 1.0303744000000001e+00 -2.2e-16 8 0.850000 9.6322816000000000e-01 9.6322815999999956e-01 4.4e-16 9 0.850000 3.1814502400000000e-01 3.1814502399999922e-01 7.8e-16 10 0.850000 -5.1782512640000000e-01 -5.1782512640000067e-01 6.7e-16 11 0.850000 -1.0431002009599999e+00 -1.0431002009600001e+00 2.2e-16 12 0.850000 -9.4251515494399996e-01 -9.4251515494399940e-01 -5.6e-16 7 0.000000 -1.5000000000000000e+01 -1.5000000000000000e+01 0 7 0.100000 3.1417983999999999e+00 3.1417984000000003e+00 -4.4e-16 7 0.200000 -1.3912447999999999e+00 -1.3912448000000008e+00 8.9e-16 7 0.300000 -1.2177792000000001e+00 -1.2177792000000005e+00 4.4e-16 7 0.400000 1.1837055999999999e+00 1.1837056000000004e+00 -4.4e-16 7 0.500000 1.0000000000000000e+00 1.0000000000000000e+00 0 7 0.600000 -8.5589760000000004e-01 -8.5589759999999959e-01 -4.4e-16 7 0.700000 -8.9050879999999999e-01 -8.9050880000000054e-01 5.6e-16 7 0.800000 8.7527679999999997e-01 8.7527680000000074e-01 -7.8e-16 7 0.900000 1.1976960000000000e-01 1.1976959999999959e-01 4.2e-16 7 1.000000 1.0000000000000000e+00 1.0000000000000000e+00 0 V_POLYNOMIAL_COEFFICIENTS_TEST V_POLYNOMIAL_COEFFICIENTS determines the Chebyshev polynomial coefficients. +1.000000 +2.000000 * x -1.000000 +4.000000 * x^2 -2.000000 * x -1.000000 +8.000000 * x^3 -4.000000 * x^2 -4.000000 * x +1.000000 +16.000000 * x^4 -8.000000 * x^3 -12.000000 * x^2 +4.000000 * x +1.000000 +32.000000 * x^5 -16.000000 * x^4 -32.000000 * x^3 +12.000000 * x^2 +6.000000 * x -1.000000 V_POLYNOMIAL_VALUE_TEST: V_POLYNOMIAL_VALUE evaluates the Chebyshev polynomial V(n,x). Tabulated Computed N X V(n,x) V(n,x) Error 0 0.800000 1.0000000000000000e+00 1.0000000000000000e+00 0 1 0.800000 5.9999999999999998e-01 6.0000000000000009e-01 -1.1e-16 2 0.800000 -4.0000000000000001e-02 -3.9999999999999813e-02 -1.9e-16 3 0.800000 -6.6400000000000003e-01 -6.6399999999999981e-01 -2.2e-16 4 0.800000 -1.0224000000000000e+00 -1.0224000000000000e+00 0 5 0.800000 -9.7184000000000004e-01 -9.7184000000000015e-01 1.1e-16 6 0.800000 -5.3254400000000002e-01 -5.3254400000000035e-01 3.3e-16 7 0.800000 1.1976960000000000e-01 1.1976959999999959e-01 4.2e-16 8 0.800000 7.2417536000000005e-01 7.2417535999999971e-01 3.3e-16 9 0.800000 1.0389109759999999e+00 1.0389109759999999e+00 0 10 0.800000 9.3808220160000000e-01 9.3808220160000022e-01 -2.2e-16 11 0.800000 4.6202054656000002e-01 4.6202054656000047e-01 -4.4e-16 12 0.800000 -1.9884932710400000e-01 -1.9884932710399938e-01 -6.1e-16 V_POLYNOMIAL_ZEROS_TEST: V_POLYNOMIAL_ZEROS computes the zeros of V(n,x); N X V(n,x) 1 0.5000 2.22045e-16 1 -0.3090 -3.33067e-16 2 0.8090 2.22045e-16 1 -0.6235 4.44089e-16 2 0.2225 -2.22045e-16 3 0.9010 1.11022e-16 1 -0.7660 -3.10862e-15 2 -0.1736 3.33067e-16 3 0.5000 -6.66134e-16 4 0.9397 7.77156e-16 1 -0.8413 3.55271e-15 2 -0.4154 -1.11022e-15 3 0.1423 -1.66533e-16 4 0.6549 -2.22045e-16 5 0.9595 -4.44089e-16 V_QUADRATURE_RULE_TEST: V_QUADRATURE_RULE computes the quadrature rule associated with V(n,x); X W 1: -0.913545 0.036214 2: -0.669131 0.138594 3: -0.309017 0.289438 4: 0.104528 0.462664 5: 0.5 0.628319 6: 0.809017 0.757759 7: 0.978148 0.828605 Use the quadrature rule to estimate: Q = Integral ( -1 <= X <= +1 ) X^E * sqrt ( 1 + x ) / sqrt ( 1-x ) dx E Q_Estimate Q_Exact 0 3.14159 3.14159 1 1.5708 1.5708 2 1.5708 1.5708 3 1.1781 1.1781 4 1.1781 1.1781 5 0.981748 0.981748 6 0.981748 0.981748 7 0.859029 0.859029 8 0.859029 0.859029 9 0.773126 0.773126 10 0.773126 0.773126 11 0.708699 0.708699 12 0.708699 0.708699 13 0.658078 0.658078 VV_PRODUCT_INTEGRAL_TEST: VV_PRODUCT_INTEGRAL computes the product integral of a pair of Chebyshev V polynomials V(i,x) and V(j,x). A(I,J) = integral ( -1 <=x <= +1 ) V(i,x) V(j,x) sqrt ( 1 + x ) / sqrt ( 1 - x ) dx 0 if i is not equal to j; pi if i = j V(i,x)*V(j,x) integral matrix: Col: 1 2 3 4 5 Row 1 : 3.14159 0 0 0 0 2 : 0 3.14159 0 0 0 3 : 0 0 3.14159 0 0 4 : 0 0 0 3.14159 0 5 : 0 0 0 0 3.14159 W_MASS_MATRIX_TEST: W_MASS_MATRIX computes the mass matrix for the Chebyshev polynomials W(i,x). A(I,J) = integral ( -1 <=x <= +1 ) W(i,x) W(j,x) sqrt(1-x)/sqrt(1+x) dx 0 if i is not equal to j; pi if i = j. W mass matrix: Col: 1 2 3 4 Row 1 : 3.14159 7.77156e-16 -5.55112e-16 -2.22045e-16 2 : 7.77156e-16 1.5708 1.11022e-16 3.33067e-16 3 :-5.55112e-16 3.33067e-16 1.5708 6.10623e-16 4 :-2.22045e-16 3.33067e-16 6.38378e-16 1.5708 W_MOMENT_TEST: W_MOMENT returns the value of integral ( -1 <=x <= +1 ) x^e * sqrt ( 1 - x ) / sqrt ( 1 + x ) dx E Integral 0 3.14159 1 -1.5708 2 1.5708 3 -1.1781 4 1.1781 5 -0.981748 6 0.981748 7 -0.859029 8 0.859029 9 -0.773126 10 0.773126 W_POLYNOMIAL_TEST: W_POLYNOMIAL evaluates the Chebyshev polynomial W(n,x). Tabulated Computed N X W(n,x) W(n,x) Error 0 0.8 1 1 0 1 0.8 2.6 2.6 0 2 0.8 3.16 3.16 0 3 0.8 2.456 2.456000000000001 -8.9e-16 4 0.8 0.7696 0.7696000000000014 -1.4e-15 5 0.8 -1.22464 -1.224639999999999 -1.3e-15 6 0.8 -2.729024 -2.729023999999999 -8.9e-16 7 0.8 -3.1417984 -3.1417984 4.4e-16 8 0.8 -2.29785344 -2.297853440000002 1.8e-15 9 0.8 -0.534767104 -0.5347671040000024 2.3e-15 10 0.8 1.4422260736 1.442226073599998 2.2e-15 11 0.8 2.84232882176 2.842328821759999 8.9e-16 12 0.8 3.105500041216 3.105500041216001 -4.4e-16 W_POLYNOMIAL_AB_TEST: W_POLYNOMIAL_AB evaluates Chebyshev polynomials WAB(n,x) shifted from [-1,+1] to the domain [A,B]. Here, we will use the new domain [0,1] and the desired maximum polynomial degree will be N = 5. Tables of T values: Col: 1 2 3 4 5 Row 1 : 1 -1 1 -1 1 2 : 1 -0.6 -0.04 0.664 -1.0224 3 : 1 -0.2 -0.76 1.112 -0.5744 4 : 1 0.2 -1.16 0.728 0.5776 5 : 1 0.6 -1.24 -0.104 1.2816 6 : 1 1 -1 -1 1 7 : 1 1.4 -0.44 -1.576 -0.1904 8 : 1 1.8 0.44 -1.448 -1.5984 9 : 1 2.2 1.64 -0.232 -1.9184 10 : 1 2.6 3.16 2.456 0.7696 11 : 1 3 5 7 9 Col: 6 Row 1 : -1 2 : 0.97184 3 : -0.42272 4 : -1.19008 5 : -0.40864 6 : 1 7 : 1.49984 8 : 0.16928 9 : -2.07008 10 : -1.22464 11 : 11 W_POLYNOMIAL_AB_VALUE_TEST: W_POLYNOMIAL_AB_VALUE evaluates the shifted Chebyshev polynomial WAB(n,x). Here, we will use the new domain [0,1]. Tabulated Computed N X01 W01(n,x) W01(n,x) Error -1 0.850000 0.0000000000000000e+00 0.0000000000000000e+00 0 0 0.850000 1.0000000000000000e+00 1.0000000000000000e+00 0 1 0.850000 2.3999999999999999e+00 2.3999999999999999e+00 0 2 0.850000 2.3599999999999999e+00 2.3599999999999999e+00 0 3 0.850000 9.0400000000000003e-01 9.0399999999999991e-01 1.1e-16 4 0.850000 -1.0944000000000000e+00 -1.0944000000000000e+00 0 5 0.850000 -2.4361600000000001e+00 -2.4361600000000001e+00 0 6 0.850000 -2.3162240000000001e+00 -2.3162240000000001e+00 0 7 0.850000 -8.0655359999999998e-01 -8.0655359999999998e-01 0 8 0.850000 1.1870489600000000e+00 1.1870489600000003e+00 -2.2e-16 9 0.850000 2.4684221439999998e+00 2.4684221440000003e+00 -4.4e-16 10 0.850000 2.2687420415999999e+00 2.2687420415999999e+00 0 11 0.850000 7.0781671424000003e-01 7.0781671423999937e-01 6.7e-16 12 0.850000 -1.2777986416639999e+00 -1.2777986416640008e+00 8.9e-16 7 0.000000 -1.0000000000000000e+00 -1.0000000000000000e+00 0 7 0.100000 -1.1976960000000000e-01 -1.1976959999999959e-01 -4.2e-16 7 0.200000 -8.7527679999999997e-01 -8.7527679999999985e-01 -1.1e-16 7 0.300000 8.9050879999999999e-01 8.9050879999999988e-01 1.1e-16 7 0.400000 8.5589760000000004e-01 8.5589759999999959e-01 4.4e-16 7 0.500000 -1.0000000000000000e+00 -1.0000000000000000e+00 0 7 0.600000 -1.1837055999999999e+00 -1.1837056000000004e+00 4.4e-16 7 0.700000 1.2177792000000001e+00 1.2177791999999990e+00 1.1e-15 7 0.800000 1.3912447999999999e+00 1.3912447999999986e+00 1.3e-15 7 0.900000 -3.1417983999999999e+00 -3.1417984000000003e+00 4.4e-16 7 1.000000 1.5000000000000000e+01 1.5000000000000000e+01 0 W_POLYNOMIAL_COEFFICIENTS_TEST W_POLYNOMIAL_COEFFICIENTS determines the Chebyshev polynomial coefficients. +1.000000 +2.000000 * x +1.000000 +4.000000 * x^2 +2.000000 * x -1.000000 +8.000000 * x^3 +4.000000 * x^2 -4.000000 * x -1.000000 +16.000000 * x^4 +8.000000 * x^3 -12.000000 * x^2 -4.000000 * x +1.000000 +32.000000 * x^5 +16.000000 * x^4 -32.000000 * x^3 -12.000000 * x^2 +6.000000 * x +1.000000 W_POLYNOMIAL_VALUE_TEST: W_POLYNOMIAL_VALUE evaluates the Chebyshev polynomial W(n,x). Tabulated Computed N X W(n,x) W(n,x) Error 0 0.800000 1.0000000000000000e+00 1.0000000000000000e+00 0 1 0.800000 2.6000000000000001e+00 2.6000000000000001e+00 0 2 0.800000 3.1600000000000001e+00 3.1600000000000001e+00 0 3 0.800000 2.4560000000000000e+00 2.4560000000000008e+00 -8.9e-16 4 0.800000 7.6959999999999995e-01 7.6960000000000139e-01 -1.4e-15 5 0.800000 -1.2246400000000000e+00 -1.2246399999999986e+00 -1.3e-15 6 0.800000 -2.7290239999999999e+00 -2.7290239999999990e+00 -8.9e-16 7 0.800000 -3.1417983999999999e+00 -3.1417984000000003e+00 4.4e-16 8 0.800000 -2.2978534399999999e+00 -2.2978534400000017e+00 1.8e-15 9 0.800000 -5.3476710400000005e-01 -5.3476710400000238e-01 2.3e-15 10 0.800000 1.4422260736000001e+00 1.4422260735999979e+00 2.2e-15 11 0.800000 2.8423288217599998e+00 2.8423288217599989e+00 8.9e-16 12 0.800000 3.1055000412160001e+00 3.1055000412160005e+00 -4.4e-16 W_POLYNOMIAL_ZEROS_TEST: W_POLYNOMIAL_ZEROS computes the zeros of W(n,x); N X W(n,x) 1 -0.5000 4.44089e-16 1 -0.8090 -3.33067e-16 2 0.3090 2.22045e-16 1 -0.9010 8.88178e-16 2 -0.2225 -3.33067e-16 3 0.6235 4.44089e-16 1 -0.9397 -6.66134e-16 2 -0.5000 1.33227e-15 3 0.1736 -5.55112e-16 4 0.7660 -8.88178e-16 1 -0.9595 4.44089e-16 2 -0.6549 -6.66134e-16 3 -0.1423 1.05471e-15 4 0.4154 0 5 0.8413 3.10862e-15 W_QUADRATURE_RULE_TEST: W_QUADRATURE_RULE computes the quadrature rule associated with W(n,x); X W 1: -0.978148 0.828605 2: -0.809017 0.757759 3: -0.5 0.628319 4: -0.104528 0.462664 5: 0.309017 0.289438 6: 0.669131 0.138594 7: 0.913545 0.036214 Use the quadrature rule to estimate: Q = Integral ( -1 <= X <= +1 ) X^E * sqrt(1-x)/sqrt(1+x) dx E Q_Estimate Q_Exact 0 3.14159 3.14159 1 -1.5708 -1.5708 2 1.5708 1.5708 3 -1.1781 -1.1781 4 1.1781 1.1781 5 -0.981748 -0.981748 6 0.981748 0.981748 7 -0.859029 -0.859029 8 0.859029 0.859029 9 -0.773126 -0.773126 10 0.773126 0.773126 11 -0.708699 -0.708699 12 0.708699 0.708699 13 -0.658078 -0.658078 WW_PRODUCT_INTEGRAL_TEST: WW_PRODUCT_INTEGRAL computes the product integral of a pair of Chebyshev W polynomials W(i,x) and W(j,x). A(I,J) = integral ( -1 <=x <= +1 ) W(i,x) W(j,x) sqrt ( 1 - x ) / sqrt ( 1 + x ) dx 0 if i is not equal to j; pi if i = j W(i,x)*W(j,x) integral matrix: Col: 1 2 3 4 5 Row 1 : 3.14159 0 0 0 0 2 : 0 3.14159 0 0 0 3 : 0 0 3.14159 0 0 4 : 0 0 0 3.14159 0 5 : 0 0 0 0 3.14159 CHEBYSHEV_POLYNOMIAL_TEST: Normal end of execution. 07-Jan-2022 18:00:42