10-Dec-2018 20:34:11 CHEBYSHEV_POLYNOMIAL_TEST: MATLAB version. Test the CHEBYSHEV_POLYNOMIAL library. 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: 0.218418 0.0477066 2: 0.956318 0.914543 3: 0.829509 0.688086 4: 0.561695 0.315502 5: 0.415307 0.17248 6: 0.0661187 0.00437169 7: 0.257578 0.0663463 8: 0.109957 0.0120905 9: 0.043829 0.00192098 10: 0.633966 0.401913 11: 0.0617272 0.00381025 12: 0.449539 0.202085 13: 0.401306 0.161047 14: 0.754673 0.569532 15: 0.797287 0.635666 16: 0.00183837 3.37961e-06 17: 0.897504 0.805514 18: 0.350752 0.123027 19: 0.0945448 0.00893871 20: 0.0136169 0.00018542 Coefficients of Chebyshev expansion of degree 4. 1: 0.375 2: 0.5 3: 0.125 4: -6.93889e-17 I X(I) Data(I) Chebyshev(X(I)) 1 0.218418 0.0477066 0.0477066 2 0.956318 0.914543 0.914543 3 0.829509 0.688086 0.688086 4 0.561695 0.315502 0.315502 5 0.415307 0.17248 0.17248 6 0.0661187 0.00437169 0.00437169 7 0.257578 0.0663463 0.0663463 8 0.109957 0.0120905 0.0120905 9 0.043829 0.00192098 0.00192098 10 0.633966 0.401913 0.401913 11 0.0617272 0.00381025 0.00381025 12 0.449539 0.202085 0.202085 13 0.401306 0.161047 0.161047 14 0.754673 0.569532 0.569532 15 0.797287 0.635666 0.635666 16 0.00183837 3.37961e-06 3.37961e-06 17 0.897504 0.805514 0.805514 18 0.350752 0.123027 0.123027 19 0.0945448 0.00893871 0.00893871 20 0.0136169 0.00018542 0.00018542 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 7.77922e-16 -5.59603e-16 -2.43611e-16 2 : 7.77156e-16 1.5708 1.16598e-16 2.65193e-16 3 :-5.55112e-16 3.43333e-16 1.5708 5.84485e-16 4 :-2.22045e-16 3.40921e-16 5.98016e-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_COEFFICIENTS_TEST T_POLYNOMIAL_COEFFICIENTS determines the Chebyshev polynomial coefficients. p(x) = 1.000000 p(x) = 1.000000 * x p(x) = 2.000000 * x^2 - 1.000000 p(x) = 4.000000 * x^3 - 3.000000 * x p(x) = 8.000000 * x^4 - 8.000000 * x^2 + 1.000000 p(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 0 0 6 0.981748 0.981748 7 0 0 8 0.859029 0.859029 9 1.11022e-16 0 10 0.773126 0.773126 11 1.66533e-16 0 12 0.708699 0.708699 13 2.77556e-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: -5.77887e-17 2: 0.880099 3: -1.11022e-15 4: -0.0391279 5: -1.22125e-15 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: -2.30327e-07 2: 0.880106 3: -8.76709e-07 4: -0.0391223 5: 2.87075e-09 6: 0.000503295 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 6 3 -0.563163 0.901462 0.975054 0.878974 0.878974 2 -1 0.123391 -0.969549 0 -0 0 0 -1 -0.484844 1 0 0 0 0 1 0.267931 1 0.267931 0.267931 0.267931 5 3 -0.197387 -0.83792 0.5614 -0.470408 -0.470408 6 1 -0.996323 0.870454 -0.996323 -0.867254 -0.867254 0 4 -0.81091 1 -0.801359 -0.801359 -0.801359 0 -1 0.681695 1 0 0 0 6 -1 -0.479394 0.989962 0 0 0 5 0 -0.296743 -0.997927 1 -0.997927 -0.997927 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 3 3 4 0 -6.27276e-15 4 2 0 0 -4.60743e-15 3 1 0 0 -4.2466e-15 5 1 2 0 -5.55112e-15 4 3 3 0 -6.27276e-15 2 3 1 0.785398 0.785398 2 1 4 0 -3.91354e-15 6 1 0 0 -6.68909e-15 3 3 0 1.5708 1.5708 3 3 1 0 -2.7478e-15 5 2 4 0 -4.38538e-15 4 3 2 0 -3.21965e-15 2 3 1 0.785398 0.785398 2 2 1 0 -1.77636e-15 5 2 1 0 -5.55112e-15 5 1 1 0 -5.66214e-15 6 3 0 0 -4.02456e-15 5 1 3 0 -3.66374e-15 5 1 4 0.785398 0.785398 5 3 2 0.785398 0.785398 TU_PRODUCT_TEST: TU_PRODUCT(I,J;X) = T(I,X) * U(J,X) I J X TI UJ TI*UJ TU_PRODUCT 6 3 -0.563163 0.901462 0.823782 0.742608 0.742608 2 -1 0.123391 -0.969549 0 -0 0 0 -1 -0.484844 1 0 0 0 0 1 0.267931 1 0.535863 0.535863 0.267931 5 3 -0.197387 -0.83792 0.728025 -0.610027 -0.610027 6 1 -0.996323 0.870454 -1.99265 -1.73451 -1.73451 0 4 -0.81091 1 0.0275857 0.0275857 0.0137928 0 -1 0.681695 1 0 0 0 6 -1 -0.479394 0.989962 0 0 0 5 0 -0.296743 -0.997927 1 -0.997927 -0.997927 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 7.60274e-16 5.02803e-17 3.13994e-16 2 : 6.66134e-16 1.5708 1.1644e-15 1.75801e-15 3 : 5.55112e-17 1.14555e-15 1.5708 2.31312e-15 4 : 2.77556e-16 1.78259e-15 2.29384e-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). p(x) = 1.000000 p(x) = 2.000000 * x p(x) = 4.000000 * x^2 - 1.000000 p(x) = 8.000000 * x^3 - 4.000000 * x p(x) = 16.000000 * x^4 - 12.000000 * x^2 + 1.000000 p(x) = 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 8.32667e-17 0 2 0.392699 0.392699 3 -5.55112e-17 0 4 0.19635 0.19635 5 -1.38778e-16 0 6 0.122718 0.122718 7 -1.66533e-16 0 8 0.0859029 0.0859029 9 -1.73472e-16 0 10 0.0644272 0.0644272 11 -1.76942e-16 0 12 0.0506214 0.0506214 13 -1.73472e-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 6 3 -0.563163 0.606464 0.823782 0.499594 0.499594 2 -1 0.123391 -0.939099 0 -0 0 0 -1 -0.484844 1 0 0 0 0 1 0.267931 1 0.535863 0.535863 0.535863 5 3 -0.197387 -0.947815 0.728025 -0.690033 -0.690033 6 1 -0.996323 6.59497 -1.99265 -13.1414 -13.1414 0 4 -0.81091 1 0.0275857 0.0275857 0.0275857 0 -1 0.681695 1 0 0 0 6 -1 -0.479394 1.06717 0 0 0 5 0 -0.296743 -1.01792 1 -1.01792 -1.01792 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 7.77922e-16 -5.59603e-16 -2.43611e-16 2 : 7.77156e-16 1.5708 1.16598e-16 2.65193e-16 3 :-5.55112e-16 3.43333e-16 1.5708 5.84485e-16 4 :-2.22045e-16 3.40921e-16 5.98016e-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. p(x) = 1.000000 p(x) = 2.000000 * x - 1.000000 p(x) = 4.000000 * x^2 - 2.000000 * x - 1.000000 p(x) = 8.000000 * x^3 - 4.000000 * x^2 - 4.000000 * x + 1.000000 p(x) = 16.000000 * x^4 - 8.000000 * x^3 - 12.000000 * x^2 + 4.000000 * x + 1.000000 p(x) = 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 7.77922e-16 -5.59603e-16 -2.43611e-16 2 : 7.77156e-16 1.5708 1.16598e-16 2.65193e-16 3 :-5.55112e-16 3.43333e-16 1.5708 5.84485e-16 4 :-2.22045e-16 3.40921e-16 5.98016e-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. p(x) = 1.000000 p(x) = 2.000000 * x + 1.000000 p(x) = 4.000000 * x^2 + 2.000000 * x - 1.000000 p(x) = 8.000000 * x^3 + 4.000000 * x^2 - 4.000000 * x - 1.000000 p(x) = 16.000000 * x^4 + 8.000000 * x^3 - 12.000000 * x^2 - 4.000000 * x + 1.000000 p(x) = 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. 10-Dec-2018 20:34:32