Sun Feb 2 20:48:33 2025 jacobi_polynomial_test(): python version: 3.10.12 numpy version: 1.26.4 jacobi_polynomial() evalutes the Jacobi polynomial and associated functions. jacobi_polynomial_test01(): j_polynomial_values() stores values of the Jacobi polynomials. j_polynomial() evaluates the polynomial. Tabulated Computed N A B X J(N,A,B,X) J(N,A,B,X) Error 0 0.000000 1.000000 0.500000 1.0000000000000000e+00 1.0000000000000000e+00 0 1 0.000000 1.000000 0.500000 2.5000000000000000e-01 2.5000000000000000e-01 0 2 0.000000 1.000000 0.500000 -3.7500000000000000e-01 -3.7500000000000000e-01 0 3 0.000000 1.000000 0.500000 -4.8437500000000000e-01 -4.8437500000000000e-01 0 4 0.000000 1.000000 0.500000 -1.3281250000000000e-01 -1.3281250000000000e-01 0 5 0.000000 1.000000 0.500000 2.7539062500000000e-01 2.7539062500000000e-01 0 5 1.000000 1.000000 0.500000 -1.6406250000000000e-01 -1.6406250000000000e-01 0 5 2.000000 1.000000 0.500000 -1.1748046875000000e+00 -1.1748046875000000e+00 0 5 3.000000 1.000000 0.500000 -2.3613281250000000e+00 -2.3613281250000000e+00 0 5 4.000000 1.000000 0.500000 -2.6162109375000000e+00 -2.6162109375000000e+00 0 5 5.000000 1.000000 0.500000 1.1718750000000000e-01 1.1718750000000000e-01 0 5 0.000000 2.000000 0.500000 4.2187500000000000e-01 4.2187500000000000e-01 0 5 0.000000 3.000000 0.500000 5.0488281250000000e-01 5.0488281250000000e-01 0 5 0.000000 4.000000 0.500000 5.0976562500000000e-01 5.0976562500000000e-01 0 5 0.000000 5.000000 0.500000 4.3066406250000000e-01 4.3066406250000000e-01 0 5 0.000000 1.000000 -1.000000 -6.0000000000000000e+00 -6.0000000000000000e+00 0 5 0.000000 1.000000 -0.800000 3.8620000000000002e-02 3.8619999999999495e-02 5.1e-16 5 0.000000 1.000000 -0.600000 8.1184000000000001e-01 8.1184000000000001e-01 0 5 0.000000 1.000000 -0.400000 3.6659999999999998e-02 3.6660000000000220e-02 -2.2e-16 5 0.000000 1.000000 -0.200000 -4.8512000000000000e-01 -4.8512000000000005e-01 5.6e-17 5 0.000000 1.000000 0.000000 -3.1250000000000000e-01 -3.1250000000000000e-01 0 5 0.000000 1.000000 0.200000 1.8912000000000001e-01 1.8912000000000004e-01 -2.8e-17 5 0.000000 1.000000 0.400000 4.0233999999999998e-01 4.0233999999999998e-01 0 5 0.000000 1.000000 0.600000 1.2160000000000001e-02 1.2160000000000028e-02 -2.8e-17 5 0.000000 1.000000 0.800000 -4.3962000000000001e-01 -4.3961999999999996e-01 -5.6e-17 5 0.000000 1.000000 1.000000 1.0000000000000000e+00 1.0000000000000000e+00 0 jacobi_polynomial_test02: j_polynomial_zeros() computes the zeros of J(n,a,b,x) Check by calling j_polynomial() there. zeros for J(1,0.5,0.5) 0: 0 J(1,0.5,0.5, z ) 0: 0 zeros for J(2,0.5,0.5) 0: -0.5 1: 0.5 J(2,0.5,0.5, z ) 0: -2.96059e-16 1: -2.96059e-16 zeros for J(3,0.5,0.5) 0: -0.707107 1: 4.83768e-17 2: 0.707107 J(3,0.5,0.5, z ) 0: -2.36848e-16 1: -1.05824e-16 2: -8.28967e-16 zeros for J(4,0.5,0.5) 0: -0.809017 1: -0.309017 2: 0.309017 3: 0.809017 J(4,0.5,0.5, z ) 0: -6.09037e-16 1: 2.03012e-16 2: 2.03012e-16 3: 4.06024e-16 zeros for J(5,0.5,0.5) 0: -0.866025 1: -0.5 2: 1.88481e-17 3: 0.5 4: 0.866025 J(5,0.5,0.5, z ) 0: 2.21058e-15 1: -5.78961e-16 2: 5.10224e-17 3: 0 4: -4.10536e-15 zeros for J(1,1.0,1.5) 0: 0.111111 J(1,1.0,1.5, z ) 0: 0 zeros for J(2,1.0,1.5) 0: -0.348215 1: 0.502061 J(2,1.0,1.5, z ) 0: -1.75443e-16 1: 1.75443e-16 zeros for J(3,1.0,1.5) 0: -0.578486 1: 0.0708943 2: 0.684062 J(3,1.0,1.5, z ) 0: 1.85503e-15 1: 2.81567e-16 2: -6.62511e-16 zeros for J(4,1.0,1.5) 0: -0.706793 1: -0.217417 2: 0.332314 3: 0.782371 J(4,1.0,1.5, z ) 0: 2.82931e-15 1: -4.50118e-16 2: -6.43025e-17 3: 1.28605e-15 zeros for J(5,1.0,1.5) 0: -0.784837 1: -0.409111 2: 0.0520736 3: 0.500669 4: 0.841205 J(5,1.0,1.5, z ) 0: 4.33093e-15 1: -1.44364e-16 2: 1.89478e-16 3: 1.51582e-15 4: 2.30983e-15 zeros for J(1,2.0,0.5) 0: -0.333333 J(1,2.0,0.5, z ) 0: 0 zeros for J(2,2.0,0.5) 0: -0.645661 1: 0.184123 J(2,2.0,0.5, z ) 0: -3.50885e-16 1: -8.77213e-17 zeros for J(3,2.0,0.5) 0: -0.780044 1: -0.212793 2: 0.463425 J(3,2.0,0.5, z ) 0: 1.19252e-15 1: 9.93766e-17 2: -2.65004e-16 zeros for J(4,2.0,0.5) 0: -0.850186 1: -0.444336 2: 0.0995069 3: 0.623586 J(4,2.0,0.5, z ) 0: -1.28605e-16 1: -6.43025e-17 2: -3.53664e-16 3: -2.5721e-15 zeros for J(5,2.0,0.5) 0: -0.891404 1: -0.58871 2: -0.156285 3: 0.31397 4: 0.722429 J(5,2.0,0.5, z ) 0: 5.05275e-15 1: -2.88728e-16 2: -1.08273e-16 3: 1.44364e-15 4: -1.73237e-15 jacobi_polynomial_test03(): j_quadrature_rule() computes the quadrature rule associated with J(n,a,b,x) X W 0: -0.810587 0.00620279 1: -0.550876 0.0619382 2: -0.230689 0.216462 3: 0.113888 0.398821 4: 0.4436 0.431011 5: 0.720815 0.259969 6: 0.91385 0.0622568 Use the quadrature rule to estimate: Q = Integral (-1