22 January 2020 08:20:40 AM DIVDIF_TEST C++ version Test DIVDIF. TEST01 DATA_TO_DIF_DISPLAY sets up a difference table and displays intermediate calculations; DIF_APPEND appends a new data point; DIF_ANTIDERIV computes the antiderivative; DIF_DERIV_TABLE computes the derivative; DIF_SHIFT_ZERO shifts all the abscissas to 0; DIF_VAL evaluates at a point; The divided difference table: 1 2 3 4 0 1 4 9 16 1 3 5 7 2 1 1 3 0 The divided difference polynomial: p(x) = 1 + ( x - 1) * ( 3 + ( x - 2) * ( 1 + ( x - 3) * ( 0 ))) DIF_APPEND can add the data (5,25) to the table. The updated divided difference polynomial: p(x) = 25 + ( x - 5) * ( 6 + ( x - 1) * ( 1 + ( x - 2) * ( -0 + ( x - 3) * ( -0 )))) DIF_VAL can evaluate the table at a point. DIF_VAL reports P(2.5) = 6.25 The divided difference table after DIF_SHIFT_ZERO: p(x) = 0 + ( x - 0) * ( 0 + ( x - 0) * ( 1 + ( x - 0) * ( 0 + ( x - 0) * ( -0 )))) The divided difference table for the derivative: p(x) = 0 + ( x - 0) * ( 2 + ( x - 0) * ( 0 + ( x - 0) * ( -0 ))) DIF_VAL reports P'(2.5) = 5 The divided difference table for the antiderivative: p(x) = 0 + ( x - 0) * ( 0 + ( x - 0) * ( 0 + ( x - 0) * ( 0.333333 + ( x - 0) * ( 0 + ( x - 0) * ( -0 ))))) DIF_VAL reports (Anti)P(2.5) = 5.20833 TEST02 Approximate Y = EXP(X) using orders 1 to 8. Original data: X Y 0 1 1 2.71828 2 7.38906 3 20.0855 4 54.5982 5 148.413 6 403.429 7 1096.63 Evaluate at X = 2.5 where EXP(X) = 12.1825 Order Approximate Y Error 1 1 -11.1825 2 5.2957 -6.88679 3 10.8316 -1.35087 4 12.417 0.234513 5 12.0765 -0.106003 6 12.252 0.069528 7 12.1264 -0.0561433 8 12.2343 0.0518261 dif_basis_test dif_basis() computes Lagrange basis polynomials in difference form. The base points: 1 1 2 2 3 3 4 4 5 5 The table of difference vectors defining the basis polynomials. Each ROW represents a polynomial. 1 -1 0.5 -0.166667 0.0416667 0 1 -1 0.5 -0.166667 0 0 0.5 -0.5 0.25 0 0 0 0.166667 -0.166667 0 0 0 0 0.0416667 Evaluate basis polynomial #3 at a set of points. X Y 1 0 1.5 -0.546875 2 0 2.5 0.703125 3 1 3.5 0.703125 4 0 4.5 -0.546875 5 0 TEST05 DIF_TO_R8POLY converts a difference table to a polynomial; DIF_SHIFT_ZERO shifts a divided difference table to 0; These are equivalent operations The divided difference table: 1 2 3 4 0 -2 2 14 40 1 4 12 26 2 4 7 3 1 The divided difference table: 1 2 3 4 0 -2 2 14 40 1 4 12 26 2 4 7 3 1 The divided difference table: p(x) = -2 + ( x - 1) * ( 4 + ( x - 2) * ( 4 + ( x - 3) * ( 1 ))) The polynomial using DIF_SHIFT_ZERO: p(x) = 1 * x ^ 3 - 2 * x ^ 2 + 3 * x - 4 The polynomial using DIF_TO_R8POLY: p(x) = 1 * x ^ 3 - 2 * x ^ 2 + 3 * x - 4 TEST06 R8POLY_ANT_COF computes the coefficients of the antiderivative of a polynomial; R8POLY_ANT_VAL evaluates the antiderivative of a polynomial; R8POLY_DER_COF computes the coefficients of the derivative of a polynomial; R8POLY_DER_VAL evaluates the derivative of a polynomial; R8POLY_PRINT prints a polynomial; R8POLY_VAL evaluates a polynomial. The initial polynomial: p(x) = 5 * x ^ 4 + 4 * x ^ 3 + 3 * x ^ 2 + 2 * x + 1 The antiderivative polynomial: p(x) = 1 * x ^ 5 + 1 * x ^ 4 + 1 * x ^ 3 + 1 * x ^ 2 + 1 * x The derivative polynomial: p(x) = 20 * x ^ 3 + 12 * x ^ 2 + 6 * x + 2 Evaluate the polynomial, antiderivative and derivative, using only the original polynomial coefficients: X P(X) Anti_P(X) P'(X) 0 1 0 2 1 15 5 40 2 129 62 222 r8poly_basis_test r8poly_basis() computes Lagrange basis polynomials in standard form. The table of difference vectors defining the basis polynomials. Each ROW represents a polynomial. 5 -6.41667 2.95833 -0.583333 0.0416667 -10 17.8333 -9.83333 2.16667 -0.166667 10 -19.5 12.25 -3 0.25 -5 10.1667 -6.83333 1.83333 -0.166667 1 -2.08333 1.45833 -0.416667 0.0416667 One basis polynomial in standard form: p(x) = 0.25 * x ^ 4 - 3 * x ^ 3 + 12.25 * x ^ 2 - 19.5 * x + 10 Evaluate basis polynomial #3 at a set of points. X Y 1 0 1.5 -0.546875 2 0 2.5 0.703125 3 1 3.5 0.703125 4 0 4.5 -0.546875 5 0 r8poly_shift_test r8poly_shift shifts polynomial coefficients. Polynomial coefficients for argument X 0 6 1 -1 2 2 SCALE = 2 SHIFT = 3 Polynomial coefficients for argument Z = SCALE * X + SHIFT: 0 12 1 -3.5 2 0.5 ncc_rule_test ncc_rule() computes closed Newton Cotes formulas; Newton-Cotes Closed Quadrature Rule: Abscissa Weight 1 -1 0.0869213 2 -0.714286 0.414005 3 -0.428571 0.153125 4 -0.142857 0.345949 5 0.142857 0.345949 6 0.428571 0.153125 7 0.714286 0.414005 8 1 0.0869213 nco_rule_test nco_rule() computes open Newton Cotes formulas. Newton-Cotes Open Quadrature Rule: Abscissa Weight 1 -0.777778 0.797768 2 -0.555556 -1.25134 3 -0.333333 2.21741 4 -0.111111 -0.763839 5 0.111111 -0.763839 6 0.333333 2.21741 7 0.555556 -1.25134 8 0.777778 0.797768 TEST18 ROOTS_TO_DIF computes a divided difference polynomial with the given roots; DIF_TO_R8POLY converts it to a standard form polynomial. The roots: 1 3 The polynomial: p(x) = 1 * x - 3 The roots: 1 3 2 1 The polynomial: p(x) = 1 * x ^ 2 - 4 * x + 3 The roots: 1 3 2 1 3 2 The polynomial: p(x) = 1 * x ^ 3 - 6 * x ^ 2 + 11 * x - 6 The roots: 1 3 2 1 3 2 4 4 The polynomial: p(x) = 1 * x ^ 4 - 10 * x ^ 3 + 35 * x ^ 2 - 50 * x + 24 roots_to_r8poly_test roots_to_r8poly() computes polynomial coefficients from roots. The roots: 1 3 The polynomial: p(x) = 1 * x - 3 The roots: 1 3 2 1 The polynomial: p(x) = 1 * x ^ 2 - 4 * x + 3 The roots: 1 3 2 1 3 2 The polynomial: p(x) = 1 * x ^ 3 - 6 * x ^ 2 + 11 * x - 6 The roots: 1 3 2 1 3 2 4 4 The polynomial: p(x) = 1 * x ^ 4 - 10 * x ^ 3 + 35 * x ^ 2 - 50 * x + 24 dif_derivk_table_test dif_derivk_table() computes the K-th derivative for a divided difference table. The divided difference polynomial P0: p(x) = 0.333333 + ( x - -2) * ( 0.0416667 + ( x - -1) * ( 0.291667 + ( x - 0) * ( 0.0833333 + ( x - 1) * ( 0.0416667 )))) Using DIF_TO_R8POLY p(x) = 0.0416667 * x ^ 4 + 0.166667 * x ^ 3 + 0.5 * x ^ 2 + 1 * x + 1 Evaluate difference tables for the function P0 and its first four derivatives, P1...P4. X P0 P1 P2 P3 P4 0 1 1 1 1 1 0.2 1.2214 1.22133 1.22 1.2 1 0.4 1.49173 1.49067 1.48 1.4 1 0.6 1.8214 1.816 1.78 1.6 1 0.8 2.2224 2.20533 2.12 1.8 1 1 2.70833 2.66667 2.5 2 1 1.2 3.2944 3.208 2.92 2.2 1 1.4 3.9974 3.83733 3.38 2.4 1 1.6 4.83573 4.56267 3.88 2.6 1 1.8 5.8294 5.392 4.42 2.8 1 2 7 6.33333 5 3 1 dif_basis_deriv_test dif_basis_deriv() computes difference tables for the first derivative of each Lagrange basis. Lagrange basis derivative polynomial coefficients: Row: 0 1 Col 0: -0.285714 0.0952381 1: 0.25 -0.166667 2: 0.0357143 0.0714286 P1'=-(2x-6)/21 p(x) = 0.0952381 * x - 0.285714 P2'=-(2x-3)/12 p(x) = - 0.166667 * x + 0.25 P3'=(2x+1)/28 p(x) = 0.0714286 * x + 0.0357143 dif_basis_derivk_test DIF_BASIS_DERIVK computes difference tables for the K-th derivative of each Lagrange basis. Lagrange basis K-th derivative polynomial coefficients: Row: 0 1 2 Col 0: 5.91667 -3.5 0.5 1: -19.6667 13 -2 2: 24.5 -18 3 3: -13.6667 11 -2 4: 2.91667 -2.5 0.5 P1''=(12x^2-84x+142)/24 p(x) = 0.5 * x ^ 2 - 3.5 * x + 5.91667 P2''=-2x^2+13x-59/3 p(x) = - 2 * x ^ 2 + 13 * x - 19.6667 P3''=3x^2-18x+49/2 p(x) = 3 * x ^ 2 - 18 * x + 24.5 P4''=-2x^2+11x-41/3 p(x) = - 2 * x ^ 2 + 11 * x - 13.6667 P5''=(6x^2-30x+35)/12 p(x) = 0.5 * x ^ 2 - 2.5 * x + 2.91667 divdif_test Normal end of execution. 22 January 2020 08:20:40 AM