19 June 2018 9:44:38.475 PM POLPAK_TEST FORTRAN90 version Test the POLPAK library. AGUD_TEST AGUD computes the inverse Gudermannian; X GUD(X) AGUD(GUD(X)) 1.00000 0.865769 1.00000 1.20000 0.985692 1.20000 1.40000 1.08725 1.40000 1.60000 1.17236 1.60000 1.80000 1.24316 1.80000 2.00000 1.30176 2.00000 2.20000 1.35009 2.20000 2.40000 1.38986 2.40000 2.60000 1.42252 2.60000 2.80000 1.44933 2.80000 3.00000 1.47130 3.00000 ALIGN_ENUM_TEST ALIGN_ENUM counts the number of possible alignments of two biological sequences. Alignment enumeration table: 0 1 2 3 4 5 6 7 8 9 10 0 1 1 1 1 1 1 1 1 1 1 1 1 1 3 5 7 9 11 13 15 17 19 21 2 1 5 13 25 41 61 85 113 145 181 221 3 1 7 25 63 129 231 377 575 833 1159 1561 4 1 9 41 129 321 681 1289 2241 3649 5641 8361 5 1 11 61 231 681 1683 3653 7183 13073 22363 36365 6 1 13 85 377 1289 3653 8989 19825 40081 75517 134245 7 1 15 113 575 2241 7183 19825 48639 108545 224143 433905 8 1 17 145 833 3649 13073 40081 108545 265729 598417 1256465 9 1 19 181 1159 5641 22363 75517 224143 598417 1462563 3317445 10 1 21 221 1561 8361 36365 134245 433905 1256465 3317445 8097453 BELL_TEST BELL computes Bell numbers. N exact C(I) computed C(I) 0 1 1 1 1 1 2 2 2 3 5 5 4 15 15 5 52 52 6 203 203 7 877 877 8 4140 4140 9 21147 21147 10 115975 115975 BELL_POLY_COEF_TEST BELL_POLY_COEF returns the coefficients of a Bell polynomial. Table of polynomial coefficients: 0: 1 1: 0 1 2: 0 1 1 3: 0 1 3 1 4: 0 1 7 6 1 5: 0 1 15 25 10 1 6: 0 1 31 90 65 15 1 7: 0 1 63 301 350 140 21 1 8: 0 1 127 966 1701 1050 266 28 1 9: 0 1 255 3025 7770 6951 2646 462 36 1 10: 0 1 511 9330 34105 42525 22827 5880 750 45 1 BENFORD_TEST BENFORD(I) is the Benford probability of the initial digit sequence I. I, BENFORD(I) 1 0.301030 2 0.176091 3 0.124939 4 0.969100E-01 5 0.791812E-01 6 0.669468E-01 7 0.579919E-01 8 0.511525E-01 9 0.457575E-01 BERNOULLI_NUMBER_TEST BERNOULLI_NUMBER computes Bernoulli numbers; I Exact Bernoulli 0 1.00000 1.00000 1 -0.500000 -0.500000 2 0.166667 0.166667 3 0.00000 0.00000 4 -0.333333E-01 -0.333333E-01 6 -0.238095E-01 0.238095E-01 8 -0.333333E-01 -0.333333E-01 10 0.757576E-01 0.757576E-01 20 -529.124 -529.124 30 0.601581E+09 0.601581E+09 BERNOULLI_NUMBER2_TEST BERNOULLI_NUMBER2 computes Bernoulli numbers; I Exact Bernoulli2 0 1.00000 1.00000 1 -0.500000 -0.500000 2 0.166667 0.166667 3 0.00000 0.00000 4 -0.333333E-01 -0.333333E-01 6 -0.238095E-01 0.238095E-01 8 -0.333333E-01 -0.333333E-01 10 0.757576E-01 0.757576E-01 20 -529.124 -529.124 30 0.601581E+09 0.601581E+09 BERNOULLI_NUMBER3_TEST BERNOULLI_NUMBER3 computes Bernoulli numbers. I Exact BERNOULLI3 0 1.00000 1.00000 1 -0.500000 -0.500000 2 0.166667 0.166667 3 0.00000 0.00000 4 -0.333333E-01 -0.333331E-01 6 -0.238095E-01 0.238095E-01 8 -0.333333E-01 -0.333333E-01 10 0.757576E-01 0.757576E-01 20 -529.124 -529.124 30 0.601581E+09 0.601581E+09 BERNOULLI_POLY_TEST BERNOULLI_POLY evaluates Bernoulli polynomials; X = 0.200000 I BX 1 -0.30000000 2 0.66666667E-02 3 0.48000000E-01 4 -0.77333333E-02 5 -0.23680000E-01 6 0.69135238E-02 7 0.24908800E-01 8 -0.10149973E-01 9 -0.45278208E-01 10 0.23326318E-01 11 0.12605002 12 -0.78146785E-01 13 -0.49797890 14 0.36043995 15 2.6487812 BERNOULLI_POLY2_TEST BERNOULLI_POLY2 evaluates Bernoulli polynomials. X = 0.200000 I BX 1 -0.30000000 2 0.66666667E-02 3 0.48000000E-01 4 -0.77331387E-02 5 -0.23679805E-01 6 0.69136254E-02 7 0.24908833E-01 8 -0.10149965E-01 9 -0.45278204E-01 10 0.23326320E-01 11 0.12605002 12 -0.78146787E-01 13 -0.49797890 14 0.36043994 15 2.6487812 BERNSTEIN_POLY_TEST: BERNSTEIN_POLY evaluates the Bernstein polynomials. N K X Exact B(N,K)(X) 0 0 0.2500 1.00000 1.00000 1 0 0.2500 0.750000 0.750000 1 1 0.2500 0.250000 0.250000 2 0 0.2500 0.562500 0.562500 2 1 0.2500 0.375000 0.375000 2 2 0.2500 0.625000E-01 0.625000E-01 3 0 0.2500 0.421875 0.421875 3 1 0.2500 0.421875 0.421875 3 2 0.2500 0.140625 0.140625 3 3 0.2500 0.156250E-01 0.156250E-01 4 0 0.2500 0.316406 0.316406 4 1 0.2500 0.421875 0.421875 4 2 0.2500 0.210938 0.210938 4 3 0.2500 0.468750E-01 0.468750E-01 4 4 0.2500 0.390625E-02 0.390625E-02 BPAB_TEST BPAB evaluates Bernstein polynomials. The Bernstein polynomials of degree 10 based on the interval from 0.00000 to 1.00000 evaluated at X = 0.300000 0 0.282475E-01 1 0.121061 2 0.233474 3 0.266828 4 0.200121 5 0.102919 6 0.367569E-01 7 0.900169E-02 8 0.144670E-02 9 0.137781E-03 10 0.590490E-05 CARDAN_POLY_TEST CARDAN_POLY evaluates a Cardan polynomial directly. Compare CARDAN_POLY_COEF + R8POLY_VAL_HORNER versus CARDAN_POLY alone. Evaluate polynomials at X = 0.250000 We use the parameter S = 0.500000 Order, Horner, Direct 0 2.00000 2.00000 1 0.250000 0.250000 2 -0.937500 -0.937500 3 -0.359375 -0.359375 4 0.378906 0.378906 5 0.274414 0.274414 6 -0.120850 -0.120850 7 -0.167419 -0.167419 8 0.185699E-01 0.185699E-01 9 0.883522E-01 0.883522E-01 10 0.128031E-01 0.128031E-01 CARDAN_POLY_COEF_TEST CARDAN_POLY_COEF returns the coefficients of a Cardan polynomial. We use the parameter S = 1.00000 Table of polynomial coefficients: 0 2. 1 0. 1. 2 -2. 0. 1. 3 0. -3. 0. 1. 4 2. 0. -4. 0. 1. 5 0. 5. 0. -5. 0. 1. 6 -2. 0. 9. 0. -6. 0. 1. 7 0. -7. 0. 14. 0. -7. 0. 1. 8 2. 0. -16. 0. 20. 0. -8. 0. 1. 9 0. 9. 0. -30. 0. 27. 0. -9. 0. 1. 10 -2. 0. 25. 0. -50. 0. 35. 0. -10. 0. 1. CARDINAL_COS_TEST CARDINAL_COS evaluates cardinal cosine functions. Ci(Tj) = Delta(i,j), where Tj = cos(pi*i/(n+1)). A simple check of all pairs should form the identity matrix. The CARDINAL_COS test matrix: 1.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 CARDINAL_SIN_TEST CARDINAL_SIN evaluates cardinal sine functions. Si(Tj) = Delta(i,j), where Tj = cos(pi*i/(n+1)). A simple check of all pairs should form the identity matrix. The CARDINAL_SIN test matrix: 1.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 -0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 -0.0 0.0 1.0 CATALAN_TEST CATALAN computes Catalan numbers. N exact C(I) computed C(I) 0 1 1 1 1 1 2 2 2 3 5 5 4 14 14 5 42 42 6 132 132 7 429 429 8 1430 1430 9 4862 4862 10 16796 16796 CATALAN_ROW_NEXT_TEST CATALAN_ROW_NEXT computes a row of Catalan's triangle. First, compute row 7: 7 1 7 27 75 165 297 429 429 Now compute rows one at a time: 0 1 1 1 1 2 1 2 2 3 1 3 5 5 4 1 4 9 14 14 5 1 5 14 28 42 42 6 1 6 20 48 90 132 132 7 1 7 27 75 165 297 429 429 8 1 8 35 110 275 572 1001 1430 1430 9 1 9 44 154 429 1001 2002 3432 4862 4862 10 1 10 54 208 637 1638 3640 7072 11934 16796 16796 CHARLIER_TEST: CHARLIER evaluates Charlier polynomials. N A X P(N,A,X) 0 0.2500 0.0000 1.00000 1 0.2500 0.0000 -0.00000 2 0.2500 0.0000 -4.00000 3 0.2500 0.0000 -36.0000 4 0.2500 0.0000 -420.000 5 0.2500 0.0000 -6564.00 0 0.2500 0.5000 1.00000 1 0.2500 0.5000 -2.00000 2 0.2500 0.5000 -10.0000 3 0.2500 0.5000 -54.0000 4 0.2500 0.5000 -474.000 5 0.2500 0.5000 -6246.00 0 0.2500 1.0000 1.00000 1 0.2500 1.0000 -4.00000 2 0.2500 1.0000 -8.00000 3 0.2500 1.0000 -8.00000 4 0.2500 1.0000 24.0000 5 0.2500 1.0000 440.000 0 0.2500 1.5000 1.00000 1 0.2500 1.5000 -6.00000 2 0.2500 1.5000 2.00000 3 0.2500 1.5000 54.0000 4 0.2500 1.5000 354.000 5 0.2500 1.5000 3030.00 0 0.2500 2.0000 1.00000 1 0.2500 2.0000 -8.00000 2 0.2500 2.0000 20.0000 3 0.2500 2.0000 84.0000 4 0.2500 2.0000 180.000 5 0.2500 2.0000 276.000 0 0.2500 2.5000 1.00000 1 0.2500 2.5000 -10.0000 2 0.2500 2.5000 46.0000 3 0.2500 2.5000 34.0000 4 0.2500 2.5000 -450.000 5 0.2500 2.5000 -3694.00 0 0.5000 0.0000 1.00000 1 0.5000 0.0000 -0.00000 2 0.5000 0.0000 -2.00000 3 0.5000 0.0000 -10.0000 4 0.5000 0.0000 -58.0000 5 0.5000 0.0000 -442.000 0 0.5000 0.5000 1.00000 1 0.5000 0.5000 -1.00000 2 0.5000 0.5000 -4.00000 3 0.5000 0.5000 -12.0000 4 0.5000 0.5000 -48.0000 5 0.5000 0.5000 -288.000 0 0.5000 1.0000 1.00000 1 0.5000 1.0000 -2.00000 2 0.5000 1.0000 -4.00000 3 0.5000 1.0000 -4.00000 4 0.5000 1.0000 4.00000 5 0.5000 1.0000 60.0000 0 0.5000 1.5000 1.00000 1 0.5000 1.5000 -3.00000 2 0.5000 1.5000 -2.00000 3 0.5000 1.5000 8.00000 4 0.5000 1.5000 44.0000 5 0.5000 1.5000 200.000 0 0.5000 2.0000 1.00000 1 0.5000 2.0000 -4.00000 2 0.5000 2.0000 2.00000 3 0.5000 2.0000 18.0000 4 0.5000 2.0000 42.0000 5 0.5000 2.0000 66.0000 0 0.5000 2.5000 1.00000 1 0.5000 2.5000 -5.00000 2 0.5000 2.5000 8.00000 3 0.5000 2.5000 20.0000 4 0.5000 2.5000 -8.00000 5 0.5000 2.5000 -192.000 0 1.0000 0.0000 1.00000 1 1.0000 0.0000 -0.00000 2 1.0000 0.0000 -1.00000 3 1.0000 0.0000 -3.00000 4 1.0000 0.0000 -9.00000 5 1.0000 0.0000 -33.0000 0 1.0000 0.5000 1.00000 1 1.0000 0.5000 -0.500000 2 1.0000 0.5000 -1.75000 3 1.0000 0.5000 -3.37500 4 1.0000 0.5000 -6.56250 5 1.0000 0.5000 -16.0312 0 1.0000 1.0000 1.00000 1 1.0000 1.0000 -1.00000 2 1.0000 1.0000 -2.00000 3 1.0000 1.0000 -2.00000 4 1.0000 1.0000 0.00000 5 1.0000 1.0000 8.00000 0 1.0000 1.5000 1.00000 1 1.0000 1.5000 -1.50000 2 1.0000 1.5000 -1.75000 3 1.0000 1.5000 0.375000 4 1.0000 1.5000 6.18750 5 1.0000 1.5000 20.1562 0 1.0000 2.0000 1.00000 1 1.0000 2.0000 -2.00000 2 1.0000 2.0000 -1.00000 3 1.0000 2.0000 3.00000 4 1.0000 2.0000 9.00000 5 1.0000 2.0000 15.0000 0 1.0000 2.5000 1.00000 1 1.0000 2.5000 -2.50000 2 1.0000 2.5000 0.250000 3 1.0000 2.5000 5.12500 4 1.0000 2.5000 6.93750 5 1.0000 2.5000 -3.15625 0 2.0000 0.0000 1.00000 1 2.0000 0.0000 -0.00000 2 2.0000 0.0000 -0.500000 3 2.0000 0.0000 -1.00000 4 2.0000 0.0000 -1.75000 5 2.0000 0.0000 -3.25000 0 2.0000 0.5000 1.00000 1 2.0000 0.5000 -0.250000 2 2.0000 0.5000 -0.812500 3 2.0000 0.5000 -1.17188 4 2.0000 0.5000 -1.41797 5 2.0000 0.5000 -1.55566 0 2.0000 1.0000 1.00000 1 2.0000 1.0000 -0.500000 2 2.0000 1.0000 -1.00000 3 2.0000 1.0000 -1.00000 4 2.0000 1.0000 -0.500000 5 2.0000 1.0000 0.750000 0 2.0000 1.5000 1.00000 1 2.0000 1.5000 -0.750000 2 2.0000 1.5000 -1.06250 3 2.0000 1.5000 -0.578125 4 2.0000 1.5000 0.582031 5 2.0000 1.5000 2.46582 0 2.0000 2.0000 1.00000 1 2.0000 2.0000 -1.00000 2 2.0000 2.0000 -1.00000 3 2.0000 2.0000 0.00000 4 2.0000 2.0000 1.50000 5 2.0000 2.0000 3.00000 0 2.0000 2.5000 1.00000 1 2.0000 2.5000 -1.25000 2 2.0000 2.5000 -0.812500 3 2.0000 2.5000 0.640625 4 2.0000 2.5000 2.01953 5 2.0000 2.5000 2.25293 0 10.0000 0.0000 1.00000 1 10.0000 0.0000 -0.00000 2 10.0000 0.0000 -0.100000 3 10.0000 0.0000 -0.120000 4 10.0000 0.0000 -0.126000 5 10.0000 0.0000 -0.128400 0 10.0000 0.5000 1.00000 1 10.0000 0.5000 -0.500000E-01 2 10.0000 0.5000 -0.152500 3 10.0000 0.5000 -0.165375 4 10.0000 0.5000 -0.160969 5 10.0000 0.5000 -0.151158 0 10.0000 1.0000 1.00000 1 10.0000 1.0000 -0.100000 2 10.0000 1.0000 -0.200000 3 10.0000 1.0000 -0.200000 4 10.0000 1.0000 -0.180000 5 10.0000 1.0000 -0.154000 0 10.0000 1.5000 1.00000 1 10.0000 1.5000 -0.150000 2 10.0000 1.5000 -0.242500 3 10.0000 1.5000 -0.224625 4 10.0000 1.5000 -0.185569 5 10.0000 1.5000 -0.142111 0 10.0000 2.0000 1.00000 1 10.0000 2.0000 -0.200000 2 10.0000 2.0000 -0.280000 3 10.0000 2.0000 -0.240000 4 10.0000 2.0000 -0.180000 5 10.0000 2.0000 -0.120000 0 10.0000 2.5000 1.00000 1 10.0000 2.5000 -0.250000 2 10.0000 2.5000 -0.312500 3 10.0000 2.5000 -0.246875 4 10.0000 2.5000 -0.165469 5 10.0000 2.5000 -0.915391E-01 CHEBY_T_POLY_TEST CHEBY_T_POLY evaluates the Chebyshev T polynomial. N X Exact F T(N)(X) 0 0.8000 1.00000 1.00000 1 0.8000 0.800000 0.800000 2 0.8000 0.280000 0.280000 3 0.8000 -0.352000 -0.352000 4 0.8000 -0.843200 -0.843200 5 0.8000 -0.997120 -0.997120 6 0.8000 -0.752192 -0.752192 7 0.8000 -0.206387 -0.206387 8 0.8000 0.421972 0.421972 9 0.8000 0.881543 0.881543 10 0.8000 0.988497 0.988497 11 0.8000 0.700051 0.700051 12 0.8000 0.131586 0.131586 CHEBY_T_POLY_COEF_TEST CHEBY_T_POLY_COEF determines the Chebyshev T polynomial coefficients. T( 0) 1.00000 T( 1) 1.00000 * x 0.00000 T( 2) 2.00000 * x** 2 0.00000 * x -1.00000 T( 3) 4.00000 * x** 3 0.00000 * x** 2 -3.00000 * x -0.00000 T( 4) 8.00000 * x** 4 0.00000 * x** 3 -8.00000 * x** 2 -0.00000 * x 1.00000 T( 5) 16.0000 * x** 5 0.00000 * x** 4 -20.0000 * x** 3 -0.00000 * x** 2 5.00000 * x 0.00000 CHEBY_T_POLY_ZERO_TEST: CHEBY_T_POLY_ZERO returns zeroes of T(N)(X). N X T(N)(X) 1 0.0000 0.612323E-16 2 0.7071 0.222045E-15 2 -0.7071 -0.222045E-15 3 0.8660 0.333067E-15 3 0.0000 -0.183697E-15 3 -0.8660 -0.333067E-15 4 0.9239 -0.222045E-15 4 0.3827 -0.222045E-15 4 -0.3827 0.111022E-15 4 -0.9239 -0.222045E-15 CHEBY_U_POLY_TEST CHEBY_U_POLY evaluates the Chebyshev U polynomial. N X Exact F U(N)(X) 0 0.8000 1.00000 1.00000 1 0.8000 1.60000 1.60000 2 0.8000 1.56000 1.56000 3 0.8000 0.896000 0.896000 4 0.8000 -0.126400 -0.126400 5 0.8000 -1.09824 -1.09824 6 0.8000 -1.63078 -1.63078 7 0.8000 -1.51101 -1.51101 8 0.8000 -0.786839 -0.786839 9 0.8000 0.252072 0.252072 10 0.8000 1.19015 1.19015 11 0.8000 1.65217 1.65217 12 0.8000 1.45333 1.45333 CHEBY_U_POLY_COEF_TEST CHEBY_U_POLY_COEF determines the Chebyshev U polynomial coefficients. U( 0) 1.00000 U( 1) 2.00000 * x 0.00000 U( 2) 4.00000 * x** 2 0.00000 * x -1.00000 U( 3) 8.00000 * x** 3 0.00000 * x** 2 -4.00000 * x -0.00000 U( 4) 16.0000 * x** 4 0.00000 * x** 3 -12.0000 * x** 2 -0.00000 * x 1.00000 U( 5) 32.0000 * x** 5 0.00000 * x** 4 -32.0000 * x** 3 -0.00000 * x** 2 6.00000 * x 0.00000 CHEBY_U_POLY_ZERO_TEST: CHEBY_U_POLY_ZERO returns zeroes of the U(N)(X). N X U(N)(X) 1 0.0000 0.122465E-15 2 0.5000 0.444089E-15 2 -0.5000 -0.888178E-15 3 0.7071 0.666134E-15 3 0.0000 -0.244929E-15 3 -0.7071 0.666134E-15 4 0.8090 0.00000 4 0.3090 -0.111022E-15 4 -0.3090 0.555112E-15 4 -0.8090 -0.888178E-15 CHEBYSHEV_DISCRETE_TEST: CHEBYSHEV_DISCRETE evaluates discrete Chebyshev polynomials. N M X T(N,M,X) 0 5 0.0000 1.00000 1 5 0.0000 -4.00000 2 5 0.0000 12.0000 3 5 0.0000 -24.0000 4 5 0.0000 24.0000 5 5 0.0000 0.00000 0 5 0.5000 1.00000 1 5 0.5000 -3.00000 2 5 0.5000 1.50000 3 5 0.5000 34.5000 4 5 0.5000 -199.125 5 5 0.5000 826.875 0 5 1.0000 1.00000 1 5 1.0000 -2.00000 2 5 1.0000 -6.00000 3 5 1.0000 48.0000 4 5 1.0000 -96.0000 5 5 1.0000 0.00000 0 5 1.5000 1.00000 1 5 1.5000 -1.00000 2 5 1.5000 -10.5000 3 5 1.5000 31.5000 4 5 1.5000 70.8750 5 5 1.5000 -354.375 0 5 2.0000 1.00000 1 5 2.0000 0.00000 2 5 2.0000 -12.0000 3 5 2.0000 -0.00000 4 5 2.0000 144.000 5 5 2.0000 0.00000 0 5 2.5000 1.00000 1 5 2.5000 1.00000 2 5 2.5000 -10.5000 3 5 2.5000 -31.5000 4 5 2.5000 70.8750 5 5 2.5000 354.375 COLLATZ_COUNT_TEST: COLLATZ_COUNT(N) counts the length of the Collatz sequence beginning with N. N COUNT(N) COUNT(N) (computed) (table) 1 1 1 2 2 2 3 8 8 4 3 3 5 6 6 6 9 9 7 17 17 8 4 4 9 20 20 10 7 7 27 112 112 50 25 25 100 26 26 200 27 27 300 17 17 400 28 28 500 111 111 600 18 18 700 83 83 800 29 29 COLLATZ_COUNT_MAX_TEST: COLLATZ_COUNT_MAX(N) returns the length of the longest Collatz sequence from 1 to N. N I_MAX J_MAX 10 9 20 100 97 119 1000 871 179 10000 6171 262 100000 77031 351 COMB_ROW_NEXT_TEST COMB_ROW computes a row of Pascal's triangle. 0 1 1 1 1 2 1 2 1 3 1 3 3 1 4 1 4 6 4 1 5 1 5 10 10 5 1 6 1 6 15 20 15 6 1 7 1 7 21 35 35 21 7 1 8 1 8 28 56 70 56 28 8 1 9 1 9 36 84 126 126 84 36 9 1 10 1 10 45 120 210 252 210 120 45 10 1 COMMUL_TEST COMMUL computes a multinomial coefficient. N = 8 Number of factors = 2 1 6 2 2 Value of coefficient = 28 N = 8 Number of factors = 3 1 2 2 2 3 4 Value of coefficient = 420 N = 13 Number of factors = 4 1 5 2 3 3 3 4 2 Value of coefficient = 720720 COMPLETE_SYMMETRIC_POLY_TEST COMPLETE_SYMMETRIC_POLY evaluates a complete symmetric. polynomial in a given set of variables X. Variable vector X: 1: 1.0000000 2: 2.0000000 3: 3.0000000 4: 4.0000000 5: 5.0000000 N\R 0 1 2 3 4 5 0 1. 0. 0. 0. 0. 0. 1 1. 1. 1. 1. 1. 1. 2 1. 3. 7. 15. 31. 63. 3 1. 6. 25. 90. 301. 966. 4 1. 10. 65. 350. 1701. 7770. 5 1. 15. 140. 1050. 6951. 42525. COS_POWER_INT_TEST: COS_POWER_INT returns values of the integral of COS(X)^N from A to B. A B N Exact Computed 0.0000 3.1416 0 3.14159 3.14159 0.0000 3.1416 1 0.00000 0.122465E-15 0.0000 3.1416 2 1.57080 1.57080 0.0000 3.1416 3 0.00000 0.122465E-15 0.0000 3.1416 4 1.17810 1.17810 0.0000 3.1416 5 0.00000 0.122465E-15 0.0000 3.1416 6 0.981748 0.981748 0.0000 3.1416 7 0.00000 0.122465E-15 0.0000 3.1416 8 0.859029 0.859029 0.0000 3.1416 9 0.00000 0.122465E-15 0.0000 3.1416 10 0.773126 0.773126 DELANNOY_TEST DELANNOY computes the Delannoy numbers A(0:M,0:N). A(M,N) counts the paths from (0,0) to (M,N). 0 1 1 1 1 1 1 1 1 1 1 1 3 5 7 9 11 13 15 17 2 1 5 13 25 41 61 85 113 145 3 1 7 25 63 129 231 377 575 833 4 1 9 41 129 321 681 1289 2241 3649 5 1 11 61 231 681 1683 3653 7183 13073 6 1 13 85 3771289 3653 8989 19825 40081 7 1 15 113 5752241 7183 19825 48639 108545 8 1 17 145 8333649 13073 40081 108545 265729 DOMINO_TILING_NUM_TEST: DOMINO_TILING_NUM returns the number of tilings of an MxN rectangle by dominoes. M N Tilings 1 1 0 2 1 1 2 2 2 3 1 0 3 2 3 3 3 0 4 1 1 4 2 5 4 3 11 4 4 36 5 1 0 5 2 8 5 3 0 5 4 95 5 5 0 6 1 1 6 2 13 6 3 41 6 4 281 6 5 1183 6 6 6728 7 1 0 7 2 21 7 3 0 7 4 781 7 5 0 7 6 31529 7 7 0 8 1 1 8 2 34 8 3 153 8 4 2245 8 5 14824 8 6 167089 8 7 1292697 8 8 12988816 EULER_NUMBER_TEST EULER_NUMBER computes Euler numbers. N exact EULER_NUMBER 0 1 1 1 0 0 2 -1 -1 4 5 5 6 -61 -61 8 1385 1385 10 -50521 -50521 12 2702765 2702765 EULER_NUMBER2_TEST EULER_NUMBER2 computes Euler numbers. N exact EULER_NUMBER2 0 1 1.00000 1 0 0.00000 2 -1 -1.00000 4 5 5.00000 6 -61 -61.0000 8 1385 1385.00 10 -50521 -50521.0 12 2702765 0.270276E+07 EULER_POLY_TEST EULER_POLY evaluates Euler polynomials. N X F(X) 0 0.500000 1.00000 1 0.500000 0.277556E-16 2 0.500000 -0.250000 3 0.500000 -0.145953E-05 4 0.500000 0.312497 5 0.500000 -0.332929E-05 6 0.500000 -0.953128 7 0.500000 -0.173264E-05 8 0.500000 5.41016 9 0.500000 -0.102449E-05 10 0.500000 -49.3369 11 0.500000 0.647439E-06 12 0.500000 659.855 13 0.500000 0.522754E-05 14 0.500000 -12168.0 15 0.500000 0.218677E-03 EULERIAN_TEST EULERIAN evaluates Eulerian numbers. 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 4 1 0 0 0 0 1 11 11 1 0 0 0 1 26 66 26 1 0 0 1 57 302 302 57 1 0 1 120 1191 2416 1191 120 1 F_HOFSTADTER_TEST F_HOFSTADTER evaluates Hofstadter's recursive F function. N F(N) 0 0 1 1 2 1 3 2 4 2 5 3 6 3 7 4 8 4 9 5 10 5 11 6 12 6 13 7 14 7 15 8 16 8 17 9 18 9 19 10 20 10 21 11 22 11 23 12 24 12 25 13 26 13 27 14 28 14 29 15 30 15 FIBONACCI_DIRECT_TEST FIBONACCI_DIRECT evalutes a Fibonacci number directly. I F(I) 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144 13 233 14 377 15 610 16 987 17 1597 18 2584 19 4181 20 6765 FIBONACCI_FLOOR_TEST FIBONACCI_FLOOR computes the largest Fibonacci number less than or equal to a given positive integer. N Fibonacci Index 1 1 2 2 2 3 3 3 4 4 3 4 5 5 5 6 5 5 7 5 5 8 8 6 9 8 6 10 8 6 11 8 6 12 8 6 13 13 7 14 13 7 15 13 7 16 13 7 17 13 7 18 13 7 19 13 7 20 13 7 FIBONACCI_RECURSIVE_TEST FIBONACCI_RECURSIVE computes the Fibonacci sequence. N F(N) 1 1 2 1 3 2 4 3 5 5 6 8 7 13 8 21 9 34 10 55 11 89 12 144 13 233 14 377 15 610 16 987 17 1597 18 2584 19 4181 20 6765 G_HOFSTADTER_TEST G_HOFSTADTER evaluates Hofstadter's recursive G function. N G(N) 0 0 1 1 2 1 3 2 4 3 5 3 6 4 7 4 8 5 9 6 10 6 11 7 12 8 13 8 14 9 15 9 16 10 17 11 18 11 19 12 20 12 21 13 22 14 23 14 24 15 25 16 26 16 27 17 28 17 29 18 30 19 GEGENBAUER_POLY_TEST: GEGENBAUER_POLY computes values of the Gegenbauer polynomials. N A X GPV GEGENBAUER 0 0.5000 0.2000 1.00000 1.00000 1 0.5000 0.2000 0.200000 0.200000 2 0.5000 0.2000 -0.440000 -0.440000 3 0.5000 0.2000 -0.280000 -0.280000 4 0.5000 0.2000 0.232000 0.232000 5 0.5000 0.2000 0.307520 0.307520 6 0.5000 0.2000 -0.805760E-01 -0.805760E-01 7 0.5000 0.2000 -0.293517 -0.293517 8 0.5000 0.2000 -0.395648E-01 -0.395648E-01 9 0.5000 0.2000 0.245971 0.245957 10 0.5000 0.2000 0.129072 0.129072 2 0.0000 0.4000 0.00000 0.00000 2 1.0000 0.4000 -0.360000 -0.360000 2 2.0000 0.4000 -0.800000E-01 -0.800000E-01 2 3.0000 0.4000 0.840000 0.840000 2 4.0000 0.4000 2.40000 2.40000 2 5.0000 0.4000 4.60000 4.60000 2 6.0000 0.4000 7.44000 7.44000 2 7.0000 0.4000 10.9200 10.9200 2 8.0000 0.4000 15.0400 15.0400 2 9.0000 0.4000 19.8000 19.8000 2 10.0000 0.4000 25.2000 25.2000 5 3.0000 -0.5000 -9.00000 9.00000 5 3.0000 -0.4000 -0.161280 -0.161280 5 3.0000 -0.3000 -6.67296 -6.67296 5 3.0000 -0.2000 -8.37504 -8.37504 5 3.0000 -0.1000 -5.52672 -5.52672 5 3.0000 0.0000 0.00000 0.00000 5 3.0000 0.1000 5.52672 5.52672 5 3.0000 0.2000 8.37504 8.37504 5 3.0000 0.3000 6.67296 6.67296 5 3.0000 0.4000 0.161280 0.161280 5 3.0000 0.5000 -9.00000 -9.00000 5 3.0000 0.6000 -15.4253 -15.4253 5 3.0000 0.7000 -9.69696 -9.69696 5 3.0000 0.8000 22.4410 22.4410 5 3.0000 0.9000 100.889 100.889 5 3.0000 1.0000 252.000 252.000 GEN_HERMITE_POLY_TEST GEN_HERMITE_POLY evaluates the generalized Hermite polynomials. Table of H(N,MU)(X) for N(max) = 10 MU = 0.00000 X = 0.00000 0 1.00000 1 0.00000 2 -2.00000 3 -0.00000 4 12.0000 5 0.00000 6 -120.000 7 -0.00000 8 1680.00 9 0.00000 10 -30240.0 Table of H(N,MU)(X) for N(max) = 10 MU = 0.00000 X = 1.00000 0 1.00000 1 2.00000 2 2.00000 3 -4.00000 4 -20.0000 5 -8.00000 6 184.000 7 464.000 8 -1648.00 9 -10720.0 10 8224.00 Table of H(N,MU)(X) for N(max) = 10 MU = 0.100000 X = 0.00000 0 1.00000 1 0.00000 2 -2.40000 3 -0.00000 4 15.3600 5 0.00000 6 -159.744 7 -0.00000 8 2300.31 9 0.00000 10 -42325.8 Table of H(N,MU)(X) for N(max) = 10 MU = 0.100000 X = 0.500000 0 1.00000 1 1.00000 2 -1.40000 3 -5.40000 4 3.56000 5 46.7600 6 9.73600 7 -551.384 8 -691.582 9 8130.56 10 20855.7 Table of H(N,MU)(X) for N(max) = 10 MU = 0.500000 X = 0.500000 0 1.00000 1 1.00000 2 -3.00000 3 -7.00000 4 17.0000 5 73.0000 6 -131.000 7 -1007.00 8 1089.00 9 17201.0 10 -4579.00 Table of H(N,MU)(X) for N(max) = 10 MU = 1.00000 X = 0.500000 0 1.00000 1 1.00000 2 -5.00000 3 -9.00000 4 41.0000 5 113.000 6 -461.000 7 -1817.00 8 6481.00 9 35553.0 10 -107029. GEN_LAGUERRE_POLY_TEST GEN_LAGUERRE_POLY evaluates the generalized Laguerre polynomials. Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.00000 X = 0.00000 0 1.00000 1 1.00000 2 1.00000 3 1.00000 4 1.00000 5 1.00000 6 1.00000 7 1.00000 8 1.00000 9 1.00000 10 1.00000 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.00000 X = 1.00000 0 1.00000 1 0.00000 2 -0.500000 3 -0.666667 4 -0.625000 5 -0.466667 6 -0.256944 7 -0.404762E-01 8 0.153993 9 0.309744 10 0.418946 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.100000 X = 0.00000 0 1.00000 1 1.10000 2 1.15500 3 1.19350 4 1.22334 5 1.24780 6 1.26860 7 1.28672 8 1.30281 9 1.31728 10 1.33046 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.100000 X = 0.500000 0 1.00000 1 0.600000 2 0.230000 3 -0.673333E-01 4 -0.289350 5 -0.442469 6 -0.535747 7 -0.578765 8 -0.580771 9 -0.550311 10 -0.495076 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.500000 X = 0.500000 0 1.00000 1 1.00000 2 0.750000 3 0.416667 4 0.729167E-01 5 -0.243750 6 -0.513715 7 -0.727703 8 -0.882836 9 -0.980303 10 -1.02388 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 1.00000 X = 0.500000 0 1.00000 1 1.50000 2 1.62500 3 1.47917 4 1.14844 5 0.702865 6 0.198720 7 -0.319620 8 -0.817983 9 -1.27090 10 -1.66028 GUD_TEST: GUD evaluates the Gudermannian function. X Exact F GUD(X) -2.0000 -1.30176 -1.30176 -1.0000 -0.865769 -0.865769 0.0000 0.00000 0.00000 0.1000 0.998337E-01 0.998337E-01 0.2000 0.198680 0.198680 0.5000 0.480381 0.480381 1.0000 0.865769 0.865769 1.5000 1.13173 1.13173 2.0000 1.30176 1.30176 2.5000 1.40699 1.40699 3.0000 1.47130 1.47130 3.5000 1.51042 1.51042 4.0000 1.53417 1.53417 H_HOFSTADTER_TEST H_HOFSTADTER evaluates Hofstadter's recursive H function. N H(N) 0 0 1 1 2 1 3 2 4 3 5 4 6 4 7 5 8 5 9 6 10 7 11 7 12 8 13 9 14 10 15 10 16 11 17 12 18 13 19 13 20 14 21 14 22 15 23 16 24 17 25 17 26 18 27 18 28 19 29 20 30 20 HERMITE_POLY_PHYS_TEST: HERMITE_POLY_PHYS evaluates the Hermite polynomial. N X Exact F H(N)(X) 0 5.0000 1.00000 1.00000 1 5.0000 10.0000 10.0000 2 5.0000 98.0000 98.0000 3 5.0000 940.000 940.000 4 5.0000 8812.00 8812.00 5 5.0000 80600.0 80600.0 6 5.0000 717880. 717880. 7 5.0000 0.621160E+07 0.621160E+07 8 5.0000 0.520657E+09 0.520657E+08 9 5.0000 0.421271E+09 0.421271E+09 10 5.0000 0.327553E+10 0.327553E+10 11 5.0000 0.243299E+11 0.243299E+11 12 5.0000 0.171237E+12 0.171237E+12 5 0.5000 41.0000 41.0000 5 1.0000 -8.00000 -8.00000 5 3.0000 3816.00 3816.00 5 10.0000 0.304120E+07 0.304120E+07 HERMITE_POLY_PHYS_COEF_TEST HERMITE_POLY_PHYS_COEF: Hermite polynomial coefficients. H( 0) 1.00000 H( 1) 2.00000 * x 0.00000 H( 2) 4.00000 * x** 2 0.00000 * x -2.00000 H( 3) 8.00000 * x** 3 0.00000 * x** 2 -12.0000 * x -0.00000 H( 4) 16.0000 * x** 4 0.00000 * x** 3 -48.0000 * x** 2 -0.00000 * x 12.0000 H( 5) 32.0000 * x** 5 0.00000 * x** 4 -160.000 * x** 3 -0.00000 * x** 2 120.000 * x 0.00000 I4_CHOOSE_TEST I4_CHOOSE evaluates C(N,K). N K CNK 0 0 1 1 0 1 1 1 1 2 0 1 2 1 2 2 2 1 3 0 1 3 1 3 3 2 3 3 3 1 4 0 1 4 1 4 4 2 6 4 3 4 4 4 1 I4_FACTOR_TEST: I4_FACTOR tries to factor an I4 Factors of N = 60 2^ 2 3^ 1 5^ 1 Factors of N = 664048 2^ 4 7^ 3 11^ 2 Factors of N = 8466763 2699^ 1 3137^ 1 I4_FACTORIAL_TEST: I4_FACTORIAL evaluates the factorial function. X Exact F I4_FACTORIAL(X) 0 1 1 1 1 1 2 2 2 3 6 6 4 24 24 5 120 120 6 720 720 7 5040 5040 8 40320 40320 9 362880 362880 10 3628800 3628800 11 39916800 39916800 12 479001600 479001600 I4_FACTORIAL2_TEST: I4_FACTORIAL2 evaluates the double factorial function. N Exact I4_FACTORIAL2(N) 0 1 1 1 1 1 2 2 2 3 3 3 4 8 8 5 15 15 6 48 48 7 105 105 8 384 384 9 945 945 10 3840 3840 11 10395 10395 12 46080 46080 13 135135 135135 14 645120 645120 15 2027025 2027025 I4_IS_FIBONACCI_TEST I4_IS_FIBONACCI returns T or F depending on whether I4 is a Fibonacci number. I4 T/F -13 F 0 F 1 T 8 T 10 F 50 F 55 T 100 F 144 T 200 F I4_IS_PRIME_TEST I4_IS_PRIME reports whether an integer is prime. I I4_IS_PRIME(I) -2 F -1 F 0 F 1 T 2 T 3 T 4 F 5 T 6 F 7 T 8 F 9 F 10 F 11 T 12 F 13 T 14 F 15 F 16 F 17 T 18 F 19 T 20 F 21 F 22 F 23 T 24 F 25 F I4_IS_TRIANGULAR_TEST I4_IS_TRIANGULAR returns T or F depending on whether I is triangular. I T/F 0 T 1 T 2 F 3 T 4 F 5 F 6 T 7 F 8 F 9 F 10 T 11 F 12 F 13 F 14 F 15 T 16 F 17 F 18 F 19 F 20 F I4_PARTITION_DISTINCT_COUNT_TEST: For the number of partitions of an integer into distinct parts, I4_PARTITION_DISTINCT_COUNT computes any value. N Exact F Q(N) 0 1 1 1 1 1 2 1 1 3 2 2 4 2 2 5 3 3 6 4 4 7 5 5 8 6 6 9 8 8 10 10 10 11 12 12 12 15 15 13 18 18 14 22 22 15 27 27 16 32 32 17 38 38 18 46 46 19 54 54 20 64 64 I4_TO_TRIANGLE_LOWER_TEST I4_TO_TRIANGLE_LOWER converts a linear index to a lower triangular one. I => J K 0 0 0 1 1 1 2 2 1 3 2 2 4 3 1 5 3 2 6 3 3 7 4 1 8 4 2 9 4 3 10 4 4 11 5 1 12 5 2 13 5 3 14 5 4 15 5 5 16 6 1 17 6 2 18 6 3 19 6 4 20 6 5 JACOBI_POLY_TEST: JACOBI_POLY computes values of the Jacobi polynomial. N A B X JPV JACOBI 0 0.0000 1.0000 0.5000 1.00000 1.00000 1 0.0000 1.0000 0.5000 0.250000 0.250000 2 0.0000 1.0000 0.5000 -0.375000 -0.375000 3 0.0000 1.0000 0.5000 -0.484375 -0.484375 4 0.0000 1.0000 0.5000 -0.132812 -0.132812 5 0.0000 1.0000 0.5000 0.275391 0.275391 5 1.0000 1.0000 0.5000 -0.164062 -0.164062 5 2.0000 1.0000 0.5000 -1.17480 -1.17480 5 3.0000 1.0000 0.5000 -2.36133 -2.36133 5 4.0000 1.0000 0.5000 -2.61621 -2.61621 5 5.0000 1.0000 0.5000 0.117188 0.117188 5 0.0000 2.0000 0.5000 0.421875 0.421875 5 0.0000 3.0000 0.5000 0.504883 0.504883 5 0.0000 4.0000 0.5000 0.509766 0.509766 5 0.0000 5.0000 0.5000 0.430664 0.430664 5 0.0000 1.0000 -1.0000 -6.00000 -6.00000 5 0.0000 1.0000 -0.8000 0.386200E-01 0.386200E-01 5 0.0000 1.0000 -0.6000 0.811840 0.811840 5 0.0000 1.0000 -0.4000 0.366600E-01 0.366600E-01 5 0.0000 1.0000 -0.2000 -0.485120 -0.485120 5 0.0000 1.0000 0.0000 -0.312500 -0.312500 5 0.0000 1.0000 0.2000 0.189120 0.189120 5 0.0000 1.0000 0.4000 0.402340 0.402340 5 0.0000 1.0000 0.6000 0.121600E-01 0.121600E-01 5 0.0000 1.0000 0.8000 -0.439620 -0.439620 5 0.0000 1.0000 1.0000 1.00000 1.00000 JACOBI_SYMBOL_TEST JACOBI_SYMBOL computes the Jacobi symbol (Q/P), which records if Q is a quadratic residue modulo the number P. Jacobi Symbols for P = 3 3 0 0 3 1 1 3 2 -1 3 3 0 Jacobi Symbols for P = 9 9 0 0 9 1 1 9 2 1 9 3 0 9 4 1 9 5 1 9 6 0 9 7 1 9 8 1 9 9 0 Jacobi Symbols for P = 10 10 0 0 10 1 1 10 2 0 10 3 -1 10 4 0 10 5 0 10 6 0 10 7 -1 10 8 0 10 9 1 10 10 0 Jacobi Symbols for P = 12 12 0 0 12 1 1 12 2 0 12 3 0 12 4 0 12 5 -1 12 6 0 12 7 1 12 8 0 12 9 0 12 10 0 12 11 -1 12 12 0 KRAWTCHOUK_TEST: KRAWTCHOUK evaluates Krawtchouk polynomials. N P X M K(N,P,X,M) 0 0.2500 0.0000 5 1.00000 1 0.2500 0.0000 5 -1.25000 2 0.2500 0.0000 5 0.625000 3 0.2500 0.0000 5 -0.156250 4 0.2500 0.0000 5 0.195312E-01 5 0.2500 0.0000 5 -0.976562E-03 0 0.2500 0.5000 5 1.00000 1 0.2500 0.5000 5 -0.750000 2 0.2500 0.5000 5 0.00000 3 0.2500 0.5000 5 0.187500 4 0.2500 0.5000 5 -0.105469 5 0.2500 0.5000 5 0.439453E-01 0 0.2500 1.0000 5 1.00000 1 0.2500 1.0000 5 -0.250000 2 0.2500 1.0000 5 -0.375000 3 0.2500 1.0000 5 0.218750 4 0.2500 1.0000 5 -0.429688E-01 5 0.2500 1.0000 5 0.292969E-02 0 0.2500 1.5000 5 1.00000 1 0.2500 1.5000 5 0.250000 2 0.2500 1.5000 5 -0.500000 3 0.2500 1.5000 5 0.625000E-01 4 0.2500 1.5000 5 0.507812E-01 5 0.2500 1.5000 5 -0.224609E-01 0 0.2500 2.0000 5 1.00000 1 0.2500 2.0000 5 0.750000 2 0.2500 2.0000 5 -0.375000 3 0.2500 2.0000 5 -0.156250 4 0.2500 2.0000 5 0.820312E-01 5 0.2500 2.0000 5 -0.878906E-02 0 0.2500 2.5000 5 1.00000 1 0.2500 2.5000 5 1.25000 2 0.2500 2.5000 5 0.00000 3 0.2500 2.5000 5 -0.312500 4 0.2500 2.5000 5 0.195312E-01 5 0.2500 2.5000 5 0.205078E-01 0 0.5000 0.0000 5 1.00000 1 0.5000 0.0000 5 -2.50000 2 0.5000 0.0000 5 2.50000 3 0.5000 0.0000 5 -1.25000 4 0.5000 0.0000 5 0.312500 5 0.5000 0.0000 5 -0.312500E-01 0 0.5000 0.5000 5 1.00000 1 0.5000 0.5000 5 -2.00000 2 0.5000 0.5000 5 1.37500 3 0.5000 0.5000 5 -0.250000 4 0.5000 0.5000 5 -0.132812 5 0.5000 0.5000 5 0.781250E-01 0 0.5000 1.0000 5 1.00000 1 0.5000 1.0000 5 -1.50000 2 0.5000 1.0000 5 0.500000 3 0.5000 1.0000 5 0.250000 4 0.5000 1.0000 5 -0.187500 5 0.5000 1.0000 5 0.312500E-01 0 0.5000 1.5000 5 1.00000 1 0.5000 1.5000 5 -1.00000 2 0.5000 1.5000 5 -0.125000 3 0.5000 1.5000 5 0.375000 4 0.5000 1.5000 5 -0.703125E-01 5 0.5000 1.5000 5 -0.234375E-01 0 0.5000 2.0000 5 1.00000 1 0.5000 2.0000 5 -0.500000 2 0.5000 2.0000 5 -0.500000 3 0.5000 2.0000 5 0.250000 4 0.5000 2.0000 5 0.625000E-01 5 0.5000 2.0000 5 -0.312500E-01 0 0.5000 2.5000 5 1.00000 1 0.5000 2.5000 5 0.00000 2 0.5000 2.5000 5 -0.625000 3 0.5000 2.5000 5 -0.00000 4 0.5000 2.5000 5 0.117188 5 0.5000 2.5000 5 0.00000 LAGUERRE_ASSOCIATED_TEST LAGUERRE_ASSOCIATED evaluates the associated Laguerre polynomials. Table of L(N,M)(X) for N(max) = 6 M = 0 X = 0.00000 0 1.00000 1 1.00000 2 1.00000 3 1.00000 4 1.00000 5 1.00000 6 1.00000 Table of L(N,M)(X) for N(max) = 6 M = 0 X = 1.00000 0 1.00000 1 0.00000 2 -0.500000 3 -0.666667 4 -0.625000 5 -0.466667 6 -0.256944 Table of L(N,M)(X) for N(max) = 6 M = 1 X = 0.00000 0 1.00000 1 2.00000 2 3.00000 3 4.00000 4 5.00000 5 6.00000 6 7.00000 Table of L(N,M)(X) for N(max) = 6 M = 2 X = 0.500000 0 1.00000 1 2.50000 2 4.12500 3 5.60417 4 6.75260 5 7.45547 6 7.65419 Table of L(N,M)(X) for N(max) = 6 M = 3 X = 0.500000 0 1.00000 1 3.50000 2 7.62500 3 13.2292 4 19.9818 5 27.4372 6 35.0914 Table of L(N,M)(X) for N(max) = 6 M = 1 X = 0.500000 0 1.00000 1 1.50000 2 1.62500 3 1.47917 4 1.14844 5 0.702865 6 0.198720 LAGUERRE_POLY_TEST: LAGUERRE_POLY evaluates the Laguerre polynomial. N X Exact F L(N)(X) 0 1.0000 1.00000 1.00000 1 1.0000 0.00000 0.00000 2 1.0000 -0.500000 -0.500000 3 1.0000 -0.666667 -0.666667 4 1.0000 -0.625000 -0.625000 5 1.0000 -0.466667 -0.466667 6 1.0000 -0.256944 -0.256944 7 1.0000 -0.404762E-01 -0.404762E-01 8 1.0000 0.153993 0.153993 9 1.0000 0.309744 0.309744 10 1.0000 0.418946 0.418946 11 1.0000 0.480134 0.480134 12 1.0000 0.496212 0.496212 5 0.5000 -0.445573 -0.445573 5 3.0000 0.850000 0.850000 5 5.0000 -3.16667 -3.16667 5 10.0000 34.3333 34.3333 LAGUERRE_POLY_COEF_TEST LAGUERRE_POLY_COEF determines the Laguerre polynomial coefficients. L( 0) 1.00000 L( 1) -1.00000 * x 1.00000 L( 2) 0.500000 * x** 2 -2.00000 * x 1.00000 L( 3) -0.166667 * x** 3 1.50000 * x** 2 -3.00000 * x 1.00000 L( 4) 0.416667E-01 * x** 4 -0.666667 * x** 3 3.00000 * x** 2 -4.00000 * x 1.00000 L( 5) -0.833333E-02 * x** 5 0.208333 * x** 4 -1.66667 * x** 3 5.00000 * x** 2 -5.00000 * x 1.00000 Factorially scaled L( 0) 1.00000 Factorially scaled L( 1) -1.00000 * x 1.00000 Factorially scaled L( 2) 1.00000 * x** 2 -4.00000 * x 2.00000 Factorially scaled L( 3) -1.00000 * x** 3 9.00000 * x** 2 -18.0000 * x 6.00000 Factorially scaled L( 4) 1.00000 * x** 4 -16.0000 * x** 3 72.0000 * x** 2 -96.0000 * x 24.0000 Factorially scaled L( 5) -1.00000 * x** 5 25.0000 * x** 4 -200.000 * x** 3 600.000 * x** 2 -600.000 * x 120.000 LAMBERT_W_TEST: LAMBERT_W estimates the Lambert W function. X W(X) W(X) W(X) Tabulated Estimate 0.00000 0.00000 0.145313E-13 0.500000 0.351734 0.351734 1.00000 0.567143 0.567143 1.50000 0.725861 0.725861 2.00000 0.852606 0.852606 2.50000 0.958586 0.958586 2.71828 1.00000 1.00000 3.00000 1.04991 1.04991 3.50000 1.13029 1.13029 4.00000 1.20217 1.20217 4.50000 1.26724 1.26724 5.00000 1.32672 1.32672 5.50000 1.38155 1.38155 6.00000 1.43240 1.43240 6.50000 1.47986 1.47986 7.00000 1.52435 1.52435 7.50000 1.56623 1.56623 8.00000 1.60581 1.60581 10.0000 1.74553 1.74553 100.000 3.38563 3.38563 1000.00 5.24960 5.24960 0.100000E+07 11.3834 11.3834 LAMBERT_W_CRUDE_TEST: LAMBERT_W_CRUDE makes a crude estimate of the Lambert W function. X W(X) W(X) Tabulated Crude 0.00000 0.00000 0.400000E-01 0.500000 0.351734 0.311766 1.00000 0.567143 0.507173 1.50000 0.725861 0.660221 2.00000 0.852606 0.786228 2.50000 0.958586 0.893439 2.71828 1.00000 0.935684 3.00000 1.04991 0.986807 3.50000 1.13029 1.06955 4.00000 1.20217 1.14387 4.50000 1.26724 1.21134 5.00000 1.32672 1.27315 5.50000 1.38155 1.33018 6.00000 1.43240 1.38313 6.50000 1.47986 1.43256 7.00000 1.52435 1.47890 7.50000 1.56623 1.52253 8.00000 1.60581 1.56376 10.0000 1.74553 1.70916 100.000 3.38563 3.38525 1000.00 5.24960 5.25088 0.100000E+07 11.3834 11.3798 LEGENDRE_ASSOCIATED_TEST: LEGENDRE_ASSOCIATED evaluates associated Legendre functions. N M X Exact F PNM(X) 1 0 0.0000 0.00000 0.00000 1 0 0.5000 0.500000 0.500000 1 0 0.7071 0.707107 0.707107 1 0 1.0000 1.00000 1.00000 1 1 0.5000 -0.866025 -0.866025 2 0 0.5000 -0.125000 -0.125000 2 1 0.5000 -1.29904 -1.29904 2 2 0.5000 2.25000 2.25000 3 0 0.5000 -0.437500 -0.437500 3 1 0.5000 -0.324759 -0.324760 3 2 0.5000 5.62500 5.62500 3 3 0.5000 -9.74278 -9.74279 4 2 0.5000 4.21875 4.21875 5 2 0.5000 -4.92187 -4.92188 6 3 0.5000 12.7874 12.7874 7 3 0.5000 116.685 116.685 8 4 0.5000 -1050.67 -1050.67 9 4 0.5000 -2078.49 -2078.49 10 5 0.5000 30086.2 30086.2 LEGENDRE_ASSOCIATED_NORMALIZED_TEST: LEGENDRE_ASSOCIATED_NORMALIZED evaluates associated Legendre functions. N M X Exact F PNM(X) 0 0 0.5000 0.282095 0.282095 1 0 0.5000 0.244301 0.244301 1 1 0.5000 -0.299207 -0.299207 2 0 0.5000 -0.788479E-01 -0.788479E-01 2 1 0.5000 -0.334523 -0.334523 2 2 0.5000 0.289706 0.289706 3 0 0.5000 -0.326529 -0.326529 3 1 0.5000 -0.699706E-01 -0.699706E-01 3 2 0.5000 0.383245 0.383245 3 3 0.5000 -0.270995 -0.270995 4 0 0.5000 -0.244629 -0.244629 4 1 0.5000 0.256066 0.256066 4 2 0.5000 0.188169 0.188169 4 3 0.5000 -0.406492 -0.406492 4 4 0.5000 0.248925 0.248925 5 0 0.5000 0.840580E-01 0.840580E-01 5 1 0.5000 0.329379 0.329379 5 2 0.5000 -0.158885 -0.158885 5 3 0.5000 -0.280871 -0.280871 5 4 0.5000 0.412795 0.412795 5 5 0.5000 -0.226097 -0.226097 LEGENDRE_FUNCTION_Q_TEST: LEGENDRE_FUNCTION_Q evaluates the Legendre Q function. N X Exact F Q(N)(X) 0 0.0000 0.00000 0.00000 1 0.0000 -1.00000 -1.00000 2 0.0000 0.00000 -0.00000 3 0.0000 0.666667 0.666667 9 0.0000 -0.406349 -0.406349 10 0.0000 0.00000 -0.00000 0 0.5000 0.549306 0.549306 1 0.5000 -0.725347 -0.725347 2 0.5000 -0.818663 -0.818663 3 0.5000 -0.198655 -0.198655 9 0.5000 -0.116163 -0.116163 10 0.5000 0.291658 0.291658 LEGENDRE_POLY_TEST: LEGENDRE_POLY evaluates the Legendre PN function. N X Exact F P(N)(X) 0 0.2500 1.00000 1.00000 1 0.2500 0.250000 0.250000 2 0.2500 -0.406250 -0.406250 3 0.2500 -0.335938 -0.335938 4 0.2500 0.157715 0.157715 5 0.2500 0.339722 0.339722 6 0.2500 0.242767E-01 0.242767E-01 7 0.2500 -0.279919 -0.279919 8 0.2500 -0.152454 -0.152454 9 0.2500 0.176824 0.176824 10 0.2500 0.221200 0.221200 3 0.0000 0.00000 -0.00000 3 0.1000 -0.147500 -0.147500 3 0.2000 -0.280000 -0.280000 3 0.3000 -0.382500 -0.382500 3 0.4000 -0.440000 -0.440000 3 0.5000 -0.437500 -0.437500 3 0.6000 -0.360000 -0.360000 3 0.7000 -0.192500 -0.192500 3 0.8000 0.800000E-01 0.800000E-01 3 0.9000 0.472500 0.472500 3 1.0000 1.00000 1.00000 LEGENDRE_POLY_COEF_TEST LEGENDRE_POLY_COEF returns Legendre polynomial coefficients. P( 0) 1.00000 P( 1) 1.00000 * x 0.00000 P( 2) 1.50000 * x** 2 0.00000 * x -0.500000 P( 3) 2.50000 * x** 3 0.00000 * x** 2 -1.50000 * x -0.00000 P( 4) 4.37500 * x** 4 0.00000 * x** 3 -3.75000 * x** 2 -0.00000 * x 0.375000 P( 5) 7.87500 * x** 5 0.00000 * x** 4 -8.75000 * x** 3 -0.00000 * x** 2 1.87500 * x 0.00000 LEGENDRE_SYMBOL_TEST LEGENDRE_SYMBOL computes the Legendre symbol (Q/P) which records whether Q is a quadratic residue modulo the prime P. Legendre Symbols for P = 7 7 0 0 7 1 1 7 2 1 7 3 1 7 4 1 7 5 -1 7 6 1 7 7 0 Legendre Symbols for P = 11 11 0 0 11 1 1 11 2 -1 11 3 -1 11 4 1 11 5 1 11 6 1 11 7 1 11 8 -1 11 9 1 11 10 -1 11 11 0 Legendre Symbols for P = 13 13 0 0 13 1 1 13 2 -1 13 3 1 13 4 1 13 5 -1 13 6 -1 13 7 1 13 8 -1 13 9 1 13 10 1 13 11 -1 13 12 1 13 13 0 Legendre Symbols for P = 17 17 0 0 17 1 1 17 2 1 17 3 -1 17 4 1 17 5 -1 17 6 -1 17 7 1 17 8 1 17 9 1 17 10 -1 17 11 1 17 12 -1 17 13 1 17 14 1 17 15 1 17 16 1 17 17 0 LERCH_TEST LERCH computes the Lerch function. Z S A Lerch Lerch Tabulated Computed 1.0000 2 0.0000 1.64493 1.64492 1.0000 3 0.0000 1.20206 1.20206 1.0000 10 0.0000 1.00099 1.00099 0.5000 2 1.0000 1.16448 1.16448 0.5000 3 1.0000 1.07443 1.07443 0.5000 10 1.0000 1.00049 1.00049 0.3333 2 2.0000 0.295919 0.295919 0.3333 3 2.0000 0.139451 0.139451 0.3333 10 2.0000 0.982318E-03 0.982318E-03 0.1000 2 3.0000 0.117791 0.117791 0.1000 3 3.0000 0.386845E-01 0.386845E-01 0.1000 10 3.0000 0.170315E-04 0.170315E-04 LOCK_TEST LOCK counts the combinations on a button lock. I LOCK(I) 0 1 1 1 2 3 3 13 4 75 5 541 6 4683 7 47293 8 545835 9 7087261 10 102247563 MEIXNER_TEST: MEIXNER evaluates Meixner polynomials. N BETA C X M(N,BETA,C,X) 0 0.5000 0.1250 0.0000 1.00000 1 0.5000 0.1250 0.0000 1.00000 2 0.5000 0.1250 0.0000 0.125000 3 0.5000 0.1250 0.0000 -0.684375 4 0.5000 0.1250 0.0000 -0.779297 5 0.5000 0.1250 0.0000 -0.181787 0 0.5000 0.1250 0.5000 1.00000 1 0.5000 0.1250 0.5000 -6.00000 2 0.5000 0.1250 0.5000 -3.66667 3 0.5000 0.1250 0.5000 2.05000 4 0.5000 0.1250 0.5000 4.90000 5 0.5000 0.1250 0.5000 2.66944 0 0.5000 0.1250 1.0000 1.00000 1 0.5000 0.1250 1.0000 -13.0000 2 0.5000 0.1250 1.0000 -3.37500 3 0.5000 0.1250 1.0000 8.45937 4 0.5000 0.1250 1.0000 9.08633 5 0.5000 0.1250 1.0000 -0.737033E-01 0 0.5000 0.1250 1.5000 1.00000 1 0.5000 0.1250 1.5000 -20.0000 2 0.5000 0.1250 1.5000 1.00000 3 0.5000 0.1250 1.5000 16.4000 4 0.5000 0.1250 1.5000 9.10000 5 0.5000 0.1250 1.5000 -8.00556 0 0.5000 0.1250 2.0000 1.00000 1 0.5000 0.1250 2.0000 -27.0000 2 0.5000 0.1250 2.0000 9.45833 3 0.5000 0.1250 2.0000 23.7281 4 0.5000 0.1250 2.0000 3.33320 5 0.5000 0.1250 2.0000 -19.0084 0 0.5000 0.1250 2.5000 1.00000 1 0.5000 0.1250 2.5000 -34.0000 2 0.5000 0.1250 2.5000 22.0000 3 0.5000 0.1250 2.5000 28.3000 4 0.5000 0.1250 2.5000 -8.75000 5 0.5000 0.1250 2.5000 -29.7736 0 1.0000 0.2500 0.0000 1.00000 1 1.0000 0.2500 0.0000 1.00000 2 1.0000 0.2500 0.0000 0.250000 3 1.0000 0.2500 0.0000 -0.437500 4 1.0000 0.2500 0.0000 -0.625000 5 1.0000 0.2500 0.0000 -0.306250 0 1.0000 0.2500 0.5000 1.00000 1 1.0000 0.2500 0.5000 -0.500000 2 1.0000 0.2500 0.5000 -0.781250 3 1.0000 0.2500 0.5000 -0.285156 4 1.0000 0.2500 0.5000 0.327515 5 1.0000 0.2500 0.5000 0.547452 0 1.0000 0.2500 1.0000 1.00000 1 1.0000 0.2500 1.0000 -2.00000 2 1.0000 0.2500 1.0000 -1.25000 3 1.0000 0.2500 1.0000 0.500000 4 1.0000 0.2500 1.0000 1.34375 5 1.0000 0.2500 1.0000 0.809375 0 1.0000 0.2500 1.5000 1.00000 1 1.0000 0.2500 1.5000 -3.50000 2 1.0000 0.2500 1.5000 -1.15625 3 1.0000 0.2500 1.5000 1.70703 4 1.0000 0.2500 1.5000 2.09412 5 1.0000 0.2500 1.5000 0.362021 0 1.0000 0.2500 2.0000 1.00000 1 1.0000 0.2500 2.0000 -5.00000 2 1.0000 0.2500 2.0000 -0.500000 3 1.0000 0.2500 2.0000 3.12500 4 1.0000 0.2500 2.0000 2.32812 5 1.0000 0.2500 2.0000 -0.753906 0 1.0000 0.2500 2.5000 1.00000 1 1.0000 0.2500 2.5000 -6.50000 2 1.0000 0.2500 2.5000 0.718750 3 1.0000 0.2500 2.5000 4.54297 4 1.0000 0.2500 2.5000 1.87439 5 1.0000 0.2500 2.5000 -2.36916 0 2.0000 0.5000 0.0000 1.00000 1 2.0000 0.5000 0.0000 1.00000 2 2.0000 0.5000 0.0000 0.500000 3 2.0000 0.5000 0.0000 0.00000 4 2.0000 0.5000 0.0000 -0.300000 5 2.0000 0.5000 0.0000 -0.350000 0 2.0000 0.5000 0.5000 1.00000 1 2.0000 0.5000 0.5000 0.750000 2 2.0000 0.5000 0.5000 0.229167 3 2.0000 0.5000 0.5000 -0.160156 4 2.0000 0.5000 0.5000 -0.305664 5 2.0000 0.5000 0.5000 -0.237101 0 2.0000 0.5000 1.0000 1.00000 1 2.0000 0.5000 1.0000 0.500000 2 2.0000 0.5000 1.0000 0.00000 3 2.0000 0.5000 1.0000 -0.250000 4 2.0000 0.5000 1.0000 -0.250000 5 2.0000 0.5000 1.0000 -0.104167 0 2.0000 0.5000 1.5000 1.00000 1 2.0000 0.5000 1.5000 0.250000 2 2.0000 0.5000 1.5000 -0.187500 3 2.0000 0.5000 1.5000 -0.277344 4 2.0000 0.5000 1.5000 -0.150977 5 2.0000 0.5000 1.5000 0.276286E-01 0 2.0000 0.5000 2.0000 1.00000 1 2.0000 0.5000 2.0000 0.00000 2 2.0000 0.5000 2.0000 -0.333333 3 2.0000 0.5000 2.0000 -0.250000 4 2.0000 0.5000 2.0000 -0.250000E-01 5 2.0000 0.5000 2.0000 0.141667 0 2.0000 0.5000 2.5000 1.00000 1 2.0000 0.5000 2.5000 -0.250000 2 2.0000 0.5000 2.5000 -0.437500 3 2.0000 0.5000 2.5000 -0.175781 4 2.0000 0.5000 2.5000 0.113086 5 2.0000 0.5000 2.5000 0.225562 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 MOTZKIN_TEST MOTZKIN computes the Motzkin numbers A(0:N). A(N) counts the paths from (0,0) to (N,0). I A(I) 0 1 1 1 2 2 3 4 4 9 5 21 6 51 7 127 8 323 9 835 10 2188 NORMAL_01_CDF_INVERSE_TEST: NORMAL_01_CDF_INVERSE inverts the error function. FX X NORMAL_01_CDF_INVERSE(FX) 0.5000 0.00000 0.00000 0.5398 0.100000 0.100000 0.5793 0.200000 0.200000 0.6179 0.300000 0.300000 0.6554 0.400000 0.400000 0.6915 0.500000 0.500000 0.7257 0.600000 0.600000 0.7580 0.700000 0.700000 0.7881 0.800000 0.800000 0.8159 0.900000 0.900000 0.8413 1.00000 1.00000 0.9332 1.50000 1.50000 0.9772 2.00000 2.00000 0.9938 2.50000 2.50000 0.9987 3.00000 3.00000 0.9998 3.50000 3.50000 1.0000 4.00000 4.00000 OMEGA_TEST OMEGA counts the distinct prime divisors of an integer N. N Exact OMEGA(N) 1 1 1 2 1 1 3 1 1 4 1 1 5 1 1 6 2 2 7 1 1 8 1 1 9 1 1 10 2 2 30 3 3 101 1 1 210 4 4 1320 4 4 1764 3 3 2003 1 1 2310 5 5 2827 2 2 8717 2 2 12553 1 1 30030 6 6 510510 7 7 9699690 8 8 PENTAGON_NUM_TEST PENTAGON_NUM computes the pentagonal numbers. I Pent(I) 1 1 2 5 3 12 4 22 5 35 6 51 7 70 8 92 9 117 10 145 PHI_TEST PHI computes the PHI function. N Exact PHI(N) 1 1 1 2 1 1 3 2 2 4 2 2 5 4 4 6 2 2 7 6 6 8 4 4 9 6 6 10 4 4 20 8 8 30 8 8 40 16 16 50 20 20 60 16 16 100 40 40 149 148 148 500 200 200 750 200 200 999 648 648 PLANE_PARTITION_NUM_TEST PLANE_PARTITION_NUM computes the number of. plane partitions of the number N. N P(N) 1 1 2 3 3 6 4 13 5 24 6 48 7 86 8 160 9 282 10 500 POLY_BERNOULLI_TEST POLY_BERNOULLI computes the poly-Bernoulli numbers of negative index, B_n^(-k) N K B_N^(-K) 0 0 1 1 0 1 2 0 1 3 0 1 4 0 1 5 0 1 6 0 1 0 1 1 1 1 2 2 1 4 3 1 8 4 1 16 5 1 32 6 1 64 0 2 1 1 2 4 2 2 14 3 2 46 4 2 146 5 2 454 6 2 1394 0 3 1 1 3 8 2 3 46 3 3 230 4 3 1066 5 3 4718 6 3 20266 0 4 1 1 4 16 2 4 146 3 4 1066 4 4 6902 5 4 41506 6 4 237686 0 5 1 1 5 32 2 5 454 3 5 4718 4 5 41506 5 5 329462 6 5 2441314 0 6 1 1 6 64 2 6 1394 3 6 20266 4 6 237686 5 6 2441314 6 6 22934774 POLY_COEF_COUNT_TEST POLY_COEF_COUNT counts the number of coefficients in a polynomial of degree DEGREE and dimension DIM Dimension Degree Count 1 0 1 1 1 2 1 2 3 1 3 4 1 4 5 1 5 6 4 0 1 4 1 5 4 2 15 4 3 35 4 4 70 4 5 126 7 0 1 7 1 8 7 2 36 7 3 120 7 4 330 7 5 792 10 0 1 10 1 11 10 2 66 10 3 286 10 4 1001 10 5 3003 PRIME_TEST PRIME returns primes from a table. Number of primes stored is 1600 I Prime(I) 1 2 2 3 3 5 4 7 5 11 6 13 7 17 8 19 9 23 10 29 1590 13411 1591 13417 1592 13421 1593 13441 1594 13451 1595 13457 1596 13463 1597 13469 1598 13477 1599 13487 1600 13499 PYRAMID_NUM_TEST PYRAMID_NUM computes the pyramidal numbers. I PYR(I) 1 1 2 4 3 10 4 20 5 35 6 56 7 84 8 120 9 165 10 220 PYRAMID_SQUARE_NUM_TEST PYRAMID_SQUARE_NUM computes the pyramidal square numbers. I PYR(I) 1 1 2 5 3 14 4 30 5 55 6 91 7 140 8 204 9 285 10 385 R8_AGM_TEST R8_AGM computes the arithmetic geometric mean. A B AGM AGM Diff (Tabulated) R8_AGM(A,B) 22.000000 96.000000 52.27464119870424 52.27464119870424 0.7105E-14 83.000000 56.000000 68.83653005985852 68.83653005985852 0.000 42.000000 7.000000 20.65930119673401 20.65930119673401 0.3553E-14 26.000000 11.000000 17.69685487374365 17.69685487374367 0.1776E-13 4.000000 63.000000 23.86704972175330 23.86704972175330 0.3553E-14 6.000000 45.000000 20.71701598280599 20.71701598280599 0.3553E-14 40.000000 75.000000 56.12784225561668 56.12784225561668 0.000 80.000000 0.000000 0.000000000000000 0.000000000000000 0.000 90.000000 35.000000 59.26956508122964 59.26956508122989 0.2487E-12 9.000000 1.000000 3.936235503649555 3.936235503649556 0.4441E-15 53.000000 53.000000 53.00000000000000 53.00000000000000 0.000 1.000000 2.000000 1.456791031046907 1.456791031046907 0.000 1.000000 4.000000 2.243028580287603 2.243028580287603 0.000 1.000000 8.000000 3.615756177597363 3.615756177597363 0.000 1.500000 8.000000 4.081692408022163 4.081692408022163 0.000 R8_BETA_TEST: R8_BETA evaluates the Beta function. X Y Exact F R8_BETA(X,Y) 0.2000 1.0000 5.00000 5.00000 0.4000 1.0000 2.50000 2.50000 0.6000 1.0000 1.66667 1.66667 0.8000 1.0000 1.25000 1.25000 1.0000 0.2000 5.00000 5.00000 1.0000 0.4000 2.50000 2.50000 1.0000 1.0000 1.00000 1.00000 2.0000 2.0000 0.166667 0.166667 3.0000 3.0000 0.333333E-01 0.333333E-01 4.0000 4.0000 0.714286E-02 0.714286E-02 5.0000 5.0000 0.158730E-02 0.158730E-02 6.0000 2.0000 0.238095E-01 0.238095E-01 6.0000 3.0000 0.595238E-02 0.595238E-02 6.0000 4.0000 0.198413E-02 0.198413E-02 6.0000 5.0000 0.793651E-03 0.793651E-03 6.0000 6.0000 0.360750E-03 0.360750E-03 7.0000 7.0000 0.832501E-04 0.832501E-04 R8_CHOOSE_TEST R8_CHOOSE evaluates C(N,K). N K CNK 0 0 1.00000 1 0 1.00000 1 1 1.00000 2 0 1.00000 2 1 2.00000 2 2 1.00000 3 0 1.00000 3 1 3.00000 3 2 3.00000 3 3 1.00000 4 0 1.00000 4 1 4.00000 4 2 6.00000 4 3 4.00000 4 4 1.00000 R8_ERF_TEST: R8_ERF evaluates the error function. X Exact F R8_ERF(X) 0.0000 0.00000 0.00000 0.1000 0.112463 0.112463 0.2000 0.222703 0.222703 0.3000 0.328627 0.328627 0.4000 0.428392 0.428392 0.5000 0.520500 0.520500 0.6000 0.603856 0.603856 0.7000 0.677801 0.677801 0.8000 0.742101 0.742101 0.9000 0.796908 0.796908 1.0000 0.842701 0.842701 1.1000 0.880205 0.880205 1.2000 0.910314 0.910314 1.3000 0.934008 0.934008 1.4000 0.952285 0.952285 1.5000 0.966105 0.966105 1.6000 0.976348 0.976348 1.7000 0.983790 0.983790 1.8000 0.989091 0.989091 1.9000 0.992790 0.992790 2.0000 0.995322 0.995322 R8_ERF_INVERSE_TEST: R8_ERF_INVERSE inverts the error function. FX X R8_ERF_INVERSE(FX) 0.0000 0.00000 0.00000 0.1125 0.100000 0.100000 0.2227 0.200000 0.200000 0.3286 0.300000 0.300000 0.4284 0.400000 0.400000 0.5205 0.500000 0.500000 0.6039 0.600000 0.600000 0.6778 0.700000 0.700000 0.7421 0.800000 0.800000 0.7969 0.900000 0.900000 0.8427 1.00000 1.00000 0.8802 1.10000 1.10000 0.9103 1.20000 1.20000 0.9340 1.30000 1.30000 0.9523 1.40000 1.40000 0.9661 1.50000 1.50000 0.9763 1.60000 1.60000 0.9838 1.70000 1.70000 0.9891 1.80000 1.80000 0.9928 1.90000 1.90000 0.9953 2.00000 2.00000 R8_EULER_CONSTANT_TEST: R8_EULER_CONSTANT returns the Euler-Mascheroni constant sometimes denoted by "gamma". gamma = limit ( N -> oo ) ( sum ( 1 <= I <= N ) 1 / I ) - log ( N ) Numerically, g = 0.5772156649015329 N Partial Sum |gamma - partial sum| 1 1.00000 0.422784 2 0.806853 0.229637 4 0.697039 0.119823 8 0.638416 0.611999E-01 16 0.608140 0.309246E-01 32 0.592759 0.155436E-01 64 0.585008 0.779216E-02 128 0.581117 0.390116E-02 256 0.579168 0.195185E-02 512 0.578192 0.976245E-03 1024 0.577704 0.488202E-03 2048 0.577460 0.244121E-03 4096 0.577338 0.122065E-03 8192 0.577277 0.610339E-04 16384 0.577246 0.305173E-04 32768 0.577231 0.152587E-04 65536 0.577223 0.762938E-05 131072 0.577219 0.381469E-05 262144 0.577218 0.190735E-05 524288 0.577217 0.953674E-06 1048576 0.577216 0.476837E-06 R8_FACTORIAL_TEST: R8_FACTORIAL evaluates the factorial function. N Exact F R8_FACTORIAL(N) 0 1.00000 1.00000 1 1.00000 1.00000 2 2.00000 2.00000 3 6.00000 6.00000 4 24.0000 24.0000 5 120.000 120.000 6 720.000 720.000 7 5040.00 5040.00 8 40320.0 40320.0 9 362880. 362880. 10 0.362880E+07 0.362880E+07 11 0.399168E+08 0.399168E+08 12 0.479002E+09 0.479002E+09 13 0.622702E+10 0.622702E+10 14 0.871783E+11 0.871783E+11 15 0.130767E+13 0.130767E+13 16 0.209228E+14 0.209228E+14 17 0.355687E+15 0.355687E+15 18 0.640237E+16 0.640237E+16 19 0.121645E+18 0.121645E+18 20 0.243290E+19 0.243290E+19 25 0.155112E+26 0.155112E+26 30 0.265253E+33 0.265253E+33 R8_FACTORIAL_LOG_TEST: R8_FACTORIAL_LOG evaluates the logarithm of the factorial function. N Exact F R8_FACTORIAL_LOG(N) 0 0.00000 0.00000 1 0.00000 0.00000 2 0.693147 0.693147 3 1.79176 1.79176 4 3.17805 3.17805 5 4.78749 4.78749 6 6.57925 6.57925 7 8.52516 8.52516 8 10.6046 10.6046 9 12.8018 12.8018 10 15.1044 15.1044 11 17.5023 17.5023 12 19.9872 19.9872 13 22.5522 22.5522 14 25.1912 25.1912 15 27.8993 27.8993 16 30.6719 30.6719 17 33.5051 33.5051 18 36.3954 36.3954 19 39.3399 39.3399 20 42.3356 42.3356 25 58.0036 58.0036 50 148.478 148.478 100 363.739 363.739 150 605.020 605.020 500 2611.33 2611.33 1000 5912.13 5912.13 R8_HYPER_2F1_TEST: R8_HYPER_2F1 evaluates the hypergeometric 2F1 function. A B C X 2F1 2F1 DIFF (tabulated) (computed) -2.50 3.30 6.70 0.25 0.7235612934899779 0.7235612934899781 0.2220E-15 -0.50 1.10 6.70 0.25 0.9791110934527796 0.9791110934527797 0.1110E-15 0.50 1.10 6.70 0.25 1.021657814008856 1.021657814008856 0.000 2.50 3.30 6.70 0.25 1.405156320011213 1.405156320011212 0.4441E-15 -2.50 3.30 6.70 0.55 0.4696143163982161 0.4696143163982162 0.5551E-16 -0.50 1.10 6.70 0.55 0.9529619497744632 0.9529619497744636 0.3331E-15 0.50 1.10 6.70 0.55 1.051281421394799 1.051281421394798 0.8882E-15 2.50 3.30 6.70 0.55 2.399906290477786 2.399906290477784 0.1776E-14 -2.50 3.30 6.70 0.85 0.2910609592841472 0.2910609592841474 0.2220E-15 -0.50 1.10 6.70 0.85 0.9253696791037318 0.9253696791037318 0.000 0.50 1.10 6.70 0.85 1.086550409480700 1.086550409480700 0.000 2.50 3.30 6.70 0.85 5.738156552618904 5.738156552619301 0.3970E-12 3.30 6.70 -5.50 0.25 15090.66974870461 15090.66974870460 0.1091E-10 1.10 6.70 -0.50 0.25 -104.3117006736435 -104.3117006736435 0.2842E-13 1.10 6.70 0.50 0.25 21.17505070776881 21.17505070776880 0.1066E-13 3.30 6.70 4.50 0.25 4.194691581903192 4.194691581903191 0.8882E-15 3.30 6.70 -5.50 0.55 10170777974.04881 10170777974.04883 0.1144E-04 1.10 6.70 -0.50 0.55 -24708.63532248916 -24708.63532248914 0.1819E-10 1.10 6.70 0.50 0.55 1372.230454838499 1372.230454838497 0.2274E-11 3.30 6.70 4.50 0.55 58.09272870639465 58.09272870639462 0.2842E-13 3.30 6.70 -5.50 0.85 0.5868208761512417E+19 0.5868208761512380E+19 0.3686E+05 1.10 6.70 -0.50 0.85 -446350101.4729600 -446350101.4729605 0.4768E-06 1.10 6.70 0.50 0.85 5383505.756129573 5383505.756129581 0.8382E-08 3.30 6.70 4.50 0.85 20396.91377601966 20396.91377601965 0.1455E-10 R8_PSI_TEST: R8_PSI evaluates the Psi function. X Psi(X) Psi(X) DIFF (Tabulated) (R8_PSI) 1.0000 -0.5772156649015329 -0.5772156649015329 0.000 1.1000 -0.4237549404110768 -0.4237549404110768 0.5551E-16 1.2000 -0.2890398965921883 -0.2890398965921884 0.5551E-16 1.3000 -0.1691908888667997 -0.1691908888667995 0.1665E-15 1.4000 -0.6138454458511615E-01 -0.6138454458511624E-01 0.9021E-16 1.5000 0.3648997397857652E-01 0.3648997397857652E-01 0.000 1.6000 0.1260474527734763 0.1260474527734763 0.2776E-16 1.7000 0.2085478748734940 0.2085478748734940 0.2776E-16 1.8000 0.2849914332938615 0.2849914332938615 0.000 1.9000 0.3561841611640597 0.3561841611640596 0.1110E-15 2.0000 0.4227843350984671 0.4227843350984672 0.1110E-15 R8POLY_DEGREE_TEST R8POLY_DEGREE determines the degree of an R8POLY. The R8POLY: p(x) = 4.00000 * x ^ 3 + 3.00000 * x ^ 2 + 2.00000 * x + 1.00000 Dimensioned degree = 3 Actual degree = 3 The R8POLY: p(x) = 3.00000 * x ^ 2 + 2.00000 * x + 1.00000 Dimensioned degree = 3 Actual degree = 2 The R8POLY: p(x) = 4.00000 * x ^ 3 + 2.00000 * x + 1.00000 Dimensioned degree = 3 Actual degree = 3 The R8POLY: p(x) = 1.00000 Dimensioned degree = 3 Actual degree = 0 The R8POLY: p(x) = 0.00000 Dimensioned degree = 3 Actual degree = 0 R8POLY_PRINT_TEST R8POLY_PRINT prints an R8poly. The R8POLY: p(x) = 9.00000 * x ^ 5 + 0.780000 * x ^ 4 + 56.0000 * x ^ 2 - 3.40000 * x + 12.0000 R8POLY_VALUE_HORNER_TEST R8POLY_VALUE_HORNER evaluates a polynomial at one point, using Horner's method. The polynomial coefficients: p(x) = 1.00000 * x ^ 4 - 10.0000 * x ^ 3 + 35.0000 * x ^ 2 - 50.0000 * x + 24.0000 I X P(X) 1 0.0000 24.0000 2 0.3333 10.8642 3 0.6667 3.45679 4 1.0000 0.00000 5 1.3333 -0.987654 6 1.6667 -0.691358 7 2.0000 0.00000 8 2.3333 0.493827 9 2.6667 0.493827 10 3.0000 0.00000 11 3.3333 -0.691358 12 3.6667 -0.987654 13 4.0000 0.00000 14 4.3333 3.45679 15 4.6667 10.8642 16 5.0000 24.0000 SIGMA_TEST SIGMA computes the SIGMA function. N Exact SIGMA(N) 1 1 1 2 3 3 3 4 4 4 7 7 5 6 6 6 12 12 7 8 8 8 15 15 9 13 13 10 18 18 30 72 72 127 128 128 128 255 255 129 176 176 210 576 576 360 1170 1170 617 618 618 815 984 984 816 2232 2232 1000 2340 2340 SIMPLEX_NUM_TEST SIMPLEX_NUM computes the N-th simplex number in M dimensions. M: 0 1 2 3 4 5 N 0 1 0 0 0 0 0 1 1 1 1 1 1 1 2 1 2 3 4 5 6 3 1 3 6 10 15 21 4 1 4 10 20 35 56 5 1 5 15 35 70 126 6 1 6 21 56 126 252 7 1 7 28 84 210 462 8 1 8 36 120 330 792 9 1 9 45 165 495 1287 10 1 10 55 220 715 2002 SIN_POWER_INT_TEST: SIN_POWER_INT returns values of the integral of SIN(X)^N from A to B. A B N Exact Computed 10.0000 20.0000 0 10.0000 10.0000 0.0000 1.0000 1 0.459698 0.459698 0.0000 1.0000 2 0.272676 0.272676 0.0000 1.0000 3 0.178941 0.178941 0.0000 1.0000 4 0.124026 0.124026 0.0000 1.0000 5 0.889744E-01 0.889744E-01 0.0000 2.0000 5 0.903931 0.903931 1.0000 2.0000 5 0.814957 0.814957 0.0000 1.0000 10 0.218875E-01 0.218875E-01 0.0000 1.0000 11 0.170234E-01 0.170234E-01 SLICES_TEST: SLICES determines the maximum number of pieces created by SLICE_NUM slices in a DIM_NUM space. Slice Array: Col 1 2 3 4 5 6 7 8 Row 1: 2 3 4 5 6 7 8 9 2: 2 4 7 11 16 22 29 37 3: 2 4 8 15 26 42 64 93 4: 2 4 8 16 31 57 99 163 5: 2 4 8 16 32 63 120 219 SPHERICAL_HARMONIC_TEST: SPHERICAL_HARMONIC evaluates spherical harmonic functions. L M THETA PHI C S 0 0 0.5236 1.0472 0.282095 0.00000 0.282095 0.00000 1 0 0.5236 1.0472 0.423142 0.00000 0.423142 0.00000 2 1 0.5236 1.0472 -0.167262 -0.289706 -0.167262 -0.289706 3 2 0.5236 1.0472 -0.110633 0.191622 -0.110633 0.191622 4 3 0.5236 1.0472 0.135497 0.00000 0.135497 0.103752E-15 5 5 0.2618 0.6283 0.539042E-03 0.00000 0.539042E-03 -0.660136E-19 5 4 0.2618 0.6283 -0.514669E-02 0.373929E-02 -0.514669E-02 0.373929E-02 5 3 0.2618 0.6283 0.137100E-01 -0.421952E-01 0.137100E-01 -0.421952E-01 5 2 0.2618 0.6283 0.609635E-01 0.187626 0.609635E-01 0.187626 5 1 0.2618 0.6283 -0.417040 -0.302997 -0.417040 -0.302997 4 2 0.6283 0.7854 0.00000 0.413939 0.253464E-16 0.413939 4 2 1.8850 0.7854 0.00000 -0.100323 -0.614301E-17 -0.100323 4 2 3.1416 0.7854 0.00000 0.00000 0.00000 0.00000 4 2 4.3982 0.7854 0.00000 -0.100323 -0.614301E-17 -0.100323 4 2 5.6549 0.7854 0.00000 0.413939 0.253464E-16 0.413939 3 -1 0.3927 0.4488 0.364121 -0.175351 0.364121 -0.175351 3 -1 0.3927 0.8976 0.251979 -0.315972 0.251979 -0.315972 3 -1 0.3927 1.3464 0.899304E-01 -0.394011 0.899304E-01 -0.394011 3 -1 0.3927 1.7952 -0.899304E-01 -0.394011 -0.899304E-01 -0.394011 3 -1 0.3927 2.2440 -0.251979 -0.315972 -0.251979 -0.315972 STIRLING1_TEST STIRLING1: Stirling numbers of first kind. Get rows 1 through 8 1 1 0 0 0 0 0 0 0 2 -1 1 0 0 0 0 0 0 3 2 -3 1 0 0 0 0 0 4 -6 11 -6 1 0 0 0 0 5 24 -50 35 -10 1 0 0 0 6 -120 274 -225 85 -15 1 0 0 7 720 -1764 1624 -735 175 -21 1 0 8 -5040 13068 -13132 6769 -1960 322 -28 1 STIRLING2_TEST STIRLING2: Stirling numbers of second kind. Get rows 1 through 8 1 1 0 0 0 0 0 0 0 2 1 1 0 0 0 0 0 0 3 1 3 1 0 0 0 0 0 4 1 7 6 1 0 0 0 0 5 1 15 25 10 1 0 0 0 6 1 31 90 65 15 1 0 0 7 1 63 301 350 140 21 1 0 8 1 127 966 1701 1050 266 28 1 TAU_TEST TAU computes the Tau function. N exact C(I) computed C(I) 1 1 1 2 2 2 3 2 2 4 3 3 5 2 2 6 4 4 7 2 2 8 4 4 9 3 3 10 4 4 23 2 2 72 12 12 126 12 12 226 4 4 300 18 18 480 24 24 521 2 2 610 8 8 832 14 14 960 28 28 TETRAHEDRON_NUM_TEST TETRAHEDRON_NUM computes the tetrahedron numbers. I TETR(I) 1 1 2 4 3 10 4 20 5 35 6 56 7 84 8 120 9 165 10 220 TRIANGLE_NUM_TEST TRIANGLE_NUM computes the triangular numbers. I TRI(I) 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 9 45 10 55 TRIANGLE_LOWER_TO_I4_TEST TRIANGLE_LOWER_TO_I4 converts a lower triangular index to a linear one. I, J ==> K 1 1 1 2 1 2 2 2 3 3 1 4 3 2 5 3 3 6 4 1 7 4 2 8 4 3 9 4 4 10 TRIBONACCI_RECURSIVE_TEST TRIBONACCI_RECURSIVE computes the Tribonacci sequence. N F(N) 1 1 2 1 3 1 4 3 5 5 6 9 7 17 8 31 9 57 10 105 11 193 12 355 13 653 14 1201 15 2209 16 4063 17 7473 18 13745 19 25281 20 46499 TRINOMIAL_TEST TRINOMIAL evaluates the trinomial coefficient: T(I,J,K) = (I+J+K)! / I! / J! / K! I J K T(I,J,K) 0 0 0 1 1 0 0 1 2 0 0 1 3 0 0 1 4 0 0 1 0 1 0 1 1 1 0 2 2 1 0 3 3 1 0 4 4 1 0 5 0 2 0 1 1 2 0 3 2 2 0 6 3 2 0 10 4 2 0 15 0 3 0 1 1 3 0 4 2 3 0 10 3 3 0 20 4 3 0 35 0 4 0 1 1 4 0 5 2 4 0 15 3 4 0 35 4 4 0 70 0 0 1 1 1 0 1 2 2 0 1 3 3 0 1 4 4 0 1 5 0 1 1 2 1 1 1 6 2 1 1 12 3 1 1 20 4 1 1 30 0 2 1 3 1 2 1 12 2 2 1 30 3 2 1 60 4 2 1 105 0 3 1 4 1 3 1 20 2 3 1 60 3 3 1 140 4 3 1 280 0 4 1 5 1 4 1 30 2 4 1 105 3 4 1 280 4 4 1 630 0 0 2 1 1 0 2 3 2 0 2 6 3 0 2 10 4 0 2 15 0 1 2 3 1 1 2 12 2 1 2 30 3 1 2 60 4 1 2 105 0 2 2 6 1 2 2 30 2 2 2 90 3 2 2 210 4 2 2 420 0 3 2 10 1 3 2 60 2 3 2 210 3 3 2 560 4 3 2 1260 0 4 2 15 1 4 2 105 2 4 2 420 3 4 2 1260 4 4 2 3150 0 0 3 1 1 0 3 4 2 0 3 10 3 0 3 20 4 0 3 35 0 1 3 4 1 1 3 20 2 1 3 60 3 1 3 140 4 1 3 280 0 2 3 10 1 2 3 60 2 2 3 210 3 2 3 560 4 2 3 1260 0 3 3 20 1 3 3 140 2 3 3 560 3 3 3 1680 4 3 3 4200 0 4 3 35 1 4 3 280 2 4 3 1260 3 4 3 4200 4 4 3 11550 0 0 4 1 1 0 4 5 2 0 4 15 3 0 4 35 4 0 4 70 0 1 4 5 1 1 4 30 2 1 4 105 3 1 4 280 4 1 4 630 0 2 4 15 1 2 4 105 2 2 4 420 3 2 4 1260 4 2 4 3150 0 3 4 35 1 3 4 280 2 3 4 1260 3 3 4 4200 4 3 4 11550 0 4 4 70 1 4 4 630 2 4 4 3150 3 4 4 11550 4 4 4 34650 V_HOFSTADTER_TEST V_HOFSTADTER evaluates Hofstadter's recursive V function. N V(N) 0 0 1 1 2 1 3 1 4 1 5 2 6 3 7 4 8 5 9 5 10 6 11 6 12 7 13 8 14 8 15 9 16 9 17 10 18 11 19 11 20 11 21 12 22 12 23 13 24 14 25 14 26 15 27 15 28 16 29 17 30 17 VIBONACCI_TEST VIBONACCI computes a Vibonacci sequence. Number of times we compute the series: 3 1 1 1 1 2 1 1 1 3 0 0 -2 4 1 -1 1 5 -1 -1 -1 6 0 0 -2 7 -1 -1 -3 8 1 1 1 9 -2 0 -2 10 -3 1 -3 11 -1 -1 5 12 4 -2 2 13 3 -3 -7 14 -7 -1 -5 15 10 -4 2 16 -3 3 7 17 -13 1 9 18 -16 2 -2 19 -3 -3 -11 20 -19 1 -9 ZECKENDORF_TEST ZECKENDORF computes the Zeckendorf decomposition of an integer N into nonconsecutive Fibonacci numbers. N Sum M Parts 1 1 2 2 3 3 4 3 1 5 5 6 5 1 7 5 2 8 8 9 8 1 10 8 2 11 8 3 12 8 3 1 13 13 14 13 1 15 13 2 16 13 3 17 13 3 1 18 13 5 19 13 5 1 20 13 5 2 21 21 22 21 1 23 21 2 24 21 3 25 21 3 1 26 21 5 27 21 5 1 28 21 5 2 29 21 8 30 21 8 1 31 21 8 2 32 21 8 3 33 21 8 3 1 34 34 35 34 1 36 34 2 37 34 3 38 34 3 1 39 34 5 40 34 5 1 41 34 5 2 42 34 8 43 34 8 1 44 34 8 2 45 34 8 3 46 34 8 3 1 47 34 13 48 34 13 1 49 34 13 2 50 34 13 3 51 34 13 3 1 52 34 13 5 53 34 13 5 1 54 34 13 5 2 55 55 56 55 1 57 55 2 58 55 3 59 55 3 1 60 55 5 61 55 5 1 62 55 5 2 63 55 8 64 55 8 1 65 55 8 2 66 55 8 3 67 55 8 3 1 68 55 13 69 55 13 1 70 55 13 2 71 55 13 3 72 55 13 3 1 73 55 13 5 74 55 13 5 1 75 55 13 5 2 76 55 21 77 55 21 1 78 55 21 2 79 55 21 3 80 55 21 3 1 81 55 21 5 82 55 21 5 1 83 55 21 5 2 84 55 21 8 85 55 21 8 1 86 55 21 8 2 87 55 21 8 3 88 55 21 8 3 1 89 89 90 89 1 91 89 2 92 89 3 93 89 3 1 94 89 5 95 89 5 1 96 89 5 2 97 89 8 98 89 8 1 99 89 8 2 100 89 8 3 ZERNIKE_POLY_TEST ZERNIKE_POLY evaluates a Zernike polynomial directly. Z1: Compute polynomial coefficients, then evaluate by Horner's method; Z2: Evaluate directly by recursion. N M Z1 Z2 0 0 1.0000000 1.0000000 1 0 0.0000000 0.0000000 1 1 0.98765432 0.98765432 2 0 0.95092212 0.95092212 2 1 0.0000000 0.0000000 2 2 0.97546106 0.97546106 3 0 0.0000000 0.0000000 3 1 0.91494634 0.91494634 3 2 0.0000000 0.0000000 3 3 0.96341833 0.96341833 4 0 0.85637930 0.85637930 4 1 0.0000000 0.0000000 4 2 0.87971393 0.87971393 4 3 0.0000000 0.0000000 4 4 0.95152428 0.95152428 5 0 0.0000000 0.0000000 5 1 0.79971364 0.79971364 5 2 0.0000000 0.0000000 5 3 0.84521200 0.84521200 5 4 0.0000000 0.0000000 5 5 0.93977706 0.93977706 ZERNIKE_POLY_COEF_TEST ZERNIKE_POLY_COEF determines the Zernike polynomial coefficients. Zernike polynomial p(x) = 0.00000 Zernike polynomial p(x) = 10.0000 * x ^ 5 - 12.0000 * x ^ 3 + 3.00000 * x Zernike polynomial p(x) = 0.00000 Zernike polynomial p(x) = 5.00000 * x ^ 5 - 4.00000 * x ^ 3 Zernike polynomial p(x) = 0.00000 Zernike polynomial p(x) = 1.00000 * x ^ 5 ZETA_M1_TEST ZETA_M1 computes the Zeta Minus One function. Requested tolerance = 0.100000E-09 N exact Zeta computed Zeta 2.00 0.644934066848 0.644934066822 2.50 0.341487257300 0.341487257233 3.00 0.202056903160 0.202056903140 3.50 0.126733867300 0.126733867303 4.00 0.823232337111E-01 0.823232337005E-01 5.00 0.369277551434E-01 0.369277551398E-01 6.00 0.173430619844E-01 0.173430619810E-01 7.00 0.834927738192E-02 0.834927738016E-02 8.00 0.407735619794E-02 0.407735619764E-02 9.00 0.200839292608E-02 0.200839282603E-02 10.00 0.994575127818E-03 0.994575127811E-03 11.00 0.494188604119E-03 0.494188604118E-03 12.00 0.246086553308E-03 0.246086553308E-03 16.00 0.152822594087E-04 0.152822594087E-04 20.00 0.953962033873E-06 0.953962033873E-06 30.00 0.931327432420E-10 0.931327432420E-09 40.00 0.909494780000E-12 0.909494784026E-12 ZETA_NAIVE_TEST ZETA_NAIVE computes the Zeta function. N exact Zeta computed Zeta 2 1.64493406685 1.64393456668 3 1.20205690316 1.20205640366 4 1.08232323371 1.08232323338 5 1.03692775514 1.03692775514 6 1.01734306198 1.01734306198 7 1.00834927738 1.00834927738 8 1.00407735620 1.00407735620 9 1.00200839293 1.00200839283 10 1.00099457513 1.00099457513 11 1.00049418860 1.00049418860 12 1.00024608655 1.00024608655 16 1.00001528226 1.00001528226 20 1.00000095396 1.00000095396 30 1.00000000093 1.00000000093 40 1.00000000000 1.00000000000 POLPAK_TEST Normal end of execution. 19 June 2018 9:44:38.552 PM