Thu Sep 13 12:59:29 2018 POLPAK_TEST Python version: 3.6.5 Test the POLPAK library. AGM_VALUES_TEST: Python version: 3.6.5 AGM_VALUES stores values of the arithmetic geometric mean function. A B AGM(A,B) 22.000000 96.000000 52.2746411987042379 83.000000 56.000000 68.8365300598585179 42.000000 7.000000 20.6593011967340097 26.000000 11.000000 17.6968548737436500 4.000000 63.000000 23.8670497217533004 6.000000 45.000000 20.7170159828059930 40.000000 75.000000 56.1278422556166845 80.000000 0.000000 0.0000000000000000 90.000000 35.000000 59.2695650812296364 9.000000 1.000000 3.9362355036495553 53.000000 53.000000 53.0000000000000000 1.000000 2.000000 1.4567910310469068 1.000000 4.000000 2.2430285802876027 1.000000 8.000000 3.6157561775973628 AGM_VALUES_TEST: Normal end of execution. BELL_VALUES_TEST: Python version: 3.6.5 BELL_VALUES returns values of the Bell numbers. N BELL(N) 0 1 1 1 2 2 3 5 4 15 5 52 6 203 7 877 8 4140 9 21147 10 115975 BELL_VALUES_TEST: Normal end of execution. BERNOULLI_NUMBER_VALUES_TEST: Python version: 3.6.5 BERNOULLI_NUMBER_VALUES returns values of the Bernoulli numbers. N BERNOULLI_NUMBER(N) 0 1 1 -0.5 2 0.166667 3 0 4 -0.0333333 6 -0.0238095 8 -0.0333333 10 0.0757576 20 -529.124 30 6.01581e+08 BERNOULLI_NUMBER_VALUES_TEST: Normal end of execution. BERNSTEIN_POLY_01_VALUES_TEST: Python version: 3.6.5 BERNSTEIN_POLY_01_VALUES stores values of Bernstein polynomials. N K X F 0 0 0.250000 1 1 0 0.250000 0.75 1 1 0.250000 0.25 2 0 0.250000 0.5625 2 1 0.250000 0.375 2 2 0.250000 0.0625 3 0 0.250000 0.421875 3 1 0.250000 0.421875 3 2 0.250000 0.140625 3 3 0.250000 0.015625 4 0 0.250000 0.31640625 4 1 0.250000 0.421875 4 2 0.250000 0.2109375 4 3 0.250000 0.046875 4 4 0.250000 0.00390625 BERNSTEIN_POLY_01_VALUES_TEST: Normal end of execution. CATALAN_VALUES_TEST: Python version: 3.6.5 CATALAN_VALUES returns values of the Catalan numbers. N C(N) 0 1 1 1 2 2 3 5 4 14 5 42 6 132 7 429 8 1430 9 4862 10 16796 CATALAN_VALUES_TEST: Normal end of execution. CHEBY_T_POLY_VALUES_TEST: Python version: 3.6.5 CHEBY_T_POLY_VALUES stores values of the Bernoulli polynomials. N X FX 0 0.800000 1 1 0.800000 0.8 2 0.800000 0.28 3 0.800000 -0.352 4 0.800000 -0.8431999999999999 5 0.800000 -0.99712 6 0.800000 -0.752192 7 0.800000 -0.2063872 8 0.800000 0.42197248 9 0.800000 0.881543168 10 0.800000 0.9884965888 11 0.800000 0.70005137408 12 0.800000 0.131585609728 CHEBY_T_POLY_VALUES_TEST: Normal end of execution. CHEBY_U_POLY_VALUES_TEST: Python version: 3.6.5 CHEBY_U_POLY_VALUES stores values of the Chebyshev U polynomials. N X FX 0 0.800000 1 1 0.800000 1.6 2 0.800000 1.56 3 0.800000 0.896 4 0.800000 -0.1264 5 0.800000 -1.09824 6 0.800000 -1.630784 7 0.800000 -1.5110144 8 0.800000 -0.78683904 9 0.800000 0.252071936 10 0.800000 1.1901541376 11 0.800000 1.65217468416 12 0.800000 1.453325357056 CHEBY_U_POLY_VALUES_TEST: Normal end of execution. COLLATZ_COUNT_VALUES_TEST: Python version: 3.6.5 COLLATZ_COUNT_VALUES returns values of the length of the Collatz sequence that starts at N. N Count 1 1 2 2 3 8 4 3 5 6 6 9 7 17 8 4 9 20 10 7 27 112 50 25 100 26 200 27 300 17 400 28 500 111 600 18 700 83 800 29 COLLATZ_COUNT_VALUES_TEST: Normal end of execution. COS_POWER_INT_VALUES_TEST: Python version: 3.6.5 COS_POWER_INT_VALUES stores values of the cosine power integral. A B N F 0.000000 3.141593 0 3.141592653589793 0.000000 3.141593 1 0 0.000000 3.141593 2 1.570796326794897 0.000000 3.141593 3 0 0.000000 3.141593 4 1.178097245096172 0.000000 3.141593 5 0 0.000000 3.141593 6 0.9817477042468103 0.000000 3.141593 7 0 0.000000 3.141593 8 0.8590292412159591 0.000000 3.141593 9 0 0.000000 3.141593 10 0.7731263170943632 COS_POWER_INT_VALUES_TEST: Normal end of execution. ERF_VALUES_TEST: Python version: 3.6.5 ERF_VALUES stores values of the error function. X ERF(X) 0.000000 0.0000000000000000 0.100000 0.1124629160182849 0.200000 0.2227025892104785 0.300000 0.3286267594591274 0.400000 0.4283923550466685 0.500000 0.5204998778130465 0.600000 0.6038560908479259 0.700000 0.6778011938374185 0.800000 0.7421009647076605 0.900000 0.7969082124228321 1.000000 0.8427007929497149 1.100000 0.8802050695740817 1.200000 0.9103139782296354 1.300000 0.9340079449406524 1.400000 0.9522851197626488 1.500000 0.9661051464753106 1.600000 0.9763483833446440 1.700000 0.9837904585907746 1.800000 0.9890905016357306 1.900000 0.9927904292352575 2.000000 0.9953222650189527 ERF_VALUES_TEST: Normal end of execution. EULER_NUMBER_VALUES_TEST: Python version: 3.6.5 EULER_NUMBER_VALUES returns values of the Euler numbers. N Euler_Number(N) 0 1 1 0 2 -1 4 5 6 -61 8 1385 10 -50521 12 2702765 EULER_NUMBER_VALUES_TEST: Normal end of execution. GAMMA_VALUES_TEST: Python version: 3.6.5 GAMMA_VALUES stores values of the Gamma function. X GAMMA(X) -0.500000 -3.5449077018110322 -0.010000 -100.5871979644108052 0.010000 99.4325851191506018 0.100000 9.5135076986687324 0.200000 4.5908437119988026 0.400000 2.2181595437576882 0.500000 1.7724538509055161 0.600000 1.4891922488128171 0.800000 1.1642297137253030 1.000000 1.0000000000000000 1.100000 0.9513507698668732 1.200000 0.9181687423997607 1.300000 0.8974706963062772 1.400000 0.8872638175030753 1.500000 0.8862269254527581 1.600000 0.8935153492876903 1.700000 0.9086387328532904 1.800000 0.9313837709802427 1.900000 0.9617658319073874 2.000000 1.0000000000000000 3.000000 2.0000000000000000 4.000000 6.0000000000000000 10.000000 362880.0000000000000000 20.000000 121645100408832000.0000000000000000 30.000000 8841761993739701898620088352768.0000000000000000 GAMMA_VALUES_TEST: Normal end of execution. GAMMA_LOG_VALUES: Python version: 3.6.5 GAMMA_LOG_VALUES stores values of the logarithm of the Gamma function. X GAMMA_LOG(X) 0.200000 1.5240638224307841 0.400000 0.7966778177017837 0.600000 0.3982338580692348 0.800000 0.1520596783998375 1.000000 0.0000000000000000 1.100000 -0.0498724412598397 1.200000 -0.0853740900033158 1.300000 -0.1081748095078604 1.400000 -0.1196129141723712 1.500000 -0.1207822376352452 1.600000 -0.1125917656967557 1.700000 -0.0958076974070659 1.800000 -0.0710838729143722 1.900000 -0.0389842759230833 2.000000 0.0000000000000000 3.000000 0.6931471805599453 4.000000 1.7917594692280550 10.000000 12.8018274800814691 20.000000 39.3398841871994946 30.000000 71.2570389671680147 GAMMA_LOG_VALUES_TEST: Normal end of execution. GEGENBAUER_POLY_VALUES_TEST: Python version: 3.6.5 GEGENBAUER_POLY_VALUES stores values of the Gegenbauer polynomials. N A X FX 0 0.500000 0.200000 1 1 0.500000 0.200000 0.2 2 0.500000 0.200000 -0.44 3 0.500000 0.200000 -0.28 4 0.500000 0.200000 0.232 5 0.500000 0.200000 0.30752 6 0.500000 0.200000 -0.08057599999999999 7 0.500000 0.200000 -0.2935168 8 0.500000 0.200000 -0.0395648 9 0.500000 0.200000 0.2459712 10 0.500000 0.200000 0.1290720256 2 0.000000 0.400000 0 2 1.000000 0.400000 -0.36 2 2.000000 0.400000 -0.08 2 3.000000 0.400000 0.84 2 4.000000 0.400000 2.4 2 5.000000 0.400000 4.6 2 6.000000 0.400000 7.44 2 7.000000 0.400000 10.92 2 8.000000 0.400000 15.04 2 9.000000 0.400000 19.8 2 10.000000 0.400000 25.2 5 3.000000 -0.500000 -9 5 3.000000 -0.400000 -0.16128 5 3.000000 -0.300000 -6.67296 5 3.000000 -0.200000 -8.37504 5 3.000000 -0.100000 -5.52672 5 3.000000 0.000000 0 5 3.000000 0.100000 5.52672 5 3.000000 0.200000 8.37504 5 3.000000 0.300000 6.67296 5 3.000000 0.400000 0.16128 5 3.000000 0.500000 -9 5 3.000000 0.600000 -15.42528 5 3.000000 0.700000 -9.696960000000001 5 3.000000 0.800000 22.44096 5 3.000000 0.900000 100.88928 5 3.000000 1.000000 252 GEGENBAUER_POLY_VALUES_TEST: Normal end of execution. GUD_VALUES: Python version: 3.6.5 GUD_VALUES stores values of the Gudermannian function. X GUD(X) -2.000000 -1.3017603360460150 -1.000000 -0.8657694832396586 0.000000 0.0000000000000000 0.100000 0.0998337487934866 0.200000 0.1986798470079397 0.500000 0.4803810791337294 1.000000 0.8657694832396586 1.500000 1.1317283452505089 2.000000 1.3017603360460150 2.500000 1.4069935689361539 3.000000 1.4713043411171931 3.500000 1.5104199075457001 4.000000 1.5341691443347329 GUD_VALUES_TEST: Normal end of execution. HERMITE_POLY_PHYS_VALUES_TEST: Python version: 3.6.5 HERMITE_POLY_PHYS_VALUES stores values of the Hermite physicist polynomials. N X FX 0 5.000000 1 1 5.000000 10 2 5.000000 98 3 5.000000 940 4 5.000000 8812 5 5.000000 80600 6 5.000000 717880 7 5.000000 6211600 8 5.000000 52065680 9 5.000000 421271200 10 5.000000 3275529760 11 5.000000 24329873600 12 5.000000 171237081280 5 0.000000 0 5 0.500000 41 5 1.000000 -8 5 3.000000 3816 5 10.000000 3041200 HERMITE_POLY_PHYS_VALUES_TEST: Normal end of execution. HYPER_2F1_VALUES_TEST: Python version: 3.6.5 HYPER_2F1_VALUES stores values of the hypergeometric function 2F1 A B C X F -2 3 6.700000 0.250000 0.7235612934899779 0 1 6.700000 0.250000 0.9791110934527796 0 1 6.700000 0.250000 1.021657814008856 2 3 6.700000 0.250000 1.405156320011213 -2 3 6.700000 0.550000 0.4696143163982161 0 1 6.700000 0.550000 0.9529619497744632 0 1 6.700000 0.550000 1.051281421394799 2 3 6.700000 0.550000 2.399906290477786 -2 3 6.700000 0.850000 0.2910609592841472 0 1 6.700000 0.850000 0.9253696791037318 0 1 6.700000 0.850000 1.0865504094807 2 3 6.700000 0.850000 5.738156552618904 3 6 -5.500000 0.250000 15090.66974870461 1 6 -0.500000 0.250000 -104.3117006736435 1 6 0.500000 0.250000 21.17505070776881 3 6 4.500000 0.250000 4.194691581903192 3 6 -5.500000 0.550000 10170777974.04881 1 6 -0.500000 0.550000 -24708.63532248916 1 6 0.500000 0.550000 1372.230454838499 3 6 4.500000 0.550000 58.09272870639465 3 6 -5.500000 0.850000 5.868208761512417e+18 1 6 -0.500000 0.850000 -446350101.47296 1 6 0.500000 0.850000 5383505.756129573 3 6 4.500000 0.850000 20396.91377601966 HYPER_2F1_VALUES_TEST: Normal end of execution. I4_FACTORIAL_VALUES_TEST: Python version: 3.6.5 I4_FACTORIAL_VALUES returns values of the integer factorial function. N I4_FACTORIAL(N) 0 1 1 1 2 2 3 6 4 24 5 120 6 720 7 5040 8 40320 9 362880 10 3628800 11 39916800 12 479001600 I4_FACTORIAL_VALUES_TEST: Normal end of execution. I4_FACTORIAL2_VALUES_TEST: Python version: 3.6.5 I4_FACTORIAL2_VALUES returns values of the double factorial function. N N!! 0 1 1 1 2 2 3 3 4 8 5 15 6 48 7 105 8 384 9 945 10 3840 11 10395 12 46080 13 135135 14 645120 15 2027025 I4_FACTORIAL2_VALUES_TEST: Normal end of execution. I4_UNIFORM_AB_TEST Python version: 3.6.5 I4_UNIFORM_AB computes pseudorandom values in an interval [A,B]. The lower endpoint A = -100 The upper endpoint B = 200 The initial seed is 123456789 1 -35 2 187 3 149 4 69 5 25 6 -81 7 -23 8 -67 9 -87 10 90 11 -82 12 35 13 20 14 127 15 139 16 -100 17 170 18 5 19 -72 20 -96 I4_UNIFORM_AB_TEST: Normal end of execution. I4MAT_PRINT_TEST: Python version: 3.6.5 Test I4MAT_PRINT, which prints an I4MAT. A 5 x 6 integer matrix: Col: 0 1 2 3 4 Row 0: 11 12 13 14 15 1: 21 22 23 24 25 2: 31 32 33 34 35 3: 41 42 43 44 45 4: 51 52 53 54 55 Col: 5 Row 0: 16 1: 26 2: 36 3: 46 4: 56 I4MAT_PRINT_TEST: Normal end of execution. I4MAT_PRINT_SOME_TEST Python version: 3.6.5 I4MAT_PRINT_SOME prints some of an I4MAT. Here is I4MAT, rows 0:2, cols 3:5: Col: 3 4 5 Row 0: 14 15 16 1: 24 25 26 2: 34 35 36 I4MAT_PRINT_SOME_TEST: Normal end of execution. I4VEC_PRINT_TEST Python version: 3.6.5 I4VEC_PRINT prints an I4VEC. Here is an I4VEC: 0 91 1 92 2 93 3 94 I4VEC_PRINT_TEST: Normal end of execution. JACOBI_POLY_VALUES_TEST: Python version: 3.6.5 JACOBI_POLY_VALUES stores values of the Jacobi polynomials. N A B X F 0 0.000000 1.000000 0.500000 1 1 0.000000 1.000000 0.500000 0.25 2 0.000000 1.000000 0.500000 -0.375 3 0.000000 1.000000 0.500000 -0.484375 4 0.000000 1.000000 0.500000 -0.1328125 5 0.000000 1.000000 0.500000 0.275390625 5 1.000000 1.000000 0.500000 -0.1640625 5 2.000000 1.000000 0.500000 -1.1748046875 5 3.000000 1.000000 0.500000 -2.361328125 5 4.000000 1.000000 0.500000 -2.6162109375 5 5.000000 1.000000 0.500000 0.1171875 5 0.000000 2.000000 0.500000 0.421875 5 0.000000 3.000000 0.500000 0.5048828125 5 0.000000 4.000000 0.500000 0.509765625 5 0.000000 5.000000 0.500000 0.4306640625 5 0.000000 1.000000 -1.000000 -6 5 0.000000 1.000000 -0.800000 0.03862 5 0.000000 1.000000 -0.600000 0.81184 5 0.000000 1.000000 -0.400000 0.03666 5 0.000000 1.000000 -0.200000 -0.48512 5 0.000000 1.000000 0.000000 -0.3125 5 0.000000 1.000000 0.200000 0.18912 5 0.000000 1.000000 0.400000 0.40234 5 0.000000 1.000000 0.600000 0.01216 5 0.000000 1.000000 0.800000 -0.43962 5 0.000000 1.000000 1.000000 1 JACOBI_POLY_VALUES_TEST: Normal end of execution. LAGUERRE_POLYNOMIAL_VALUES_TEST: Python version: 3.6.5 LAGUERRE_POLYNOMIAL_VALUES stores values of the Laguerre polynomials. N X L(N)(X) 0 1.000000 1.0000000000000000 1 1.000000 0.0000000000000000 2 1.000000 -0.5000000000000000 3 1.000000 -0.6666666666666667 4 1.000000 -0.6250000000000000 5 1.000000 -0.4666666666666667 6 1.000000 -0.2569444444444444 7 1.000000 -0.0404761904761905 8 1.000000 0.1539930555555556 9 1.000000 0.3097442680776014 10 1.000000 0.4189459325396825 11 1.000000 0.4801341790925124 12 1.000000 0.4962122235082305 5 0.500000 -0.4455729166666667 5 3.000000 0.8500000000000000 5 5.000000 -3.1666666666666670 5 10.000000 34.3333333333333286 LAGUERRE_POLYNOMIAL_VALUES_TEST: Normal end of execution. LEGENDRE_ASSOCIATED_VALUES_TEST: Python version: 3.6.5 LEGENDRE_ASSOCIATED_VALUES stores values of the associated Legendre function. N M X F 1 0 0.000000 0 2 0 0.000000 -0.5 3 0 0.000000 0 4 0 0.000000 0.375 5 0 0.000000 0 1 1 0.500000 -0.8660254037844386 2 1 0.500000 -1.299038105676658 3 1 0.500000 -0.3247595264191645 4 1 0.500000 1.353164693413185 3 0 0.200000 -0.28 3 1 0.200000 1.175755076535925 3 2 0.200000 2.88 3 3 0.200000 -14.10906091843111 4 2 0.250000 -3.955078125 5 2 0.250000 -9.99755859375 6 3 0.250000 82.65311444100485 7 3 0.250000 20.24442836815152 8 4 0.250000 -423.7997531890869 9 4 0.250000 1638.320624828339 10 5 0.250000 -20256.87389227225 LEGENDRE_ASSOCIATED_VALUES_TEST: Normal end of execution. LEGENDRE_ASSOCIATED_NORMALIZED_SPHERE_VALUES_TEST: Python version: 3.6.5 LEGENDRE_ASSOCIATED_NORMALIZED_SPHERE_VALUES stores values of the associated Legendre function normalized for the surface of a sphere. N M X F 0 0 0.500000 0.2820947917738781 1 0 0.500000 0.24430125595146 1 1 0.500000 -0.2992067103010745 2 0 0.500000 -0.07884789131313 2 1 0.500000 -0.3345232717786446 2 2 0.500000 0.2897056515173922 3 0 0.500000 -0.326529291016351 3 1 0.500000 -0.06997056236064664 3 2 0.500000 0.3832445536624809 3 3 0.500000 -0.2709948227475519 4 0 0.500000 -0.24462907724141 4 1 0.500000 0.2560660384200185 4 2 0.500000 0.1881693403754876 4 3 0.500000 -0.4064922341213279 4 4 0.500000 0.2489246395003027 5 0 0.500000 0.0840580442633982 5 1 0.500000 0.3293793022891428 5 2 0.500000 -0.1588847984307093 5 3 0.500000 -0.2808712959945307 5 4 0.500000 0.4127948151484925 5 5 0.500000 -0.2260970318780046 LEGENDRE_ASSOCIATED_NORMALIZED_SPHERE_VALUES_TEST: Normal end of execution. LEGENDRE_FUNCTION_Q_VALUES_TEST: Python version: 3.6.5 LEGENDRE_FUNCTION_Q_VALUES stores values of the Legendre Q function N X F 0 0.250000 0.2554128118829953 1 0.250000 -0.9361467970292512 2 0.250000 -0.4787614548274669 3 0.250000 0.4246139251747229 4 0.250000 0.5448396833845414 5 0.250000 -0.0945132826167347 6 0.250000 -0.4973516573531213 7 0.250000 -0.1499018843853194 8 0.250000 0.3649161918783626 9 0.250000 0.3055676545072885 10 0.250000 -0.1832799367995643 3 0.000000 0.6666666666666667 3 0.100000 0.626867202876333 3 0.200000 0.5099015515315237 3 0.300000 0.3232754180589764 3 0.400000 0.08026113738148187 3 0.500000 -0.1986547714794823 3 0.600000 -0.4828663183349136 3 0.700000 -0.7252886849144387 3 0.800000 -0.8454443502398846 3 0.900000 -0.6627096245052618 LEGENDRE_FUNCTION_Q_VALUES_TEST: Normal end of execution. LEGENDRE_POLY_VALUES_TEST: Python version: 3.6.5 LEGENDRE_POLY_VALUES stores values of the Legendre polynomials. N X F 0 0.250000 1 1 0.250000 0.25 2 0.250000 -0.40625 3 0.250000 -0.3359375 4 0.250000 0.15771484375 5 0.250000 0.3397216796875 6 0.250000 0.0242767333984375 7 0.250000 -0.2799186706542969 8 0.250000 -0.1524540185928345 9 0.250000 0.1768244206905365 10 0.250000 0.2212002165615559 3 0.000000 0 3 0.100000 -0.1475 3 0.200000 -0.28 3 0.300000 -0.3825 3 0.400000 -0.44 3 0.500000 -0.4375 3 0.600000 -0.36 3 0.700000 -0.1925 3 0.800000 0.08 3 0.900000 0.4725 3 1.000000 1 LEGENDRE_POLY_VALUES_TEST: Normal end of execution. LERCH_VALUES_TEST: Python version: 3.6.5 LERCH_VALUES stores values of the Lerch function. Z S A F 1.000000 2 0.000000 1.6449340668482260 1.000000 3 0.000000 1.2020569031595940 1.000000 10 0.000000 1.0009945751278180 0.500000 2 1.000000 1.1644810529300249 0.500000 3 1.000000 1.0744263872160800 0.500000 10 1.000000 1.0004926412120141 0.333333 2 2.000000 0.2959190697935714 0.333333 3 2.000000 0.1394507503935608 0.333333 10 2.000000 0.0009823175058446 0.100000 2 3.000000 0.1177910993911311 0.100000 3 3.000000 0.0386844792229896 0.100000 10 3.000000 0.0000170314961419 LERCH_VALUES_TEST: Normal end of execution. MERTENS_VALUES_TEST: Python version: 3.6.5 MERTENS_VALUES stores values of the MERTENS function. N MERTENS(N) 1 1 2 0 3 -1 4 -1 5 -2 6 -1 7 -2 8 -2 9 -2 10 -1 11 -2 12 -2 100 1 1000 2 10000 -23 MERTENS_VALUES_TEST: Normal end of execution. MOEBIUS_VALUES_TEST: Python version: 3.6.5 MOEBIUS_VALUES stores values of the MOEBIUS function. N MOEBIUS(N) 1 1 2 -1 3 -1 4 0 5 -1 6 1 7 -1 8 0 9 0 10 1 11 -1 12 0 13 -1 14 1 15 1 16 0 17 -1 18 0 19 -1 20 0 MOEBIUS_VALUES_TEST: Normal end of execution. NORMAL_01_CDF_INVERSE_TEST: Python version: 3.6.5 NORMAL_01_CDF_INVERSE inverts the error function. FX X NORMAL_01_CDF_INVERSE(FX) 0.5 0 0 0.539828 0.1 0.09999999999999999 0.57926 0.2 0.1999999999999999 0.617911 0.3 0.2999999999999998 0.655422 0.4 0.4 0.691462 0.5 0.4999999999999998 0.725747 0.6 0.6000000000000016 0.758036 0.7 0.6999999999999998 0.788145 0.8 0.7999999999999998 0.81594 0.9 0.9 0.841345 1 1 0.933193 1.5 1.5 0.97725 2 2 0.99379 2.5 2.500000000000004 0.99865 3 2.999999999999997 0.999767 3.5 3.499999999999983 0.999968 4 4 NORMAL_01_CDF_INVERSE_TEST Normal end of execution. NORMAL_01_CDF_VALUES_TEST: Python version: 3.6.5 NORMAL_01_CDF_VALUES stores values of the unit normal CDF. X NORMAL_01_CDF(X) 0.000000 0.5000000000000000 0.100000 0.5398278372770290 0.200000 0.5792597094391030 0.300000 0.6179114221889526 0.400000 0.6554217416103242 0.500000 0.6914624612740131 0.600000 0.7257468822499270 0.700000 0.7580363477769270 0.800000 0.7881446014166033 0.900000 0.8159398746532405 1.000000 0.8413447460685429 1.500000 0.9331927987311419 2.000000 0.9772498680518208 2.500000 0.9937903346742240 3.000000 0.9986501019683699 3.500000 0.9997673709209645 4.000000 0.9999683287581669 NORMAL_01_CDF_VALUES_TEST: Normal end of execution. OMEGA_VALUES_TEST: Python version: 3.6.5 OMEGA_VALUES stores values of the OMEGA function. N OMEGA(N) 1 0 2 1 3 1 4 1 5 1 6 2 7 1 8 1 9 1 10 2 30 3 101 1 210 4 1320 4 1764 3 2003 1 2310 5 2827 2 8717 2 12553 1 30030 6 510510 7 9699690 8 OMEGA_VALUES_TEST: Normal end of execution. PARTITION_DISTINCT_COUNT_VALUES_TEST: Python version: 3.6.5 PARTITION_DISTINCT_COUNT_VALUES returns values of the integer partition count function for distinct parts N P(N) 0 1 1 1 2 1 3 2 4 2 5 3 6 4 7 5 8 6 9 8 10 10 11 12 12 15 13 18 14 22 15 27 16 32 17 38 18 46 19 54 20 64 PARTITION_DISTINCT_COUNT_VALUES_TEST: Normal end of execution. PHI_VALUES_TEST: Python version: 3.6.5 PHI_VALUES stores values of the PHI function. N PHI(N) 1 1 2 1 3 2 4 2 5 4 6 2 7 6 8 4 9 6 10 4 20 8 30 8 40 16 50 20 60 16 100 40 149 148 500 200 750 200 999 648 PHI_VALUES_TEST: Normal end of execution. PSI_VALUES_TEST: Python version: 3.6.5 PSI_VALUES stores values of the PSI function. X PSI(X) 0.100000 -10.4237549404110794 0.200000 -5.2890398965921879 0.300000 -3.5025242222001332 0.400000 -2.5613845445851160 0.500000 -1.9635100260214231 0.600000 -1.5406192138931900 0.700000 -1.2200235536979349 0.800000 -0.9650085667061385 0.900000 -0.7549269499470515 1.000000 -0.5772156649015329 1.100000 -0.4237549404110768 1.200000 -0.2890398965921883 1.300000 -0.1691908888667997 1.400000 -0.0613845445851161 1.500000 0.0364899739785765 1.600000 0.1260474527734763 1.700000 0.2085478748734940 1.800000 0.2849914332938615 1.900000 0.3561841611640597 2.000000 0.4227843350984671 PSI_VALUES_TEST: Normal end of execution. R8_FACTORIAL_VALUES_TEST: Python version: 3.6.5 R8_FACTORIAL_VALUES returns values of the real factorial function. N R8_FACTORIAL(N) 0 1 1 1 2 2 3 6 4 24 5 120 6 720 7 5040 8 40320 9 362880 10 3.6288e+06 11 3.99168e+07 12 4.79002e+08 13 6.22702e+09 14 8.71783e+10 15 1.30767e+12 16 2.09228e+13 17 3.55687e+14 18 6.40237e+15 19 1.21645e+17 20 2.4329e+18 25 1.55112e+25 50 3.04141e+64 100 9.33262e+157 150 5.71338e+262 R8_FACTORIAL_VALUES_TEST: Normal end of execution. R8_FACTORIAL_LOG_VALUES_TEST: Python version: 3.6.5 R8_FACTORIAL_LOG_VALUES returns values of the log factorial function. N R8_FACTORIAL_LOG(N) 0 0 1 0 2 0.693147 3 1.79176 4 3.17805 5 4.78749 6 6.57925 7 8.52516 8 10.6046 9 12.8018 10 15.1044 11 17.5023 12 19.9872 13 22.5522 14 25.1912 15 27.8993 16 30.6719 17 33.5051 18 36.3954 19 39.3399 20 42.3356 25 58.0036 50 148.478 100 363.739 150 605.02 500 2611.33 1000 5912.13 R8_FACTORIAL_LOG_VALUES_TEST: Normal end of execution. R8_GAMMA_TEST: Python version: 3.6.5 R8_GAMMA evaluates the Gamma function. X GAMMA(X) R8_GAMMA(X) -0.5 -3.544907701811032 -3.544907701811032 -0.01 -100.5871979644108 -100.5871979644108 0.01 99.4325851191506 99.4325851191506 0.1 9.513507698668732 9.513507698668731 0.2 4.590843711998803 4.590843711998803 0.4 2.218159543757688 2.218159543757688 0.5 1.772453850905516 1.772453850905516 0.6 1.489192248812817 1.489192248812817 0.8 1.164229713725303 1.164229713725303 1 1 1 1.1 0.9513507698668732 0.9513507698668732 1.2 0.9181687423997607 0.9181687423997607 1.3 0.8974706963062772 0.8974706963062772 1.4 0.8872638175030753 0.8872638175030754 1.5 0.8862269254527581 0.8862269254527581 1.6 0.8935153492876903 0.8935153492876903 1.7 0.9086387328532904 0.9086387328532904 1.8 0.9313837709802427 0.9313837709802427 1.9 0.9617658319073874 0.9617658319073874 2 1 1 3 2 2 4 6 6 10 362880 362880 20 1.21645100408832e+17 1.216451004088321e+17 30 8.841761993739702e+30 8.841761993739751e+30 R8_GAMMA_TEST Normal end of execution. R8_GAMMA_LOG_TEST: Python version: 3.6.5 R8_GAMMA_LOG evaluates the logarithm of the Gamma function. X GAMMA_LOG(X) R8_GAMMA_LOG(X) 0.2 1.524063822430784 1.524063822430784 0.4 0.7966778177017837 0.7966778177017837 0.6 0.3982338580692348 0.3982338580692349 0.8 0.1520596783998375 0.1520596783998376 1 0 0 1.1 -0.04987244125983972 -0.04987244125983976 1.2 -0.08537409000331583 -0.08537409000331585 1.3 -0.1081748095078604 -0.1081748095078605 1.4 -0.1196129141723712 -0.1196129141723713 1.5 -0.1207822376352452 -0.1207822376352453 1.6 -0.1125917656967557 -0.1125917656967558 1.7 -0.09580769740706586 -0.09580769740706586 1.8 -0.07108387291437215 -0.07108387291437215 1.9 -0.03898427592308333 -0.03898427592308337 2 0 0 3 0.6931471805599453 0.6931471805599454 4 1.791759469228055 1.791759469228055 10 12.80182748008147 12.80182748008147 20 39.33988418719949 39.33988418719949 30 71.25703896716801 71.257038967168 R8_GAMMA_LOG_TEST Normal end of execution. R8_HUGE_TEST Python version: 3.6.5 R8_HUGE returns a "huge" R8; R8_HUGE = 1.79769e+308 R8_HUGE_TEST Normal end of execution. R8_MOP_TEST Python version: 3.6.5 R8_MOP evaluates (-1.0)^I4 as an R8. I4 R8_MOP(I4) -57 -1.0 92 1.0 66 1.0 12 1.0 -17 -1.0 -87 -1.0 -49 -1.0 -78 1.0 -92 1.0 27 -1.0 R8_MOP_TEST Normal end of execution. R8_NINT_TEST Python version: 3.6.5 R8_NINT produces the nearest integer. X R8_NINT(X) -5.631634 -6 9.126352 10 6.590185 7 1.233909 2 -1.693858 -2 -8.677625 -9 -4.848444 -5 -7.800864 -8 -9.123420 -10 2.679314 3 R8_NINT_TEST Normal end of execution. R8_PI_TEST Python version: 3.6.5 R8_PI returns the value of PI. R8_PI = 3.1415926535897931 4 * Atan(1) = 3.1415926535897931 np.pi = 3.1415926535897931 R8_PI_TEST Normal end of execution. R8_UNIFORM_01_TEST Python version: 3.6.5 R8_UNIFORM_01 produces a sequence of random values. Using random seed 123456789 SEED R8_UNIFORM_01(SEED) 469049721 0.218418 2053676357 0.956318 1781357515 0.829509 1206231778 0.561695 891865166 0.415307 141988902 0.066119 553144097 0.257578 236130416 0.109957 94122056 0.043829 1361431000 0.633966 Verify that the sequence can be restarted. Set the seed back to its original value, and see that we generate the same sequence. SEED R8_UNIFORM_01(SEED) 469049721 0.218418 2053676357 0.956318 1781357515 0.829509 1206231778 0.561695 891865166 0.415307 141988902 0.066119 553144097 0.257578 236130416 0.109957 94122056 0.043829 1361431000 0.633966 R8_UNIFORM_01_TEST Normal end of execution. R8_UNIFORM_AB_TEST Python version: 3.6.5 R8_UNIFORM_AB returns random values in a given range: [ A, B ] For this problem: A = 10.000000 B = 20.000000 12.184183 19.563176 18.295092 15.616954 14.153071 10.661187 12.575778 11.099568 10.438290 16.339657 R8_UNIFORM_AB_TEST Normal end of execution R8POLY_DEGREE_TEST Python version: 3.6.5 R8POLY_DEGREE determines the degree of an R8POLY. The R8POLY: p(x) = 4 * x^3 + 3 * x^2 + 2 * x + 1 Dimensioned degree = 3, Actual degree = 3 The R8POLY: p(x) = 0 * x^3 + 3 * x^2 + 2 * x + 1 Dimensioned degree = 3, Actual degree = 2 The R8POLY: p(x) = 4 * x^3 + 2 * x + 1 Dimensioned degree = 3, Actual degree = 3 The R8POLY: p(x) = 0 * x^3 + 1 Dimensioned degree = 3, Actual degree = 0 The R8POLY: p(x) = 0 * x^3 Dimensioned degree = 3, Actual degree = 0 R8POLY_DEGREE_TEST: Normal end of execution. R8POLY_PRINT_TEST Python version: 3.6.5 R8POLY_PRINT prints an R8POLY. The R8POLY: p(x) = 9 * x^5 + 0.78 * x^4 + 56 * x^2 - 3.4 * x + 12 R8POLY_PRINT_TEST: Normal end of execution. R8POLY_VALUE_HORNER_TEST Python version: 3.6.5 R8POLY_VALUE_HORNER evaluates a polynomial at a point using Horners method. The polynomial coefficients: p(x) = 1 * x^4 - 10 * x^3 + 35 * x^2 - 50 * x + 24 I X P(X) 0 0.0000 24 1 0.3333 10.8642 2 0.6667 3.45679 3 1.0000 0 4 1.3333 -0.987654 5 1.6667 -0.691358 6 2.0000 0 7 2.3333 0.493827 8 2.6667 0.493827 9 3.0000 0 10 3.3333 -0.691358 11 3.6667 -0.987654 12 4.0000 0 13 4.3333 3.45679 14 4.6667 10.8642 15 5.0000 24 R8POLY_VALUE_HORNER_TEST: Normal end of execution. R8VEC_PRINT_TEST Python version: 3.6.5 R8VEC_PRINT prints an R8VEC. Here is an R8VEC: 0: 123.456 1: 5e-06 2: -1e+06 3: 3.14159 R8VEC_PRINT_TEST: Normal end of execution. SIGMA_VALUES_TEST: Python version: 3.6.5 SIGMA_VALUES stores values of the SIGMA function. N SIGMA(N) 1 1 2 3 3 4 4 7 5 6 6 12 7 8 8 15 9 13 10 18 30 72 127 128 128 255 129 176 210 576 360 1170 617 618 815 984 816 2232 1000 2340 SIGMA_VALUES_TEST: Normal end of execution. SIN_POWER_INT_VALUES_TEST: Python version: 3.6.5 SIN_POWER_INT_VALUES stores values of the cosine power integral. A B N F 10.000000 20.000000 0 10 0.000000 1.000000 1 0.4596976941318603 0.000000 1.000000 2 0.2726756432935796 0.000000 1.000000 3 0.1789405625488581 0.000000 1.000000 4 0.1240255653152068 0.000000 1.000000 5 0.08897439645157594 0.000000 2.000000 5 0.9039312384814995 1.000000 2.000000 5 0.8149568420299235 0.000000 1.000000 10 0.02188752242172985 0.000000 1.000000 11 0.01702343937406933 SIN_POWER_INT_VALUES_TEST: Normal end of execution. SPHERICAL_HARMONIC_VALUES_TEST: Python version: 3.6.5 SPHERICAL_HARMONIC_VALUES stores values of the SPHERICAL_HARMONIC function. L M THETA PHI YR YI 0 0 0.523599 1.047198 0.2820947917738781 0.0000000000000000 1 0 0.523599 1.047198 0.4231421876608172 0.0000000000000000 2 1 0.523599 1.047198 -0.1672616358893223 -0.2897056515173922 3 2 0.523599 1.047198 -0.1106331731112457 0.1916222768312404 4 3 0.523599 1.047198 0.1354974113737760 0.0000000000000000 5 5 0.261799 0.628319 0.0005390423109044 0.0000000000000000 5 4 0.261799 0.628319 -0.0051466904429519 0.0037392894852833 5 3 0.261799 0.628319 0.0137100436134949 -0.0421951755232080 5 2 0.261799 0.628319 0.0609635202226554 0.1876264225575173 5 1 0.261799 0.628319 -0.4170400640977983 -0.3029973424491321 4 2 0.628319 0.785398 0.0000000000000000 0.4139385503112256 4 2 1.884956 0.785398 0.0000000000000000 -0.1003229830187463 4 2 3.141593 0.785398 0.0000000000000000 0.0000000000000000 4 2 4.398230 0.785398 0.0000000000000000 -0.1003229830187463 4 2 5.654867 0.785398 0.0000000000000000 0.4139385503112256 3 -1 0.392699 0.448799 0.3641205966137958 -0.1753512375142586 3 -1 0.392699 0.897598 0.2519792711195075 -0.3159720118970196 3 -1 0.392699 1.346397 0.0899303606570430 -0.3940106541811563 3 -1 0.392699 1.795196 -0.0899303606570430 -0.3940106541811563 3 -1 0.392699 2.243995 -0.2519792711195075 -0.3159720118970196 SPHERICAL_HARMONIC_VALUES_TEST: Normal end of execution. TIMESTAMP_TEST: Python version: 3.6.5 TIMESTAMP prints a timestamp of the current date and time. Thu Sep 13 12:59:29 2018 TIMESTAMP_TEST: Normal end of execution. AGUD_TEST Python version: 3.6.5 AGUD evaluates the inverse Gudermannian function. X GUD(X) AGUD(GUD(X)) 1.000000 0.865769 1.000000 1.200000 0.985692 1.200000 1.400000 1.087250 1.400000 1.600000 1.172359 1.600000 1.800000 1.243161 1.800000 2.000000 1.301760 2.000000 2.200000 1.350090 2.200000 2.400000 1.389856 2.400000 2.600000 1.422521 2.600000 2.800000 1.449326 2.800000 3.000000 1.471304 3.000000 AGUD_TEST Normal end of execution. ALIGN_ENUM_TEST Python version: 3.6.5 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 1 1 1 1 1 1 1 1 1 1 1 1 3 5 7 9 11 13 15 17 19 21 1 5 13 25 41 61 85 113 145 181 221 1 7 25 63 129 231 377 575 833 1159 1561 1 9 41 129 321 681 1289 2241 3649 5641 8361 1 11 61 231 681 1683 3653 7183 13073 22363 36365 1 13 85 377 1289 3653 8989 19825 40081 75517 134245 1 15 113 575 2241 7183 19825 48639 108545 224143 433905 1 17 145 833 3649 13073 40081 108545 265729 598417 1256465 1 19 181 1159 5641 22363 75517 224143 598417 1462563 3317445 1 21 221 1561 8361 36365 134245 433905 1256465 3317445 8097453 ALIGN_ENUM_TEST Normal end of execution. BELL_TEST Python version: 3.6.5 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_TEST Normal end of execution. BELL_POLY_COEF_TEST Python version: 3.6.5 BELL_POLY_COEF returns the coefficients of a Bell polynomial. Table of polyomial 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 BELL_POLY_COEF_TEST Normal end of execution. BENFORD_TEST Python version: 3.6.5 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 0.096910 5 0.079181 6 0.066947 7 0.057992 8 0.051153 9 0.045757 BENFORD_TEST Normal end of execution. BERNOULLI_NUMBER_TEST Python version: 3.6.5 BERNOULLI_NUMBER computes Bernoulli numbers; I Exact Bernoulli 0 1.000000e+00 1.000000e+00 1 -5.000000e-01 -5.000000e-01 2 1.666667e-01 1.666667e-01 3 0.000000e+00 0.000000e+00 4 -3.333333e-02 -3.333333e-02 6 -2.380952e-02 2.380952e-02 8 -3.333333e-02 -3.333333e-02 10 7.575758e-02 7.575758e-02 20 -5.291242e+02 -5.291242e+02 30 6.015809e+08 6.015808e+08 BERNOULLI_NUMBER_TEST Normal end of execution. BERNOULLI_NUMBER2_TEST Python version: 3.6.5 BERNOULLI_NUMBER2 computes Bernoulli numbers; I Exact Bernoulli 0 1.000000e+00 1.000000e+00 1 -5.000000e-01 -5.000000e-01 2 1.666667e-01 1.666667e-01 3 0.000000e+00 0.000000e+00 4 -3.333333e-02 -3.333333e-02 6 -2.380952e-02 2.380949e-02 8 -3.333333e-02 -3.333332e-02 10 7.575758e-02 7.575757e-02 20 -5.291242e+02 -5.291242e+02 30 6.015809e+08 6.015809e+08 BERNOULLI_NUMBER2_TEST Normal end of execution. BERNOULLI_NUMBER3_TEST Python version: 3.6.5 BERNOULLI_NUMBER3 computes Bernoulli numbers; I Exact Bernoulli 0 1.000000e+00 1.000000e+00 1 -5.000000e-01 -5.000000e-01 2 1.666667e-01 1.666667e-01 3 0.000000e+00 0.000000e+00 4 -3.333333e-02 -3.333314e-02 6 -2.380952e-02 2.380951e-02 8 -3.333333e-02 -3.333333e-02 10 7.575758e-02 7.575757e-02 20 -5.291242e+02 -5.291242e+02 30 6.015809e+08 6.015809e+08 BERNOULLI_NUMBER3_TEST Normal end of execution. BERNOULLI_POLY_TEST Python version: 3.6.5 BERNOULLI_POLY computes Bernoulli polynomials; X = 0.2 I B(I,X) 1 -0.3 2 0.00666667 3 0.048 4 -0.00773333 5 -0.02368 6 0.00691352 7 0.0249088 8 -0.01015 9 -0.0452782 10 0.0233263 11 0.12605 12 -0.0781468 13 -0.497979 14 0.36044 15 2.64878 BERNOULLI_POLY_TEST Normal end of execution. BERNOULLI_POLY2_TEST Python version: 3.6.5 BERNOULLI_POLY2 computes Bernoulli polynomials; X = 0.2 I B(I,X) 1 -0.3 2 0.00666667 3 0.048 4 -0.00773314 5 -0.0236798 6 0.00691363 7 0.0249088 8 -0.01015 9 -0.0452782 10 0.0233263 11 0.12605 12 -0.0781468 13 -0.497979 14 0.36044 15 2.64878 BERNOULLI_POLY2_TEST Normal end of execution. BERNSTEIN_POLY_TEST Python version: 3.6.5 BERNSTEIN_POLY computes Bernstein polynomials; N K X Exact B(N,K)(X) 0 0 0.25 1 1 1 0 0.25 0.75 0.75 1 1 0.25 0.25 0.25 2 0 0.25 0.5625 0.5625 2 1 0.25 0.375 0.375 2 2 0.25 0.0625 0.0625 3 0 0.25 0.421875 0.421875 3 1 0.25 0.421875 0.421875 3 2 0.25 0.140625 0.140625 3 3 0.25 0.015625 0.015625 4 0 0.25 0.316406 0.316406 4 1 0.25 0.421875 0.421875 4 2 0.25 0.210938 0.210938 4 3 0.25 0.046875 0.046875 4 4 0.25 0.00390625 0.00390625 BERNSTEIN_POLY_TEST Normal end of execution. BPAB_TEST Python version: 3.6.5 BPAB computes Bernstein polynomials; The Bernstein polynomials of degree 10 based on the interval [0.000000,1.000000] evaluated at X = 0.3 I Bern(I,X) 0 0.0282475 1 0.121061 2 0.233474 3 0.266828 4 0.200121 5 0.102919 6 0.0367569 7 0.00900169 8 0.0014467 9 0.000137781 10 5.9049e-06 BPAB_TEST Normal end of execution. CARDAN_POLY_TEST Python version: 3.6.5 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 2 1 0.25 0.25 2 -0.9375 -0.9375 3 -0.359375 -0.359375 4 0.378906 0.378906 5 0.274414 0.274414 6 -0.12085 -0.12085 7 -0.167419 -0.167419 8 0.0185699 0.0185699 9 0.0883522 0.0883522 10 0.0128031 0.0128031 CARDAN_POLY_TEST Normal end of execution. CARDAN_POLY_COEF_TEST Python version: 3.6.5 CARDAN_POLY_COEF returns the coefficients of a Cardan polynomial. We use the parameter S = 1 Table of polyomial coefficients: 0: 2.000000 1: 0.000000 1.000000 2: -2.000000 0.000000 1.000000 3: 0.000000 -3.000000 0.000000 1.000000 4: 2.000000 0.000000 -4.000000 0.000000 1.000000 5: 0.000000 5.000000 0.000000 -5.000000 0.000000 1.000000 6: -2.000000 0.000000 9.000000 0.000000 -6.000000 0.000000 1.000000 7: 0.000000 -7.000000 0.000000 14.000000 0.000000 -7.000000 0.000000 1.000000 8: 2.000000 0.000000 -16.000000 0.000000 20.000000 0.000000 -8.000000 0.000000 1.000000 9: 0.000000 9.000000 0.000000 -30.000000 0.000000 27.000000 0.000000 -9.000000 0.000000 1.000000 10: -2.000000 0.000000 25.000000 0.000000 -50.000000 0.000000 35.000000 0.000000 -10.000000 0.000000 1.000000 CARDAN_POLY_COEF_TEST Normal end of execution. CARDINAL_COS_TEST Python version: 3.6.5 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: 1.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 CARDINAL_COS_TEST Normal end of execution. CARDINAL_SIN_TEST Python version: 3.6.5 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: 1.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 -0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 -0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 -0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 -0.00 0.00 1.00 CARDINAL_SIN_TEST Normal end of execution. CATALAN_TEST Python version: 3.6.5 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_TEST Normal end of execution. CATALAN_ROW_NEXT_TEST Python version: 3.6.5 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 CATALAN_ROW_NEXT_TEST Normal end of execution. CHARLIER_TEST: Python version: 3.6.5 CHARLIER evaluates Charlier polynomials. N A X P(N,A,X) 0 0.250000 0.000000 1.000000 1 0.250000 0.000000 -0.000000 2 0.250000 0.000000 -4.000000 3 0.250000 0.000000 -36.000000 4 0.250000 0.000000 -420.000000 5 0.250000 0.000000 -6564.000000 0 0.250000 0.500000 1.000000 1 0.250000 0.500000 -2.000000 2 0.250000 0.500000 -10.000000 3 0.250000 0.500000 -54.000000 4 0.250000 0.500000 -474.000000 5 0.250000 0.500000 -6246.000000 0 0.250000 1.000000 1.000000 1 0.250000 1.000000 -4.000000 2 0.250000 1.000000 -8.000000 3 0.250000 1.000000 -8.000000 4 0.250000 1.000000 24.000000 5 0.250000 1.000000 440.000000 0 0.250000 1.500000 1.000000 1 0.250000 1.500000 -6.000000 2 0.250000 1.500000 2.000000 3 0.250000 1.500000 54.000000 4 0.250000 1.500000 354.000000 5 0.250000 1.500000 3030.000000 0 0.250000 2.000000 1.000000 1 0.250000 2.000000 -8.000000 2 0.250000 2.000000 20.000000 3 0.250000 2.000000 84.000000 4 0.250000 2.000000 180.000000 5 0.250000 2.000000 276.000000 0 0.250000 2.500000 1.000000 1 0.250000 2.500000 -10.000000 2 0.250000 2.500000 46.000000 3 0.250000 2.500000 34.000000 4 0.250000 2.500000 -450.000000 5 0.250000 2.500000 -3694.000000 0 0.500000 0.000000 1.000000 1 0.500000 0.000000 -0.000000 2 0.500000 0.000000 -2.000000 3 0.500000 0.000000 -10.000000 4 0.500000 0.000000 -58.000000 5 0.500000 0.000000 -442.000000 0 0.500000 0.500000 1.000000 1 0.500000 0.500000 -1.000000 2 0.500000 0.500000 -4.000000 3 0.500000 0.500000 -12.000000 4 0.500000 0.500000 -48.000000 5 0.500000 0.500000 -288.000000 0 0.500000 1.000000 1.000000 1 0.500000 1.000000 -2.000000 2 0.500000 1.000000 -4.000000 3 0.500000 1.000000 -4.000000 4 0.500000 1.000000 4.000000 5 0.500000 1.000000 60.000000 0 0.500000 1.500000 1.000000 1 0.500000 1.500000 -3.000000 2 0.500000 1.500000 -2.000000 3 0.500000 1.500000 8.000000 4 0.500000 1.500000 44.000000 5 0.500000 1.500000 200.000000 0 0.500000 2.000000 1.000000 1 0.500000 2.000000 -4.000000 2 0.500000 2.000000 2.000000 3 0.500000 2.000000 18.000000 4 0.500000 2.000000 42.000000 5 0.500000 2.000000 66.000000 0 0.500000 2.500000 1.000000 1 0.500000 2.500000 -5.000000 2 0.500000 2.500000 8.000000 3 0.500000 2.500000 20.000000 4 0.500000 2.500000 -8.000000 5 0.500000 2.500000 -192.000000 0 1.000000 0.000000 1.000000 1 1.000000 0.000000 -0.000000 2 1.000000 0.000000 -1.000000 3 1.000000 0.000000 -3.000000 4 1.000000 0.000000 -9.000000 5 1.000000 0.000000 -33.000000 0 1.000000 0.500000 1.000000 1 1.000000 0.500000 -0.500000 2 1.000000 0.500000 -1.750000 3 1.000000 0.500000 -3.375000 4 1.000000 0.500000 -6.562500 5 1.000000 0.500000 -16.031250 0 1.000000 1.000000 1.000000 1 1.000000 1.000000 -1.000000 2 1.000000 1.000000 -2.000000 3 1.000000 1.000000 -2.000000 4 1.000000 1.000000 0.000000 5 1.000000 1.000000 8.000000 0 1.000000 1.500000 1.000000 1 1.000000 1.500000 -1.500000 2 1.000000 1.500000 -1.750000 3 1.000000 1.500000 0.375000 4 1.000000 1.500000 6.187500 5 1.000000 1.500000 20.156250 0 1.000000 2.000000 1.000000 1 1.000000 2.000000 -2.000000 2 1.000000 2.000000 -1.000000 3 1.000000 2.000000 3.000000 4 1.000000 2.000000 9.000000 5 1.000000 2.000000 15.000000 0 1.000000 2.500000 1.000000 1 1.000000 2.500000 -2.500000 2 1.000000 2.500000 0.250000 3 1.000000 2.500000 5.125000 4 1.000000 2.500000 6.937500 5 1.000000 2.500000 -3.156250 0 2.000000 0.000000 1.000000 1 2.000000 0.000000 -0.000000 2 2.000000 0.000000 -0.500000 3 2.000000 0.000000 -1.000000 4 2.000000 0.000000 -1.750000 5 2.000000 0.000000 -3.250000 0 2.000000 0.500000 1.000000 1 2.000000 0.500000 -0.250000 2 2.000000 0.500000 -0.812500 3 2.000000 0.500000 -1.171875 4 2.000000 0.500000 -1.417969 5 2.000000 0.500000 -1.555664 0 2.000000 1.000000 1.000000 1 2.000000 1.000000 -0.500000 2 2.000000 1.000000 -1.000000 3 2.000000 1.000000 -1.000000 4 2.000000 1.000000 -0.500000 5 2.000000 1.000000 0.750000 0 2.000000 1.500000 1.000000 1 2.000000 1.500000 -0.750000 2 2.000000 1.500000 -1.062500 3 2.000000 1.500000 -0.578125 4 2.000000 1.500000 0.582031 5 2.000000 1.500000 2.465820 0 2.000000 2.000000 1.000000 1 2.000000 2.000000 -1.000000 2 2.000000 2.000000 -1.000000 3 2.000000 2.000000 0.000000 4 2.000000 2.000000 1.500000 5 2.000000 2.000000 3.000000 0 2.000000 2.500000 1.000000 1 2.000000 2.500000 -1.250000 2 2.000000 2.500000 -0.812500 3 2.000000 2.500000 0.640625 4 2.000000 2.500000 2.019531 5 2.000000 2.500000 2.252930 0 10.000000 0.000000 1.000000 1 10.000000 0.000000 -0.000000 2 10.000000 0.000000 -0.100000 3 10.000000 0.000000 -0.120000 4 10.000000 0.000000 -0.126000 5 10.000000 0.000000 -0.128400 0 10.000000 0.500000 1.000000 1 10.000000 0.500000 -0.050000 2 10.000000 0.500000 -0.152500 3 10.000000 0.500000 -0.165375 4 10.000000 0.500000 -0.160969 5 10.000000 0.500000 -0.151158 0 10.000000 1.000000 1.000000 1 10.000000 1.000000 -0.100000 2 10.000000 1.000000 -0.200000 3 10.000000 1.000000 -0.200000 4 10.000000 1.000000 -0.180000 5 10.000000 1.000000 -0.154000 0 10.000000 1.500000 1.000000 1 10.000000 1.500000 -0.150000 2 10.000000 1.500000 -0.242500 3 10.000000 1.500000 -0.224625 4 10.000000 1.500000 -0.185569 5 10.000000 1.500000 -0.142111 0 10.000000 2.000000 1.000000 1 10.000000 2.000000 -0.200000 2 10.000000 2.000000 -0.280000 3 10.000000 2.000000 -0.240000 4 10.000000 2.000000 -0.180000 5 10.000000 2.000000 -0.120000 0 10.000000 2.500000 1.000000 1 10.000000 2.500000 -0.250000 2 10.000000 2.500000 -0.312500 3 10.000000 2.500000 -0.246875 4 10.000000 2.500000 -0.165469 5 10.000000 2.500000 -0.091539 CHARLIER_TEST Normal end of execution. CHEBY_T_POLY_TEST Python version: 3.6.5 CHEBY_T_POLY evaluates the Chebyshev T polynomial. N X Exact F T(N)(X) 0 0.8 1 1 1 0.8 0.8 0.8 2 0.8 0.28 0.28 3 0.8 -0.352 -0.352 4 0.8 -0.8432 -0.8432 5 0.8 -0.99712 -0.99712 6 0.8 -0.752192 -0.752192 7 0.8 -0.206387 -0.206387 8 0.8 0.421972 0.421972 9 0.8 0.881543 0.881543 10 0.8 0.988497 0.988497 11 0.8 0.700051 0.700051 12 0.8 0.131586 0.131586 CHEBY_T_POLY_TEST Normal end of execution. CHEBY_T_POLY_COEF_TEST Python version: 3.6.5 CHEBY_T_POLY_COEF determines the Chebyshev T polynomial coefficients. T(0) 1.000000 T(1) 1.000000 * x 0.000000 T(2) 2.000000 * x^2 0.000000 * x -1.000000 T(3) 4.000000 * x^3 0.000000 * x^2 -3.000000 * x -0.000000 T(4) 8.000000 * x^4 0.000000 * x^3 -8.000000 * x^2 -0.000000 * x 1.000000 T(5) 16.000000 * x^5 0.000000 * x^4 -20.000000 * x^3 -0.000000 * x^2 5.000000 * x 0.000000 CHEBY_T_POLY_COEF_TEST Normal end of execution. CHEBY_T_POLY_ZERO_TEST: Python version: 3.6.5 CHEBY_T_POLY_ZERO returns zeroes of T(N,X). N X T(N,X) 1 6.12323e-17 6.12323e-17 2 0.707107 2.22045e-16 2 -0.707107 -2.22045e-16 3 0.866025 3.33067e-16 3 6.12323e-17 -1.83697e-16 3 -0.866025 -3.33067e-16 4 0.92388 -2.22045e-16 4 0.382683 -2.22045e-16 4 -0.382683 1.11022e-16 4 -0.92388 -2.22045e-16 CHEBY_T_POLY_ZERO_TEST Normal end of execution. CHEBY_U_POLY_TEST Python version: 3.6.5 CHEBY_U_POLY evaluates the Chebyshev U polynomial. N X Exact F U(N)(X) 0 0.8 1 1 1 0.8 1.6 1.6 2 0.8 1.56 1.56 3 0.8 0.896 0.896 4 0.8 -0.1264 -0.1264 5 0.8 -1.09824 -1.09824 6 0.8 -1.63078 -1.63078 7 0.8 -1.51101 -1.51101 8 0.8 -0.786839 -0.786839 9 0.8 0.252072 0.252072 10 0.8 1.19015 1.19015 11 0.8 1.65217 1.65217 12 0.8 1.45333 1.45333 CHEBY_U_POLY_TEST Normal end of execution. CHEBY_U_POLY_COEF_TEST Python version: 3.6.5 CHEBY_U_POLY_COEF determines the Chebyshev U polynomial coefficients. U(0) 1.000000 U(1) 2.000000 * x 0.000000 U(2) 4.000000 * x^2 0.000000 * x -1.000000 U(3) 8.000000 * x^3 0.000000 * x^2 -4.000000 * x -0.000000 U(4) 16.000000 * x^4 0.000000 * x^3 -12.000000 * x^2 -0.000000 * x 1.000000 U(5) 32.000000 * x^5 0.000000 * x^4 -32.000000 * x^3 -0.000000 * x^2 6.000000 * x 0.000000 CHEBY_U_POLY_COEF_TEST Normal end of execution. CHEBYSHEV_DISCRETE_TEST Python version: 3.6.5 CHEBY_T_POLY evaluates the Chebyshev T polynomial. N M X T(N,M,X) 0 5 0 1 1 5 0 -4 2 5 0 12 3 5 0 -24 4 5 0 24 5 5 0 0 0 5 0.5 1 1 5 0.5 -3 2 5 0.5 1.5 3 5 0.5 34.5 4 5 0.5 -199.125 5 5 0.5 826.875 0 5 1 1 1 5 1 -2 2 5 1 -6 3 5 1 48 4 5 1 -96 5 5 1 0 0 5 1.5 1 1 5 1.5 -1 2 5 1.5 -10.5 3 5 1.5 31.5 4 5 1.5 70.875 5 5 1.5 -354.375 0 5 2 1 1 5 2 0 2 5 2 -12 3 5 2 -0 4 5 2 144 5 5 2 0 0 5 2.5 1 1 5 2.5 1 2 5 2.5 -10.5 3 5 2.5 -31.5 4 5 2.5 70.875 5 5 2.5 354.375 CHEBYSHEV_DISCRETE_TEST Normal end of execution. COLLATZ_COUNT_TEST Python version: 3.6.5 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_TEST Normal end of execution. COMB_ROW_NEXT_TEST Python version: 3.6.5 COMB_ROW_NEXT computes the next row of Pascals 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 COMB_ROW_NEXT_TEST Normal end of execution. COMMUL_TEST Python version: 3.6.5 COMMUL computes a multinomial coefficient. N = 8 Number of factors = 2 0 6 1 2 Value of coefficient = 28 N = 8 Number of factors = 3 0 2 1 2 2 4 Value of coefficient = 420 N = 13 Number of factors = 4 0 5 1 3 2 3 3 2 Value of coefficient = 720720 COMMUL_TEST: Normal end of execution. COMPLETE_SYMMETRIC_POLY_TEST Python version: 3.6.5 COMPLETE_SYMMETRIC_POLY evaluates a complete symmetric polynomial in a given set of variables X. Variable vector X: 0: 1 1: 2 2: 3 3: 4 4: 5 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 COMPLETE_SYMMETRIC_POLY_TEST: Normal end of execution. COS_POWER_INT_TEST Python version: 3.6.5 COS_POWER_INT returns values of the integral of COS(X)^N from A to B. A B N Exact Computed 0.000000 3.141593 0 3.141593e+00 3.141593e+00 0.000000 3.141593 1 0.000000e+00 1.224647e-16 0.000000 3.141593 2 1.570796e+00 1.570796e+00 0.000000 3.141593 3 0.000000e+00 1.224647e-16 0.000000 3.141593 4 1.178097e+00 1.178097e+00 0.000000 3.141593 5 0.000000e+00 1.224647e-16 0.000000 3.141593 6 9.817477e-01 9.817477e-01 0.000000 3.141593 7 0.000000e+00 1.224647e-16 0.000000 3.141593 8 8.590292e-01 8.590292e-01 0.000000 3.141593 9 0.000000e+00 1.224647e-16 0.000000 3.141593 10 7.731263e-01 7.731263e-01 COS_POWER_INT_TEST Normal end of execution. DELANNOY_TEST Python version: 3.6.5 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 85 377 1289 3653 8989 19825 40081 7 1 15 113 575 2241 7183 19825 48639 108545 8 1 17 145 833 3649 13073 40081 108545 265729 DELANNOY_TEST Normal end of execution. DOMINO_TILING_NUM_TEST: Python version: 3.6.5 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 DOMINO_TILING_NUM_TEST: Normal end of execution. EULER_NUMBER_TEST Python version: 3.6.5 EULER_NUMBER computes Euler numbers; I Exact Euler 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_NUMBER_TEST Normal end of execution. EULER_NUMBER2_TEST Python version: 3.6.5 EULER_NUMBER2 computes Euler numbers; I Exact Euler 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 Normal end of execution. EULER_POLY_TEST Python version: 3.6.5 EULER_POLY computes Euler polynomials; X = 0.5 N X F(X) 1 0.500000 2.77556e-17 2 0.500000 -0.25 3 0.500000 -1.45953e-06 4 0.500000 0.312497 5 0.500000 -3.32929e-06 6 0.500000 -0.953128 7 0.500000 -1.73264e-06 8 0.500000 5.41016 9 0.500000 -1.02449e-06 10 0.500000 -49.3369 11 0.500000 6.47439e-07 12 0.500000 659.855 13 0.500000 5.22754e-06 14 0.500000 -12168 15 0.500000 0.000218677 EULER_POLY_TEST Normal end of execution. EULERIAN_TEST Python version: 3.6.5 EULERIAN computes 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 EULERIAN_TEST Normal end of execution. F_HOFSTADTER_TEST Python version: 3.6.5 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 F_HOFSTADTER_TEST Normal end of execution. FIBONACCI_DIRECT_TEST Python version: 3.6.5 FIBONACCI_DIRECT computes a Fibonacci number directly; I F(I) 0 0 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_DIRECT_TEST Normal end of execution. FIBONACCI_FLOOR_TEST Python version: 3.6.5 FIBONACCI_FLOOR computes the largest Fibonacci number less than or equal to N. N Fibonacci Index 0 0 0 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_FLOOR_TEST Normal end of execution. FIBONACCI_RECURSIVE_TEST Python version: 3.6.5 FIBONACCI_RECURSIVE computes Fibonacci numbers recursively; The Fibonacci numbers: 0 0 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_RECURSIVE_TEST Normal end of execution. G_HOFSTADTER_TEST Python version: 3.6.5 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 G_HOFSTADTER_TEST Normal end of execution. GEGENBAUER_POLY_TEST Python version: 3.6.5 GEGENBAUER_POLY computes values of the Gegenbauer polynomial. N A X GPV GEGENBAUER 0 0.5 0.2 1 1 1 0.5 0.2 0.2 0.2 2 0.5 0.2 -0.44 -0.1 3 0.5 0.2 -0.28 0.0666667 4 0.5 0.2 0.232 -0.05 5 0.5 0.2 0.30752 0.04 6 0.5 0.2 -0.080576 -0.0333333 7 0.5 0.2 -0.293517 0.0285714 8 0.5 0.2 -0.0395648 -0.025 9 0.5 0.2 0.245971 0.0222222 10 0.5 0.2 0.129072 -0.02 2 0 0.4 0 0 2 1 0.4 -0.36 -0.8 2 2 0.4 -0.08 -3.2 2 3 0.4 0.84 -7.2 2 4 0.4 2.4 -12.8 2 5 0.4 4.6 -20 2 6 0.4 7.44 -28.8 2 7 0.4 10.92 -39.2 2 8 0.4 15.04 -51.2 2 9 0.4 19.8 -64.8 2 10 0.4 25.2 -80 5 3 -0.5 -9 -75.6 5 3 -0.4 -0.16128 -60.48 5 3 -0.3 -6.67296 -45.36 5 3 -0.2 -8.37504 -30.24 5 3 -0.1 -5.52672 -15.12 5 3 0 0 0 5 3 0.1 5.52672 15.12 5 3 0.2 8.37504 30.24 5 3 0.3 6.67296 45.36 5 3 0.4 0.16128 60.48 5 3 0.5 -9 75.6 5 3 0.6 -15.4253 90.72 5 3 0.7 -9.69696 105.84 5 3 0.8 22.441 120.96 5 3 0.9 100.889 136.08 5 3 1 252 151.2 GEGENBAUER_POLY_TEST Normal end of execution. GEN_HERMITE_POLY_TEST Python version: 3.6.5 GEN_HERMITE_POLY evaluates the generalized Hermite polynomial. Table of H(N,MU)(X) for N(max) = 10 MU = 0.000000 X = 0.000000 0 1.000000 1 0.000000 2 -2.000000 3 -0.000000 4 12.000000 5 0.000000 6 -120.000000 7 -0.000000 8 1680.000000 9 0.000000 10 -30240.000000 Table of H(N,MU)(X) for N(max) = 10 MU = 0.000000 X = 1.000000 0 1.000000 1 2.000000 2 2.000000 3 -4.000000 4 -20.000000 5 -8.000000 6 184.000000 7 464.000000 8 -1648.000000 9 -10720.000000 10 8224.000000 Table of H(N,MU)(X) for N(max) = 10 MU = 0.100000 X = 0.000000 0 1.000000 1 0.000000 2 -2.400000 3 -0.000000 4 15.360000 5 0.000000 6 -159.744000 7 -0.000000 8 2300.313600 9 0.000000 10 -42325.770240 Table of H(N,MU)(X) for N(max) = 10 MU = 0.100000 X = 0.500000 0 1.000000 1 1.000000 2 -1.400000 3 -5.400000 4 3.560000 5 46.760000 6 9.736000 7 -551.384000 8 -691.582400 9 8130.561600 10 20855.677760 Table of H(N,MU)(X) for N(max) = 10 MU = 0.500000 X = 0.500000 0 1.000000 1 1.000000 2 -3.000000 3 -7.000000 4 17.000000 5 73.000000 6 -131.000000 7 -1007.000000 8 1089.000000 9 17201.000000 10 -4579.000000 Table of H(N,MU)(X) for N(max) = 10 MU = 1.000000 X = 0.500000 0 1.000000 1 1.000000 2 -5.000000 3 -9.000000 4 41.000000 5 113.000000 6 -461.000000 7 -1817.000000 8 6481.000000 9 35553.000000 10 -107029.000000 GEN_HERMITE_POLY_TEST Normal end of execution. GEN_LAGUERRE_POLY_TEST Python version: 3.6.5 GEN_LAGUERRE_POLY evaluates the generalized Laguerre polynomial. Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.000000 X = 0.000000 0 1.000000 1 1.000000 2 1.000000 3 1.000000 4 1.000000 5 1.000000 6 1.000000 7 1.000000 8 1.000000 9 1.000000 10 0.000000 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.000000 X = 1.000000 0 1.000000 1 0.000000 2 -0.500000 3 -0.666667 4 -0.625000 5 -0.466667 6 -0.256944 7 -0.040476 8 0.153993 9 0.309744 10 0.000000 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.100000 X = 0.000000 0 1.000000 1 1.100000 2 1.155000 3 1.193500 4 1.223337 5 1.247804 6 1.268601 7 1.286724 8 1.302808 9 1.317284 10 0.000000 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.100000 X = 0.500000 0 1.000000 1 0.600000 2 0.230000 3 -0.067333 4 -0.289350 5 -0.442469 6 -0.535747 7 -0.578765 8 -0.580771 9 -0.550311 10 0.000000 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 0.500000 X = 0.500000 0 1.000000 1 1.000000 2 0.750000 3 0.416667 4 0.072917 5 -0.243750 6 -0.513715 7 -0.727703 8 -0.882836 9 -0.980303 10 0.000000 Table of L(N,ALPHA)(X) for N(max) = 10 ALPHA = 1.000000 X = 0.500000 0 1.000000 1 1.500000 2 1.625000 3 1.479167 4 1.148438 5 0.702865 6 0.198720 7 -0.319620 8 -0.817983 9 -1.270902 10 0.000000 GEN_LAGUERRE_POLY_TEST Normal end of execution. GUD_TEST Python version: 3.6.5 GUD evaluates the Gudermannian function. X Exact F GUD(X) -2.000000 -1.3017603360460150 -1.3017603360460150 0 -1.000000 -0.8657694832396586 -0.8657694832396586 0 0.000000 0.0000000000000000 0.0000000000000000 0 0.100000 0.0998337487934866 0.0998337487934866 0 0.200000 0.1986798470079397 0.1986798470079397 2.776e-17 0.500000 0.4803810791337294 0.4803810791337295 5.551e-17 1.000000 0.8657694832396586 0.8657694832396586 0 1.500000 1.1317283452505089 1.1317283452505091 2.22e-16 2.000000 1.3017603360460150 1.3017603360460150 0 2.500000 1.4069935689361539 1.4069935689361537 2.22e-16 3.000000 1.4713043411171931 1.4713043411171927 4.441e-16 3.500000 1.5104199075457001 1.5104199075457003 2.22e-16 4.000000 1.5341691443347329 1.5341691443347332 2.22e-16 GUD_TEST Normal end of execution. H_HOFSTADTER_TEST Python version: 3.6.5 H_HOFSTADTER evaluates Hofstadter's recursive G function. N G(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 H_HOFSTADTER_TEST Normal end of execution. HAIL_TEST Python version: 3.6.5 HAIL(I) computes the length of the hail sequence for I, also known as the 3*N+1 sequence. I HAIL(I)) 0 0 1 0 2 1 3 7 4 2 5 5 6 8 7 16 8 3 9 19 10 6 11 14 12 9 13 9 14 17 15 17 16 4 17 12 18 20 19 20 20 7 HAIL_TEST Normal end of execution. HERMITE_POLY_PHYS_TEST Python version: 3.6.5 HERMITE_POLY_PHYS computes the Hermite physicist polynomials; N X Exact F H(N)(X) 0 5.000000 1 1 1 5.000000 10 10 2 5.000000 98 98 3 5.000000 940 940 4 5.000000 8812 8812 5 5.000000 80600 80600 6 5.000000 717880 717880 7 5.000000 6.2116e+06 6.2116e+06 8 5.000000 5.20657e+07 5.20657e+07 9 5.000000 4.21271e+08 4.21271e+08 10 5.000000 3.27553e+09 3.27553e+09 11 5.000000 2.43299e+10 2.43299e+10 12 5.000000 1.71237e+11 1.71237e+11 5 0.000000 0 0 5 0.500000 41 41 5 1.000000 -8 -8 5 3.000000 3816 3816 5 10.000000 3.0412e+06 3.0412e+06 HERMITE_POLY_PHYS_TEST Normal end of execution. HERMITE_POLY_PHYS_COEF_TEST Python version: 3.6.5 HERMITE_POLY_PHYS_COEF determines the Hermite physicist's polynomial coefficients. H(0) 1.000000 H(1) 2.000000 * x 0.000000 H(2) 4.000000 * x^2 0.000000 * x -2.000000 H(3) 8.000000 * x^3 0.000000 * x^2 -12.000000 * x -0.000000 H(4) 16.000000 * x^4 0.000000 * x^3 -48.000000 * x^2 -0.000000 * x 12.000000 H(5) 32.000000 * x^5 0.000000 * x^4 -160.000000 * x^3 -0.000000 * x^2 120.000000 * x 0.000000 HERMITE_POLY_PHYS_COEF_TEST Normal end of execution. I4_CHOOSE_TEST Python version: 3.6.5 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_CHOOSE_TEST: Normal end of execution. I4_FACTOR_TEST Python version: 3.6.5 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_FACTOR_TEST: Normal end of execution. I4_FACTORIAL_TEST Python version: 3.6.5 I4_FACTORIAL evaluates the factorial function. N Exact I4_FACTORIAL(N) 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_FACTORIAL_TEST Normal end of execution. I4_FACTORIAL2_TEST Python version: 3.6.5 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_FACTORIAL2_TEST Normal end of execution. I4_IS_FIBONACCI_TEST I4_IS_FIBONACCI returns T or F depending on whether I4 is a Fibonacci number. I4 T/F -13 False 0 False 1 True 8 True 10 False 50 False 55 True 100 False 144 True 200 False I4_IS_FIBONACCI_TEST Normal end of execution. I4_IS_PRIME_TEST Python version: 3.6.5 I4_IS_PRIME reports whether an I4 is prime. I I4_IS_PRIME(I) -2 False -1 False 0 False 1 False 2 True 3 True 4 False 5 True 6 False 7 True 8 False 9 False 10 False 11 True 12 False 13 True 14 False 15 False 16 False 17 True 18 False 19 True 20 False 21 False 22 False 23 True 24 False 25 False I4_IS_PRIME_TEST Normal end of execution. I4_IS_TRIANGULAR_TEST Python version: 3.6.5 I4_IS_TRIANGULAR reports whether an I4 is prime. I I4_IS_TRIANGULAR(I) 0 True 1 True 2 False 3 True 4 False 5 False 6 True 7 False 8 False 9 False 10 True 11 False 12 False 13 False 14 False 15 True 16 False 17 False 18 False 19 False 20 False I4_IS_TRIANGULAR_TEST Normal end of execution. I4_PARTITION_DISTINCT_COUNT_TEST Python version: 3.6.5 I4_CHOOSE evaluates C(N,K). 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_PARTITION_DISTINCT_COUNT_TEST: Normal end of execution. I4_TO_TRIANGLE_LOWER_TEST Python version: 3.6.5 I4_TO_TRIANGLE_LOWER converts a linear index to a lower triangular one. K ==> ( I J ) 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 I4_TO_TRIANGLE_LOWER_TEST Normal end of execution. I4_TO_TRIANGLE_UPPER_TEST Python version: 3.6.5 I4_TO_TRIANGLE_UPPER converts a linear index to an upper triangular one. K ==> ( I J ) 1 1 1 2 1 2 3 2 2 4 1 3 5 2 3 6 3 3 7 1 4 8 2 4 9 3 4 10 4 4 11 1 5 12 2 5 13 3 5 14 4 5 15 5 5 16 1 6 17 2 6 18 3 6 19 4 6 20 5 6 I4_TO_TRIANGLE_UPPER_TEST Normal end of execution. JACOBI_POLY_TEST Python version: 3.6.5 JACOBI_POLY computes values of the Jacobi polynomial. N A B X GPV JACOBI 0 0 1 0.5 1 1 1 0 1 0.5 0.25 0.25 2 0 1 0.5 -0.375 -0.375 3 0 1 0.5 -0.484375 -0.484375 4 0 1 0.5 -0.132812 -0.132812 5 0 1 0.5 0.275391 0.275391 5 1 1 0.5 -0.164062 -0.164062 5 2 1 0.5 -1.1748 -1.1748 5 3 1 0.5 -2.36133 -2.36133 5 4 1 0.5 -2.61621 -2.61621 5 5 1 0.5 0.117188 0.117188 5 0 2 0.5 0.421875 0.421875 5 0 3 0.5 0.504883 0.504883 5 0 4 0.5 0.509766 0.509766 5 0 5 0.5 0.430664 0.430664 5 0 1 -1 -6 -6 5 0 1 -0.8 0.03862 0.03862 5 0 1 -0.6 0.81184 0.81184 5 0 1 -0.4 0.03666 0.03666 5 0 1 -0.2 -0.48512 -0.48512 5 0 1 0 -0.3125 -0.3125 5 0 1 0.2 0.18912 0.18912 5 0 1 0.4 0.40234 0.40234 5 0 1 0.6 0.01216 0.01216 5 0 1 0.8 -0.43962 -0.43962 5 0 1 1 1 1 JACOBI_POLY_TEST Normal end of execution. JACOBI_SYMBOL_TEST Python version: 3.6.5 JACOBI_SYMBOL computes the Jacobi symbol (Q/P) which records whether 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 JACOBI_SYMBOL_TEST Normal end of execution. KRAWTCHOUK_TEST Python version: 3.6.5 KRAWTCHOUK computes Krawtchouk polynomials; N P X M K(N,P,X,M) 0 0.250000 0.000000 5 1.000000 1 0.250000 0.000000 5 -1.250000 2 0.250000 0.000000 5 0.625000 3 0.250000 0.000000 5 -0.156250 4 0.250000 0.000000 5 0.019531 5 0.250000 0.000000 5 -0.000977 0 0.250000 0.500000 5 1.000000 1 0.250000 0.500000 5 -0.750000 2 0.250000 0.500000 5 0.000000 3 0.250000 0.500000 5 0.187500 4 0.250000 0.500000 5 -0.105469 5 0.250000 0.500000 5 0.043945 0 0.250000 1.000000 5 1.000000 1 0.250000 1.000000 5 -0.250000 2 0.250000 1.000000 5 -0.375000 3 0.250000 1.000000 5 0.218750 4 0.250000 1.000000 5 -0.042969 5 0.250000 1.000000 5 0.002930 0 0.250000 1.500000 5 1.000000 1 0.250000 1.500000 5 0.250000 2 0.250000 1.500000 5 -0.500000 3 0.250000 1.500000 5 0.062500 4 0.250000 1.500000 5 0.050781 5 0.250000 1.500000 5 -0.022461 0 0.250000 2.000000 5 1.000000 1 0.250000 2.000000 5 0.750000 2 0.250000 2.000000 5 -0.375000 3 0.250000 2.000000 5 -0.156250 4 0.250000 2.000000 5 0.082031 5 0.250000 2.000000 5 -0.008789 0 0.250000 2.500000 5 1.000000 1 0.250000 2.500000 5 1.250000 2 0.250000 2.500000 5 0.000000 3 0.250000 2.500000 5 -0.312500 4 0.250000 2.500000 5 0.019531 5 0.250000 2.500000 5 0.020508 0 0.500000 0.000000 5 1.000000 1 0.500000 0.000000 5 -2.500000 2 0.500000 0.000000 5 2.500000 3 0.500000 0.000000 5 -1.250000 4 0.500000 0.000000 5 0.312500 5 0.500000 0.000000 5 -0.031250 0 0.500000 0.500000 5 1.000000 1 0.500000 0.500000 5 -2.000000 2 0.500000 0.500000 5 1.375000 3 0.500000 0.500000 5 -0.250000 4 0.500000 0.500000 5 -0.132812 5 0.500000 0.500000 5 0.078125 0 0.500000 1.000000 5 1.000000 1 0.500000 1.000000 5 -1.500000 2 0.500000 1.000000 5 0.500000 3 0.500000 1.000000 5 0.250000 4 0.500000 1.000000 5 -0.187500 5 0.500000 1.000000 5 0.031250 0 0.500000 1.500000 5 1.000000 1 0.500000 1.500000 5 -1.000000 2 0.500000 1.500000 5 -0.125000 3 0.500000 1.500000 5 0.375000 4 0.500000 1.500000 5 -0.070312 5 0.500000 1.500000 5 -0.023438 0 0.500000 2.000000 5 1.000000 1 0.500000 2.000000 5 -0.500000 2 0.500000 2.000000 5 -0.500000 3 0.500000 2.000000 5 0.250000 4 0.500000 2.000000 5 0.062500 5 0.500000 2.000000 5 -0.031250 0 0.500000 2.500000 5 1.000000 1 0.500000 2.500000 5 0.000000 2 0.500000 2.500000 5 -0.625000 3 0.500000 2.500000 5 -0.000000 4 0.500000 2.500000 5 0.117188 5 0.500000 2.500000 5 0.000000 KRAWTCHOUK_TEST Normal end of execution. LAGUERRE_ASSOCIATED_TEST Python version: 3.6.5 LAGUERRE_ASSOCIATED evaluates the associated Laguerre polynomials; Table of L(N,M)(X) for N(max) = 6 M = 0 X = 0.000000 0 1 1 1 2 1 3 1 4 1 5 1 6 1 Table of L(N,M)(X) for N(max) = 6 M = 0 X = 1.000000 0 1 1 0 2 -0.5 3 -0.666667 4 -0.625 5 -0.466667 6 -0.256944 Table of L(N,M)(X) for N(max) = 6 M = 1 X = 0.000000 0 1 1 2 2 3 3 4 4 5 5 6 6 7 Table of L(N,M)(X) for N(max) = 6 M = 2 X = 0.500000 0 1 1 2.5 2 4.125 3 5.60417 4 6.7526 5 7.45547 6 7.65419 Table of L(N,M)(X) for N(max) = 6 M = 3 X = 0.500000 0 1 1 3.5 2 7.625 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 1 1.5 2 1.625 3 1.47917 4 1.14844 5 0.702865 6 0.19872 LAGUERRE_ASSOCIATED_TEST Normal end of execution. LAGUERRE_POLY_TEST Python version: 3.6.5 LAGUERRE_POLY computes Laguerre polynomials; N X Exact F L(N)(X) 0 1.000000 1.000000 1.000000 1 1.000000 0.000000 0.000000 2 1.000000 -0.500000 -0.500000 3 1.000000 -0.666667 -0.666667 4 1.000000 -0.625000 -0.625000 5 1.000000 -0.466667 -0.466667 6 1.000000 -0.256944 -0.256944 7 1.000000 -0.040476 -0.040476 8 1.000000 0.153993 0.153993 9 1.000000 0.309744 0.309744 10 1.000000 0.418946 0.418946 11 1.000000 0.480134 0.480134 12 1.000000 0.496212 0.496212 5 0.500000 -0.445573 -0.445573 5 3.000000 0.850000 0.850000 5 5.000000 -3.166667 -3.166667 5 10.000000 34.333333 34.333333 LAGUERRE_POLY_TEST Normal end of execution. LAGUERRE_POLY_COEF_TEST Python version: 3.6.5 LAGUERRE_POLY_COEF determines the Laguerre polynomial coefficients. L(0) 1.000000 L(1) -1.000000 * x 1.000000 L(2) 0.500000 * x^2 -2.000000 * x 1.000000 L(3) -0.166667 * x^3 1.500000 * x^2 -3.000000 * x 1.000000 L(4) 0.041667 * x^4 -0.666667 * x^3 3.000000 * x^2 -4.000000 * x 1.000000 L(5) -0.008333 * x^5 0.208333 * x^4 -1.666667 * x^3 5.000000 * x^2 -5.000000 * x 1.000000 Factorially scaled L(0) 1.000000 Factorially scaled L(1) -1.000000 * x 1.000000 Factorially scaled L(2) 1.000000 * x^2 -4.000000 * x 2.000000 Factorially scaled L(3) -1.000000 * x^3 9.000000 * x^2 -18.000000 * x 6.000000 Factorially scaled L(4) 1.000000 * x^4 -16.000000 * x^3 72.000000 * x^2 -96.000000 * x 24.000000 Factorially scaled L(5) -1.000000 * x^5 25.000000 * x^4 -200.000000 * x^3 600.000000 * x^2 -600.000000 * x 120.000000 LAGUERRE_POLY_COEF_TEST Normal end of execution. LEGENDRE_ASSOCIATED_TEST Python version: 3.6.5 LEGENDRE_ASSOCIATED evaluates the associated Legendre functions; N M X Exact F PNM(X) 1 0 0.000000 0.000000 0.000000 2 0 0.000000 -0.500000 -0.500000 3 0 0.000000 0.000000 -0.000000 4 0 0.000000 0.375000 0.375000 5 0 0.000000 0.000000 0.000000 1 1 0.500000 -0.866025 -0.866025 2 1 0.500000 -1.299038 -1.299038 3 1 0.500000 -0.324760 -0.324760 4 1 0.500000 1.353165 1.353165 3 0 0.200000 -0.280000 -0.280000 3 1 0.200000 1.175755 1.175755 3 2 0.200000 2.880000 2.880000 3 3 0.200000 -14.109061 -14.109061 4 2 0.250000 -3.955078 -3.955078 5 2 0.250000 -9.997559 -9.997559 6 3 0.250000 82.653114 82.653114 7 3 0.250000 20.244428 20.244428 8 4 0.250000 -423.799753 -423.799753 9 4 0.250000 1638.320625 1638.320625 10 5 0.250000 -20256.873892 -20256.873892 LEGENDRE_ASSOCIATED_TEST Normal end of execution. LEGENDRE_ASSOCIATED_NORMALIZED_TEST Python version: 3.6.5 LEGENDRE_ASSOCIATED_NORMALIZED evaluates the associated Legendre functions; N M X Exact F PNM(X) 0 0 0.500000 0.282095 0.282095 1 0 0.500000 0.244301 0.244301 1 1 0.500000 -0.299207 -0.299207 2 0 0.500000 -0.078848 -0.078848 2 1 0.500000 -0.334523 -0.334523 2 2 0.500000 0.289706 0.289706 3 0 0.500000 -0.326529 -0.326529 3 1 0.500000 -0.069971 -0.069971 3 2 0.500000 0.383245 0.383245 3 3 0.500000 -0.270995 -0.270995 4 0 0.500000 -0.244629 -0.244629 4 1 0.500000 0.256066 0.256066 4 2 0.500000 0.188169 0.188169 4 3 0.500000 -0.406492 -0.406492 4 4 0.500000 0.248925 0.248925 5 0 0.500000 0.084058 0.084058 5 1 0.500000 0.329379 0.329379 5 2 0.500000 -0.158885 -0.158885 5 3 0.500000 -0.280871 -0.280871 5 4 0.500000 0.412795 0.412795 5 5 0.500000 -0.226097 -0.226097 LEGENDRE_ASSOCIATED_NORMALIZED_TEST Normal end of execution. LEGENDRE_FUNCTION_Q_TEST Python version: 3.6.5 LEGENDRE_FUNCTION_Q computes Legendre QN functions N X Exact F Q(N)(X) 0 0.250000 0.255413 0.255413 1 0.250000 -0.936147 -0.936147 2 0.250000 -0.478761 0.468073 3 0.250000 0.424614 -0.312049 4 0.250000 0.544840 0.234037 5 0.250000 -0.094513 -0.187229 6 0.250000 -0.497352 0.156024 7 0.250000 -0.149902 -0.133735 8 0.250000 0.364916 0.117018 9 0.250000 0.305568 -0.104016 10 0.250000 -0.183280 0.093615 3 0.000000 0.666667 -0.333333 3 0.100000 0.626867 -0.329989 3 0.200000 0.509902 -0.319818 3 0.300000 0.323275 -0.302381 3 0.400000 0.080261 -0.276847 3 0.500000 -0.198655 -0.241782 3 0.600000 -0.482866 -0.194704 3 0.700000 -0.725289 -0.130963 3 0.800000 -0.845444 -0.040370 3 0.900000 -0.662710 0.108333 LEGENDRE_FUNCTION_Q_TEST Normal end of execution. LEGENDRE_POLY_TEST Python version: 3.6.5 LEGENDRE_POLY computes Legendre polynomials; N X Exact F L(N)(X) 0 0.250000 1.000000 1.000000 1 0.250000 0.250000 0.250000 2 0.250000 -0.406250 -0.406250 3 0.250000 -0.335938 -0.335938 4 0.250000 0.157715 0.157715 5 0.250000 0.339722 0.339722 6 0.250000 0.024277 0.024277 7 0.250000 -0.279919 -0.279919 8 0.250000 -0.152454 -0.152454 9 0.250000 0.176824 0.176824 10 0.250000 0.221200 0.221200 3 0.000000 0.000000 -0.000000 3 0.100000 -0.147500 -0.147500 3 0.200000 -0.280000 -0.280000 3 0.300000 -0.382500 -0.382500 3 0.400000 -0.440000 -0.440000 3 0.500000 -0.437500 -0.437500 3 0.600000 -0.360000 -0.360000 3 0.700000 -0.192500 -0.192500 3 0.800000 0.080000 0.080000 3 0.900000 0.472500 0.472500 3 1.000000 1.000000 1.000000 LEGENDRE_POLY_TEST Normal end of execution. LEGENDRE_POLY_COEF_TEST Python version: 3.6.5 LEGENDRE_POLY_COEF determines the Legendre polynomial coefficients. L(0) 1.000000 L(1) 1.000000 * x 0.000000 L(2) 1.500000 * x^2 0.000000 * x -0.500000 L(3) 2.500000 * x^3 0.000000 * x^2 -1.500000 * x -0.000000 L(4) 4.375000 * x^4 0.000000 * x^3 -3.750000 * x^2 -0.000000 * x 0.375000 L(5) 7.875000 * x^5 0.000000 * x^4 -8.750000 * x^3 -0.000000 * x^2 1.875000 * x 0.000000 LEGENDRE_POLY_COEF_TEST Normal end of execution. LEGENDRE_SYMBOL_TEST Python version: 3.6.5 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 LEGENDRE_SYMBOL_TEST_TEST Normal end of execution. LERCH_TEST Python version: 3.6.5 LERCH evaluates the Lerch function; Z S A Lerch Lerch Tabulated Computed 1 2 0 1.64493 1.64492 1 3 0 1.20206 1.20206 1 10 0 1.00099 1.00099 0.5 2 1 1.16448 1.16448 0.5 3 1 1.07443 1.07443 0.5 10 1 1.00049 1.00049 0.333333 2 2 0.295919 0.295919 0.333333 3 2 0.139451 0.139451 0.333333 10 2 0.000982318 0.000982318 0.1 2 3 0.117791 0.117791 0.1 3 3 0.0386845 0.0386845 0.1 10 3 1.70315e-05 1.70315e-05 LERCH_TEST Normal end of execution. LOCK_TEST Python version: 3.6.5 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 LOCK_TEST Normal end of execution. MEIXNER_TEST: Python version: 3.6.5 MEIXNER evaluates Meixner polynomials. N BETA C X M(N,BETA,C,X) 0 0.5 0.125 0 1 1 0.5 0.125 0 1 2 0.5 0.125 0 0.125 3 0.5 0.125 0 -0.684375 4 0.5 0.125 0 -0.779297 5 0.5 0.125 0 -0.181787 0 0.5 0.125 0.5 1 1 0.5 0.125 0.5 -6 2 0.5 0.125 0.5 -3.66667 3 0.5 0.125 0.5 2.05 4 0.5 0.125 0.5 4.9 5 0.5 0.125 0.5 2.66944 0 0.5 0.125 1 1 1 0.5 0.125 1 -13 2 0.5 0.125 1 -3.375 3 0.5 0.125 1 8.45937 4 0.5 0.125 1 9.08633 5 0.5 0.125 1 -0.0737033 0 0.5 0.125 1.5 1 1 0.5 0.125 1.5 -20 2 0.5 0.125 1.5 1 3 0.5 0.125 1.5 16.4 4 0.5 0.125 1.5 9.1 5 0.5 0.125 1.5 -8.00556 0 0.5 0.125 2 1 1 0.5 0.125 2 -27 2 0.5 0.125 2 9.45833 3 0.5 0.125 2 23.7281 4 0.5 0.125 2 3.3332 5 0.5 0.125 2 -19.0084 0 0.5 0.125 2.5 1 1 0.5 0.125 2.5 -34 2 0.5 0.125 2.5 22 3 0.5 0.125 2.5 28.3 4 0.5 0.125 2.5 -8.75 5 0.5 0.125 2.5 -29.7736 0 1 0.25 0 1 1 1 0.25 0 1 2 1 0.25 0 0.25 3 1 0.25 0 -0.4375 4 1 0.25 0 -0.625 5 1 0.25 0 -0.30625 0 1 0.25 0.5 1 1 1 0.25 0.5 -0.5 2 1 0.25 0.5 -0.78125 3 1 0.25 0.5 -0.285156 4 1 0.25 0.5 0.327515 5 1 0.25 0.5 0.547452 0 1 0.25 1 1 1 1 0.25 1 -2 2 1 0.25 1 -1.25 3 1 0.25 1 0.5 4 1 0.25 1 1.34375 5 1 0.25 1 0.809375 0 1 0.25 1.5 1 1 1 0.25 1.5 -3.5 2 1 0.25 1.5 -1.15625 3 1 0.25 1.5 1.70703 4 1 0.25 1.5 2.09412 5 1 0.25 1.5 0.362021 0 1 0.25 2 1 1 1 0.25 2 -5 2 1 0.25 2 -0.5 3 1 0.25 2 3.125 4 1 0.25 2 2.32812 5 1 0.25 2 -0.753906 0 1 0.25 2.5 1 1 1 0.25 2.5 -6.5 2 1 0.25 2.5 0.71875 3 1 0.25 2.5 4.54297 4 1 0.25 2.5 1.87439 5 1 0.25 2.5 -2.36916 0 2 0.5 0 1 1 2 0.5 0 1 2 2 0.5 0 0.5 3 2 0.5 0 0 4 2 0.5 0 -0.3 5 2 0.5 0 -0.35 0 2 0.5 0.5 1 1 2 0.5 0.5 0.75 2 2 0.5 0.5 0.229167 3 2 0.5 0.5 -0.160156 4 2 0.5 0.5 -0.305664 5 2 0.5 0.5 -0.237101 0 2 0.5 1 1 1 2 0.5 1 0.5 2 2 0.5 1 0 3 2 0.5 1 -0.25 4 2 0.5 1 -0.25 5 2 0.5 1 -0.104167 0 2 0.5 1.5 1 1 2 0.5 1.5 0.25 2 2 0.5 1.5 -0.1875 3 2 0.5 1.5 -0.277344 4 2 0.5 1.5 -0.150977 5 2 0.5 1.5 0.0276286 0 2 0.5 2 1 1 2 0.5 2 0 2 2 0.5 2 -0.333333 3 2 0.5 2 -0.25 4 2 0.5 2 -0.025 5 2 0.5 2 0.141667 0 2 0.5 2.5 1 1 2 0.5 2.5 -0.25 2 2 0.5 2.5 -0.4375 3 2 0.5 2.5 -0.175781 4 2 0.5 2.5 0.113086 5 2 0.5 2.5 0.225562 MEIXNER_TEST Normal end of execution. MERTENS_TEST Python version: 3.6.5 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 MERTENS_TEST Normal end of execution. MOEBIUS_TEST Python version: 3.6.5 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 MOEBIUS_TEST Normal end of execution. MOTZKIN_TEST Python version: 3.6.5 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 MOTZKIN_TEST Normal end of execution. OMEGA_TEST Python version: 3.6.5 OMEGA counts the distinct prime divisors of an integer N. N Exact OMEGA(N) 1 0 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 OMEGA_TEST Normal end of execution. PENTAGON_NUM_TEST Python version: 3.6.5 PENTAGON_NUM computes the pentagonal numbers. N PENTAGON_NUM(N) 1 1 2 5 3 12 4 22 5 35 6 51 7 70 8 92 9 117 10 145 PENTAGON_NUM_TEST Normal end of execution. PHI_TEST Python version: 3.6.5 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 PHI_TEST Normal end of execution. PLANE_PARTITION_NUM_TEST Python version: 3.6.5 PLANE_PARTITION_NUM computes the number of plane partitions of an integer. 1 1 2 3 3 6 4 13 5 24 6 48 7 86 8 160 9 282 10 500 PLANE_PARTITION_NUM_TEST: Normal end of execution. POLY_BERNOULLI_TEST Python version: 3.6.5 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_BERNOULLI_TEST: Normal end of execution. POLY_COEF_COUNT_TEST Python version: 3.6.5 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 POLY_COEF_COUNT_TEST: Normal end of execution. PRIME_TEST Python version: 3.6.5 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 PRIME_TEST Normal end of execution. PYRAMID_NUM_TEST Python version: 3.6.5 PYRAMID_NUM computes the pyramidal numbers. 1 1 2 4 3 10 4 20 5 35 6 56 7 84 8 120 9 165 10 220 PYRAMID_NUM_TEST: Normal end of execution. PYRAMID_SQUARE_NUM_TEST Python version: 3.6.5 PYRAMID_SQUARE_NUM computes the pyramidal square numbers. 1 1 2 5 3 14 4 30 5 55 6 91 7 140 8 204 9 285 10 385 PYRAMID_SQUARE_NUM_TEST: Normal end of execution. R8_AGM_TEST Python version: 3.6.5 R8_AGM computes the arithmetic geometric mean. A B AGM AGM Diff (Tabulated) R8_AGM(A,B) 22.000000 96.000000 52.2746411987042379 52.2746411987042450 7.105e-15 83.000000 56.000000 68.8365300598585179 68.8365300598585179 0 42.000000 7.000000 20.6593011967340097 20.6593011967340061 3.553e-15 26.000000 11.000000 17.6968548737436500 17.6968548737436677 1.776e-14 4.000000 63.000000 23.8670497217533004 23.8670497217533040 3.553e-15 6.000000 45.000000 20.7170159828059930 20.7170159828059894 3.553e-15 40.000000 75.000000 56.1278422556166845 56.1278422556166845 0 80.000000 0.000000 0.0000000000000000 0.0000000000000000 0 90.000000 35.000000 59.2695650812296364 59.2695650812298851 2.487e-13 9.000000 1.000000 3.9362355036495553 3.9362355036495558 4.441e-16 53.000000 53.000000 53.0000000000000000 53.0000000000000000 0 1.000000 2.000000 1.4567910310469068 1.4567910310469068 0 1.000000 4.000000 2.2430285802876027 2.2430285802876027 0 1.000000 8.000000 3.6157561775973628 3.6157561775973628 0 R8_AGM_TEST Normal end of execution. R8_BETA_TEST: Python version: 3.6.5 R8_BETA evaluates the Beta function. X Y BETA(X,Y) R8_BETA(X,Y) 0.2 1 5 4.999999999999999 0.4 1 2.5 2.5 0.6 1 1.666666666666667 1.666666666666667 0.8 1 1.25 1.25 1 0.2 5 4.999999999999999 1 0.4 2.5 2.5 1 1 1 1 2 2 0.1666666666666667 0.1666666666666667 3 3 0.03333333333333333 0.03333333333333333 4 4 0.007142857142857143 0.007142857142857143 5 5 0.001587301587301587 0.001587301587301587 6 2 0.02380952380952381 0.02380952380952381 6 3 0.005952380952380952 0.005952380952380952 6 4 0.001984126984126984 0.001984126984126984 6 5 0.0007936507936507937 0.0007936507936507937 6 6 0.0003607503607503608 0.0003607503607503605 7 7 8.325008325008325e-05 8.32500832500834e-05 R8_BETA_TEST Normal end of execution. R8_CHOOSE_TEST Python version: 3.6.5 R8_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 5 0 1 5 1 5 5 2 10 5 3 10 5 4 5 5 5 1 R8_CHOOSE_TEST Normal end of execution. R8_ERF_TEST: Python version: 3.6.5 R8_ERF evaluates the error function. X ERF(X) R8_ERF(X) 0 0 0 0.1 0.1124629160182849 0.1124629160182849 0.2 0.2227025892104785 0.2227025892104785 0.3 0.3286267594591274 0.3286267594591273 0.4 0.4283923550466685 0.4283923550466684 0.5 0.5204998778130465 0.5204998778130465 0.6 0.6038560908479259 0.6038560908479259 0.7 0.6778011938374185 0.6778011938374184 0.8 0.7421009647076605 0.7421009647076605 0.9 0.7969082124228321 0.7969082124228322 1 0.8427007929497149 0.8427007929497148 1.1 0.8802050695740817 0.8802050695740817 1.2 0.9103139782296354 0.9103139782296354 1.3 0.9340079449406524 0.9340079449406524 1.4 0.9522851197626488 0.9522851197626487 1.5 0.9661051464753106 0.9661051464753108 1.6 0.976348383344644 0.976348383344644 1.7 0.9837904585907746 0.9837904585907746 1.8 0.9890905016357306 0.9890905016357308 1.9 0.9927904292352575 0.9927904292352574 2 0.9953222650189527 0.9953222650189527 R8_ERF_TEST Normal end of execution. R8_ERF_INVERSE_TEST: Python version: 3.6.5 R8_ERF_INVERSE inverts the error function. FX X R8_ERF_INVERSE(FX) 0 0 0 0.112463 0.1 0.09999999999999995 0.222703 0.2 0.2000000000000001 0.328627 0.3 0.2999999999999999 0.428392 0.4 0.4000000000000002 0.5205 0.5 0.5 0.603856 0.6 0.6 0.677801 0.7 0.7000000000000002 0.742101 0.8 0.8 0.796908 0.9 0.8999999999999996 0.842701 1 1 0.880205 1.1 1.1 0.910314 1.2 1.2 0.934008 1.3 1.3 0.952285 1.4 1.400000000000001 0.966105 1.5 1.5 0.976348 1.6 1.6 0.98379 1.7 1.700000000000001 0.989091 1.8 1.800000000000001 0.99279 1.9 1.899999999999997 0.995322 2 2.000000000000004 R8_ERF_INVERSE_TEST Normal end of execution. R8_EULER_CONSTANT_TEST: Python version: 3.6.5 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 0.422784 2 0.806853 0.229637 4 0.697039 0.119823 8 0.638416 0.0611999 16 0.60814 0.0309246 32 0.592759 0.0155436 64 0.585008 0.00779216 128 0.581117 0.00390116 256 0.579168 0.00195185 512 0.578192 0.000976245 1024 0.577704 0.000488202 2048 0.57746 0.000244121 4096 0.577338 0.000122065 8192 0.577277 6.10339e-05 16384 0.577246 3.05173e-05 32768 0.577231 1.52587e-05 65536 0.577223 7.62938e-06 131072 0.577219 3.81469e-06 262144 0.577218 1.90735e-06 524288 0.577217 9.53674e-07 1048576 0.577216 4.76837e-07 R8_EULER_CONSTANT_TEST Normal end of execution. R8_FACTORIAL_TEST Python version: 3.6.5 R8_FACTORIAL evaluates the factorial function. N Exact Computed 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 13 6227020800 6227020800 14 87178291200 87178291200 15 1307674368000 1307674368000 16 20922789888000 20922789888000 17 355687428096000 355687428096000 18 6402373705728000 6402373705728000 19 1.21645100408832e+17 1.21645100408832e+17 20 2.43290200817664e+18 2.43290200817664e+18 25 1.551121004333099e+25 1.551121004333099e+25 50 3.041409320171338e+64 3.041409320171338e+64 100 9.332621544394415e+157 9.33262154439441e+157 150 5.713383956445855e+262 5.71338395644585e+262 R8_FACTORIAL_TEST Normal end of execution. R8_FACTORIAL_LOG_TEST Python version: 3.6.5 R8_FACTORIAL_LOG evaluates the factorial log function. N Exact Computed 0 0 0 1 0 0 2 0.6931471805599453 0.6931471805599454 3 1.791759469228055 1.791759469228055 4 3.178053830347946 3.178053830347945 5 4.787491742782046 4.787491742782045 6 6.579251212010101 6.579251212010101 7 8.525161361065415 8.525161361065413 8 10.60460290274525 10.60460290274525 9 12.80182748008147 12.80182748008147 10 15.10441257307552 15.10441257307552 11 17.50230784587389 17.50230784587389 12 19.98721449566189 19.98721449566189 13 22.55216385312342 22.55216385312342 14 25.19122118273868 25.19122118273868 15 27.89927138384089 27.89927138384089 16 30.67186010608067 30.67186010608067 17 33.50507345013689 33.50507345013689 18 36.39544520803305 36.39544520803305 19 39.33988418719949 39.33988418719949 20 42.33561646075349 42.33561646075349 25 58.00360522298052 58.00360522298052 50 148.477766951773 148.477766951773 100 363.7393755555635 363.7393755555635 150 605.0201058494237 605.0201058494237 500 2611.330458460156 2611.330458460156 1000 5912.128178488163 5912.128178488163 R8_FACTORIAL_LOG_TEST Normal end of execution. R8_GAMMA_TEST: Python version: 3.6.5 R8_GAMMA evaluates the Gamma function. X GAMMA(X) R8_GAMMA(X) -0.5 -3.544907701811032 -3.544907701811032 -0.01 -100.5871979644108 -100.5871979644108 0.01 99.4325851191506 99.4325851191506 0.1 9.513507698668732 9.513507698668731 0.2 4.590843711998803 4.590843711998803 0.4 2.218159543757688 2.218159543757688 0.5 1.772453850905516 1.772453850905516 0.6 1.489192248812817 1.489192248812817 0.8 1.164229713725303 1.164229713725303 1 1 1 1.1 0.9513507698668732 0.9513507698668732 1.2 0.9181687423997607 0.9181687423997607 1.3 0.8974706963062772 0.8974706963062772 1.4 0.8872638175030753 0.8872638175030754 1.5 0.8862269254527581 0.8862269254527581 1.6 0.8935153492876903 0.8935153492876903 1.7 0.9086387328532904 0.9086387328532904 1.8 0.9313837709802427 0.9313837709802427 1.9 0.9617658319073874 0.9617658319073874 2 1 1 3 2 2 4 6 6 10 362880 362880 20 1.21645100408832e+17 1.216451004088321e+17 30 8.841761993739702e+30 8.841761993739751e+30 R8_GAMMA_TEST Normal end of execution. R8_GAMMA_LOG_TEST: Python version: 3.6.5 R8_GAMMA_LOG evaluates the logarithm of the Gamma function. X GAMMA_LOG(X) R8_GAMMA_LOG(X) 0.2 1.524063822430784 1.524063822430784 0.4 0.7966778177017837 0.7966778177017837 0.6 0.3982338580692348 0.3982338580692349 0.8 0.1520596783998375 0.1520596783998376 1 0 0 1.1 -0.04987244125983972 -0.04987244125983976 1.2 -0.08537409000331583 -0.08537409000331585 1.3 -0.1081748095078604 -0.1081748095078605 1.4 -0.1196129141723712 -0.1196129141723713 1.5 -0.1207822376352452 -0.1207822376352453 1.6 -0.1125917656967557 -0.1125917656967558 1.7 -0.09580769740706586 -0.09580769740706586 1.8 -0.07108387291437215 -0.07108387291437215 1.9 -0.03898427592308333 -0.03898427592308337 2 0 0 3 0.6931471805599453 0.6931471805599454 4 1.791759469228055 1.791759469228055 10 12.80182748008147 12.80182748008147 20 39.33988418719949 39.33988418719949 30 71.25703896716801 71.257038967168 R8_GAMMA_LOG_TEST Normal end of execution. R8_HYPER_2F1_TEST Python version: 3.6.5 R8_HYPER_2F1 evaluates the hypergeometric 2F1 function. A B C X 2F1 2F1 DIFF (tabulated) (computed) -2.5 3.3 6.7 0.25 0.723561 0.723561 2.22045e-16 -0.5 1.1 6.7 0.25 0.979111 0.979111 1.11022e-16 0.5 1.1 6.7 0.25 1.02166 1.02166 0 2.5 3.3 6.7 0.25 1.40516 1.40516 4.44089e-16 -2.5 3.3 6.7 0.55 0.469614 0.469614 5.55112e-17 -0.5 1.1 6.7 0.55 0.952962 0.952962 3.33067e-16 0.5 1.1 6.7 0.55 1.05128 1.05128 8.88178e-16 2.5 3.3 6.7 0.55 2.39991 2.39991 1.77636e-15 -2.5 3.3 6.7 0.85 0.291061 0.291061 2.22045e-16 -0.5 1.1 6.7 0.85 0.92537 0.92537 0 0.5 1.1 6.7 0.85 1.08655 1.08655 0 2.5 3.3 6.7 0.85 5.73816 5.73816 3.97016e-13 3.3 6.7 -5.5 0.25 15090.7 15090.7 1.09139e-11 1.1 6.7 -0.5 0.25 -104.312 -104.312 2.84217e-14 1.1 6.7 0.5 0.25 21.1751 21.1751 1.06581e-14 3.3 6.7 4.5 0.25 4.19469 4.19469 8.88178e-16 3.3 6.7 -5.5 0.55 1.01708e+10 1.01708e+10 1.14441e-05 1.1 6.7 -0.5 0.55 -24708.6 -24708.6 1.81899e-11 1.1 6.7 0.5 0.55 1372.23 1372.23 2.27374e-12 3.3 6.7 4.5 0.55 58.0927 58.0927 2.84217e-14 3.3 6.7 -5.5 0.85 5.86821e+18 5.86821e+18 36864 1.1 6.7 -0.5 0.85 -4.4635e+08 -4.4635e+08 4.76837e-07 1.1 6.7 0.5 0.85 5.38351e+06 5.38351e+06 8.3819e-09 3.3 6.7 4.5 0.85 20396.9 20396.9 1.45519e-11 R8_HYPER_2F1_TEST Normal end of execution. R8_PSI_TEST: Python version: 3.6.5 R8_PSI evaluates the PSI function. X PSI(X) R8_PSI(X) 0.1 -10.42375494041108 -10.42375494041108 0.2 -5.289039896592188 -5.289039896592188 0.3 -3.502524222200133 -3.502524222200133 0.4 -2.561384544585116 -2.561384544585116 0.5 -1.963510026021423 -1.963510026021424 0.6 -1.54061921389319 -1.540619213893191 0.7 -1.220023553697935 -1.220023553697935 0.8 -0.9650085667061385 -0.9650085667061382 0.9 -0.7549269499470515 -0.7549269499470511 1 -0.5772156649015329 -0.5772156649015329 1.1 -0.4237549404110768 -0.4237549404110768 1.2 -0.2890398965921883 -0.2890398965921884 1.3 -0.1691908888667997 -0.1691908888667995 1.4 -0.06138454458511615 -0.06138454458511624 1.5 0.03648997397857652 0.03648997397857652 1.6 0.1260474527734763 0.1260474527734763 1.7 0.208547874873494 0.208547874873494 1.8 0.2849914332938615 0.2849914332938615 1.9 0.3561841611640597 0.3561841611640596 2 0.4227843350984671 0.4227843350984672 R8_PSI_TEST Normal end of execution. SIGMA_TEST Python version: 3.6.5 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 SIGMA_TEST Normal end of execution. SIMPLEX_NUM_TEST Python version: 3.6.5 SIMPLEX_NUM computes the N-th simplex number in M dimensions. M: 0 1 2 3 4 5 N 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 SIMPLEX_NUM_TEST: Normal end of execution. SIN_POWER_INT_TEST Python version: 3.6.5 SIN_POWER_INT returns values of the integral of SIN(X)^N from A to B. A B N Exact Computed 10.000000 20.000000 0 1.000000e+01 1.000000e+01 0.000000 1.000000 1 4.596977e-01 4.596977e-01 0.000000 1.000000 2 2.726756e-01 2.726756e-01 0.000000 1.000000 3 1.789406e-01 1.789406e-01 0.000000 1.000000 4 1.240256e-01 1.240256e-01 0.000000 1.000000 5 8.897440e-02 8.897440e-02 0.000000 2.000000 5 9.039312e-01 9.039312e-01 1.000000 2.000000 5 8.149568e-01 8.149568e-01 0.000000 1.000000 10 2.188752e-02 2.188752e-02 0.000000 1.000000 11 1.702344e-02 1.702344e-02 SIN_POWER_INT_TEST Normal end of execution. SLICES_TEST: Python version: 3.6.5 SLICES determines the maximum number of pieces created by SLICE_NUM slices in a DIM_NUM space. SLICES 1 2 3 4 5 6 7 8 DIM 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 SLICES_TEST: Normal end of execution. SPHERICAL_HARMONIC_TEST Python version: 3.6.5 SPHERICAL_HARMONIC evaluats the spherical harmonic function; L M THETA PHI YR YI 0 0 0.523599 1.047198 0.282095 0.000000 0.282095 0.000000 1 0 0.523599 1.047198 0.423142 0.000000 0.423142 0.000000 2 1 0.523599 1.047198 -0.167262 -0.289706 -0.167262 -0.289706 3 2 0.523599 1.047198 -0.110633 0.191622 -0.110633 0.191622 4 3 0.523599 1.047198 0.135497 0.000000 0.135497 0.000000 5 5 0.261799 0.628319 0.000539 0.000000 0.000539 -0.000000 5 4 0.261799 0.628319 -0.005147 0.003739 -0.005147 0.003739 5 3 0.261799 0.628319 0.013710 -0.042195 0.013710 -0.042195 5 2 0.261799 0.628319 0.060964 0.187626 0.060964 0.187626 5 1 0.261799 0.628319 -0.417040 -0.302997 -0.417040 -0.302997 4 2 0.628319 0.785398 0.000000 0.413939 0.000000 0.413939 4 2 1.884956 0.785398 0.000000 -0.100323 -0.000000 -0.100323 4 2 3.141593 0.785398 0.000000 0.000000 0.000000 0.000000 4 2 4.398230 0.785398 0.000000 -0.100323 -0.000000 -0.100323 4 2 5.654867 0.785398 0.000000 0.413939 0.000000 0.413939 3 -1 0.392699 0.448799 0.364121 -0.175351 0.364121 -0.175351 3 -1 0.392699 0.897598 0.251979 -0.315972 0.251979 -0.315972 3 -1 0.392699 1.346397 0.089930 -0.394011 0.089930 -0.394011 3 -1 0.392699 1.795196 -0.089930 -0.394011 -0.089930 -0.394011 3 -1 0.392699 2.243995 -0.251979 -0.315972 -0.251979 -0.315972 SPHERICAL_HARMONIC_TEST Normal end of execution. STIRLING1_TEST: Python version: 3.6.5 Test STIRLING1, which returns Stirling numbers of the first kind. Stirling1 matrix: Col: 0 1 2 3 4 Row 0: 1 0 0 0 0 1: -1 1 0 0 0 2: 2 -3 1 0 0 3: -6 11 -6 1 0 4: 24 -50 35 -10 1 5: -120 274 -225 85 -15 6: 720 -1764 1624 -735 175 7: -5040 13068 -13132 6769 -1960 Col: 5 6 7 Row 0: 0 0 0 1: 0 0 0 2: 0 0 0 3: 0 0 0 4: 0 0 0 5: 1 0 0 6: -21 1 0 7: 322 -28 1 STIRLING1_TEST: Normal end of execution. STIRLING2_TEST: Python version: 3.6.5 Test STIRLING2, which returns Stirling numbers of the second kind. Stirling2 matrix: Col: 0 1 2 3 4 Row 0: 1 0 0 0 0 1: 1 1 0 0 0 2: 1 3 1 0 0 3: 1 7 6 1 0 4: 1 15 25 10 1 5: 1 31 90 65 15 6: 1 63 301 350 140 7: 1 127 966 1701 1050 Col: 5 6 7 Row 0: 0 0 0 1: 0 0 0 2: 0 0 0 3: 0 0 0 4: 0 0 0 5: 1 0 0 6: 21 1 0 7: 266 28 1 STIRLING2_TEST: Normal end of execution. TAU_TEST Python version: 3.6.5 TAU computes the TAU function. N Exact TAU(N) 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 TAU_TEST Normal end of execution. TETRAHEDRON_NUM_TEST Python version: 3.6.5 TETRAHEDRON_NUM computes the triangular numbers. 1 1 2 4 3 10 4 20 5 35 6 56 7 84 8 120 9 165 10 220 TETRAHEDRON_NUM_TEST: Normal end of execution. TRIANGLE_NUM_TEST Python version: 3.6.5 TRIANGLE_NUM computes the triangular numbers. 1 1 2 3 3 6 4 10 5 15 6 21 7 28 8 36 9 45 10 55 TRIANGLE_NUM_TEST: Normal end of execution. TRIANGLE_LOWER_TO_I4_TEST Python version: 3.6.5 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 TRIANGLE_LOWER_TO_I4_TEST: Normal end of execution. TRIANGLE_UPPER_TO_I4_TEST Python version: 3.6.5 TRIANGLE_UPPER_TO_I4 converts an upper triangular index to a linear one. ( I, J ) ==> K 1 1 1 2 1 2 2 2 3 3 1 3 3 2 4 3 3 6 4 1 4 4 2 5 4 3 7 4 4 10 TRIANGLE_UPPER_TO_I4_TEST: Normal end of execution. TRIBONACCI_RECURSIVE_TEST Python version: 3.6.5 TRIBONACCI_RECURSIVE computes Tribonacci numbers recursively; The Tribonacci numbers: 0 0 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 TRIBONACCI_RECURSIVE_TEST Normal end of execution. TRINOMIAL_TEST Python version: 3.6.5 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 TRINOMIAL_TEST Normal end of execution. V_HOFSTADTER_TEST Python version: 3.6.5 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 V_HOFSTADTER_TEST Normal end of execution. VIBONACCI_TEST Python version: 3.6.5 VIBONACCI computes a Vibonacci sequence. Compute the series 3 times. 0: 1 1 1 1: 1 1 1 2: 0 0 -2 3: 1 -1 1 4: -1 -1 -1 5: 0 0 -2 6: -1 -1 -3 7: 1 1 1 8: -2 0 -2 9: -3 1 -3 10: -1 -1 5 11: 4 -2 2 12: 3 -3 -7 13: -7 -1 -5 14: 10 -4 2 15: -3 3 7 16: -13 1 9 17: -16 2 -2 18: -3 -3 -11 19: -19 1 -9 VIBONACCI_TEST: Normal end of execution. ZECKENDORF_TEST Python version: 3.6.5 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 ZECKENDORF_TEST Normal end of execution. ZERNIKE_POLY_TEST Python version: 3.6.5 ZERNIKE_POLY evaluates a Zernike polynomial directly. Table of polynomial coefficients: N M 0 0 1.000000 1 0 0.000000 0.000000 1 1 0.000000 1.000000 2 0 -1.000000 0.000000 2.000000 2 1 0.000000 0.000000 0.000000 2 2 0.000000 0.000000 1.000000 3 0 0.000000 0.000000 0.000000 0.000000 3 1 0.000000 -2.000000 0.000000 3.000000 3 2 0.000000 0.000000 0.000000 0.000000 3 3 0.000000 0.000000 0.000000 1.000000 4 0 1.000000 0.000000 -6.000000 0.000000 6.000000 4 1 0.000000 0.000000 0.000000 0.000000 0.000000 4 2 0.000000 0.000000 -3.000000 0.000000 4.000000 4 3 0.000000 0.000000 0.000000 0.000000 0.000000 4 4 0.000000 0.000000 0.000000 0.000000 1.000000 5 0 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 5 1 0.000000 3.000000 0.000000 -12.000000 0.000000 10.000000 5 2 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 5 3 0.000000 0.000000 0.000000 -4.000000 0.000000 5.000000 5 4 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 5 5 0.000000 0.000000 0.000000 0.000000 0.000000 1.000000 Z1: Compute polynomial coefficients, then evaluate by Horner's method; Z2: Evaluate directly by recursion. N M Z1 Z2 0 0 1.000000 1.000000 1 0 0.000000 0.000000 1 1 0.987654 0.987654 2 0 0.950922 0.950922 2 1 0.000000 0.000000 2 2 0.975461 0.975461 3 0 0.000000 0.000000 3 1 0.914946 0.914946 3 2 0.000000 0.000000 3 3 0.963418 0.963418 4 0 0.856379 0.856379 4 1 0.000000 0.000000 4 2 0.879714 0.879714 4 3 0.000000 0.000000 4 4 0.951524 0.951524 5 0 0.000000 0.000000 5 1 0.799714 0.799714 5 2 0.000000 0.000000 5 3 0.845212 0.845212 5 4 0.000000 0.000000 5 5 0.939777 0.939777 ZERNIKE_POLY_TEST Normal end of execution. ZERNIKE_POLY_COEF_TEST Python version: 3.6.5 ZERNIKE_POLY_COEF determines the Zernike polynomial coefficients. Zernike polynomial: p(x) = 0 * x^5 Zernike polynomial: p(x) = 10 * x^5 - 12 * x^3 + 3 * x Zernike polynomial: p(x) = 0 * x^5 Zernike polynomial: p(x) = 5 * x^5 - 4 * x^3 Zernike polynomial: p(x) = 0 * x^5 Zernike polynomial: p(x) = 1 * x^5 ZERNIKE_POLY_COEF_TEST Normal end of execution. ZETA_M1_TEST: Python version: 3.6.5 ZETA_M1 evaluates the Riemann Zeta function minus 1. Relative accuracy requested is TOL = 1e-10 P ZETA_M1(P) ZETA_M1(P) tabulated computed 2 6.4493406684822641e-01 6.4493406682176657e-01 2.5 3.4148725730000001e-01 3.4148725723309625e-01 3 2.0205690315959429e-01 2.0205690314033142e-01 3.5 1.2673386730000000e-01 1.2673386730311337e-01 4 8.2323233711138186e-02 8.2323233700504289e-02 5 3.6927755143369927e-02 3.6927755139800719e-02 6 1.7343061984449140e-02 1.7343061981036332e-02 7 8.3492773819228271e-03 8.3492773801574424e-03 8 4.0773561979443396e-03 4.0773561976435498e-03 9 2.0083929260822143e-03 2.0083928260336546e-03 10 9.9457512781808526e-04 9.9457512781058409e-04 11 4.9418860411946453e-04 4.9418860411834780e-04 12 2.4608655330804832e-04 2.4608655330788710e-04 16 1.5282259408651871e-05 1.5282259408651813e-05 20 9.5396203387280002e-07 9.5396203387279621e-07 30 9.3132743242000005e-11 9.3132743241966817e-10 40 9.0949477999999997e-13 9.0949478402638904e-13 ZETA_M1_TEST Normal end of execution. ZETA_NAIVE_TEST: Python version: 3.6.5 ZETA evaluates the Riemann Zeta function using a naive approach. N ZETA(N) ZETA_NAIVE(N) tabulate computed 2 1.644934066848226 1.644834071848065 3 1.202056903159594 1.202056898160098 4 1.082323233711138 1.082323233710861 5 1.03692775514337 1.036927755143338 6 1.017343061984449 1.017343061984441 7 1.008349277381923 1.008349277381921 8 1.004077356197944 1.004077356197943 9 1.002008392926082 1.002008392826082 10 1.000994575127818 1.000994575127818 11 1.000494188604119 1.000494188604119 12 1.000246086553308 1.000246086553308 16 1.000015282259409 1.000015282259408 20 1.000000953962034 1.000000953962034 30 1.000000000931327 1.000000000931327 40 1.000000000000909 1.000000000000909 ZETA_NAIVE_TEST Normal end of execution. POLPAK_TEST: Normal end of execution. Thu Sep 13 12:59:30 2018