13 April 2015 10:09:06 AM
POLPAK_PRB
C++ version
Test the POLPAK library.
AGUD_TEST
AGUD computes the inverse Gudermannian;
X GUD(X) AGUD(GUD(X))
1 0.865769 1
1.2 0.985692 1.2
1.4 1.08725 1.4
1.6 1.17236 1.6
1.8 1.24316 1.8
2 1.30176 2
2.2 1.35009 2.2
2.4 1.38986 2.4
2.6 1.42252 2.6
2.8 1.44933 2.8
3 1.4713 3
ALIGN_ENUM_TEST
ALIGN_ENUM counts the number of possible
alignments of two biological sequences.
Alignment enumeration table:
0 1 2 3 4 5
0 1 1 1 1 1 1
1 1 3 5 7 9 11
2 1 5 13 25 41 61
3 1 7 25 63 129 231
4 1 9 41 129 321 681
5 1 11 61 231 681 1683
6 1 13 85 377 1289 3653
7 1 15 113 575 2241 7183
8 1 17 145 833 3649 13073
9 1 19 181 1159 5641 22363
10 1 21 221 1561 8361 36365
6 7 8 9 10
0 1 1 1 1 1
1 13 15 17 19 21
2 85 113 145 181 221
3 377 575 833 1159 1561
4 1289 2241 3649 5641 8361
5 3653 7183 13073 22363 36365
6 8989 19825 40081 75517 134245
7 19825 48639 108545 224143 433905
8 40081 108545 265729 598417 1256465
9 75517 224143 598417 1462563 3317445
10 134245 433905 1256465 3317445 8097453
BELL_TEST
BELL computes Bell numbers.
N exact C(I) computed C(I)
0 1 1
1 1 1
2 2 2
3 5 5
4 15 15
5 52 52
6 203 203
7 877 877
8 4140 4140
9 21147 21147
10 115975 115975
BENFORD_TEST
BENFORD(I) is the Benford probability of the
initial digit sequence I.
I BENFORD(I)
1 0.30103
2 0.176091
3 0.124939
4 0.09691
5 0.0791812
6 0.0669468
7 0.0579919
8 0.0511525
9 0.0457575
BERNOULLI_NUMBER_TEST
BERNOULLI_NUMBER computes Bernoulli numbers;
I Exact BERNOULLI_NUMBER
0 1 1
1 -0.5 -0.5
2 0.166667 0.166667
3 0 0
4 -0.0333333 -0.0333333
6 -0.0238095 0.0238095
8 -0.0333333 -0.0333333
10 0.0757576 0.0757576
20 -529.124 -529.124
30 6.01581e+08 6.01581e+08
BERNOULLI_NUMBER2_TEST
BERNOULLI_NUMBER2 computes Bernoulli numbers;
I Exact BERNOULLI_NUMBER2
0 1 1
1 -0.5 -0.5
2 0.166667 0.166667
3 0 0
4 -0.0333333 -0.0333333
6 -0.0238095 0.0238095
8 -0.0333333 -0.0333333
10 0.0757576 0.0757576
20 -529.124 -529.124
30 6.01581e+08 6.01581e+08
BERNOULLI_NUMBER3_TEST
BERNOULLI_NUMBER3 computes Bernoulli numbers.
I Exact BERNOULLI_NUMBER3
0 1 1
1 -0.5 -0.5
2 0.166667 0.166667
3 0 0
4 -0.0333333 -0.0333331
6 -0.0238095 0.0238095
8 -0.0333333 -0.0333333
10 0.0757576 0.0757576
20 -529.124 -529.124
30 6.01581e+08 6.01581e+08
BERNOULLI_POLY_TEST
BERNOULLI_POLY evaluates Bernoulli polynomials;
X = 0.2
I BX
1 -0.3
2 0.00666667
3 1.8547e+06
4 525499
5 1.25096e+07
6 -0.94882
7 35.5459
8 16.3941
9 -17139.4
10 -418.656
11 29482.6
12 15985.3
13 -5.54642e+07
14 -871872
15 1.09544e+08
BERNOULLI_POLY2_TEST
BERNOULLI_POLY2 evaluates Bernoulli polynomials.
X = 0.2
I BX
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
BERNSTEIN_POLY_TEST:
BERNSTEIN_POLY evaluates the 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
BPAB_TEST
BPAB evaluates Bernstein polynomials.
The Bernstein polynomials of degree 10
based on the interval from 0
to 1
evaluated at X = 0.3
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
CARDAN_POLY_TEST
CARDAN_POLY evaluates a Cardan polynomial directly.
Compare CARDAN_POLY_COEF + R8POLY_VALUE_HORNER
versus CARDAN_POLY alone.
Evaluate polynomials at X = 0.25
We use the parameter S = 0.5
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_COEF_TEST
CARDAN_POLY_COEF returns the coefficients of a
Cardan polynomial.
We use the parameter S = 1
Table of polynomial coefficients:
0 2
1 0 1
2 -2 0 1
3 0 -3 0 1
4 2 0 -4 0 1
5 0 5 0 -5 0 1
6 -2 0 9 0 -6 0 1
7 0 -7 0 14 0 -7 0 1
8 2 0 -16 0 20 0 -8 0 1
9 0 9 0 -30 0 27 0 -9 0 1
10 -2 0 25 0 -50 0 35 0 -10 0 1
CARDINAL_COS_TEST
CARDINAL_COS evaluates cardinal cosine functions.
Ci(Tj) = Delta(i,j), where Tj = cos(pi*i/(n+1)).
A simple check of all pairs should form the identity matrix.
The CARDINAL_COS test matrix:
1 3.87588e-17 -3.8087e-17 3.6957e-17 -3.53525e-17 3.32498e-17 -3.06162e-17 2.7408e-17 -2.35683e-17 1.90224e-17 -1.36726e-17 2.68781e-17 -3.7494e-33
0 1 1.02156e-16 -8.36449e-17 7.58758e-17 -6.97041e-17 6.33924e-17 -5.63412e-17 4.82323e-17 -3.88197e-17 2.78539e-17 -5.47043e-17 7.62877e-33
-0 -2.64398e-17 1 1.36226e-16 -9.65848e-17 8.11722e-17 -7.0705e-17 6.1345e-17 -5.17597e-17 4.1285e-17 -2.94604e-17 5.76875e-17 -8.03719e-33
0 1.02054e-17 -6.42177e-17 1 1.70697e-16 -1.09948e-16 8.65956e-17 -7.14377e-17 5.8574e-17 -4.59243e-17 3.24365e-17 -6.31671e-17 8.78539e-33
-0 -5.66903e-18 2.78816e-17 -1.0453e-16 1 2.04362e-16 -1.22465e-16 9.09354e-17 -7.0705e-17 5.38036e-17 -3.73543e-17 7.20914e-17 -9.9984e-33
0 3.73543e-18 -1.68071e-17 4.82924e-17 -1.46581e-16 1 2.36584e-16 -1.33305e-16 9.31777e-17 -6.72378e-17 4.53636e-17 -8.62879e-17 1.1914e-32
-0 -2.73453e-18 1.17842e-17 -3.06162e-17 7.0705e-17 -1.90435e-16 1 2.66609e-16 -1.4141e-16 9.18485e-17 -5.89208e-17 1.09409e-16 -1.49976e-32
0 2.15665e-18 -9.07271e-18 2.24126e-17 -4.65889e-17 9.52176e-17 -2.36584e-16 1 2.93162e-16 -1.44877e-16 8.40356e-17 -1.49455e-16 2.02347e-32
-0 -1.80183e-18 7.47086e-18 -1.79345e-17 3.53525e-17 -6.49539e-17 1.22465e-16 -2.86107e-16 1 3.1359e-16 -1.39408e-16 2.26819e-16 -2.99952e-32
0 1.57878e-18 -6.48731e-18 1.53081e-17 -2.9287e-17 5.10269e-17 -8.65956e-17 1.53927e-16 -3.41394e-16 1 3.21089e-16 -4.08319e-16 5.1205e-32
-0 -1.44182e-18 5.89208e-18 -1.37617e-17 2.58798e-17 -4.38178e-17 7.0705e-17 -1.13641e-16 1.9317e-16 -4.08679e-16 1 1.05786e-15 -1.11944e-31
0 1.36726e-18 -5.57078e-18 1.29399e-17 -2.41162e-17 4.02437e-17 -6.33924e-17 9.75858e-17 -1.51752e-16 2.50935e-16 -5.10778e-16 1 4.40146e-31
-0 -6.71783e-19 2.73453e-18 -6.34082e-18 1.17842e-17 -1.95772e-17 3.06162e-17 -4.65497e-17 7.0705e-17 -1.10871e-16 1.90435e-16 -1.55074e-15 1
CARDINAL_SIN_TEST
CARDINAL_SIN evaluates cardinal sine functions.
Si(Tj) = Delta(i,j), where Tj = cos(pi*i/(n+1)).
A simple check of all pairs should form the identity matrix.
The CARDINAL_SIN test matrix:
1 0 -0 0 -0 0 -0 0 -0 0 -0 0 -0
0 1 5.28796e-17 -3.06162e-17 2.26761e-17 -1.86772e-17 1.64072e-17 -1.50966e-17 1.44146e-17 -1.4209e-17 1.44182e-17 -5.47043e-17 1.61228e-17
-0 -5.10778e-17 1 9.63266e-17 -5.57633e-17 4.20178e-17 -3.53525e-17 3.17545e-17 -2.98835e-17 2.91929e-17 -2.94604e-17 1.11444e-16 -3.28143e-17
0 2.78816e-17 -9.08176e-17 1 1.39374e-16 -8.04874e-17 6.12323e-17 -5.2296e-17 4.78254e-17 -4.59243e-17 4.58722e-17 -1.72576e-16 5.07265e-17
-0 -1.8969e-17 4.82924e-17 -1.28023e-16 1 1.83226e-16 -1.06058e-16 8.15305e-17 -7.0705e-17 6.58957e-17 -6.46996e-17 2.41223e-16 -7.0705e-17
0 1.39408e-17 -3.24689e-17 6.59687e-17 -1.6349e-16 1 2.28522e-16 -1.33305e-16 1.03926e-16 -9.18485e-17 8.76357e-17 -3.22031e-16 9.39705e-17
-0 -1.05654e-17 2.35683e-17 -4.32978e-17 8.16431e-17 -1.97153e-16 1 2.76014e-16 -1.63286e-16 1.29893e-16 -1.17842e-16 4.22723e-16 -1.22465e-16
0 8.04874e-18 -1.75271e-17 3.06162e-17 -5.19631e-17 9.52176e-17 -2.28522e-16 1 3.26979e-16 -1.97906e-16 1.62344e-16 -5.57774e-16 1.59599e-16
-0 -6.02904e-18 1.29399e-17 -2.19652e-17 3.53525e-17 -5.82361e-17 1.06058e-16 -2.56516e-16 1 3.84068e-16 -2.41462e-16 7.5895e-16 -2.12115e-16
0 4.31331e-18 -9.17444e-18 1.53081e-17 -2.39127e-17 3.73543e-17 -6.12323e-17 1.12682e-16 -2.78747e-16 1 4.54088e-16 -1.11555e-15 2.95656e-16
-0 -2.78539e-18 5.89208e-18 -9.73096e-18 1.49417e-17 -2.26818e-17 3.53525e-17 -5.88249e-17 1.11527e-16 -2.8898e-16 1 2.04363e-15 -4.57044e-16
0 1.36726e-18 -2.88365e-18 4.73634e-18 -7.20732e-18 1.07833e-17 -1.64072e-17 2.6148e-17 -4.53523e-17 9.18485e-17 -2.64398e-16 1 9.30212e-16
-0 -6.35731e-34 1.33953e-33 -2.19635e-33 3.3328e-33 -4.96418e-33 7.4988e-33 -1.18036e-32 1.99968e-32 -3.84038e-32 9.32863e-32 -1.46752e-30 1
CATALAN_TEST
CATALAN computes Catalan numbers.
N exact C(I) computed C(I)
0 1 1
1 1 1
2 2 2
3 5 5
4 14 14
5 42 42
6 132 132
7 429 429
8 1430 1430
9 4862 4862
10 16796 16796
CATALAN_ROW_NEXT_TEST
CATALAN_ROW_NEXT computes a row of Catalan''s triangle.
First, compute row 7:
7 1 7 27 75 165 297 429 429
Now compute rows consecutively, one at a time:
8 1
1 1 1
2 1 2 2
3 1 3 5 5
4 1 4 9 14 14
5 1 5 14 28 42 42
6 1 6 20 48 90 132 132
7 1 7 27 75 165 297 429 429
8 1 8 35 110 275 572 1001 1430 1430
9 1 9 44 154 429 1001 2002 3432 4862 4862
10 1 10 54 208 637 1638 3640 7072 11934 16796 16796
CHARLIER_TEST:
CHARLIER evaluates Charlier polynomials.
N A X P(N,A,X)
0 0.25 0 1
1 0.25 0 -0
2 0.25 0 -4
3 0.25 0 -36
4 0.25 0 -420
5 0.25 0 -6564
0 0.25 0.5 1
1 0.25 0.5 -2
2 0.25 0.5 -10
3 0.25 0.5 -54
4 0.25 0.5 -474
5 0.25 0.5 -6246
0 0.25 1 1
1 0.25 1 -4
2 0.25 1 -8
3 0.25 1 -8
4 0.25 1 24
5 0.25 1 440
0 0.25 1.5 1
1 0.25 1.5 -6
2 0.25 1.5 2
3 0.25 1.5 54
4 0.25 1.5 354
5 0.25 1.5 3030
0 0.25 2 1
1 0.25 2 -8
2 0.25 2 20
3 0.25 2 84
4 0.25 2 180
5 0.25 2 276
0 0.25 2.5 1
1 0.25 2.5 -10
2 0.25 2.5 46
3 0.25 2.5 34
4 0.25 2.5 -450
5 0.25 2.5 -3694
0 0.5 0 1
1 0.5 0 -0
2 0.5 0 -2
3 0.5 0 -10
4 0.5 0 -58
5 0.5 0 -442
0 0.5 0.5 1
1 0.5 0.5 -1
2 0.5 0.5 -4
3 0.5 0.5 -12
4 0.5 0.5 -48
5 0.5 0.5 -288
0 0.5 1 1
1 0.5 1 -2
2 0.5 1 -4
3 0.5 1 -4
4 0.5 1 4
5 0.5 1 60
0 0.5 1.5 1
1 0.5 1.5 -3
2 0.5 1.5 -2
3 0.5 1.5 8
4 0.5 1.5 44
5 0.5 1.5 200
0 0.5 2 1
1 0.5 2 -4
2 0.5 2 2
3 0.5 2 18
4 0.5 2 42
5 0.5 2 66
0 0.5 2.5 1
1 0.5 2.5 -5
2 0.5 2.5 8
3 0.5 2.5 20
4 0.5 2.5 -8
5 0.5 2.5 -192
0 1 0 1
1 1 0 -0
2 1 0 -1
3 1 0 -3
4 1 0 -9
5 1 0 -33
0 1 0.5 1
1 1 0.5 -0.5
2 1 0.5 -1.75
3 1 0.5 -3.375
4 1 0.5 -6.5625
5 1 0.5 -16.0312
0 1 1 1
1 1 1 -1
2 1 1 -2
3 1 1 -2
4 1 1 0
5 1 1 8
0 1 1.5 1
1 1 1.5 -1.5
2 1 1.5 -1.75
3 1 1.5 0.375
4 1 1.5 6.1875
5 1 1.5 20.1562
0 1 2 1
1 1 2 -2
2 1 2 -1
3 1 2 3
4 1 2 9
5 1 2 15
0 1 2.5 1
1 1 2.5 -2.5
2 1 2.5 0.25
3 1 2.5 5.125
4 1 2.5 6.9375
5 1 2.5 -3.15625
0 2 0 1
1 2 0 -0
2 2 0 -0.5
3 2 0 -1
4 2 0 -1.75
5 2 0 -3.25
0 2 0.5 1
1 2 0.5 -0.25
2 2 0.5 -0.8125
3 2 0.5 -1.17188
4 2 0.5 -1.41797
5 2 0.5 -1.55566
0 2 1 1
1 2 1 -0.5
2 2 1 -1
3 2 1 -1
4 2 1 -0.5
5 2 1 0.75
0 2 1.5 1
1 2 1.5 -0.75
2 2 1.5 -1.0625
3 2 1.5 -0.578125
4 2 1.5 0.582031
5 2 1.5 2.46582
0 2 2 1
1 2 2 -1
2 2 2 -1
3 2 2 0
4 2 2 1.5
5 2 2 3
0 2 2.5 1
1 2 2.5 -1.25
2 2 2.5 -0.8125
3 2 2.5 0.640625
4 2 2.5 2.01953
5 2 2.5 2.25293
0 10 0 1
1 10 0 -0
2 10 0 -0.1
3 10 0 -0.12
4 10 0 -0.126
5 10 0 -0.1284
0 10 0.5 1
1 10 0.5 -0.05
2 10 0.5 -0.1525
3 10 0.5 -0.165375
4 10 0.5 -0.160969
5 10 0.5 -0.151158
0 10 1 1
1 10 1 -0.1
2 10 1 -0.2
3 10 1 -0.2
4 10 1 -0.18
5 10 1 -0.154
0 10 1.5 1
1 10 1.5 -0.15
2 10 1.5 -0.2425
3 10 1.5 -0.224625
4 10 1.5 -0.185569
5 10 1.5 -0.142111
0 10 2 1
1 10 2 -0.2
2 10 2 -0.28
3 10 2 -0.24
4 10 2 -0.18
5 10 2 -0.12
0 10 2.5 1
1 10 2.5 -0.25
2 10 2.5 -0.3125
3 10 2.5 -0.246875
4 10 2.5 -0.165469
5 10 2.5 -0.0915391
CHEBY_T_POLY_TEST:
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_COEF_TEST
CHEBY_T_POLY_COEF determines the polynomial coefficients
of the Chebyshev polynomial T(n,x).
T(0,x)
1
T(1,x)
1 * x
T(2,x)
2 * x^2
-1
T(3,x)
4 * x^3
-3 * x
T(4,x)
8 * x^4
-8 * x^2
1
T(5,x)
16 * x^5
-20 * x^3
5 * x
CHEBY_T_POLY_ZERO_TEST:
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_U_POLY_TEST:
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_COEF_TEST
CHEBY_U_POLY_COEF determines the polynomial coefficients
of the Chebyshev polynomial U(n,x).
U(0,x)
1
U(1,x)
2 * x
U(2,x)
4 * x^2
-1
U(3,x)
8 * x^3
-4 * x
U(4,x)
16 * x^4
-12 * x^2
1
U(5,x)
32 * x^5
-32 * x^3
6 * x
CHEBY_U_POLY_ZERO_TEST:
CHEBY_U_POLY_ZERO returns zeroes of U(N,X).
N X U(N,X)
1 6.12323e-17 1.22465e-16
2 0.5 4.44089e-16
2 -0.5 -8.88178e-16
3 0.707107 6.66134e-16
3 6.12323e-17 -2.44929e-16
3 -0.707107 6.66134e-16
4 0.809017 0
4 0.309017 -1.11022e-16
4 -0.309017 5.55112e-16
4 -0.809017 -8.88178e-16
CHEBYSHEV_DISCRETE_TEST:
CHEBYSHEV_DISCRETE evaluates discrete Chebyshev polynomials.
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
COLLATZ_COUNT_TEST:
COLLATZ_COUNT(N) counts the length of the
Collatz sequence beginning with N.
N COUNT(N) COUNT(N)
(computed) (table)
1 1 1
2 2 2
3 8 8
4 3 3
5 6 6
6 9 9
7 17 17
8 4 4
9 20 20
10 7 7
27 112 112
50 25 25
100 26 26
200 27 27
300 17 17
400 28 28
500 111 111
600 18 18
700 83 83
800 29 29
COLLATZ_COUNT_MAX_TEST:
COLLATZ_COUNT_MAX(N) returns the length of the
longest Collatz sequence from 1 to N.
N I_MAX J_MAX
10 9 20
100 97 119
1000 871 179
10000 6171 262
100000 77031 351
COMB_ROW_NEXT_TEST
COMB_ROW_NEXT computes the next row of Pascal's triangle.
0 1
1 1 1
2 1 2 1
3 1 3 3 1
4 1 4 6 4 1
5 1 5 10 10 5 1
6 1 6 15 20 15 6 1
7 1 7 21 35 35 21 7 1
8 1 8 28 56 70 56 28 8 1
9 1 9 36 84 126 126 84 36 9 1
10 1 10 45 120 210 252 210 120 45 10 1
COMMUL_TEST
COMMUL computes a multinomial coefficient.
N = 8
Number of factors = 0x7fffbadc14d0
0 6
1 2
Value of coefficient = 28
N = 8
Number of factors = 0x7fffbadc14d0
0 2
1 2
2 4
Value of coefficient = 28
N = 13
Number of factors = 0x7fffbadc14d0
0 5
1 3
2 3
3 2
Value of coefficient = 720720
COMPLETE_SYMMETRIC_POLY_TEST
COMPLETE_SYMMETRIC_POLY evaluates a complete symmetric.
polynomial in a given set of variables X.
Variable vector X:
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
COS_POWER_INT_TEST:
COS_POWER_INT computes the integral of the N-th power
of the cosine function.
A B N Exact Computed
0 3.14159 0 3.14159 3.14159
0 3.14159 1 0 1.22465e-16
0 3.14159 2 1.5708 1.5708
0 3.14159 3 0 1.22465e-16
0 3.14159 4 1.1781 1.1781
0 3.14159 5 0 1.22465e-16
0 3.14159 6 0.981748 0.981748
0 3.14159 7 0 1.22465e-16
0 3.14159 8 0.859029 0.859029
0 3.14159 9 0 1.22465e-16
0 3.14159 10 0.773126 0.773126
EULER_NUMBER_TEST
EULER_NUMBER computes Euler numbers.
N exact EULER_NUMBER
0 1 1
1 0 0
2 -1 -1
4 5 5
6 -61 -61
8 1385 1385
10 -50521 -50521
12 2702765 2702765
EULER_NUMBER2_TEST
EULER_NUMBER2 computes Euler numbers.
N exact EULER_NUMBER2
0 1 1
1 0 0
2 -1 -1
4 5 5
6 -61 -61
8 1385 1385
10 -50521 -50521
12 2702765 2.70276e+06
EULER_POLY_TEST
EULER_POLY evaluates Euler polynomials.
N X F(X)
0 0.5 1
1 0.5 2.77556e-17
2 0.5 -0.25
3 0.5 -1.45953e-06
4 0.5 0.312497
5 0.5 -3.32929e-06
6 0.5 -0.953128
7 0.5 -1.73264e-06
8 0.5 5.41016
9 0.5 -1.02449e-06
10 0.5 -49.3369
11 0.5 6.47439e-07
12 0.5 659.855
13 0.5 5.22754e-06
14 0.5 -12168
15 0.5 0.000218677
EULERIAN_TEST
EULERIAN evaluates Eulerian numbers.
1 0 0 0 0 0 0
1 1 0 0 0 0 0
1 4 1 0 0 0 0
1 11 11 1 0 0 0
1 26 66 26 1 0 0
1 57 302 302 57 1 0
1 120 1191 2416 1191 120 1
F_HOFSTADTER_TEST
F_HOFSTADTER evaluates Hofstadter's recursive
F function.
N F(N)
0 0
1 1
2 1
3 2
4 2
5 3
6 3
7 4
8 4
9 5
10 5
11 6
12 6
13 7
14 7
15 8
16 8
17 9
18 9
19 10
20 10
21 11
22 11
23 12
24 12
25 13
26 13
27 14
28 14
29 15
30 15
FIBONACCI_DIRECT_TEST
FIBONACCI_DIRECT evalutes a Fibonacci number directly.
1 1
2 1
3 2
4 3
5 5
6 8
7 13
8 21
9 34
10 55
11 89
12 144
13 233
14 377
15 610
16 987
17 1597
18 2584
19 4181
20 6765
FIBONACCI_FLOOR_TEST
FIBONACCI_FLOOR computes the largest Fibonacci number
less than or equal to a given positive integer.
N Fibonacci Index
1 1 2
2 2 3
3 3 4
4 3 4
5 5 5
6 5 5
7 5 5
8 8 6
9 8 6
10 8 6
11 8 6
12 8 6
13 13 7
14 13 7
15 13 7
16 13 7
17 13 7
18 13 7
19 13 7
20 13 7
FIBONACCI_RECURSIVE_TEST
FIBONACCI_RECURSIVE computes the Fibonacci sequence.
1 1
2 1
3 2
4 3
5 5
6 8
7 13
8 21
9 34
10 55
11 89
12 144
13 233
14 377
15 610
16 987
17 1597
18 2584
19 4181
20 6765
G_HOFSTADTER_TEST
G_HOFSTADTER evaluates Hofstadter's recursive
G function.
N G(N)
0 0
1 1
2 1
3 2
4 3
5 3
6 4
7 4
8 5
9 6
10 6
11 7
12 8
13 8
14 9
15 9
16 10
17 11
18 11
19 12
20 12
21 13
22 14
23 14
24 15
25 16
26 16
27 17
28 17
29 18
30 19
GEGENBAUER_POLY_TEST
GEGENBAUER_POLY evaluates the Gegenbauer polynomials.
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 -0.44
3 0.5 0.2 -0.28 -0.28
4 0.5 0.2 0.232 0.232
5 0.5 0.2 0.30752 0.30752
6 0.5 0.2 -0.080576 -0.080576
7 0.5 0.2 -0.293517 -0.293517
8 0.5 0.2 -0.0395648 -0.0395648
9 0.5 0.2 0.245971 0.245957
10 0.5 0.2 0.129072 0.129072
2 0 0.4 0 0
2 1 0.4 -0.36 -0.36
2 2 0.4 -0.08 -0.08
2 3 0.4 0.84 0.84
2 4 0.4 2.4 2.4
2 5 0.4 4.6 4.6
2 6 0.4 7.44 7.44
2 7 0.4 10.92 10.92
2 8 0.4 15.04 15.04
2 9 0.4 19.8 19.8
2 10 0.4 25.2 25.2
5 3 -0.5 -9 9
5 3 -0.4 -0.16128 -0.16128
5 3 -0.3 -6.67296 -6.67296
5 3 -0.2 -8.37504 -8.37504
5 3 -0.1 -5.52672 -5.52672
5 3 0 0 0
5 3 0.1 5.52672 5.52672
5 3 0.2 8.37504 8.37504
5 3 0.3 6.67296 6.67296
5 3 0.4 0.16128 0.16128
5 3 0.5 -9 -9
5 3 0.6 -15.4253 -15.4253
5 3 0.7 -9.69696 -9.69696
5 3 0.8 22.441 22.441
5 3 0.9 100.889 100.889
5 3 1 252 252
GEN_HERMITE_POLY_TEST
GEN_HERMITE_POLY evaluates the generalized Hermite
polynomial.
Table of H(N,MU)(X) for
N(max) = 10
MU = 0
X = 0
0 1
1 0
2 -2
3 -0
4 12
5 0
6 -120
7 -0
8 1680
9 0
10 -30240
Table of H(N,MU)(X) for
N(max) = 10
MU = 0
X = 1
0 1
1 2
2 2
3 -4
4 -20
5 -8
6 184
7 464
8 -1648
9 -10720
10 8224
Table of H(N,MU)(X) for
N(max) = 10
MU = 0.1
X = 0
0 1
1 0
2 -2.4
3 -0
4 15.36
5 0
6 -159.744
7 -0
8 2300.31
9 0
10 -42325.8
Table of H(N,MU)(X) for
N(max) = 10
MU = 0.1
X = 0.5
0 1
1 1
2 -1.4
3 -5.4
4 3.56
5 46.76
6 9.736
7 -551.384
8 -691.582
9 8130.56
10 20855.7
Table of H(N,MU)(X) for
N(max) = 10
MU = 0.5
X = 0.5
0 1
1 1
2 -3
3 -7
4 17
5 73
6 -131
7 -1007
8 1089
9 17201
10 -4579
Table of H(N,MU)(X) for
N(max) = 10
MU = 1
X = 0.5
0 1
1 1
2 -5
3 -9
4 41
5 113
6 -461
7 -1817
8 6481
9 35553
10 -107029
GEN_LAGUERRE_POLY_TEST
GEN_LAGUERRE_POLY evaluates the generalized Laguerre
functions.
Table of L(N,ALPHA)(X) for
N(max) = 10
ALPHA = 0
X = 0
0 1
1 1
2 1
3 1
4 1
5 1
6 1
7 1
8 1
9 1
10 1
Table of L(N,ALPHA)(X) for
N(max) = 10
ALPHA = 0
X = 1
0 1
1 0
2 -0.5
3 -0.666667
4 -0.625
5 -0.466667
6 -0.256944
7 -0.0404762
8 0.153993
9 0.309744
10 0.418946
Table of L(N,ALPHA)(X) for
N(max) = 10
ALPHA = 0.1
X = 0
0 1
1 1.1
2 1.155
3 1.1935
4 1.22334
5 1.2478
6 1.2686
7 1.28672
8 1.30281
9 1.31728
10 1.33046
Table of L(N,ALPHA)(X) for
N(max) = 10
ALPHA = 0.1
X = 0.5
0 1
1 0.6
2 0.23
3 -0.0673333
4 -0.28935
5 -0.442469
6 -0.535747
7 -0.578765
8 -0.580771
9 -0.550311
10 -0.495076
Table of L(N,ALPHA)(X) for
N(max) = 10
ALPHA = 0.5
X = 0.5
0 1
1 1
2 0.75
3 0.416667
4 0.0729167
5 -0.24375
6 -0.513715
7 -0.727703
8 -0.882836
9 -0.980303
10 -1.02388
Table of L(N,ALPHA)(X) for
N(max) = 10
ALPHA = 1
X = 0.5
0 1
1 1.5
2 1.625
3 1.47917
4 1.14844
5 0.702865
6 0.19872
7 -0.31962
8 -0.817983
9 -1.2709
10 -1.66028
GUD_TEST:
GUD evaluates the Gudermannian function.
X Exact F GUD(X)
-2 -1.30176 -1.30176
-1 -0.865769 -0.865769
0 0 0
0.1 0.0998337 0.0998337
0.2 0.19868 0.19868
0.5 0.480381 0.480381
1 0.865769 0.865769
1.5 1.13173 1.13173
2 1.30176 1.30176
2.5 1.40699 1.40699
3 1.4713 1.4713
3.5 1.51042 1.51042
4 1.53417 1.53417
HAIL_TEST
HAIL(I) computes the length of the hail sequence
for I, also known as the 3*N+1 sequence.
I, HAIL(I)
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
H_HOFSTADTER_TEST
H_HOFSTADTER evaluates Hofstadter's recursive
H function.
N H(N)
0 0
1 1
2 1
3 2
4 3
5 4
6 4
7 5
8 5
9 6
10 7
11 7
12 8
13 9
14 10
15 10
16 11
17 12
18 13
19 13
20 14
21 14
22 15
23 16
24 17
25 17
26 18
27 18
28 19
29 20
30 20
HERMITE_POLY_PHYS_TEST:
HERMITE_POLY_PHYS evaluates the physicist's Hermite polynomial.
N X Exact F H(N)(X)
0 5 1 1
1 5 10 10
2 5 98 98
3 5 940 940
4 5 8812 8812
5 5 80600 80600
6 5 717880 717880
7 5 6.2116e+06 6.2116e+06
8 5 5.20657e+08 5.20657e+07
9 5 4.21271e+08 4.21271e+08
10 5 3.27553e+09 3.27553e+09
11 5 2.43299e+10 2.43299e+10
12 5 1.71237e+11 1.71237e+11
5 0.5 41 41
5 1 -8 -8
5 3 3816 3816
5 10 3.0412e+06 3.0412e+06
HERMITE_POLY_PHYS_COEF_TEST
HERMITE_POLY_PHYS_COEF: physicist's Hermite polynomial coefficients.
H(0)
1
H(1)
2 * x
0
H(2)
4 * x^2
0 * x
-2
H(3)
8 * x^3
0 * x^2
-12 * x
-0
H(4)
16 * x^4
0 * x^3
-48 * x^2
-0 * x
12
H(5)
32 * x^5
0 * x^4
-160 * x^3
-0 * x^2
120 * x
0
I4_CHOOSE_TEST
I4_CHOOSE evaluates C(N,K).
N K CNK
0 0 1
1 0 1
1 1 1
2 0 1
2 1 2
2 2 1
3 0 1
3 1 3
3 2 3
3 3 1
4 0 1
4 1 4
4 2 6
4 3 4
4 4 1
I4_FACTOR_TEST:
I4_FACTOR tries to factor an I4
Factors of N = 60
2^2
3^1
5^1
Factors of N = 664048
2^4
7^3
11^2
Factors of N = 8466763
2699^1
3137^1
I4_FACTORIAL_TEST:
I4_FACTORIAL evaluates the factorial function.
X Exact F I4_FACTORIAL(X)
0 1 1
1 1 1
2 2 2
3 6 6
4 24 24
5 120 120
6 720 720
7 5040 5040
8 40320 40320
9 362880 362880
10 3628800 3628800
11 39916800 39916800
12 479001600 479001600
I4_FACTORIAL2_TEST:
I4_FACTORIAL2 evaluates the double factorial function.
N Exact I4_FACTORIAL2(N)
0 1 1
1 1 1
2 2 2
3 3 3
4 8 8
5 15 15
6 48 48
7 105 105
8 384 384
9 945 945
10 3840 3840
11 10395 10395
12 46080 46080
13 135135 135135
14 645120 645120
15 2027025 2027025
I4_IS_TRIANGULAR_TEST
I4_IS_TRIANGULAR returns 0 or 1 depending on
whether I is triangular.
I => 0/1
0 1
1 0
2 1
3 0
4 0
5 1
6 0
7 0
8 0
9 1
10 0
11 0
12 0
13 0
14 1
15 0
16 0
17 0
18 0
19 0
20 1
I4_PARTITION_DISTINCT_COUNT_TEST:
For the number of partitions of an integer
into distinct parts,
I4_PARTITION_DISTINCT_COUNT computes any value.
N Exact F Q(N)
0 1 1
1 1 0
2 1 2
3 2 0
4 2 3
5 3 1
6 4 5
7 5 1
8 6 8
9 8 3
10 10 12
11 12 4
12 15 18
13 18 7
14 22 27
15 27 10
16 32 38
17 38 16
18 46 54
19 54 22
20 64 76
I4_TO_TRIANGLE_TEST
I4_TO_TRIANGLE converts a linear index to a
triangular one.
K => I J
0 0 0
1 1 0
2 1 1
3 2 0
4 2 1
5 2 2
6 3 0
7 3 1
8 3 2
9 3 3
10 4 0
11 4 1
12 4 2
13 4 3
14 4 4
15 5 0
16 5 1
17 5 2
18 5 3
19 5 4
20 5 5
JACOBI_POLY_TEST
JACOBI_POLY evaluates the Jacobi polynomials.
the Jacobi polynomials.
N A B X JPV 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_SYMBOL_TEST
JACOBI_SYMBOL computes the Jacobi symbol
(Q/P), which records if Q is a quadratic
residue modulo the number P.
Jacobi Symbols for P = 3
3 0 0
3 1 1
3 2 -1
3 3 0
Jacobi Symbols for P = 9
9 0 0
9 1 1
9 2 1
9 3 0
9 4 1
9 5 1
9 6 0
9 7 1
9 8 1
9 9 0
Jacobi Symbols for P = 10
10 0 0
10 1 1
10 2 0
10 3 -1
10 4 0
10 5 0
10 6 0
10 7 -1
10 8 0
10 9 1
10 10 0
Jacobi Symbols for P = 12
12 0 0
12 1 1
12 2 0
12 3 0
12 4 0
12 5 -1
12 6 0
12 7 1
12 8 0
12 9 0
12 10 0
12 11 -1
12 12 0
KRAWTCHOUK_TEST:
KRAWTCHOUK evaluates Krawtchouk polynomials.
N P X M K(N,P,X,M)
0 0.25 0 5 1
1 0.25 0 5 -1.25
2 0.25 0 5 0.625
3 0.25 0 5 -0.15625
4 0.25 0 5 0.0195312
5 0.25 0 5 -0.000976562
0 0.25 0.5 5 1
1 0.25 0.5 5 -0.75
2 0.25 0.5 5 0
3 0.25 0.5 5 0.1875
4 0.25 0.5 5 -0.105469
5 0.25 0.5 5 0.0439453
0 0.25 1 5 1
1 0.25 1 5 -0.25
2 0.25 1 5 -0.375
3 0.25 1 5 0.21875
4 0.25 1 5 -0.0429688
5 0.25 1 5 0.00292969
0 0.25 1.5 5 1
1 0.25 1.5 5 0.25
2 0.25 1.5 5 -0.5
3 0.25 1.5 5 0.0625
4 0.25 1.5 5 0.0507812
5 0.25 1.5 5 -0.0224609
0 0.25 2 5 1
1 0.25 2 5 0.75
2 0.25 2 5 -0.375
3 0.25 2 5 -0.15625
4 0.25 2 5 0.0820312
5 0.25 2 5 -0.00878906
0 0.25 2.5 5 1
1 0.25 2.5 5 1.25
2 0.25 2.5 5 0
3 0.25 2.5 5 -0.3125
4 0.25 2.5 5 0.0195312
5 0.25 2.5 5 0.0205078
0 0.5 0 5 1
1 0.5 0 5 -2.5
2 0.5 0 5 2.5
3 0.5 0 5 -1.25
4 0.5 0 5 0.3125
5 0.5 0 5 -0.03125
0 0.5 0.5 5 1
1 0.5 0.5 5 -2
2 0.5 0.5 5 1.375
3 0.5 0.5 5 -0.25
4 0.5 0.5 5 -0.132812
5 0.5 0.5 5 0.078125
0 0.5 1 5 1
1 0.5 1 5 -1.5
2 0.5 1 5 0.5
3 0.5 1 5 0.25
4 0.5 1 5 -0.1875
5 0.5 1 5 0.03125
0 0.5 1.5 5 1
1 0.5 1.5 5 -1
2 0.5 1.5 5 -0.125
3 0.5 1.5 5 0.375
4 0.5 1.5 5 -0.0703125
5 0.5 1.5 5 -0.0234375
0 0.5 2 5 1
1 0.5 2 5 -0.5
2 0.5 2 5 -0.5
3 0.5 2 5 0.25
4 0.5 2 5 0.0625
5 0.5 2 5 -0.03125
0 0.5 2.5 5 1
1 0.5 2.5 5 0
2 0.5 2.5 5 -0.625
3 0.5 2.5 5 -0
4 0.5 2.5 5 0.117188
5 0.5 2.5 5 0
LAGUERRE_ASSOCIATED_TEST
LAGUERRE_ASSOCIATED evaluates the associated Laguerre
polynomials.
Table of L(N,M)(X) for
N(max) = 6
M = 0
X = 0
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
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
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 = 0.5
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 = 0.5
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 = 0.5
0 1
1 1.5
2 1.625
3 1.47917
4 1.14844
5 0.702865
6 0.19872
LAGUERRE_POLY_COEF:
LAGUERRE_POLY evaluates the Laguerre polynomial.
N X Exact F L(N)(X)
0 1 1 1
1 1 0 0
2 1 -0.5 -0.5
3 1 -0.666667 -0.666667
4 1 -0.625 -0.625
5 1 -0.466667 -0.466667
6 1 -0.256944 -0.256944
7 1 -0.0404762 -0.0404762
8 1 0.153993 0.153993
9 1 0.309744 0.309744
10 1 0.418946 0.418946
11 1 0.480134 0.480134
12 1 0.496212 0.496212
5 0.5 -0.445573 -0.445573
5 3 0.85 0.85
5 5 -3.16667 -3.16667
5 10 34.3333 34.3333
LAGUERRE_POLY_COEF_TEST
LAGUERRE_POLY_COEF determines Laguerre
polynomial coefficients.
L(0)
1
L(1)
-1 * x
1
L(2)
0.5 * x^2
-2 * x
1
L(3)
-0.166667 * x^3
1.5 * x^2
-3 * x
1
L(4)
0.0416667 * x^4
-0.666667 * x^3
3 * x^2
-4 * x
1
L(5)
-0.00833333 * x^5
0.208333 * x^4
-1.66667 * x^3
5 * x^2
-5 * x
1
Factorially scaled L(0)
1
Factorially scaled L(1)
-1 * x
1
Factorially scaled L(2)
1 * x^2
-4 * x
2
Factorially scaled L(3)
-1 * x^3
9 * x^2
-18 * x
6
Factorially scaled L(4)
1 * x^4
-16 * x^3
72 * x^2
-96 * x
24
Factorially scaled L(5)
-1 * x^5
25 * x^4
-200 * x^3
600 * x^2
-600 * x
120
LEGENDRE_POLY_TEST:
LEGENDRE_POLY evaluates the Legendre PN function.
N X Exact F P(N)(X)
0 0.25 1 1
1 0.25 0.25 0.25
2 0.25 -0.40625 -0.40625
3 0.25 -0.335938 -0.335938
9 0.25 0.176824 0.176824
10 0.25 0.2212 0.2212
3 0 0 -0
3 0.1 -0.1475 -0.1475
3 0.2 -0.28 -0.28
3 0.3 -0.3825 -0.3825
3 0.4 -0.44 -0.44
3 0.5 -0.4375 -0.4375
3 1 1 1
LEGENDRE_POLY_COEF_TEST
LEGENDRE_POLY_COEF determines the Legendre P
polynomial coefficients.
P(0)
1
P(1)
1 * x
0
P(2)
1.5 * x^2
0 * x
-0.5
P(3)
2.5 * x^3
0 * x^2
-1.5 * x
-0
P(4)
4.375 * x^4
0 * x^3
-3.75 * x^2
-0 * x
0.375
P(5)
7.875 * x^5
0 * x^4
-8.75 * x^3
-0 * x^2
1.875 * x
0
LEGENDRE_ASSOCIATED_TEST:
LEGENDRE_ASSOCIATED evaluates associated Legendre functions.
N M X Exact F PNM(X)
1 0 0 0 0
1 0 0.5 0.5 0.5
1 0 0.707107 0.707107 0.707107
1 0 1 1 1
1 1 0.5 -0.866025 -0.866025
2 0 0.5 -0.125 -0.125
2 1 0.5 -1.29904 -1.29904
2 2 0.5 2.25 2.25
3 0 0.5 -0.4375 -0.4375
3 1 0.5 -0.324759 -0.32476
3 2 0.5 5.625 5.625
3 3 0.5 -9.74278 -9.74279
4 2 0.5 4.21875 4.21875
5 2 0.5 -4.92187 -4.92188
6 3 0.5 12.7874 12.7874
7 3 0.5 116.685 116.685
8 4 0.5 -1050.67 -1050.67
9 4 0.5 -2078.49 -2078.49
10 5 0.5 30086.2 30086.2
LEGENDRE_ASSOCIATED_NORMALIZED_TEST:
LEGENDRE_ASSOCIATED_NORMALIZED evaluates
normalized associated Legendre functions.
N M X Exact F PNM(X)
0 0 0.5 0.282095 0.282095
1 0 0.5 0.244301 0.244301
1 1 0.5 -0.299207 -0.299207
2 0 0.5 -0.0788479 -0.0788479
2 1 0.5 -0.334523 -0.334523
2 2 0.5 0.289706 0.289706
3 0 0.5 -0.326529 -0.326529
3 1 0.5 -0.0699706 -0.0699706
3 2 0.5 0.383245 0.383245
3 3 0.5 -0.270995 -0.270995
4 0 0.5 -0.244629 -0.244629
4 1 0.5 0.256066 0.256066
4 2 0.5 0.188169 0.188169
4 3 0.5 -0.406492 -0.406492
4 4 0.5 0.248925 0.248925
5 0 0.5 0.084058 0.084058
5 1 0.5 0.329379 0.329379
5 2 0.5 -0.158885 -0.158885
5 3 0.5 -0.280871 -0.280871
5 4 0.5 0.412795 0.412795
5 5 0.5 -0.226097 -0.226097
LEGENDRE_FUNCTION_Q_TEST:
LEGENDRE_FUNCTION_Q evaluates the Legendre Q function.
N X Exact F Q(N)(X)
0 0 0 0
1 0 -1 -1
2 0 0 -0
3 0 0.666667 0.666667
9 0 -0.406349 -0.406349
10 0 0 -0
0 0.5 0.549306 0.549306
1 0.5 -0.725347 -0.725347
2 0.5 -0.818663 -0.818663
3 0.5 -0.198655 -0.198655
9 0.5 -0.116163 -0.116163
10 0.5 0.291658 0.291658
LEGENDRE_SYMBOL_TEST
LEGENDRE_SYMBOL computes the Legendre
symbol (Q/P) which records whether Q is
a quadratic residue modulo the prime P.
Legendre Symbols for P = 7
7 0 0
7 1 1
7 2 1
7 3 -1
7 4 1
7 5 -1
7 6 -1
7 7 0
Legendre Symbols for P = 11
11 0 0
11 1 1
11 2 -1
11 3 1
11 4 1
11 5 1
11 6 -1
11 7 -1
11 8 -1
11 9 1
11 10 -1
11 11 0
Legendre Symbols for P = 13
13 0 0
13 1 1
13 2 -1
13 3 1
13 4 1
13 5 -1
13 6 -1
13 7 -1
13 8 -1
13 9 1
13 10 1
13 11 -1
13 12 1
13 13 0
Legendre Symbols for P = 17
17 0 0
17 1 1
17 2 1
17 3 -1
17 4 1
17 5 -1
17 6 -1
17 7 -1
17 8 1
17 9 1
17 10 -1
17 11 -1
17 12 -1
17 13 1
17 14 -1
17 15 1
17 16 1
17 17 0
LERCH_TEST:
LERCH 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
LGAMMA_TEST:
LGAMMA is a C math library function which evaluates
the logarithm of the Gamma function.
X Exact F LGAMMA(X)
0.2 1.52406 1.52406
0.4 0.796678 0.796678
0.6 0.398234 0.398234
0.8 0.15206 0.15206
1 0 0
1.1 -0.0498725 -0.0498724
1.2 -0.0853741 -0.0853741
1.3 -0.108175 -0.108175
1.4 -0.119613 -0.119613
1.5 -0.120782 -0.120782
1.6 -0.112592 -0.112592
1.7 -0.0958077 -0.0958077
1.8 -0.0710839 -0.0710839
1.9 -0.0389843 -0.0389843
2 0 0
10 12.8018 12.8018
20 39.3399 39.3399
30 71.257 71.257
LOCK_TEST
LOCK counts the combinations on a button lock.
I, LOCK(I)
0 1
1 1
2 3
3 13
4 75
5 541
6 4683
7 47293
8 545835
9 7087261
10 102247563
MEIXNER_TEST:
MEIXNER evaluates Meixner polynomials.
N BETA C X M(N,BETA,C,X)
0 0.5 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
MERTENS_TEST
MERTENS computes the Mertens function.
N Exact MERTENS(N)
1 1 1
2 0 0
3 -1 -1
4 -1 -1
5 -2 -2
6 -1 -1
7 -2 -2
8 -2 -2
9 -2 -2
10 -1 -1
11 -2 -2
12 -2 -2
100 1 1
1000 2 2
10000 -23 -23
MOEBIUS_TEST
MOEBIUS computes the Moebius function.
N Exact MOEBIUS(N)
1 1 1
2 -1 -1
3 -1 -1
4 0 0
5 -1 -1
6 1 1
7 -1 -1
8 0 0
9 0 0
10 1 1
11 -1 -1
12 0 0
13 -1 -1
14 1 1
15 1 1
16 0 0
17 -1 -1
18 0 0
19 -1 -1
20 0 0
MOTZKIN_TEST
MOTZKIN computes the Motzkin numbers A(0:N).
A(N) counts the paths from (0,0) to (N,0).
I, A(I)
0 1
1 1
2 2
3 4
4 9
5 21
6 51
7 127
8 323
9 835
10 2188
NORMAL_01_CDF_INVERSE_TEST:
NORMAL_01_CDF_INVERSE inverts the normal 01 CDF.
FX X NORMAL_01_CDF_INVERSE(FX)
0.5 0 0
0.539828 0.1 0.1
0.57926 0.2 0.2
0.617911 0.3 0.3
0.655422 0.4 0.4
0.691462 0.5 0.5
0.725747 0.6 0.6
0.758036 0.7 0.7
0.788145 0.8 0.8
0.81594 0.9 0.9
0.841345 1 1
0.933193 1.5 1.5
0.97725 2 2
0.99379 2.5 2.5
0.99865 3 3
0.999767 3.5 3.5
0.999968 4 4
OMEGA_TEST
OMEGA computes the OMEGA function.
N Exact OMEGA(N)
1 1 1
2 1 1
3 1 1
4 1 1
5 1 1
6 2 2
7 1 1
8 1 1
9 1 1
10 2 2
30 3 3
101 1 1
210 4 4
1320 4 4
1764 3 3
2003 1 1
2310 5 5
2827 2 2
8717 2 2
12553 1 1
30030 6 6
510510 7 7
9699690 8 8
PENTAGON_NUM_TEST
PENTAGON_NUM computes the pentagonal numbers.
1 1
2 5
3 12
4 22
5 35
6 51
7 70
8 92
9 117
10 145
PHI_TEST
PHI computes the PHI function.
N Exact PHI(N)
1 1 1
2 1 1
3 2 2
4 2 2
5 4 4
6 2 2
7 6 6
8 4 4
9 6 6
10 4 4
20 8 8
30 8 8
40 16 16
50 20 20
60 16 16
100 40 40
149 148 148
500 200 200
750 200 200
999 648 648
PLANE_PARTITION_NUM_TEST
PLANE_PARTITION_NUM computes the number of plane
partitions of an integer.
1 1
2 3
3 6
4 13
5 24
6 48
7 86
8 160
9 282
10 500
POLY_BERNOULLI_TEST
POLY_BERNOULLI computes the poly-Bernoulli numbers
of negative index, B_n^(-k)
N K B_N^(-K)
0 0 1
1 0 0
2 0 0
3 0 0
4 0 0
5 0 0
6 0 0
0 1 1
1 1 2
2 1 4
3 1 8
4 1 16
5 1 32
6 1 64
0 2 1
1 2 4
2 2 14
3 2 46
4 2 146
5 2 454
6 2 1394
0 3 1
1 3 8
2 3 46
3 3 230
4 3 1066
5 3 4718
6 3 20266
0 4 1
1 4 16
2 4 146
3 4 1066
4 4 6902
5 4 41506
6 4 237686
0 5 1
1 5 32
2 5 454
3 5 4718
4 5 41506
5 5 329462
6 5 2441314
0 6 1
1 6 64
2 6 1394
3 6 20266
4 6 237686
5 6 2441314
6 6 22934774
POLY_COEF_COUNT_TEST
POLY_COEF_COUNT counts the number of coefficients
in a polynomial of degree DEGREE and dimension DIM.
Dimension Degree Count
1 0 1
1 1 2
1 2 3
1 3 4
1 4 5
1 5 6
4 0 1
4 1 5
4 2 15
4 3 35
4 4 70
4 5 126
7 0 1
7 1 8
7 2 36
7 3 120
7 4 330
7 5 792
10 0 1
10 1 11
10 2 66
10 3 286
10 4 1001
10 5 3003
PRIME_TEST
PRIME returns primes from a table.
Number of primes stored is 1600
I Prime(I)
1 2
2 3
3 5
4 7
5 11
6 13
7 17
8 19
9 23
10 29
1590 13411
1591 13417
1592 13421
1593 13441
1594 13451
1595 13457
1596 13463
1597 13469
1598 13477
1599 13487
1600 13499
PYRAMID_NUM_TEST
PYRAMID_NUM computes the pyramidal numbers.
1 1
2 4
3 10
4 20
5 35
6 56
7 84
8 120
9 165
10 220
PYRAMID_SQUARE_NUM_TEST
PYRAMID_SQUARE_NUM computes the pyramidal square numbers.
1 1
2 5
3 14
4 30
5 55
6 91
7 140
8 204
9 285
10 385
R8_AGM_TEST
R8_AGM computes the arithmetic geometric mean.
A B AGM AGM Diff
(Tabulated) R8_AGM(A,B)
22 96 52.27464119870424 52.27464119870424 7.10543e-15
83 56 68.83653005985852 68.83653005985852 0
42 7 20.65930119673401 20.65930119673401 3.55271e-15
26 11 17.69685487374365 17.69685487374367 1.77636e-14
4 63 23.8670497217533 23.8670497217533 3.55271e-15
6 45 20.71701598280599 20.71701598280599 3.55271e-15
40 75 56.12784225561668 56.12784225561668 0
80 0 0 0 0
90 35 59.26956508122964 59.26956508122989 2.4869e-13
9 1 3.936235503649555 3.936235503649556 4.44089e-16
53 53 53 53 0
1 2 1.456791031046907 1.456791031046907 0
1 4 2.243028580287603 2.243028580287603 0
1 8 3.615756177597363 3.615756177597363 0
1.5 8 4.081692408022163 4.081692408022163 0
R8_BETA_TEST:
R8_BETA evaluates the Beta function.
X Y Exact F R8_BETA(X,Y)
0.2 1 5 5
0.4 1 2.5 2.5
0.6 1 1.66667 1.66667
0.8 1 1.25 1.25
1 0.2 5 5
1 0.4 2.5 2.5
1 1 1 1
2 2 0.166667 0.166667
3 3 0.0333333 0.0333333
4 4 0.00714286 0.00714286
5 5 0.0015873 0.0015873
6 2 0.0238095 0.0238095
6 3 0.00595238 0.00595238
6 4 0.00198413 0.00198413
6 5 0.000793651 0.000793651
6 6 0.00036075 0.00036075
7 7 8.32501e-05 8.32501e-05
R8_CHOOSE_TEST
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
R8_ERF_TEST:
R8_ERF evaluates the error function.
X Exact F R8_ERF(X)
0 0 0
0.1 0.112463 0.112463
0.2 0.222703 0.222703
0.3 0.328627 0.328627
0.4 0.428392 0.428392
0.5 0.5205 0.5205
0.6 0.603856 0.603856
0.7 0.677801 0.677801
0.8 0.742101 0.742101
0.9 0.796908 0.796908
1 0.842701 0.842701
1.1 0.880205 0.880205
1.2 0.910314 0.910314
1.3 0.934008 0.934008
1.4 0.952285 0.952285
1.5 0.966105 0.966105
1.6 0.976348 0.976348
1.7 0.98379 0.98379
1.8 0.989091 0.989091
1.9 0.99279 0.99279
2 0.995322 0.995322
R8_ERF_INVERSE_TEST
R8_ERF_INVERSE inverts the error function.
FX X1 X2
0 0 0
0.112463 0.1 0.1
0.222703 0.2 0.2
0.328627 0.3 0.3
0.428392 0.4 0.4
0.5205 0.5 0.5
0.603856 0.6 0.6
0.677801 0.7 0.7
0.742101 0.8 0.8
0.796908 0.9 0.9
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 1.4
0.966105 1.5 1.5
0.976348 1.6 1.6
0.98379 1.7 1.7
0.989091 1.8 1.8
0.99279 1.9 1.9
0.995322 2 2
R8_EULER_CONSTANT_TEST:
R8_EULER_CONSTANT returns the Euler-Mascheroni constant
sometimes denoted by 'gamma'.
gamma = limit ( N -> oo ) ( sum ( 1 <= I <= N ) 1 / I ) - log ( N )
Numerically, g = 0.577216
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_FACTORIAL_TEST:
R8_FACTORIAL evaluates the factorial function.
N Exact F R8_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 3.6288e+06 3.6288e+06
11 3.99168e+07 3.99168e+07
12 4.79002e+08 4.79002e+08
13 6.22702e+09 6.22702e+09
14 8.71783e+10 8.71783e+10
15 1.30767e+12 1.30767e+12
16 2.09228e+13 2.09228e+13
17 3.55687e+14 3.55687e+14
18 6.40237e+15 6.40237e+15
19 1.21645e+17 1.21645e+17
20 2.4329e+18 2.4329e+18
25 1.55112e+25 1.55112e+25
30 2.65253e+32 2.65253e+32
R8_FACTORIAL_LOG_TEST:
R8_FACTORIAL_LOG evaluates the logarithm of the
factorial function.
N Exact F R8_FACTORIAL_LOG(N)
0 0 0
1 0 0
2 0.693147 0.693147
3 1.79176 1.79176
4 3.17805 3.17805
5 4.78749 4.78749
6 6.57925 6.57925
7 8.52516 8.52516
8 10.6046 10.6046
9 12.8018 12.8018
10 15.1044 15.1044
11 17.5023 17.5023
12 19.9872 19.9872
13 22.5522 22.5522
14 25.1912 25.1912
15 27.8993 27.8993
16 30.6719 30.6719
17 33.5051 33.5051
18 36.3954 36.3954
19 39.3399 39.3399
20 42.3356 42.3356
25 58.0036 58.0036
50 148.478 148.478
100 363.739 363.739
150 605.02 605.02
500 2611.33 2611.33
1000 5912.13 5912.13
R8_HYPER_2F1_TEST:
R8_HYPER_2F1 evaluates the hypergeometric function 2F1.
A B C X 2F1 2F1 DIFF
(tabulated) (computed)
-2.5 3.3 6.7 0.25 0.7235612934899779 0.7235612934899781 2.22e-16
-0.5 1.1 6.7 0.25 0.9791110934527796 0.9791110934527797 1.11e-16
0.5 1.1 6.7 0.25 1.021657814008856 1.021657814008856 0
2.5 3.3 6.7 0.25 1.405156320011213 1.405156320011212 4.441e-16
-2.5 3.3 6.7 0.55 0.4696143163982161 0.4696143163982162 5.551e-17
-0.5 1.1 6.7 0.55 0.9529619497744632 0.9529619497744636 3.331e-16
0.5 1.1 6.7 0.55 1.051281421394799 1.051281421394798 8.882e-16
2.5 3.3 6.7 0.55 2.399906290477786 2.399906290477784 1.776e-15
-2.5 3.3 6.7 0.85 0.2910609592841472 0.2910609592841471 5.551e-17
-0.5 1.1 6.7 0.85 0.9253696791037318 0.9253696791037308 9.992e-16
0.5 1.1 6.7 0.85 1.0865504094807 1.086550409480699 8.882e-16
2.5 3.3 6.7 0.85 5.738156552618904 5.738156552619273 3.686e-13
3.3 6.7 -5.5 0.25 15090.66974870461 15090.6697487046 1.091e-11
1.1 6.7 -0.5 0.25 -104.3117006736435 -104.3117006736435 2.842e-14
1.1 6.7 0.5 0.25 21.17505070776881 21.1750507077688 1.066e-14
3.3 6.7 4.5 0.25 4.194691581903192 4.194691581903191 8.882e-16
3.3 6.7 -5.5 0.55 10170777974.04881 10170777974.04883 1.144e-05
1.1 6.7 -0.5 0.55 -24708.63532248916 -24708.63532248914 1.819e-11
1.1 6.7 0.5 0.55 1372.230454838499 1372.230454838497 2.274e-12
3.3 6.7 4.5 0.55 58.09272870639465 58.09272870639462 2.842e-14
3.3 6.7 -5.5 0.85 5.868208761512417e+18 5.868208761512401e+18 1.638e+04
1.1 6.7 -0.5 0.85 -446350101.47296 -446350101.4729614 1.431e-06
1.1 6.7 0.5 0.85 5383505.756129573 5383505.756129585 1.211e-08
3.3 6.7 4.5 0.85 20396.91377601966 20396.91377601966 3.638e-12
R8_PSI_TEST:
R8_PSI evaluates the Psi function.
X Psi(X) Psi(X) DIFF
(Tabulated) (R8_PSI)
1 -0.5772156649015329 -0.5772156649015329 0
1.1 -0.4237549404110768 -0.4237549404110768 5.551e-17
1.2 -0.2890398965921883 -0.2890398965921884 5.551e-17
1.3 -0.1691908888667997 -0.1691908888667995 1.665e-16
1.4 -0.06138454458511615 -0.06138454458511624 9.021e-17
1.5 0.03648997397857652 0.03648997397857652 0
1.6 0.1260474527734763 0.1260474527734763 2.776e-17
1.7 0.208547874873494 0.208547874873494 2.776e-17
1.8 0.2849914332938615 0.2849914332938615 0
1.9 0.3561841611640597 0.3561841611640596 1.11e-16
2 0.4227843350984671 0.4227843350984672 1.11e-16
R8POLY_DEGREE_TEST
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) = 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) = 1
Dimensioned degree = 3, Actual degree = 0
The R8POLY:
p(x) = 0
Dimensioned degree = 3, Actual degree = 0
R8POLY_PRINT_TEST
R8POLY_PRINT prints an R8POLY.
The R8POLY:
p(x) = 9 * x^5
+0.78 * x^4
+56 * x^2
-3.4 * x
+2
R8POLY_VALUE_HORNER_TEST
R8POLY_VALUE_HORNER evaluates a polynomial at
one point, using Horner's method.
The polynomial coefficients:
p(x) = 1 * x^4
-10 * x^3
+35 * x^2
-50 * x
+24
I X P(X)
0 0 24
1 0.3333 10.86
2 0.6667 3.457
3 1 0
4 1.333 -0.9877
5 1.667 -0.6914
6 2 0
7 2.333 0.4938
8 2.667 0.4938
9 3 0
10 3.333 -0.6914
11 3.667 -0.9877
12 4 0
13 4.333 3.457
14 4.667 10.86
15 5 24
SIGMA_TEST
SIGMA computes the SIGMA function.
N Exact SIGMA(N)
1 1 1
2 3 3
3 4 4
4 7 7
5 6 6
6 12 12
7 8 8
8 15 15
9 13 13
10 18 18
30 72 72
127 128 128
128 255 255
129 176 176
210 576 576
360 1170 1170
617 618 618
815 984 984
816 2232 2232
1000 2340 2340
SIMPLEX_NUM_TEST
SIMPLEX_NUM computes the N-th simplex number
in M dimensions.
M: 0 1 2 3 4 5
N
0 1 0 0 0 0 0
1 1 1 1 1 1 1
2 1 2 3 4 5 6
3 1 3 6 10 15 21
4 1 4 10 20 35 56
5 1 5 15 35 70 126
6 1 6 21 56 126 252
7 1 7 28 84 210 462
8 1 8 36 120 330 792
9 1 9 45 165 495 1287
10 1 10 55 220 715 2002
SIN_POWER_INT_TEST:
SIN_POWER_INT computes the integral of the N-th power
of the sine function.
A B N Exact Computed
10 20 0 10 10
0 1 1 0.4597 0.4597
0 1 2 0.2727 0.2727
0 1 3 0.1789 0.1789
0 1 4 0.124 0.124
0 1 5 0.08897 0.08897
0 2 5 0.9039 0.9039
1 2 5 0.815 0.815
0 1 10 0.02189 0.02189
0 1 11 0.01702 0.01702
SLICE_TEST:
SLICE determines the maximum number of pieces created
by SLICE_NUM slices in a DIM_NUM space.
Slice Array:
Col: 0 1 2 3 4 5 6 7
Row
0: 2 3 4 5 6 7 8 9
1: 2 4 7 11 16 22 29 37
2: 2 4 8 15 26 42 64 93
3: 2 4 8 16 31 57 99 163
4: 2 4 8 16 32 63 120 219
SPHERICAL_HARMONIC_TEST:
SPHERICAL_HARMONIC evaluates spherical harmonic functions.
N M THETA PHI YR YI
0 0 0.5236 1.047 0.2821 0
0.2821 0
1 0 0.5236 1.047 0.4231 0
0.4231 0
2 1 0.5236 1.047 -0.1673 -0.2897
-0.1673 -0.2897
3 2 0.5236 1.047 -0.1106 0.1916
-0.1106 0.1916
4 3 0.5236 1.047 0.1355 0
0.1355 1.038e-16
5 5 0.2618 0.6283 0.000539 0
0.000539 -6.601e-20
5 4 0.2618 0.6283 -0.005147 0.003739
-0.005147 0.003739
5 3 0.2618 0.6283 0.01371 -0.0422
0.01371 -0.0422
5 2 0.2618 0.6283 0.06096 0.1876
0.06096 0.1876
5 1 0.2618 0.6283 -0.417 -0.303
-0.417 -0.303
4 2 0.6283 0.7854 0 0.4139
2.535e-17 0.4139
4 2 1.885 0.7854 0 -0.1003
-6.143e-18 -0.1003
4 2 3.142 0.7854 0 0
0 0
4 2 4.398 0.7854 0 -0.1003
-6.143e-18 -0.1003
4 2 5.655 0.7854 0 0.4139
2.535e-17 0.4139
3 -1 0.3927 0.4488 0.3641 -0.1754
0.3641 -0.1754
3 -1 0.3927 0.8976 0.252 -0.316
0.252 -0.316
3 -1 0.3927 1.346 0.08993 -0.394
0.08993 -0.394
3 -1 0.3927 1.795 -0.08993 -0.394
-0.08993 -0.394
3 -1 0.3927 2.244 -0.252 -0.316
-0.252 -0.316
STIRLING1_TEST
STIRLING1: Stirling numbers of first kind.
Get rows 1 through 8
1 1 0 0 0 0 0 0 0
2 -1 1 0 0 0 0 0 0
3 2 -3 1 0 0 0 0 0
4 -6 11 -6 1 0 0 0 0
5 24 -50 35 -10 1 0 0 0
6 -120 274 -225 85 -15 1 0 0
7 720 -1764 1624 -735 175 -21 1 0
8 -5040 13068 -13132 6769 -1960 322 -28 1
STIRLING2_TEST
STIRLING2: Stirling numbers of second kind.
Get rows 1 through 8
1 1 0 0 0 0 0 0 0
2 1 1 0 0 0 0 0 0
3 1 3 1 0 0 0 0 0
4 1 7 6 1 0 0 0 0
5 1 15 25 10 1 0 0 0
6 1 31 90 65 15 1 0 0
7 1 63 301 350 140 21 1 0
8 1 127 966 1701 1050 266 28 1
TAU_TEST
TAU computes the Tau function.
N exact C(I) computed C(I)
1 1 1
2 2 2
3 2 2
4 3 3
5 2 2
6 4 4
7 2 2
8 4 4
9 3 3
10 4 4
23 2 2
72 12 12
126 12 12
226 4 4
300 18 18
480 24 24
521 2 2
610 8 8
832 14 14
960 28 28
TETRAHEDRON_NUM_TEST
TETRAHEDRON_NUM computes the tetrahedron numbers.
1 1
2 4
3 10
4 20
5 35
6 56
7 84
8 120
9 165
10 220
TRIANGLE_NUM_TEST
TRIANGLE_NUM computes the triangular numbers.
1 1
2 3
3 6
4 10
5 15
6 21
7 28
8 36
9 45
10 55
TRIANGLE_TO_I4_TEST
TRIANGLE_TO_I4 converts a triangular index to a
linear one.
I J ==> K
0 0 0
1 0 1
1 1 2
2 0 3
2 1 4
2 2 5
3 0 6
3 1 7
3 2 8
3 3 9
4 0 10
4 1 11
4 2 12
4 3 13
4 4 14
TRINOMIAL_TEST
TRINOMIAL evaluates the trinomial coefficient:
T(I,J,K) = (I+J+K)! / I! / J! / K!
I J K T(I,J,K)
0 0 0 1
1 0 0 1
2 0 0 1
3 0 0 1
4 0 0 1
0 1 0 1
1 1 0 2
2 1 0 3
3 1 0 4
4 1 0 5
0 2 0 1
1 2 0 3
2 2 0 6
3 2 0 10
4 2 0 15
0 3 0 1
1 3 0 4
2 3 0 10
3 3 0 20
4 3 0 35
0 4 0 1
1 4 0 5
2 4 0 15
3 4 0 35
4 4 0 70
0 0 1 1
1 0 1 2
2 0 1 3
3 0 1 4
4 0 1 5
0 1 1 2
1 1 1 6
2 1 1 12
3 1 1 20
4 1 1 30
0 2 1 3
1 2 1 12
2 2 1 30
3 2 1 60
4 2 1 105
0 3 1 4
1 3 1 20
2 3 1 60
3 3 1 140
4 3 1 280
0 4 1 5
1 4 1 30
2 4 1 105
3 4 1 280
4 4 1 630
0 0 2 1
1 0 2 3
2 0 2 6
3 0 2 10
4 0 2 15
0 1 2 3
1 1 2 12
2 1 2 30
3 1 2 60
4 1 2 105
0 2 2 6
1 2 2 30
2 2 2 90
3 2 2 210
4 2 2 420
0 3 2 10
1 3 2 60
2 3 2 210
3 3 2 560
4 3 2 1260
0 4 2 15
1 4 2 105
2 4 2 420
3 4 2 1260
4 4 2 3150
0 0 3 1
1 0 3 4
2 0 3 10
3 0 3 20
4 0 3 35
0 1 3 4
1 1 3 20
2 1 3 60
3 1 3 140
4 1 3 280
0 2 3 10
1 2 3 60
2 2 3 210
3 2 3 560
4 2 3 1260
0 3 3 20
1 3 3 140
2 3 3 560
3 3 3 1680
4 3 3 4200
0 4 3 35
1 4 3 280
2 4 3 1260
3 4 3 4200
4 4 3 11550
0 0 4 1
1 0 4 5
2 0 4 15
3 0 4 35
4 0 4 70
0 1 4 5
1 1 4 30
2 1 4 105
3 1 4 280
4 1 4 630
0 2 4 15
1 2 4 105
2 2 4 420
3 2 4 1260
4 2 4 3150
0 3 4 35
1 3 4 280
2 3 4 1260
3 3 4 4200
4 3 4 11550
0 4 4 70
1 4 4 630
2 4 4 3150
3 4 4 11550
4 4 4 34650
V_HOFSTADTER_TEST
V_HOFSTADTER evaluates Hofstadter's recursive
V function.
N V(N)
0 0
1 1
2 1
3 1
4 1
5 2
6 3
7 4
8 5
9 5
10 6
11 6
12 7
13 8
14 8
15 9
16 9
17 10
18 11
19 11
20 11
21 12
22 12
23 13
24 14
25 14
26 15
27 15
28 16
29 17
30 17
VIBONACCI_TEST
VIBONACCI computes a Vibonacci sequence.
We compute the series 3 times.
I V1 V2 V3
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
ZECKENDORF_TEST
ZECKENDORF computes the Zeckendorf decomposition of
an integer N into nonconsecutive Fibonacci numbers.
N Sum M Parts
1 1
2 2
3 3
4 3 1
5 5
6 5 1
7 5 2
8 8
9 8 1
10 8 2
11 8 3
12 8 3 1
13 13
14 13 1
15 13 2
16 13 3
17 13 3 1
18 13 5
19 13 5 1
20 13 5 2
21 21
22 21 1
23 21 2
24 21 3
25 21 3 1
26 21 5
27 21 5 1
28 21 5 2
29 21 8
30 21 8 1
31 21 8 2
32 21 8 3
33 21 8 3 1
34 34
35 34 1
36 34 2
37 34 3
38 34 3 1
39 34 5
40 34 5 1
41 34 5 2
42 34 8
43 34 8 1
44 34 8 2
45 34 8 3
46 34 8 3 1
47 34 13
48 34 13 1
49 34 13 2
50 34 13 3
51 34 13 3 1
52 34 13 5
53 34 13 5 1
54 34 13 5 2
55 55
56 55 1
57 55 2
58 55 3
59 55 3 1
60 55 5
61 55 5 1
62 55 5 2
63 55 8
64 55 8 1
65 55 8 2
66 55 8 3
67 55 8 3 1
68 55 13
69 55 13 1
70 55 13 2
71 55 13 3
72 55 13 3 1
73 55 13 5
74 55 13 5 1
75 55 13 5 2
76 55 21
77 55 21 1
78 55 21 2
79 55 21 3
80 55 21 3 1
81 55 21 5
82 55 21 5 1
83 55 21 5 2
84 55 21 8
85 55 21 8 1
86 55 21 8 2
87 55 21 8 3
88 55 21 8 3 1
89 89
90 89 1
91 89 2
92 89 3
93 89 3 1
94 89 5
95 89 5 1
96 89 5 2
97 89 8
98 89 8 1
99 89 8 2
100 89 8 3
ZERNIKE_POLY_TEST
ZERNIKE_POLY evaluates a Zernike polynomial directly.
Table of polynomial coefficients:
N M
0 0 1
1 0 0 0
1 1 0 1
2 0 -1 0 2
2 1 0 0 0
2 2 0 0 1
3 0 0 0 0 0
3 1 0 -2 0 3
3 2 0 0 0 0
3 3 0 0 0 1
4 0 1 0 -6 0 6
4 1 0 0 0 0 0
4 2 0 0 -3 0 4
4 3 0 0 0 0 0
4 4 0 0 0 0 1
5 0 0 0 0 0 0 0
5 1 0 3 0 -12 0 10
5 2 0 0 0 0 0 0
5 3 0 0 0 -4 0 5
5 4 0 0 0 0 0 0
5 5 0 0 0 0 0 1
Z1: Compute polynomial coefficients,
then evaluate by Horner's method;
Z2: Evaluate directly by recursion.
N M Z1 Z2
0 0 1 1
1 0 0 0
1 1 0.9877 0.9877
2 0 0.9509 0.9509
2 1 0 0
2 2 0.9755 0.9755
3 0 0 0
3 1 0.9149 0.9149
3 2 0 0
3 3 0.9634 0.9634
4 0 0.8564 0.8564
4 1 0 0
4 2 0.8797 0.8797
4 3 0 0
4 4 0.9515 0.9515
5 0 0 0
5 1 0.7997 0.7997
5 2 0 0
5 3 0.8452 0.8452
5 4 0 0
5 5 0.9398 0.9398
ZERNIKE_POLY_COEF_TEST
ZERNIKE_POLY_COEF determines the Zernike
polynomial coefficients.
Zernike polynomial
p(x) = 0
Zernike polynomial
p(x) = 10 * x^5
-12 * x^3
+3 * x
Zernike polynomial
p(x) = 0
Zernike polynomial
p(x) = 5 * x^5
-4 * x^3
Zernike polynomial
p(x) = 0
Zernike polynomial
p(x) = 1 * x^5
ZETA_TEST
ZETA computes the Zeta function.
N exact Zeta computed Zeta
2 1.645 1.644
3 1.202 1.202
4 1.082 1.082
5 1.037 1.037
6 1.017 1.017
7 1.008 1.008
8 1.004 1.004
9 1.002 1.002
10 1.001 1.001
11 1 1
12 1 1
16 1 1
20 1 1
30 1 1
40 1 1
POLPAK_PRB
Normal end of execution.
13 April 2015 10:09:07 AM