# Addons/math/misc

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## Contents

- 1 math/misc
- 1.1 Installation
- 1.2 Scripts
- 1.2.1 amoeba
- 1.2.2 bigpi
- 1.2.3 brent
- 1.2.4 contfrac
- 1.2.5 det
- 1.2.6 fermat
- 1.2.7 gamesolver
- 1.2.8 gcd
- 1.2.9 integer
- 1.2.10 integrat
- 1.2.11 jacobi
- 1.2.12 legendre
- 1.2.13 linear
- 1.2.14 makemat
- 1.2.15 matfacto
- 1.2.16 mathutil
- 1.2.17 matutil
- 1.2.18 mean
- 1.2.19 numbers
- 1.2.20 pollard
- 1.2.21 poly
- 1.2.22 primutil
- 1.2.23 quatern
- 1.2.24 rsa
- 1.2.25 simplex
- 1.2.26 simplexnr
- 1.2.27 spline
- 1.2.28 svd

# math/misc

You can browse the scripts available in this addon using Trac.

## Installation

Use JAL/Package Manager.

Load the individual scripts in the addon as follows:

load 'math/misc/name_of_script'

## Scripts

amoeba 1.0.0 Nelder-Mead multi-dimensional minimization, aka the amoeba method bigpi 1.0.0 Calculate several digits of pi brent 1.0.0 Brent's method in J contfrac 1.0.0 Continued fraction utilities det 1.0.0 Definitions for determinants fermat 1.0.0 Fermat factorization gamesolver 1.0.0 Find optimal mixed strategies for 2-person games gcd 1.0.0 Calculate GCD integer 1.0.0 Various integer definitions integrat 1.0.0 Various methods for numeric integration jacobi 1.0.0 Jacobi's method for eigenvalues and vectors legendre 1.0.0 Legendre symbol and quadratic residues linear 1.0.0 Solve linear equations makemat 1.0.0 Make various matrices matfacto 1.0.0 Matrix factorization mathutil 1.0.0 Math utilities matutil 1.0.0 Matrix utilities mean 1.0.0 Various means numbers 1.0.0 Various number definitions pollard 1.0.0 Pollard factorizations poly 1.0.0 Polynomial functions primutil 1.0.0 Primes - prime testing programs quatern 1.0.0 Definitions for quaternions rsa 1.0.0 RSA encryption simplex 1.0.0 Simplex method simplexnr 1.0.0 Simplex method spline 1.0.0 Spline utilities svd 1.0.0 Singular value decomposition

### amoeba

Nelder-Mead multi-dimensional minimization, aka the amoeba method

Henry H. Rich, October 2009

`amoeba`c Nelder-Mead multi-dimensional minimization

### bigpi

Calculate several digits of pi

from Borwein

`bigpi`v Calculate pi to different levels of of precision

### brent

Brent's method in J

(from J. Patrick Harrington)

`brent`a Adverb to solve f(x) = y0 by Brent's method.

### contfrac

Continued fraction utilities

`contfrac`v create continued fraction `contfracx`v expand continued fraction

### det

Definitions for determinants

`det`v Determinants of a matrix by recursive expansion of minors `detm`v Determinants of square matrix by Gauss elimination

### fermat

Fermat factorization

`fermatfactor`v Find factor of n near square root of n using Fermat's method

### gamesolver

Find optimal mixed strategies for 2-person games

Henry H. Rich (HenryHRich@nc.rr.com), April 2005

`solvegame`v Find optimal mixed strategies for 2-person games

### gcd

Calculate GCD

`gcd`v Greatest common denominator

### integer

Various integer definitions

`inta`v Augmented integers `inte`v Extended integers `ints`v Symmetric integers `intm`v Minus integers `intn`v Normal integers `intr`v Reflexive integers `jint`v Complex integers `jints`c symmetric ints

### integrat

Various methods for numeric integration

`integrate`c Numeric integration by Aitken extrapolation on Gauss integrals `simpson`c Numeric integration by Simpson's method `adapt`c Numeric integration by adaptive quadrature

### jacobi

Jacobi's method for eigenvalues and vectors

`jacobi`v Calculate eigenvalues and vectors using Jacobi's method

### legendre

Legendre symbol and quadratic residues

`quadres`v Quadratic residues of p `quadrec`v Quadratic reciprocity `legendre`v Legendre symbol (n/p) for integer n, odd prime p

### linear

Solve linear equations

`gauss_elimination`v Gauss elimination (partial pivoting) `gauss_jordan`v Gauss-Jordan elimination (full pivoting) `gauss_seidel`v Solves Ax=B for matrix A, vector B `jacobi_iteration`v Solves Ax=B for matrix A, vector B

### makemat

Make various matrices

`bandmat`v Band matrix in position x from diagonal y `cidmat`v Counter identity matrix of size y `diagmat`v Diagonal matrix with y on diagonal `hilbertmat`v Hilbert matrix of size y `ltmat`v Lower triangular matrix (of 1's) of size y `utmat`v Upper triangular matrix (of 1's) of size y

### matfacto

Matrix factorization

`choleski`v Choleski decomposition of matrix y `lud`v LU decomposition of matrix y `qrd`v QR decomposition of matrix y

### mathutil

Math utilities

`det`v Determinant of matrix y `mp`v Matrix product of x and y `powermod`a x (n powermod) y computes n|x^y `randomint`v Random integer in range 0, <: 10^y `randomintd`v Random integer with y digits `timesmod`a x (n timesmod) y computes n|x*y

### matutil

Matrix utilities

`diag`v Diagonal of matrix `invsut`v Invert square upper-triangular matrix `minors`v minors of matrix `band`v b band M - zero all but x bands of matrix y `cond`v Condition number of matrix `pivot`v Pivot at row, column `coldrop`v Drop cols from M `coltake`v Take cols from M `rowdrop`v Drop rows from M `rowtake`v Take rows from M `rowswap`v Swap rows i and j `rowscale`v Multiply row i by n `rowshear`v Multiply row j by n and add to row i

### mean

Various means

`arithmean`v Arithmetic mean of y `geomean`v Geometric mean of y `harmean`v Harmonic mean of y `commonmean`v Common mean of y

### numbers

Various number definitions

`bell`v Number of ways of partitioning y things into subsets `bernoulli`v Bernoulli numbers from 0 to y `catalan`v Catalan numbers `cycles`v Table of Stirling cycle numbers (S1) `cycle`v Number of ways of partitioning n items into k cycles `eulers`v Table of Eulerian numbers `euler`v Number of permutations of size n with k ascents `fermat`v Fermat numbers `fibonacci`v first y+1 Fibonacci numbers `lucas`v first y+1 Lucas numbers `fibbinet`v Calculates nth Fibonacci number using Binet formula `subsets`v Table of Stirling subset numbers (S2) `subset`v Number of ways of partitioning n items into k sets `tangent`v tangent numbers from 0 to y

### pollard

Pollard factorizations

`pollardrho`v Pollard rho factorization `pollardpm1`v Pollard p-1 factorizaton

### poly

Polynomial functions

`chebyshev_tp`v Evaluate Chebyshev T polynomial of order n at x `chebyshev_up`v Evaluate Chebyshev U polynomial of order n at x `chebyshev_tpc`v Returns coefficients of Chebyshev T polynomial of order n `chebyshev_upc`v Returns coefficients of Chebyshev U polynomial of order n `legendre_pc`v Returns coefficients of legendre polynomial of order n

### primutil

Primes - prime testing programs

`prevprime`v Previous prime number to y `nextprime`v Next prime number after y `primeq`v Test for prime `primesto`v Primes up to y `carmichaelq`v Test composite for Carmichael number `inversep`v Inverse mod p `mersenneq`v Test if prime p generates a mersenne prime `primepowersto`v Generate prime powers up to y `smallprimefactors`v Get small primefactors from number

### quatern

Definitions for quaternions

real quaternions:

- stored as real arrays with last dimension of length 4

`qncon`v Conjugate `qnmul`v Multiplication `qnrec`v Reciprocal `qndivl`v Division (left quotient) `qndivr`v Division (right quotient) `qndivml`v Division (using matrix divide, left quotient) `qndivmr`v Division (using matrix divide, right quotient) `qnpolar`v Convert to/from polar form

### rsa

RSA encryption

method:

P & Q are primes

E is exponent, relatively prime to (P-1)*(Q-1)

D is the inverse of E mod (P-1)*(Q-1)

here:

the public key is: PQ,E (i.e. a pair of numbers)

the private key is: D

then:

msg (PQ powermod) E encodes msg

msg (PQ powermod) D decodes msg

### simplex

Simplex method

see also math/misc/simplexnr for the simplex method following

"Numerical Recipes in C"

example description thanks to Henry H. Rich

`simplex`v Simplex method

### simplexnr

Simplex method

see also math/misc/simplex

This implementation follows Numerical Recipes in C, 2/e,

section 10.8, except that we let the user specify the type of

each constraint rather than expecting the order <: >: =

and we use a similar but refined way of handling degenerate pivots

Henry H. Rich (HenryHRich@nc.rr.com), June 2001

`simplexnr`v Run simplex method

### spline

Spline utilities

`cubicspline`v Calculate cubic spline `interspline`v Interpolate spline `freespline`v Calculate spline `aprxspline`v Approximate by spline

### svd

Singular value decomposition

for real matrices only

`svd`v Singular value decomposition of matrix y