Addons/math/misc
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math/misc
You can browse the scripts available in this addon using github.
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
trig
Trigonometric functions
arccos v arc cosine arccosh v arc cosh arcsin v arc sine arcsinh v arc sinh arctan v arc tangent arctanh v arc tanh cos v cosine cosd v cos in degrees cosh v hyperbolic cosine dfr v degrees from radians indegrees a convert function on radians to use degrees pi n pi rfd v radians from degrees sin v sine sind v sin in degrees sinh v hyperbolic sine tan v tangent tand v tan in degrees tanh v hyperbolic tangent