Chapter 1 Introduction

Chapter 2 Approximation

Chapter 3 Differential Equations

Chapter 4 Eigensystems

Chapter 5 Transforms

Chapter 6 Linear Algebra

Chapter 7 Optimization

Chapter 8 Partial Diffrential Equations

Chapter 9 Integration

Chapter 10 Random Numbers

Chapter 11 Roots

Chapter 12 Special Functions

Chapter 13 Statistics

Chapter 14 Utility Functions

Index

Title

Page

CHAPTER 1

1-2

INTRODUCTION TO SCIMATH

1-2

Installation

1-2

General information on using SciMath Functions

1-2

Single and Double Precision Functions

1-3

Error Handling

1-3

Function description page

1-3

Multidimensional Arrays

1-4

CHAPTER 2

2-2

APPROXIMATION

2-2

Introduction

2-2

cifrnl Differentiate a cubic spline

2-5

cinpol Interpolate using cubic spline

2-6

cleval Evaluate a cubic spline

2-7

csplit Cubic spline fit

2-8

msmshd Computes locally uniform mesh, made of discrete points

2-9

mspudp Piecewise uniform mesh for a set of discrete points

2-10

sbeval Derivative basis spline computation

2-11

sderiv Selected derivative, basis spline computation

2-12

sdeval Compute spline and derivative

2-13

sinteg Basis spline integration

2-14

smeval Compute spline

2-15

sntgdp Compute spline and selected derivatives

2-16

snzspl Basis spline computation

2-17

spdfit Apply a B-spline fit to derivatives of a function

2-18

spdisc Create B-spline mesh for fitting discrete data points

2-19

spffit Apply a B-spline fit to a function

2-20

spldpt Generate uniform mesh of discrete points

2-22

splhdi Uniform variation diminishing 3-dimensional spline

2-23

splodi Uniform variation diminishing 1-dimensional spline

2-25

spltdi Uniform variation diminishing 2-dimensional spline

2-26

splums Generate uniform mesh for a B-spline

2-28

spumsh Computes locally uniform mesh for a B spline

2-29

spwnms Piecewise uniform mesh for a B-spline

2-30

srrabs Compute absolute error in B spline fit to a function

2-31

srrasc Compute absolute error on B spline in selected intervals

2-32

srrest Computing estimate error in B spline fit using mesh refinement

2-33

srrsca Computing estimate error in B spline in selected intervals

2-34

ssqfit Least Squares B-spline fit, discrete data

2-35

ssqwfi Weighted Least Squares B-spline fit, discrete data

2-36

svaltn Compute spline and selected derivatives with more user control

2-37

uncini Best uniform approximation with initial approximation

2-39

uuncap Approximation of a mesh, best uniform approximation

2-41

CHAPTER 3

3-2

DIFFERENTIAL EQUATIONS

3-2

Introduction

3-2

odeivp Stiff ODE (Ordinary differential equation) initial value problem

3-3

odnivp Initial Value Problem, Ordinary Differential Equation Solver

3-5

odvmod Initial Value Problem, ODE Solver (Second Version)

3-7

CHAPTER 4

4-2

EIGENSYSTEMS

4-2

Introduction

4-2

eigbal Create a balanced matrix with equal eigenvalues

4-3

eighes Create a balanced matrix with equal eigenvalues

4-4

eighmt Compute eigenvalues of a Hessenberg matrix

4-5

eighos Reduction of a real symmetric matrix, Housholder method

4-6

eigsmt Compute eigenvectors and eigenvalues of a symmetric matrix

4-7

eigsur Sort eigenvalues

4-8

eigtri Eigenvalues and eigenvectors of a symmetric tridiagonal matrix

4-9

mtgenc Complex general eigenvalue problem solver

4-10

mtigen Eigenvectors and eigenvalues of a general real matrix

4-11

CHAPTER 5

5-2

TRANSFORMS

5-2

Introduction

5-2

fftcmu Initialization for fftltp

5-3

fftcpx Inverse Fast Fourier Transform, complex data

5-4

fftdat Fast Fourier Transform, real data

5-5

fftinv Inverse Fast Fourier Transform, Real Data

5-6

fftitm Initialization for fftult

5-7

fftltp Complex data multiple Fourier transform

5-8

fftmpx Fast Fourier Transform, general case

5-9

fftmul Real data multiple Fast Fourier Transform

5-10

fftplx Fast Fourier Transform, complex data

5-11

fftult Half-Complex data multiple Fast Fourier Transform

5-12

CHAPTER 6

6-3

LINEAR ALGEBRA

6-3

Introduction

6-3

aramua Multiply array by k and add to another array

6-8

arasum Add elements of an array

6-9

arcamu Complex version of arsuma

6-10

armmax Largest element of an array

6-11

arplrt Plane rotate a vector

6-12

arrcop Copy an array to the other

6-13

arrdot Dot product of two arrays

6-14

arrexc Exchange two arrays

6-15

arrgiv Givens plane rotation

6-16

arscal Scale an array

6-17

arsuma Sum of absolute values if an array

6-18

cagsum Computes magnitude of real part plus magnitude of imaginary part

6-19

ccscal Scale a complex array

6-20

cecmul Scale a complex array ( second variation

6-21

cecsum Copy a complex array to another

6-22

ceswch Exchange two complex arrays

6-23

corvec Largest element of a complex array

6-24

cotvec Plane rotation to a complex vector

6-25

cplrot Givens rotation for a complex vector

6-26

ctprct Dot product of two complex arrays

6-27

ctprod Dot product of two complex arrays, conjugate input

6-28

ctsqls Complex linear equations, Least Squares solution

6-29

culsum Multiply complex array by complex number z and add to another array

6-30

lesqso Least squares solution of linear equations

6-31

linqrs QR decomposition

6-33

linqso Linear system solver using QR decomposition

6-34

lnchbk Linear system solver using Cholesky backsubstitution

6-35

lnchol Cholesky decomposition

6-36

mlunum LU numerical decomposition of sparse matrix with input function

6-37

msucon LU decomposition of a sparse matrix with condition estimation and input function

6-38

mtarsm Multiplication of symmetric matrix and vector

6-40

mtbano Norm of a banded unsymmetric matrix

6-41

mtbdec Banded unsymmetric matrix decomposition

6-42

mtbdes Symmetric band positive definite matrix LDL decomposition

6-43

mtcmul Multiply matrix and vector

6-44

mtcomp Symmetric band positive definite matrix LDL decomposition

6-45

mtcond Linear system solution (banded), with condition estimation

6-46

mtecon Performs LU decomposition of general matrix with condition estimation

6-47

mtforw Sparse linear system forward-back solution

6-48

mtgcon Solution of general linear system with condition estimation

6-49

mtgenm Performs LU decomposition of a general matrix

6-50

mtlcem Sparse matrix LU decomposition with condition estimation, input matrix

6-51

mtlins Sparse linear system solution with input function

6-52

mtlnsy Symmetric linear system solver with condition estimation

6-53

mtlond Condition estimation of LDL decomposition

6-54

mtlsbl Lower triangular band, linear system solution

6-55

mtlsbs Linear system solver (banded)

6-56

mtlslm Solution of lower triangular linear system

6-57

mtlssp Definite band positive linear system solution with condition estimation

6-58

mtlude Banded unsymmetric matrix and condition estimation, LU decomposition

6-59

mtmfor Lower triangular linear system solution (band symmetric matrix

6-60

mtmlss Linear system solution for band positive definite system

6-61

mtmult Matrix-vector multiplication for banded positive definite matrix

6-62

mtnges Solution of general linear system

6-63

mtnlum Performs LU decomposition of a general matrix

6-64

mtnnor General matrix norm

6-65

mtorde Row/Column ordering of a sparse matrix with input function

6-66

mtposn Band positive definite matrix norm

6-67

mtqsol Least squares solution

6-68

mtrlss Solution of lower triangular linear system

6-69

mtsbup Upper triangular band linear system solution

6-70

mtslnf Sparse linear system solver (forward-back solution

6-71

mtslns Sparse linear system solver, forward-solution (modified

6-72

mtslsq Least squares solution and Singular Value Decomposition

6-73

mtslus Sparse matrix symbolic LU decomposition with input function

6-75

mtsmdm Symmetric matrix MDM^T decomposition

6-76

mtsmul Vector-sparse matrix multiplication, function input

6-77

mtspam Vector multiplication of sparse matrix, input matrix

6-78

mtspld Sparse matrix LU decomposition with input matrix

6-79

mtsyce Symmetric matrix decomposition with condition estimation

6-81

mtsydc Symmetric matrix decomposition

6-82

mtsyfb Symmetric matrix with forward-back solution

6-83

mtsyln Symmetric linear system solver

6-84

mtsynr Symmetric matrix norm

6-85

mtudec Banded unsymmetric matrix, LU decomposition

6-86

mtudeu LU decomposition of a sparse matrix with input function

6-87

mtvmul Matrix vector multiplication (banded)

6-89

mtymbp Definite upper triangular linear system solution, positive band

6-90

vadmax Element index of the largest magnitude element of a vector

6-91

vadmin Element index of the smallest magnitude element of a vector

6-92

vaemax Element index of the maximum element of a vector

6-93

vaemin Element index of the minimum element of a vector

6-94

vaxmax Element index of the largest magnitude element of a complex vector

6-95

vaxmin Element index of the smallest magnitude element of a complex vector

6-96

veucln Compute Euclidean norm of a vector

6-97

vinmax Element index of the maximum element of a integer vector

6-98

vinmin Element index of the minimum element of a integer vector

6-99

CHAPTER 7

7-2

OPTIMIZATION

7-2

Introduction

7-2

fminim Finds local minima

7-4

menghb Simple bounds minimization with gradient and Hessian

7-5

menlqb Simple bounds nonlinear least squares

7-7

menmin Minimization of a function

7-9

mennlj Nonlinear least squares with Jacobian

7-10

mensbg Simple bounds minimization of a function with gradient

7-12

mensbm Simple bounds minimization of a function

7-14

mgengh Simple bounds minimization of a function with gradient and Hessian

7-15

mgengm Minimization of a function with gradient

7-17

mgenjb Simple bounds nonlinear least squares with Jacobian

7-19

mgenlq Nonlinear least squares

7-21

mgennm Unconstrained optimization, Nelder-Mead algorithm

7-22

mgnldb Simple bounds separable nonlinear least squares with derivatives

7-23

mgnlqb Simple bounds separable nonlinear least squares

7-25

mignld Separable nonlinear least squares with derivatives

7-27

mignlq No constraint separable nonlinear least squares

7-29

mtineq Linear programming

7-31

opquaf Compute local minimum using quadratic programming

7-32

vsblty General linear equality and inequality constraints

7-33

CHAPTER 8

8-2

PARTIAL DIFFERENTIAL EQUATIONS

8-2

Introduction

8-2

pdeovx Solution of elliptic PDE, overrelaxation method

8-3

pdemlg Solution of elliptic PDE, multigrid method

8-5

pdenlm Solution of nonlinear elliptic PDE, multigrid method

8-7

CHAPTER 9

9-2

INTEGRATION

9-2

Introduction

9-2

qahere Weights and Abscissas of Gauss-Hermite quadrature with the weight of

9-3

qasxaw Weights and Abscissas of Gauss quadrature with the weight of xa

9-4

qaulqe Weights and Abscissas of Gauss-Laguerre quadrature with the weight of e^-x

9-5

qauslq Weights and Abscissas of Gauss-Legendre quadrature

9-6

qdegps Piecewise smooth function integrator

9-7

qdntgr Integration using ralative error

9-8

qdtegi Integration of a set of integrals

9-9

qdtgbc Main integration function with boundary conditions

9-10

qgausq Weights and Abscissas of Gauss quadrature

9-11

qsexaw Weights and Abscissas of Gauss quadrature with weighting of xa e^-x

9-12

qubspl Quadrature using Cubic Spline

9-14

quslog Weights and Abscissas of Gauss quadrature with weighting of log(1/x)

9-15

sntegr Integration using B-splines

9-16

CHAPTER 10

10-2

RANDOM NUMBERS

10-2

Introduction

10-2

binran Binomial distribution random deviate generator

10-3

gamdis Gamma distributed random deviate generator

10-4

posran Poisson distributed random deviate generator

10-5

raarit Random deviate and bit pattern generator

10-6

rainit Initial seed generator

10-7

ranbit Randon bit generator

10-8

ranexp Exponentially distributed random deviate generator

10-9

ranksm Uniform random number generator, Knith method

10-10

ranlec Uniform random number generator with long period sequence and shuffle

10-11

ranpmr Minimal standard random number generator

10-12

ranpsh Minimal standard random number generator with shuffle

10-13

rarvar Generate Gaussian deviate

10-14

rnddvt Generate uniform random deviate

10-15

CHAPTER 11

11-2

ROOTS

11-2

Introduction

11-2

czerop Zeros of complex polynomials

11-3

rsreal Real single root within an interval

11-4

rterop Compute complex zeros of polynomials

11-5

rtller Real/Complex root of a function

11-6

rzernl Solves nonlinear systems

11-7

rzrnlj Solves nonlinear systems using Jacobian

11-8

CHAPTER 12

12-2

SPECIAL FUNCTIONS

12-2

Introduction

12-2

acoshh Hyperbolic cosine 12-3

arccos Arc cosine

12-4

arcsin Arc sine

12-5

arsinh Hyperbolic arc sine

12-6

artanh Hyperbolic arc tangent

12-7

beinrt I, modified real argument Bessel functions of integer order

12-8

beintc I, modified Bessel functions of complex integer order and argument

12-9

beintr J, real argument Bessel fuctions of integer order

12-10

bejntc J, complex argument Bessel functions of integer order

12-11

catlog Complex natural logarithm

12-12

coshhh Hyperbolic cosine

12-13

cpontl Complex exponential: e^(r+jm)

12-14

gammaa Gammaa function (real)

12-15

sinhhh Hyperbolic sine

12-16

tangnt Tangent

12-17

tanhhh Hyperbolic tangent 12-18

CHAPTER 13

13-2

STATISTICS

13-2

Introduction

13-2

stchio Performs chi-square test for the case of difference between

13-3

stchit Performs chi-square test for the case of difference between two

13-4

stcken Contingency analysis (Kendall's tau)

13-5

stcore Correlation between two sets of data (Pearson's method)

13-6

stfvar Performs F test for difference of variances 13-7

stgssm Generate Golay-Savitzky coefficients for smoothing

13-8

stkend Correlation for two sets of data (Kendall's tau)

13-9

stksdd Kolmogorov-Smirnov test for two sets of data

13-10

stksmd Kolmogorov-Smirnov test for data and model

13-11

stksmf Kolmorov-Smirnov main probability function

13-12

stkstd Two dimensional Kolmogorov-Smirnov test, data and data

13-13

stmomt Computes moments of data

13-14

strcor Rank correlation for two sets of data (Spearman's method)

13-15

ststst Computes difference of means (Student's test)

13-16

sttaba Chi-s contingency table analysis

13-17

sttabt Entropy measure for contingency table analysis

13-18

sttpxd Performs Student's test for the case of paired data 13-19

sttvar Student's test of means for unequal variances

13-20

stvars Computes variance and mean of data

13-21

stcomp Fit to a straight line (x,y composition)

13-22

stline Fits data to a straight line (least absolute deviation method) 13-23

stlsqr Least-squares data fit to a straight line

13-24

CHAPTER 14

14-2

UTILITY FUNCTIONS

14-2

Introduction

14-2

antsym Unsymmetrize an array

14-3

arcpyd Initialize a number of floating point array elements

14-4

arcpyi Initialize a number of integer array elements

14-5

arhopr Rearrange Hollerith data using input permutation

14-6

arrsym Transfom a vector into a symmetric form

14-7

arshdp Hollerith data passive sort

14-8

arshol Hollerith data sort

14-9

artlws Get n'th smallest element in an array

14-10

contch Converts base 10 number to machine base

14-11

fltdec Decompose a floating point number

14-12

flttbt Convert a floating point number to base 10

14-13

genrep Generate floating point number

14-14

getpol Orthogonal polynomial evaluation

14-15

polccs Chebyshev polynomial evaluation

14-16

poltrs Trigonometric polynomial evaluation

14-17

vepbrn Move backward a real array

14-18

vepfin Move forward an integer array

14-19

vepfrn Move forward an array

14-20

vetest Test vector: if monotone increasing or decreasing

14-21

veybin Move backward an integer array

14-22

vncdec Test if array is strictly monotone increasing/decreasing

14-24

 Index