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ASL Basic Functions Vol.5 (for C)

Contents
INTRODUCTION SPECIAL FUNCTIONS SORTING AND RANKING ROOTS OF EQUATIONS EXTREMUM PROBLEMS AND OPTIMIZATION

Chapter 1 INTRODUCTION

1.1
OVERVIEW

1.1.1
Introduction to The Advanced Scientific Library ASL C interface

1.1.2
Distinctive Characteristics of ASL C interface

1.2
KINDS OF LIBRARIES

1.3
ORGANIZATION

1.3.1
Introduction

1.3.2
Organization of Function Description

1.3.3
Contents of Each Item

1.4
FUNCTION NAMES

1.5
NOTES

Chapter 2 SPECIAL FUNCTIONS

2.1
INTRODUCTION

2.1.1
Notes

2.1.2
Algorithms Used

2.1.2.1
Bessel Functions

2.1.2.2
Modified Bessel Functions

2.1.2.3
Spherical Bessel Functions

2.1.2.4
Functions Related To Bessel Functions

2.1.2.5
Gamma Functions

2.1.2.6
Functions Related To The Gamma Function

2.1.2.7
Elliptic Functions And Elliptic Integrals

2.1.2.8
Indefinite Integrals Of Elementary Functions

2.1.2.9
Associated Legendre Functions

2.1.2.10
Orthogonal Polynomials

2.1.2.11
Mathieu functions of integer orders

2.1.2.12
Langevin function

2.1.2.13
Gauss=Legendre integration formula

2.1.2.14
Zero points of Bessel Functions

2.1.2.15
Positive zero points of the second kind Bessel function

2.1.2.16
Zeta function of Positive definite quadratic form x² + a y²

2.1.2.17
Di-log function

2.1.2.18
Debye function

2.1.2.19
Normalized Spherical Harmonics

2.1.2.20
Hurwitz Zeta function for a real variable

2.1.2.21
The functions related to the error function

2.1.2.22
Coefficient Calculation Method

2.1.2.23
Method of Calculating Related Special Functions

2.1.3
Reference Bibliography

2.2
BESSEL FUNCTIONS

2.2.1
ASL_wibj0x, ASL_vibj0x
Bessel Function of the 1st Kind (Order 0)

2.2.2
ASL_wiby0x, ASL_viby0x
Bessel Function of the 2nd Kind (Order 0)

2.2.3
ASL_wibj1x, ASL_vibj1x
Bessel Function of the 1st Kind (Order 1)

2.2.4
ASL_wiby1x, ASL_viby1x
Bessel Function of the 2nd Kind (Order 1)

2.2.5
ASL_dibjnx, ASL_ribjnx
Bessel Function of the 1st Kind (Integer Order)

2.2.6
ASL_dibynx, ASL_ribynx
Bessel Function of the 2nd Kind (Integer Order)

2.2.7
ASL_dibjmx, ASL_ribjmx
Bessel Function of the 1st Kind (Real Number Order)

2.2.8
ASL_dibymx, ASL_ribymx
Bessel Function of the 2nd Kind (Real Number Order)

2.2.9
ASL_zibjnz, ASL_cibjnz
Bessel Function of the 1st Kind with Complex Variable (Integer Order)

2.2.10
ASL_zibynz, ASL_cibynz
Bessel Function of the 2nd Kind with Complex Variable (Integer Order)

2.3
ZERO POINTS OF THE BESSEL FUNCTIONS

2.3.1
ASL_dizbs0, ASL_rizbs0
Positive Zero Points of the Bessel Function of the 1st Kind (Order 0)

2.3.2
ASL_dizbs1, ASL_rizbs1
Positive Zero Points of the Bessel Function of the 1st Kind (Order 1)

2.3.3
ASL_dizbsn, ASL_rizbsn
Positive Zero Points of Bessel Function of the 1st Kind (Integer Order)

2.3.4
ASL_dizbyn, ASL_rizbyn
Positive Zero Points of the Second Kind Bessel Function

2.3.5
ASL_dizbsl, ASL_rizbsl
Positive Zero Points of the Function aJ₀ (α) +αJ₁ (α)

2.4
MODIFIED BESSEL FUNCTIONS

2.4.1
ASL_wibi0x, ASL_vibi0x
Modified Bessel Function of the 1st Kind (Order 0)

2.4.2
ASL_wibk0x, ASL_vibk0x
Modified Bessel Function of the 2nd Kind (Order 0)

2.4.3
ASL_wibi1x, ASL_vibi1x
Modified Bessel Function of the 1st Kind (Order 1)

2.4.4
ASL_wibk1x, ASL_vibk1x
Modified Bessel Function of the 2nd Kind (Order 1)

2.4.5
ASL_dibinx, ASL_ribinx
Modified Bessel Function of the 1st Kind (Integer Order)

2.4.6
ASL_dibknx, ASL_ribknx
Modified Bessel Function of the 2nd Kind (Integer Order)

2.4.7
ASL_dibimx, ASL_ribimx
Modified Bessel Function of the 1st Kind (Real Number Order)

2.4.8
ASL_dibkmx, ASL_ribkmx
Modified Bessel Function of the 2nd Kind (Real Number Order)

2.4.9
ASL_zibinz, ASL_cibinz
Modified Bessel Function of the 1st Kind with Complex Variable (Integer Order)

2.4.10
ASL_zibknz, ASL_cibknz
Modified Bessel Function of the 2nd Kind with Complex Variable (Integer Order)

2.5
SPHERICAL BESSEL FUNCTIONS

2.5.1
ASL_dibsjn, ASL_ribsjn
Spherical Bessel Function of the 1st Kind (Integer Order)

2.5.2
ASL_dibsyn, ASL_ribsyn
Spherical Bessel Function of the 2nd Kind (Integer Order)

2.5.3
ASL_dibsin, ASL_ribsin
Modified Spherical Bessel Function of the 1st Kind (Integer Order)

2.5.4
ASL_dibskn, ASL_ribskn
Modified Spherical Bessel Function of the 2nd Kind (Integer Order)

2.6
FUNCTIONS RELATED TO BESSEL FUNCTIONS

2.6.1
ASL_zibh1n, ASL_cibh1n
Hankel Function of the 1st Kind

2.6.2
ASL_zibh2n, ASL_cibh2n
Hankel Function of the 2nd Kind

2.6.3
ASL_dibber, ASL_ribber
Kelvin Function ber_n (x)

2.6.4
ASL_dibbei, ASL_ribbei
Kelvin Function bei_n (x)

2.6.5
ASL_dibker, ASL_ribker
Kelvin Function ker_n (x)

2.6.6
ASL_dibkei, ASL_ribkei
Kelvin Function kei_n (x)

2.6.7
ASL_wibh0x, ASL_vibh0x
Struve Function (Order 0)

2.6.8
ASL_wibh1x, ASL_vibh1x
Struve Function (Order 1)

2.6.9
ASL_wibhy0, ASL_vibhy0
Difference of Struve Function (Order 0) and Bessel Function of the 2nd Kind (Order 0)

2.6.10
ASL_wibhy1, ASL_vibhy1
Difference of Struve Function (Order 1) and Bessel Function of the 2nd Kind (Order 1)

2.6.11
ASL_dibaix, ASL_ribaix
Airy Function Ai (x)

2.6.12
ASL_dibbix, ASL_ribbix
Airy Function Bi (x)

2.6.13
ASL_dibaid, ASL_ribaid
Derived Airy Function Ai' (x)

2.6.14
ASL_dibbid, ASL_ribbid
Derived Airy Function Bi' (x)

2.7
GAMMA FUNCTIONS

2.7.1
ASL_wigamx, ASL_vigamx
Gamma Function with Real Variable

2.7.2
ASL_wiglgx, ASL_viglgx
Logarithmic Gamma Function with Real Variable

2.7.3
ASL_digig1, ASL_rigig1
Incomplete Gamma Function of the 1st Kind

2.7.4
ASL_digig2, ASL_rigig2
Incomplete Gamma Function of the 2nd Kind

2.7.5
ASL_zigamz, ASL_cigamz
Gamma Function with Complex Variable

2.7.6
ASL_ziglgz, ASL_ciglgz
Logarithmic Gamma Function with Complex Variable

2.8
FUNCTIONS RELATED TO THE GAMMA FUNCTION

2.8.1
ASL_wigdig, ASL_vigdig
Digamma Function

2.8.2
ASL_wigbet, ASL_vigbet
Beta Function

2.9
ELLIPTIC FUNCTIONS AND ELLIPTIC INTEGRALS

2.9.1
ASL_wieci1, ASL_vieci1
Complete Elliptic Integral of the 1st Kind

2.9.2
ASL_wieci2, ASL_vieci2
Complete Elliptic Integral of the 2nd Kind

2.9.3
ASL_dieii1, ASL_rieii1
Incomplete Elliptic Integral of the 1st Kind

2.9.4
ASL_dieii2, ASL_rieii2
Incomplete Elliptic Integral of the 2nd Kind

2.9.5
ASL_dieii3, ASL_rieii3
Incomplete Modified Elliptic Integral

2.9.6
ASL_dieii4, ASL_rieii4
Incomplete Elliptic Integral of The Weierstrass Type

2.9.7
ASL_wiejac, ASL_viejac
Elliptic Functions of Jacobi

2.9.8
ASL_wienmq, ASL_vienmq
Nome q and Complete Elliptic Integrals

2.9.9
ASL_wiethe, ASL_viethe
Elliptic Theta Function

2.9.10
ASL_wiejzt, ASL_viejzt
Zeta Function of Jacobi

2.9.11
ASL_wiejep, ASL_viejep
Epsilon Function of Jacobi

2.9.12
ASL_wiejte, ASL_viejte
Theta Function of Jacobi

2.9.13
ASL_wiepai, ASL_viepai
Pi Function

2.10
INDEFINITE INTEGRALS OF ELEMENTARY FUNCTIONS

2.10.1
ASL_wiiexp, ASL_viiexp
Exponential Integral

2.10.2
ASL_wiilog, ASL_viilog
Logarithmic Integral

2.10.3
ASL_diisin, ASL_riisin
Sine Integral

2.10.4
ASL_diicos, ASL_riicos
Cosine Integral

2.10.5
ASL_wiifsi, ASL_viifsi
Fresnel Sine Integral

2.10.6
ASL_wiifco, ASL_viifco
Fresnel Cosine Integral

2.10.7
ASL_wiidaw, ASL_viidaw
Dawson Integral

2.10.8
ASL_wiicnd, ASL_viicnd
Normal Distribution Function

2.10.9
ASL_wiicnc, ASL_viicnc
Complementary Normal Distribution Function

2.11
THE FUNCTIONS RELATED TO THE ERROR FUNCTIONS

2.11.1
ASL_wierrf, ASL_vierrf
Error Function

2.11.2
ASL_wierfc, ASL_vierfc
Co-Error Function

2.11.3
ASL_diierf, ASL_riierf
Inverse of Co-Error Function

2.11.4
ASL_jiierf, ASL_iiierf
Error Function for Complex Arguments

2.12
ASSOCIATED LEGENDRE FUNCTIONS

2.12.1
ASL_dileg1, ASL_rileg1
Associated Legendre Function of the 1st Kind

2.12.2
ASL_dileg2, ASL_rileg2
Associated Legendre Function of the 2nd Kind

2.13
ORTHOGONAL POLYNOMIALS

2.13.1
ASL_diople, ASL_riople
Legendre Polynomial

2.13.2
ASL_dizglw, ASL_rizglw
Gauss=Legendre Formula

2.13.3
ASL_diopla, ASL_riopla
Laguerre Polynomial

2.13.4
ASL_diophe, ASL_riophe
Hermite Polynomial

2.13.5
ASL_diopch, ASL_riopch
Chebyshev Polynomial

2.13.6
ASL_diopc2, ASL_riopc2
Chebyshev Function of the 2nd Kind

2.13.7
ASL_diopgl, ASL_riopgl
Generalized Laguerre Polynomial

2.14
MATHIEU FUNCTIONS

2.14.1
ASL_dimtce, ASL_rimtce
Mathieu Functions of Integer Orders ce_n (x,q)

2.14.2
ASL_dimtse, ASL_rimtse
Mathieu Functions of Integer Orders se_n (x,q)

2.15
OTHER SPECIAL FUNCTIONS

2.15.1
ASL_wixsps, ASL_vixsps
Di-Log Function

2.15.2
ASL_widbey, ASL_vidbey
Debye Function

2.15.3
ASL_winplg, ASL_vinplg
Spherical Harmonic Function

2.15.4
ASL_wixsla, ASL_vixsla
Langevin Function

2.15.5
ASL_wixzta, ASL_vixzta
Hurwitz Zeta Function

2.15.6
ASL_dixeps, ASL_rixeps
Zeta Function of the Positive Definite Quadratic Form x² + a y²

Chapter 3 SORTING AND RANKING

3.1
INTRODUCTION

3.1.1
Algorithms Used

3.1.1.1
Sorting

3.1.1.2
Ranking of a list of data

3.1.1.3
Top-N extraction

3.1.1.4
Merging two sorted lists of data

3.1.1.5
Merging two sorted list of pairwise data

3.1.2
Reference Bibliography

3.2
SORTING

3.2.1
ASL_dssta1, ASL_rssta1
Sorting a List of Data

3.2.2
ASL_dssta2, ASL_rssta2
Sorting a List of Pairwise Data

3.3
RANKING

3.3.1
ASL_dsstra, ASL_rsstra
Ranking of a List of Data

3.3.2
ASL_dsstpt, ASL_rsstpt
Top-N Extraction

3.4
MERGING

3.4.1
ASL_dsmgon, ASL_rsmgon
Merging Two Sorted Lists of Data

3.4.2
ASL_dsmgpa, ASL_rsmgpa
Merging Two Sorted Lists of Pairwise Data

Chapter 4 ROOTS OF EQUATIONS

4.1
INTRODUCTION

4.1.1
Notes

4.1.2
Algorithms Used

4.1.2.1
Roots of a real coefficient algebraic equation

4.1.2.2
The roots of complex coefficient algebraic equations

4.1.2.3
The roots of real functions (initial value specified; derivative definition required)

4.1.2.4
The roots of real functions (initial value specified; derivative definition not required)

4.1.2.5
The roots of real functions (interval specification; derivative definition not required)

4.1.2.6
All the roots of real functions (interval specification; derivative definition not required)

4.1.2.7
The roots of complex functions (initial value specified; derivative definition not required)

4.1.2.8
The roots of a set of simultaneous nonlinear equations (Jacobian matrix definition optional)

4.1.2.9
The roots of a set of simultaneous nonlinear equations (Jacobian matrix definition not required)

4.1.3
Reference Bibliography

4.2
ALGEBRAIC EQUATIONS

4.2.1
ASL_dlarha, ASL_rlarha
The Roots of Real Coefficient Algebraic Equations

4.2.2
ASL_zlacha, ASL_clacha
The Roots of Complex Coefficient Algebraic Equations

4.3
NONLINEAR EQUATIONS

4.3.1
ASL_dlnrds, ASL_rlnrds
A Root of a Real Function (Initial Value Specified; Derivative Definition Required)

4.3.2
ASL_dlnris, ASL_rlnris
A Root of a Real Function (Initial Value Specified; Derivative Definition Not Required)

4.3.3
ASL_dlnrss, ASL_rlnrss
A Root of a Real Function (Interval Specified; Derivative Definition Not Required)

4.3.4
ASL_dlnrsa, ASL_rlnrsa
All Roots of a Real Function (Interval Specified; Derivative Definition Not Required)

4.3.5
ASL_zlncis, ASL_clncis
A Root of a Complex Function (Initial Value Specified; Derivative Definition Not Required)

4.4
SETS OF SIMULTANEOUS NONLINEAR EQUATIONS

4.4.1
ASL_dlsrds, ASL_rlsrds
A Root of a Set of Simultaneous Nonlinear Functions (Jacobian Matrix Optional)

4.4.2
ASL_dlsris, ASL_rlsris
A Root of a Set of Simultaneous Nonlinear Functions (Jacobian Matrix Definition Not Required)

Chapter 5 EXTREMUM PROBLEMS AND OPTIMIZATION

5.1
INTRODUCTION

5.1.1
Notes

5.1.2
Algorithms Used

5.1.2.1
Minimization of a function of one variable

5.1.2.2
Minimization of a function of many variables

5.1.2.3
Nonlinear least square method

5.1.2.4
Minimization of a constrained linear function of several variables (linear constraints)

5.1.2.5
Minimization of a constrained linear function of several variables including 0-1 variables

5.1.2.6
Minimization of cost for flow in a network

5.1.2.7
Minimization of cost for project scheduling

5.1.2.8
Minimization of cost for transportation from supply place to demand place

5.1.2.9
Minimization of a constrained quadratic function of several variables (linear constraints)

5.1.2.10
Minimization of a generalized convex quadratic function of several variables (linear constraints)

5.1.2.11
Minimization of an unconstrained 0-1 quadratic function of several variables

5.1.2.12
Minimization of a constrained function of several variables

5.1.2.13
Minimization of the distance between two nodes in a network

5.1.3
Reference Bibliography

5.2
MINIMIZATION OF A FUNCTION OF ONE VARIABLE WITHOUT CONSTRAINTS

5.2.1
ASL_dmuusn, ASL_rmuusn
Minimization of a Function of One Variable

5.3
MINIMIZATION OF A FUNCTION OF MANY VARIABLES WITHOUT CONSTRAINTS

5.3.1
ASL_dmumqn, ASL_rmumqn
Minimization of a Function of Many Variables (Derivative Definition Unnecessary)

5.3.2
ASL_dmumqg, ASL_rmumqg
Minimization of a Function of Many Variables (Derivative Definition Required)

5.4
MINIMIZATION OF THE SUM OF THE SQUARES OF A FUNCTION WITHOUT CONSTRAINTS

5.4.1
ASL_dmussn, ASL_rmussn
Nonlinear Least Squares Method (Derivative Definition Unnecessary)

5.5
MINIMIZATION OF A FUNCTION OF ONE VARIABLE WITH CONSTRAINTS

5.5.1
ASL_dmcusn, ASL_rmcusn
Minimization of a Function of One Variable (Interval Specified)

5.6
MINIMIZATION OF A CONSTRAINED LINEAR FUNCTION OF SEVERAL VARIABLES (LINEAR PROGRAMMING)

5.6.1
ASL_dmclsn, ASL_rmclsn
Minimization of a Linear Function of Several Variables (Linear Constraints)

5.6.2
ASL_dmclaf, ASL_rmclaf
Minimization of a Function of Many Variables (Linear Constraint Given by a Real Irregular Sparse Matrix)

5.6.3
ASL_dmclmz, ASL_rmclmz
Minimization of a Constrained Linear Function of Several Variables Including 0-1 Variables (Mixed 0-1 Programming)

5.6.4
ASL_dmclmc, ASL_rmclmc
Minimization of Cost for Flow in a Network (Minimal-Cost Flow Problem)

5.6.5
ASL_dmclcp, ASL_rmclcp
Minimization of Cost for Project Scheduling (Project Scheduling Problem)

5.6.6
ASL_dmcltp, ASL_rmcltp
Minimization of Cost for Transportation from Supply Place to Demand Place (Transportation Problem)

5.7
MINIMIZATION OF A QUADRATIC FUNCTION OF SEVERAL VARIABLES (QUADRATIC PROGRAMMING)

5.7.1
ASL_dmcqsn, ASL_rmcqsn
Minimization of a Constrained Convex Quadratic Function of Several Variables (Linear Constraints)

5.7.2
ASL_dmcqlm, ASL_rmcqlm
Minimization of a Generalized Convex Quadratic Function of Several Variables (Linear Constraints)

5.7.3
ASL_dmcqaz, ASL_rmcqaz
Minimization of an Unconstrained 0-1 Quadratic Function of Several Variables (Unconstrained 0-1 Quadratic Programming Problem)

5.8
MINIMIZATION OF A CONSTRAINED FUNCTION OF SEVERAL VARIABLES (NONLINEAR PROGRAMMING)

5.8.1
ASL_dmsqpm, ASL_rmsqpm
Minimization of a Constrained Function of Several Variables (Nonlinear Constraints)

5.9
DISTANCE MINIMIZATION ON A GRAPH (SHORTEST PATH PROBLEM)

5.9.1
ASL_dmsp1m, ASL_rmsp1m
Distance Minimization for a Given Node to the Other Node on a Graph

5.9.2
ASL_dmspmm, ASL_rmspmm
Distance Minimization for All Sets of Two Nodes on a Graph

5.9.3
ASL_dmsp11, ASL_rmsp11
Distance Minimization for Two Nodes on a Graph

Appendix

Appendix A
GLOSSARY

Appendix B
MACHINE CONSTANTS USED IN ASL C INTERFACE

B.1
Units for Determining Error

B.2
Maximum and Minimum Values of Floating Point Data