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ASL Basic Functions Vol.5 (for Fortran)

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

1.1.2
Distinctive Characteristics of ASL

1.2
KINDS OF LIBRARIES

1.3
ORGANIZATION

1.3.1
Introduction

1.3.2
Organization of Subroutine Description

1.3.3
Contents of Each Item

1.4
SUBROUTINE 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
WIBJ0X, VIBJ0X
Bessel Function of the 1st Kind (Order 0)

2.2.2
WIBY0X, VIBY0X
Bessel Function of the 2nd Kind (Order 0)

2.2.3
WIBJ1X, VIBJ1X
Bessel Function of the 1st Kind (Order 1)

2.2.4
WIBY1X, VIBY1X
Bessel Function of the 2nd Kind (Order 1)

2.2.5
DIBJNX, RIBJNX
Bessel Function of the 1st Kind (Integer Order)

2.2.6
DIBYNX, RIBYNX
Bessel Function of the 2nd Kind (Integer Order)

2.2.7
DIBJMX, RIBJMX
Bessel Function of the 1st Kind (Real Number Order)

2.2.8
DIBYMX, RIBYMX
Bessel Function of the 2nd Kind (Real Number Order)

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

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

2.3
ZERO POINTS OF THE BESSEL FUNCTIONS

2.3.1
DIZBS0, RIZBS0
Positive Zero Points of the Bessel Function of the 1st Kind (Order 0)

2.3.2
DIZBS1, RIZBS1
Positive Zero Points of the Bessel Function of the 1st Kind (Order 1)

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

2.3.4
DIZBYN, RIZBYN
Positive Zero Points of the Second Kind Bessel Function

2.3.5
DIZBSL, RIZBSL
Positive Zero Points of the Function aJ₀ (α) +αJ₁ (α)

2.4
MODIFIED BESSEL FUNCTIONS

2.4.1
WIBI0X, VIBI0X
Modified Bessel Function of the 1st Kind (Order 0)

2.4.2
WIBK0X, VIBK0X
Modified Bessel Function of the 2nd Kind (Order 0)

2.4.3
WIBI1X, VIBI1X
Modified Bessel Function of the 1st Kind (Order 1)

2.4.4
WIBK1X, VIBK1X
Modified Bessel Function of the 2nd Kind (Order 1)

2.4.5
DIBINX, RIBINX
Modified Bessel Function of the 1st Kind (Integer Order)

2.4.6
DIBKNX, RIBKNX
Modified Bessel Function of the 2nd Kind (Integer Order)

2.4.7
DIBIMX, RIBIMX
Modified Bessel Function of the 1st Kind (Real Number Order)

2.4.8
DIBKMX, RIBKMX
Modified Bessel Function of the 2nd Kind (Real Number Order)

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

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

2.5
SPHERICAL BESSEL FUNCTIONS

2.5.1
DIBSJN, RIBSJN
Spherical Bessel Function of the 1st Kind (Integer Order)

2.5.2
DIBSYN, RIBSYN
Spherical Bessel Function of the 2nd Kind (Integer Order)

2.5.3
DIBSIN, RIBSIN
Modified Spherical Bessel Function of the 1st Kind (Integer Order)

2.5.4
DIBSKN, RIBSKN
Modified Spherical Bessel Function of the 2nd Kind (Integer Order)

2.6
FUNCTIONS RELATED TO BESSEL FUNCTIONS

2.6.1
ZIBH1N, CIBH1N
Hankel Function of the 1st Kind

2.6.2
ZIBH2N, CIBH2N
Hankel Function of the 2nd Kind

2.6.3
DIBBER, RIBBER
Kelvin Function ber_n (x)

2.6.4
DIBBEI, RIBBEI
Kelvin Function bei_n (x)

2.6.5
DIBKER, RIBKER
Kelvin Function ker_n (x)

2.6.6
DIBKEI, RIBKEI
Kelvin Function kei_n (x)

2.6.7
WIBH0X, VIBH0X
Struve Function (Order 0)

2.6.8
WIBH1X, VIBH1X
Struve Function (Order 1)

2.6.9
WIBHY0, VIBHY0
Difference of Struve Function (Order 0) and Bessel Function of the 2nd Kind (Order 0)

2.6.10
WIBHY1, VIBHY1
Difference of Struve Function (Order 1) and Bessel Function of the 2nd Kind (Order 1)

2.6.11
DIBAIX, RIBAIX
Airy Function Ai (x)

2.6.12
DIBBIX, RIBBIX
Airy Function Bi (x)

2.6.13
DIBAID, RIBAID
Derived Airy Function Ai' (x)

2.6.14
DIBBID, RIBBID
Derived Airy Function Bi' (x)

2.7
GAMMA FUNCTIONS

2.7.1
WIGAMX, VIGAMX
Gamma Function with Real Variable

2.7.2
WIGLGX, VIGLGX
Logarithmic Gamma Function with Real Variable

2.7.3
DIGIG1, RIGIG1
Incomplete Gamma Function of the 1st Kind

2.7.4
DIGIG2, RIGIG2
Incomplete Gamma Function of the 2nd Kind

2.7.5
ZIGAMZ, CIGAMZ
Gamma Function with Complex Variable

2.7.6
ZIGLGZ, CIGLGZ
Logarithmic Gamma Function with Complex Variable

2.8
FUNCTIONS RELATED TO THE GAMMA FUNCTION

2.8.1
WIGDIG, VIGDIG
Digamma Function

2.8.2
WIGBET, VIGBET
Beta Function

2.9
ELLIPTIC FUNCTIONS AND ELLIPTIC INTEGRALS

2.9.1
WIECI1, VIECI1
Complete Elliptic Integral of the 1st Kind

2.9.2
WIECI2, VIECI2
Complete Elliptic Integral of the 2nd Kind

2.9.3
DIEII1, RIEII1
Incomplete Elliptic Integral of the 1st Kind

2.9.4
DIEII2, RIEII2
Incomplete Elliptic Integral of the 2nd Kind

2.9.5
DIEII3, RIEII3
Incomplete Modified Elliptic Integral

2.9.6
DIEII4, RIEII4
Incomplete Elliptic Integral of The Weierstrass Type

2.9.7
WIEJAC, VIEJAC
Elliptic Functions of Jacobi

2.9.8
WIENMQ, VIENMQ
Nome q and Complete Elliptic Integrals

2.9.9
WIETHE, VIETHE
Elliptic Theta Function

2.9.10
WIEJZT, VIEJZT
Zeta Function of Jacobi

2.9.11
WIEJEP, VIEJEP
Epsilon Function of Jacobi

2.9.12
WIEJTE, VIEJTE
Theta Function of Jacobi

2.9.13
WIEPAI, VIEPAI
Pi Function

2.10
INDEFINITE INTEGRALS OF ELEMENTARY FUNCTIONS

2.10.1
WIIEXP, VIIEXP
Exponential Integral

2.10.2
WIILOG, VIILOG
Logarithmic Integral

2.10.3
DIISIN, RIISIN
Sine Integral

2.10.4
DIICOS, RIICOS
Cosine Integral

2.10.5
WIIFSI, VIIFSI
Fresnel Sine Integral

2.10.6
WIIFCO, VIIFCO
Fresnel Cosine Integral

2.10.7
WIIDAW, VIIDAW
Dawson Integral

2.10.8
WIICND, VIICND
Normal Distribution Function

2.10.9
WIICNC, VIICNC
Complementary Normal Distribution Function

2.11
THE FUNCTIONS RELATED TO THE ERROR FUNCTIONS

2.11.1
WIERRF, VIERRF
Error Function

2.11.2
WIERFC, VIERFC
Co-Error Function

2.11.3
DIIERF, RIIERF
Inverse of Co-Error Function

2.11.4
JIIERF, IIIERF
Error Function for Complex Arguments

2.12
ASSOCIATED LEGENDRE FUNCTIONS

2.12.1
DILEG1, RILEG1
Associated Legendre Function of the 1st Kind

2.12.2
DILEG2, RILEG2
Associated Legendre Function of the 2nd Kind

2.13
ORTHOGONAL POLYNOMIALS

2.13.1
DIOPLE, RIOPLE
Legendre Polynomial

2.13.2
DIZGLW, RIZGLW
Gauss=Legendre Formula

2.13.3
DIOPLA, RIOPLA
Laguerre Polynomial

2.13.4
DIOPHE, RIOPHE
Hermite Polynomial

2.13.5
DIOPCH, RIOPCH
Chebyshev Polynomial

2.13.6
DIOPC2, RIOPC2
Chebyshev Function of the 2nd Kind

2.13.7
DIOPGL, RIOPGL
Generalized Laguerre Polynomial

2.14
MATHIEU FUNCTIONS

2.14.1
DIMTCE, RIMTCE
Mathieu Functions of Integer Orders ce_n (x,q)

2.14.2
DIMTSE, RIMTSE
Mathieu Functions of Integer Orders se_n (x,q)

2.15
OTHER SPECIAL FUNCTIONS

2.15.1
WIXSPS, VIXSPS
Di-Log Function

2.15.2
WIDBEY, VIDBEY
Debye Function

2.15.3
WINPLG, VINPLG
Spherical Harmonic Function

2.15.4
WIXSLA, VIXSLA
Langevin Function

2.15.5
WIXZTA, VIXZTA
Hurwitz Zeta Function

2.15.6
DIXEPS, 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
DSSTA1, RSSTA1
Sorting a List of Data

3.2.2
DSSTA2, RSSTA2
Sorting a List of Pairwise Data

3.3
RANKING

3.3.1
DSSTRA, RSSTRA
Ranking of a List of Data

3.3.2
DSSTPT, RSSTPT
Top-N Extraction

3.4
MERGING

3.4.1
DSMGON, RSMGON
Merging Two Sorted Lists of Data

3.4.2
DSMGPA, 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
DLARHA, RLARHA
The Roots of Real Coefficient Algebraic Equations

4.2.2
ZLACHA, CLACHA
The Roots of Complex Coefficient Algebraic Equations

4.3
NONLINEAR EQUATIONS

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

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

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

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

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

4.4
SETS OF SIMULTANEOUS NONLINEAR EQUATIONS

4.4.1
DLSRDS, RLSRDS
A Root of a Set of Simultaneous Nonlinear Functions (Jacobian Matrix Optional)

4.4.2
DLSRIS, 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
DMUUSN, RMUUSN
Minimization of a Function of One Variable

5.3
MINIMIZATION OF A FUNCTION OF MANY VARIABLES WITHOUT CONSTRAINTS

5.3.1
DMUMQN, RMUMQN
Minimization of a Function of Many Variables (Derivative Definition Unnecessary)

5.3.2
DMUMQG, 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
DMUSSN, RMUSSN
Nonlinear Least Squares Method (Derivative Definition Unnecessary)

5.5
MINIMIZATION OF A FUNCTION OF ONE VARIABLE WITH CONSTRAINTS

5.5.1
DMCUSN, 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
DMCLSN, RMCLSN
Minimization of a Linear Function of Several Variables (Linear Constraints)

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

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

5.6.4
DMCLMC, RMCLMC
Minimization of Cost for Flow in a Network (Minimal-Cost Flow Problem)

5.6.5
DMCLCP, RMCLCP
Minimization of Cost for Project Scheduling (Project Scheduling Problem)

5.6.6
DMCLTP, 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
DMCQSN, RMCQSN
Minimization of a Constrained Convex Quadratic Function of Several Variables (Linear Constraints)

5.7.2
DMCQLM, RMCQLM
Minimization of a Generalized Convex Quadratic Function of Several Variables (Linear Constraints)

5.7.3
DMCQAZ, 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
DMSQPM, RMSQPM
Minimization of a Constrained Function of Several Variables (Nonlinear Constraints)

5.9
DISTANCE MINIMIZATION ON A GRAPH (SHORTEST PATH PROBLEM)

5.9.1
DMSP1M, RMSP1M
Distance Minimization for a Given Node to the Other Node on a Graph

5.9.2
DMSPMM, RMSPMM
Distance Minimization for All Sets of Two Nodes on a Graph

5.9.3
DMSP11, RMSP11
Distance Minimization for Two Nodes on a Graph

Appendix

Appendix A
GLOSSARY

Appendix B
MACHINE CONSTANTS USED IN ASL

B.1
Units for Determining Error

B.2
Maximum and Minimum Values of Floating Point Data