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ASL Basic Functions Vol.2 (for Fortran)

Contents
INTRODUCTION SIMULTANEOUS LINEAR EQUATIONS (DIRECT METHOD)

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 SIMULTANEOUS LINEAR EQUATIONS (DIRECT METHOD)

2.1
INTRODUCTION

2.1.1
Methods of using subroutines

2.1.2
Notes

2.1.3
Algorithms Used

2.1.3.1
Crout Method

2.1.3.2
Cholesky method

2.1.3.3
Modified Cholesky method

2.1.3.4
Gauss method

2.1.3.5
Levinson method

2.1.3.6
Vandermonde matrix

2.1.3.7
Cyclic Reduction Method

2.1.3.8
Calculating the inverse matrix

2.1.3.9
Calculating the determinant

2.1.3.10
Improving the solution

2.1.3.11
Precise estimate of the approximate solution

2.1.3.12
Condition Number

2.1.4
Reference Bibliography

2.2
REAL MATRIX (TWO-DIMENSIONAL ARRAY TYPE)

2.2.1
DBGMSM, RBGMSM
Simultaneous Linear Equations with Multiple Right-Hand Sides (Real Matrix)

2.2.2
DBGMSL, RBGMSL
Simultaneous Linear Equations (Real Matrix)

2.2.3
DBGMLU, RBGMLU
LU Decomposition of a Real Matrix

2.2.4
DBGMLC, RBGMLC
LU Decomposition and Condition Number of a Real Matrix

2.2.5
DBGMLS, RBGMLS
Simultaneous Linear Equations (LU-Decomposed Real Matrix)

2.2.6
DBGMMS, RBGMMS
Simultaneous Linear Equations with Multiple Right-Hand Sides (LU-Decomposed Real Matrix)

2.2.7
DBGMDI, RBGMDI
Determinant and Inverse Matrix of a Real Matrix

2.2.8
DBGMLX, RBGMLX
Improving the Solution of Simultaneous Linear Equations (Real Matrix)

2.3
COMPLEX MATRIX (TWO DIMENSIONAL ARRAY TYPE) (REAL ARGUMENT TYPE)

2.3.1
ZBGMSM, CBGMSM
Simultaneous Linear Equations with Multiple Right-Hand Sides (Complex Matrix)

2.3.2
ZBGMSL, CBGMSL
Simultaneous Linear Equations (Complex Matrix)

2.3.3
ZBGMLU, CBGMLU
LU Decomposition of a Complex Matrix

2.3.4
ZBGMLC, CBGMLC
LU Decomposition and Condition Number of a Complex Matrix

2.3.5
ZBGMLS, CBGMLS
Simultaneous Linear Equations (LU-Decomposed Complex Matrix)

2.3.6
ZBGMMS, CBGMMS
Simultaneous Linear Equations with Multiple Right-Hand Sides (LU-Decomposed Complex Matrix)

2.3.7
ZBGMDI, CBGMDI
Determinant and Inverse Matrix of a Complex Matrix

2.3.8
ZBGMLX, CBGMLX
Improving the Solution of Simultaneous Linear Equations (Complex Matrix)

2.4
COMPLEX MATRIX (TWO-DIMENSIONAL ARRAY TYPE)
(COMPLEX ARGUMENT TYPE)

2.4.1
ZBGNSM, CBGNSM
Simultaneous Linear Equations with Multiple Right-Hand Sides (Complex Matrix)

2.4.2
ZBGNSL, CBGNSL
Simultaneous Linear Equations (Complex Matrix)

2.4.3
ZBGNLU, CBGNLU
LU Decomposition of a Complex Matrix

2.4.4
ZBGNLC, CBGNLC
LU Decomposition and Condition Number of a Complex Matrix

2.4.5
ZBGNLS, CBGNLS
Simultaneous Linear Equations (LU-Decomposed Complex Matrix)

2.4.6
ZBGNMS, CBGNMS
Simultaneous Linear Equations with Multiple Right-Hand Sides (LU-Decomposed Complex Matrix)

2.4.7
ZBGNDI, CBGNDI
Determinant and Inverse Matrix of a Complex Matrix

2.4.8
ZBGNLX, CBGNLX
Improving the Solution of Simultaneous Linear Equations (Complex Matrix)

2.5
POSITIVE SYMMETRIC MATRIX (TWO-DIMENSIONAL ARRAY TYPE) (UPPER TRIANGULAR TYPE)

2.5.1
DBPDSL, RBPDSL
Simultaneous Linear Equations (Positive Symmetric Matrix)

2.5.2
DBPDUU, RBPDUU
LL^T Decomposition of a Positive Symmetric Matrix

2.5.3
DBPDUC, RBPDUC
LL^T Decomposition and Condition Number of a Positive Symmetric Matrix

2.5.4
DBPDLS, RBPDLS
Simultaneous Linear Equations (LL^T-Decomposed Positive Symmetric Matrix)

2.5.5
DBPDDI, RBPDDI
Determinant and Inverse Matrix of a Positive Symmetric Matrix

2.5.6
DBPDLX, RBPDLX
Improving the Solution of Simultaneous Linear Equations (Positive Symmetric Matrix)

2.6
REAL SYMMETRIC MATRIX (TWO-DIMENSIONAL ARRAY TYPE) (UPPER TRIANGULAR TYPE)

2.6.1
DBSPSL, RBSPSL
Simultaneous Linear Equations (Real Symmetric Matrix)

2.6.2
DBSPUD, RBSPUD
LDL^T Decomposition of a Real Symmetric Matrix

2.6.3
DBSPUC, RBSPUC
LDL^T Decomposition and Condition Number of a Real Symmetric Matrix

2.6.4
DBSPLS, RBSPLS
Simultaneous Linear Equations (LDL^T-Decomposed Real Symmetric Matrix)

2.6.5
DBSPMS, RBSPMS
Simultaneous Linear Equations with Multiple Right-Hand Sides (LDL^T decomposed Real Matrix)

2.6.6
DBSPDI, RBSPDI
Determinant and Inverse Matrix of a Real Symmetric Matrix

2.6.7
DBSPLX, RBSPLX
Improving the Solution of Simultaneous Linear Equations (Real Symmetric Matrix)

2.7
REAL SYMMETRIC MATRIX (TWO-DIMENSIONAL ARRAY TYPE) (UPPER TRIANGULAR TYPE) (NO PIVOTING)

2.7.1
DBSMSL, RBSMSL
Simultaneous Linear Equations (Real Symmetric Matrix) (No Pivoting)

2.7.2
DBSMUD, RBSMUD
LDL^T Decomposition of a Real Symmetric Matrix (No Pivoting)

2.7.3
DBSMUC, RBSMUC
LDL^T Decomposition and Condition Number of a Real Symmetric Matrix (No Pivoting)

2.7.4
DBSMLS, RBSMLS
Simultaneous Linear Equations (LDL^T-Decomposed Real Symmetric Matrix) (No Pivoting)

2.7.5
DBSMMS, RBSMMS
Simultaneous Linear Equations with Multiple Right-Hand Sides (LDL^T-Decomposed Real Matrix) (No Pivoting)

2.7.6
DBSMDI, RBSMDI
Determinant and Inverse Matrix of a Real Symmetric Matrix (No Pivoting)

2.7.7
DBSMLX, RBSMLX
Improving the Solution of Simultaneous Linear Equations (Real Symmetric Matrix) (No Pivoting)

2.8
REAL SYMMETRIC MATRIX (TWO-DIMENSIONAL ARRAY TYPE, LOWER TRIANGULAR TYPE) (NO PIVOTING)

2.8.1
DBSNSL, RBSNSL
Simultaneous Linear Equations (Real Symmetric Matrix) (No Pivoting)

2.8.2
DBSNUD, RBSNUD
U^TDU Decomposition of a Real Symmetric Matrix (No Pivoting)

2.8.3
DBSNLS, RBSNLS
Simultaneous Linear Equations (U^TDU-Decomposed Real Symmetric Matrix) (No Pivoting)

2.9
HERMITIAN MATRIX (TWO-DIMENSIONAL ARRAY TYPE) (UPPER TRIANGULAR TYPE) (REAL ARGUMENT TYPE)

2.9.1
ZBHPSL, CBHPSL
Simultaneous Linear Equations (Hermitian Matrix)

2.9.2
ZBHPUD, CBHPUD
LDL^* Decomposition of a Hermitian Matrix

2.9.3
ZBHPUC, CBHPUC
LDL^* Decomposition and Condition Number of a Hermitian Matrix

2.9.4
ZBHPLS, CBHPLS
Simultaneous Linear Equations (LDL^*-Decomposed Hermitian Matrix)

2.9.5
ZBHPMS, CBHPMS
Simultaneous Linear Equations with Multiple Right-Hand Sides (LDL^*-Decomposed Hermitian Matrix)

2.9.6
ZBHPDI, CBHPDI
Determinant and Inverse Matrix of a Hermitian Matrix

2.9.7
ZBHPLX, CBHPLX
Improving the Solution of Simultaneous Linear Equations (Hermitian Matrix)

2.10
HERMITIAN MATRIX (TWO-DIMENSIONAL ARRAY TYPE) (UPPER TRIANGULAR TYPE) (REAL ARGUMENT TYPE) (NO PIVOTING)

2.10.1
ZBHRSL, CBHRSL
Simultaneous Linear Equations (Hermitian Matrix) (No Pivoting)

2.10.2
ZBHRUD, CBHRUD
LDL^* Decomposition of a Hermitian Matrix (No Pivoting)

2.10.3
ZBHRUC, CBHRUC
LDL^* Decomposition and Condition Number of a Hermitian Matrix (No Pivoting)

2.10.4
ZBHRLS, CBHRLS
Simultaneous Linear Equations (LDL^*-Decomposed Hermitian Matrix) (No Pivoting)

2.10.5
ZBHRMS, CBHRMS
Simultaneous Linear Equations with Multiple Right-Hand Sides (LDL^*-Decomposed Hermitian Matrix) (No Pivoting)

2.10.6
ZBHRDI, CBHRDI
Determinant and Inverse Matrix of a Hermitian Matrix (No Pivoting)

2.10.7
ZBHRLX, CBHRLX
Improving the Solution of Simultaneous Linear Equations (Hermitian Matrix) (No Pivoting)

2.11
HERMITIAN MATRIX (TWO-DIMENSIONAL ARRAY TYPE) (UPPER TRIANGULAR TYPE) (COMPLEX ARGUMENT TYPE)

2.11.1
ZBHFSL, CBHFSL
Simultaneous Linear Equations (Hermitian Matrix)

2.11.2
ZBHFUD, CBHFUD
LDL^* Decomposition of a Hermitian Matrix

2.11.3
ZBHFUC, CBHFUC
LDL^* Decomposition and Condition Number of a Hermitian Matrix

2.11.4
ZBHFLS, CBHFLS
Simultaneous Linear Equations (LDL^*-Decomposed Hermitian Matrix)

2.11.5
ZBHFMS, CBHFMS
Simultaneous Linear Equations with Multiple Right-Hand Sides (LDL^*-Decomposed Hermitian Matrix)

2.11.6
ZBHFDI, CBHFDI
Determinant and Inverse Matrix of a Hermitian Matrix

2.11.7
ZBHFLX, CBHFLX
Improving the Solution of Simultaneous Linear Equations (Hermitian Matrix)

2.12
HERMITIAN MATRIX (TWO-DIMENSIONAL ARRAY TYPE) (UPPER TRIANGULAR TYPE) (COMPLEX ARGUMENT TYPE) (NO PIVOTING)

2.12.1
ZBHESL, CBHESL
Simultaneous Linear Equations (Hermitian Matrix) (No Pivoting)

2.12.2
ZBHEUD, CBHEUD
LDL^* Decomposition of a Hermitian Matrix (No Pivoting)

2.12.3
ZBHEUC, CBHEUC
LDL^* Decomposition and Condition Number of a Hermitian Matrix (No Pivoting)

2.12.4
ZBHELS, CBHELS
Simultaneous Linear Equations (LDL^*-Decomposed Hermitian Matrix) (No Pivoting)

2.12.5
ZBHEMS, CBHEMS
Simultaneous Linear Equations with Multiple Right-Hand Sides (LDL^*-Decomposed Hermitian Matrix) (No Pivoting)

2.12.6
ZBHEDI, CBHEDI
Determinant and Inverse Matrix of a Hermitian Matrix (No Pivoting)

2.12.7
ZBHELX, CBHELX
Improving the Solution of Simultaneous Linear Equations (Hermitian Matrix) (No Pivoting)

2.13
REAL BAND MATRIX (BAND TYPE)

2.13.1
DBBDSL, RBBDSL
Simultaneous Linear Equations (Real Band Matrix)

2.13.2
DBBDLU, RBBDLU
LU Decomposition of a Real Band Matrix

2.13.3
DBBDLC, RBBDLC
LU Decomposition and Condition Number of a Real Band Matrix

2.13.4
DBBDLS, RBBDLS
Simultaneous Linear Equations (LU-Decomposed Real Band Matrix)

2.13.5
DBBDDI, RBBDDI
Determinant of a Real Band Matrix

2.13.6
DBBDLX, RBBDLX
Improving the Solution of Simultaneous Linear Equations (Real Band Matrix)

2.14
POSITIVE SYMMETRIC BAND MATRIX (SYMMETRIC BAND TYPE)

2.14.1
DBBPSL, RBBPSL
Simultaneous Linear Equations (Positive Symmetric Band Matrix)

2.14.2
DBBPUU, RBBPUU
LL^T Decomposition of a Positive Symmetric Band Matrix

2.14.3
DBBPUC, RBBPUC
LL^T Decomposition and Condition Number of a Positive Symmetric Band Matrix

2.14.4
DBBPLS, RBBPLS
Simultaneous Linear Equations (LL^T-Decomposed Positive Symmetric Band Matrix)

2.14.5
DBBPDI, RBBPDI
Determinant of a Positive Symmetric Band Matrix

2.14.6
DBBPLX, RBBPLX
Improving the Solution of Simultaneous Linear Equations (Positive Symmetric Band Matrix)

2.15
REAL TRIDIAGONAL MATRIX (VECTOR TYPE)

2.15.1
DBTDSL, RBTDSL
Simultaneous Linear Equations (Real Tridiagonal Matrix)

2.15.2
DBTPSL, RBTPSL
Simultaneous Linear Equations (Positive Symmetric Tridiagonal Matrix)

2.16
REAL TRIDIAGONAL MATRIX (VECTOR TYPE)

2.16.1
WBTDSL
Simultaneous Linear Equations (Real Tridiagonal Matrix)

2.16.2
WBTDLS
Simultaneous Linear Equations (Real Tridiagonal Matrix after Reduction Operations)

2.17
FIXED COEFFICIENT REAL TRIDIAGONAL MATRIX
(SCALAR TYPE)

2.17.1
WBTCSL
Simultaneous Linear Equations (Fixed Coefficient Real Tridiagonal Matrix)

2.17.2
WBTCLS
Simultaneous Linear Equations (Fixed Coefficient Real Tridiagonal Matrix after Reduction Operations)

2.18
VANDERMONDE MATRIX AND TOEPLITZ MATRIX

2.18.1
DBTOSL, RBTOSL
Simultaneous Linear Equations (Toeplitz Matrix)

2.18.2
DBTSSL, RBTSSL
Simultaneous Linear Equations (Symmetric Toeplitz Matrix)

2.18.3
DBVMSL, RBVMSL
Simultaneous Linear Equations (Vandermonde Matrix)

2.19
REAL UPPER TRIANGULAR MATRIX
(TWO-DIMENSIONAL ARRAY TYPE)

2.19.1
DBTUSL, RBTUSL
Simultaneous Linear Equations (Real Upper Triangular Matrix)

2.19.2
DBTUCO, RBTUCO
Condition Number of a Real Upper Triangular Matrix

2.19.3
DBTUDI, RBTUDI
Determinant and Inverse Matrix of a Real Upper Triangular Matrix

2.20
REAL LOWER TRIANGULAR MATRIX
(TWO-DIMENSIONAL ARRAY TYPE)

2.20.1
DBTLSL, RBTLSL
Simultaneous Linear Equations (Real Lower Triangular Matrix)

2.20.2
DBTLCO, RBTLCO
Condition Number of a Real Lower Triangular Matrix

2.20.3
DBTLDI, RBTLDI
Determinant and Inverse Matrix of a Real Lower Triangular Matrix

Appendix

Appendix A
GLOSSARY

Appendix B
METHODS OF HANDLING ARRAY DATA

B.1
Methods of handling array data corresponding to matrix

B.2
Data storage modes

B.2.1
Real matrix (two-dimensional array type)

B.2.2
Complex matrix

B.2.3
Real symmetric matrix and positive symmetric matrix

B.2.4
Hermitian matrix

B.2.5
Real band matrix

B.2.6
Real symmetric band matrix and positive symmetric matrix (symmetric band type)

B.2.7
Real tridiagonal matrix (vector type)

B.2.8
Real symmetric tridiagonal matrix and positive symmetric tridiagonal matrix (vector type)

B.2.9
Fixed coefficient real tridiagonal matrix (scalar type)

B.2.10
Triangular matrix

B.2.11
Random sparse matrix (For symmetric matrix only)

B.2.12
Random sparse matrix

Appendix C
MACHINE CONSTANTS USED IN ASL

C.1
Units for Determining Error

C.2
Maximum and Minimum Values of Floating Point Data