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ASL Basic Functions Vol.2 (for C)

Contents
INTRODUCTION SIMULTANEOUS LINEAR EQUATIONS (DIRECT METHOD)

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 SIMULTANEOUS LINEAR EQUATIONS (DIRECT METHOD)

2.1
INTRODUCTION

2.1.1
Methods of using functions

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
ASL_dbgmsm, ASL_rbgmsm
Simultaneous Linear Equations with Multiple Right-Hand Sides (Real Matrix)

2.2.2
ASL_dbgmsl, ASL_rbgmsl
Simultaneous Linear Equations (Real Matrix)

2.2.3
ASL_dbgmlu, ASL_rbgmlu
LU Decomposition of a Real Matrix

2.2.4
ASL_dbgmlc, ASL_rbgmlc
LU Decomposition and Condition Number of a Real Matrix

2.2.5
ASL_dbgmls, ASL_rbgmls
Simultaneous Linear Equations (LU-Decomposed Real Matrix)

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

2.2.7
ASL_dbgmdi, ASL_rbgmdi
Determinant and Inverse Matrix of a Real Matrix

2.2.8
ASL_dbgmlx, ASL_rbgmlx
Improving the Solution of Simultaneous Linear Equations (Real Matrix)

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

2.3.1
ASL_zbgmsm, ASL_cbgmsm
Simultaneous Linear Equations with Multiple Right-Hand Sides (Complex Matrix)

2.3.2
ASL_zbgmsl, ASL_cbgmsl
Simultaneous Linear Equations (Complex Matrix)

2.3.3
ASL_zbgmlu, ASL_cbgmlu
LU Decomposition of a Complex Matrix

2.3.4
ASL_zbgmlc, ASL_cbgmlc
LU Decomposition and Condition Number of a Complex Matrix

2.3.5
ASL_zbgmls, ASL_cbgmls
Simultaneous Linear Equations (LU-Decomposed Complex Matrix)

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

2.3.7
ASL_zbgmdi, ASL_cbgmdi
Determinant and Inverse Matrix of a Complex Matrix

2.3.8
ASL_zbgmlx, ASL_cbgmlx
Improving the Solution of Simultaneous Linear Equations (Complex Matrix)

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

2.4.1
ASL_zbgnsm, ASL_cbgnsm
Simultaneous Linear Equations with Multiple Right-Hand Sides (Complex Matrix)

2.4.2
ASL_zbgnsl, ASL_cbgnsl
Simultaneous Linear Equations (Complex Matrix)

2.4.3
ASL_zbgnlu, ASL_cbgnlu
LU Decomposition of a Complex Matrix

2.4.4
ASL_zbgnlc, ASL_cbgnlc
LU Decomposition and Condition Number of a Complex Matrix

2.4.5
ASL_zbgnls, ASL_cbgnls
Simultaneous Linear Equations (LU-Decomposed Complex Matrix)

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

2.4.7
ASL_zbgndi, ASL_cbgndi
Determinant and Inverse Matrix of a Complex Matrix

2.4.8
ASL_zbgnlx, ASL_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
ASL_dbpdsl, ASL_rbpdsl
Simultaneous Linear Equations (Positive Symmetric Matrix)

2.5.2
ASL_dbpduu, ASL_rbpduu
LL^T Decomposition of a Positive Symmetric Matrix

2.5.3
ASL_dbpduc, ASL_rbpduc
LL^T Decomposition and Condition Number of a Positive Symmetric Matrix

2.5.4
ASL_dbpdls, ASL_rbpdls
Simultaneous Linear Equations (LL^T-Decomposed Positive Symmetric Matrix)

2.5.5
ASL_dbpddi, ASL_rbpddi
Determinant and Inverse Matrix of a Positive Symmetric Matrix

2.5.6
ASL_dbpdlx, ASL_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
ASL_dbspsl, ASL_rbspsl
Simultaneous Linear Equations (Real Symmetric Matrix)

2.6.2
ASL_dbspud, ASL_rbspud
LDL^T Decomposition of a Real Symmetric Matrix

2.6.3
ASL_dbspuc, ASL_rbspuc
LDL^T Decomposition and Condition Number of a Real Symmetric Matrix

2.6.4
ASL_dbspls, ASL_rbspls
Simultaneous Linear Equations (LDL^T-Decomposed Real Symmetric Matrix)

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

2.6.6
ASL_dbspdi, ASL_rbspdi
Determinant and Inverse Matrix of a Real Symmetric Matrix

2.6.7
ASL_dbsplx, ASL_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
ASL_dbsmsl, ASL_rbsmsl
Simultaneous Linear Equations (Real Symmetric Matrix) (No Pivoting)

2.7.2
ASL_dbsmud, ASL_rbsmud
LDL^T Decomposition of a Real Symmetric Matrix (No Pivoting)

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

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

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

2.7.6
ASL_dbsmdi, ASL_rbsmdi
Determinant and Inverse Matrix of a Real Symmetric Matrix (No Pivoting)

2.7.7
ASL_dbsmlx, ASL_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
ASL_dbsnsl, ASL_rbsnsl
Simultaneous Linear Equations (Real Symmetric Matrix) (No Pivoting)

2.8.2
ASL_dbsnud, ASL_rbsnud
U^TDU Decomposition of a Real Symmetric Matrix (No Pivoting)

2.8.3
ASL_dbsnls, ASL_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
ASL_zbhpsl, ASL_cbhpsl
Simultaneous Linear Equations (Hermitian Matrix)

2.9.2
ASL_zbhpud, ASL_cbhpud
LDL^* Decomposition of a Hermitian Matrix

2.9.3
ASL_zbhpuc, ASL_cbhpuc
LDL^* Decomposition and Condition Number of a Hermitian Matrix

2.9.4
ASL_zbhpls, ASL_cbhpls
Simultaneous Linear Equations (LDL^*-Decomposed Hermitian Matrix)

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

2.9.6
ASL_zbhpdi, ASL_cbhpdi
Determinant and Inverse Matrix of a Hermitian Matrix

2.9.7
ASL_zbhplx, ASL_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
ASL_zbhrsl, ASL_cbhrsl
Simultaneous Linear Equations (Hermitian Matrix) (No Pivoting)

2.10.2
ASL_zbhrud, ASL_cbhrud
LDL^* Decomposition of a Hermitian Matrix (No Pivoting)

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

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

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

2.10.6
ASL_zbhrdi, ASL_cbhrdi
Determinant and Inverse Matrix of a Hermitian Matrix (No Pivoting)

2.10.7
ASL_zbhrlx, ASL_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
ASL_zbhfsl, ASL_cbhfsl
Simultaneous Linear Equations (Hermitian Matrix)

2.11.2
ASL_zbhfud, ASL_cbhfud
LDL^* Decomposition of a Hermitian Matrix

2.11.3
ASL_zbhfuc, ASL_cbhfuc
LDL^* Decomposition and Condition Number of a Hermitian Matrix

2.11.4
ASL_zbhfls, ASL_cbhfls
Simultaneous Linear Equations (LDL^*-Decomposed Hermitian Matrix)

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

2.11.6
ASL_zbhfdi, ASL_cbhfdi
Determinant and Inverse Matrix of a Hermitian Matrix

2.11.7
ASL_zbhflx, ASL_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
ASL_zbhesl, ASL_cbhesl
Simultaneous Linear Equations (Hermitian Matrix) (No Pivoting)

2.12.2
ASL_zbheud, ASL_cbheud
LDL^* Decomposition of a Hermitian Matrix (No Pivoting)

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

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

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

2.12.6
ASL_zbhedi, ASL_cbhedi
Determinant and Inverse Matrix of a Hermitian Matrix (No Pivoting)

2.12.7
ASL_zbhelx, ASL_cbhelx
Improving the Solution of Simultaneous Linear Equations (Hermitian Matrix) (No Pivoting)

2.13
REAL BAND MATRIX (BAND TYPE)

2.13.1
ASL_dbbdsl, ASL_rbbdsl
Simultaneous Linear Equations (Real Band Matrix)

2.13.2
ASL_dbbdlu, ASL_rbbdlu
LU Decomposition of a Real Band Matrix

2.13.3
ASL_dbbdlc, ASL_rbbdlc
LU Decomposition and Condition Number of a Real Band Matrix

2.13.4
ASL_dbbdls, ASL_rbbdls
Simultaneous Linear Equations (LU-Decomposed Real Band Matrix)

2.13.5
ASL_dbbddi, ASL_rbbddi
Determinant of a Real Band Matrix

2.13.6
ASL_dbbdlx, ASL_rbbdlx
Improving the Solution of Simultaneous Linear Equations (Real Band Matrix)

2.14
POSITIVE SYMMETRIC BAND MATRIX (SYMMETRIC BAND TYPE)

2.14.1
ASL_dbbpsl, ASL_rbbpsl
Simultaneous Linear Equations (Positive Symmetric Band Matrix)

2.14.2
ASL_dbbpuu, ASL_rbbpuu
LL^T Decomposition of a Positive Symmetric Band Matrix

2.14.3
ASL_dbbpuc, ASL_rbbpuc
LL^T Decomposition and Condition Number of a Positive Symmetric Band Matrix

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

2.14.5
ASL_dbbpdi, ASL_rbbpdi
Determinant of a Positive Symmetric Band Matrix

2.14.6
ASL_dbbplx, ASL_rbbplx
Improving the Solution of Simultaneous Linear Equations (Positive Symmetric Band Matrix)

2.15
REAL TRIDIAGONAL MATRIX (VECTOR TYPE)

2.15.1
ASL_dbtdsl, ASL_rbtdsl
Simultaneous Linear Equations (Real Tridiagonal Matrix)

2.15.2
ASL_dbtpsl, ASL_rbtpsl
Simultaneous Linear Equations (Positive Symmetric Tridiagonal Matrix)

2.16
REAL TRIDIAGONAL MATRIX (VECTOR TYPE)

2.16.1
ASL_wbtdsl
Simultaneous Linear Equations (Real Tridiagonal Matrix)

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

2.17
FIXED COEFFICIENT REAL TRIDIAGONAL MATRIX
(SCALAR TYPE)

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

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

2.18
VANDERMONDE MATRIX AND TOEPLITZ MATRIX

2.18.1
ASL_dbtosl, ASL_rbtosl
Simultaneous Linear Equations (Toeplitz Matrix)

2.18.2
ASL_dbtssl, ASL_rbtssl
Simultaneous Linear Equations (Symmetric Toeplitz Matrix)

2.18.3
ASL_dbvmsl, ASL_rbvmsl
Simultaneous Linear Equations (Vandermonde Matrix)

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

2.19.1
ASL_dbtusl, ASL_rbtusl
Simultaneous Linear Equations (Real Upper Triangular Matrix)

2.19.2
ASL_dbtuco, ASL_rbtuco
Condition Number of a Real Upper Triangular Matrix

2.19.3
ASL_dbtudi, ASL_rbtudi
Determinant and Inverse Matrix of a Real Upper Triangular Matrix

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

2.20.1
ASL_dbtlsl, ASL_rbtlsl
Simultaneous Linear Equations (Real Lower Triangular Matrix)

2.20.2
ASL_dbtlco, ASL_rbtlco
Condition Number of a Real Lower Triangular Matrix

2.20.3
ASL_dbtldi, ASL_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 INTERFACE

C.1
Units for Determining Error

C.2
Maximum and Minimum Values of Floating Point Data