Engineering and Scientific Subroutine Library for AIX Version 3 Release 3: Guide and Reference

Overview of the Linear Algebraic Equation Subroutines

This section describes the subroutines in each of the four linear algebraic equation areas:

Dense linear algebraic equations (Table 83)
Banded linear algebraic equations (Table 84)
Sparse linear algebraic equations (Table 85)
Linear least squares (Table 86)

Note:: Some of the linear algebraic equations were designed in accordance with the Level 2 BLAS, Level 3 BLAS, and LAPACK de facto standard. If these subprograms do not comply with the standard as approved, IBM will consider updating them to do so. If IBM updates these subprograms, the updates could require modifications of the calling application program. For details on the Level 2 and 3 BLAS, see references [32] and [34]. For details on the LAPACK routines, see reference [8].

Dense Linear Algebraic Equation Subroutines

The dense linear algebraic equation subroutines provide solutions to linear systems of equations for both real and complex general matrices and their transposes, positive definite real symmetric and complex Hermitian matrices, real symmetric indefinite matrices and triangular matrices. Some of these subroutines correspond to the Level 2 BLAS, Level 3 BLAS, and LAPACK routines described in references [32], [34], and [8].

Table 83. List of Dense Linear Algebraic Equation Subroutines

Banded Linear Algebraic Equation Subroutines

The banded linear algebraic equation subroutines provide solutions to linear systems of equations for real general band matrices, real positive definite symmetric band matrices, real or complex general tridiagonal matrices, real positive definite symmetric tridiagonal matrices, and real or complex triangular band matrices.

Table 84. List of Banded Linear Algebraic Equation Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
General Band Matrix Factorization	SGBF	DGBF	SGBF and DGBF--General Band Matrix Factorization
General Band Matrix Solve	SGBS	DGBS	SGBS and DGBS--General Band Matrix Solve
Positive Definite Symmetric Band Matrix Factorization	SPBF SPBCHF	DPBF DPBCHF	SPBF, DPBF, SPBCHF, and DPBCHF--Positive Definite Symmetric Band Matrix Factorization
Positive Definite Symmetric Band Matrix Solve	SPBS SPBCHS	DPBS DPBCHS	SPBS, DPBS, SPBCHS, and DPBCHS--Positive Definite Symmetric Band Matrix Solve
General Tridiagonal Matrix Factorization	SGTF	DGTF	SGTF and DGTF--General Tridiagonal Matrix Factorization
General Tridiagonal Matrix Solve	SGTS	DGTS	SGTS and DGTS--General Tridiagonal Matrix Solve
General Tridiagonal Matrix Combined Factorization and Solve with No Pivoting	SGTNP CGTNP	DGTNP ZGTNP	SGTNP, DGTNP, CGTNP, and ZGTNP--General Tridiagonal Matrix Combined Factorization and Solve with No Pivoting
General Tridiagonal Matrix Factorization with No Pivoting	SGTNPF CGTNPF	DGTNPF ZGTNPF	SGTNPF, DGTNPF, CGTNPF, and ZGTNPF--General Tridiagonal Matrix Factorization with No Pivoting
General Tridiagonal Matrix Solve with No Pivoting	SGTNPS CGTNPS	DGTNPS ZGTNPS	SGTNPS, DGTNPS, CGTNPS, and ZGTNPS--General Tridiagonal Matrix Solve with No Pivoting
Positive Definite Symmetric Tridiagonal Matrix Factorization	SPTF	DPTF	SPTF and DPTF--Positive Definite Symmetric Tridiagonal Matrix Factorization
Positive Definite Symmetric Tridiagonal Matrix Solve	SPTS	DPTS	SPTS and DPTS--Positive Definite Symmetric Tridiagonal Matrix Solve
Triangular Band Equation Solve	STBSV^ø CTBSV^ø	DTBSV^ø ZTBSV^ø	STBSV, DTBSV, CTBSV, and ZTBSV--Triangular Band Equation Solve
^ø Level 2 BLAS

Sparse Linear Algebraic Equation Subroutines

The sparse linear algebraic equation subroutines provide direct and iterative solutions to linear systems of equations both for general sparse matrices and their transposes and for sparse symmetric matrices.

Table 85. List of Sparse Linear Algebraic Equation Subroutines

Descriptive Name	Long- Precision Subroutine	Page
General Sparse Matrix Factorization Using Storage by Indices, Rows, or Columns	DGSF	DGSF--General Sparse Matrix Factorization Using Storage by Indices, Rows, or Columns
General Sparse Matrix or Its Transpose Solve Using Storage by Indices, Rows, or Columns	DGSS	DGSS--General Sparse Matrix or Its Transpose Solve Using Storage by Indices, Rows, or Columns
General Sparse Matrix or Its Transpose Factorization, Determinant, and Solve Using Skyline Storage Mode	DGKFS	DGKFS--General Sparse Matrix or Its Transpose Factorization, Determinant, and Solve Using Skyline Storage Mode
Symmetric Sparse Matrix Factorization, Determinant, and Solve Using Skyline Storage Mode	DSKFS	DSKFS--Symmetric Sparse Matrix Factorization, Determinant, and Solve Using Skyline Storage Mode
Iterative Linear System Solver for a General or Symmetric Sparse Matrix Stored by Rows	DSRIS	DSRIS--Iterative Linear System Solver for a General or Symmetric Sparse Matrix Stored by Rows
Sparse Positive Definite or Negative Definite Symmetric Matrix Iterative Solve Using Compressed-Matrix Storage Mode	DSMCG^§	DSMCG--Sparse Positive Definite or Negative Definite Symmetric Matrix Iterative Solve Using Compressed-Matrix Storage Mode
Sparse Positive Definite or Negative Definite Symmetric Matrix Iterative Solve Using Compressed-Diagonal Storage Mode	DSDCG	DSDCG--Sparse Positive Definite or Negative Definite Symmetric Matrix Iterative Solve Using Compressed-Diagonal Storage Mode
General Sparse Matrix Iterative Solve Using Compressed-Matrix Storage Mode	DSMGCG^§	DSMGCG--General Sparse Matrix Iterative Solve Using Compressed-Matrix Storage Mode
General Sparse Matrix Iterative Solve Using Compressed-Diagonal Storage Mode	DSDGCG	DSDGCG--General Sparse Matrix Iterative Solve Using Compressed-Diagonal Storage Mode
^§ These subroutines are provided only for migration from earlier releases of ESSL and are not intended for use in new programs. Use DSRIS instead.

Linear Least Squares Subroutines

The linear least squares subroutines provide least squares solutions to linear systems of equations for real general matrices. |Three methods are provided: one that uses the singular value |decomposition; one that uses a QR decomposition with column |pivoting; and another that uses a QR decomposition without column |pivoting. Some of these subroutines correspond to the LAPACK routines |described in reference [8].

Table 86. List of Linear Least Squares Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
Singular Value Decomposition for a General Matrix	SGESVF	DGESVF	SGESVF and DGESVF--Singular Value Decomposition for a General Matrix
Linear Least Squares Solution for a General Matrix Using the Singular Value Decomposition	SGESVS	DGESVS	SGESVS and DGESVS--Linear Least Squares Solution for a General Matrix Using the Singular Value Decomposition
General Matrix QR Factorization		DGEQRF^°	DGEQRF--General Matrix QR Factorization
Linear Least Squares Solution for a General Matrix		DGELS^°	DGELS--Linear Least Squares Solution for a General Matrix
Linear Least Squares Solution for a General Matrix with Column Pivoting	SGELLS	DGELLS	SGELLS and DGELLS--Linear Least Squares Solution for a General Matrix with Column Pivoting
^° LAPACK

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Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
General Matrix Factorization	SGEF CGEF SGETRF^° CGETRF^°	DGEF ZGEF DGETRF^° ZGETRF^° DGEFP^§	SGEF, DGEF, CGEF, and ZGEF--General Matrix Factorization SGETRF, DGETRF, CGETRF and ZGETRF--General Matrix Factorization
General Matrix, Its Transpose, or Its Conjugate Transpose Solve	SGES CGES	DGES ZGES	SGES, DGES, CGES, and ZGES--General Matrix, Its Transpose, or Its Conjugate Transpose Solve
General Matrix, Its Transpose, or Its Conjugate Transpose Multiple Right-Hand Side Solve	SGESM CGESM SGETRS^° CGETRS^°	DGESM ZGESM DGETRS^° ZGETRS^°	SGESM, DGESM, CGESM, and ZGESM--General Matrix, Its Transpose, or Its Conjugate Transpose Multiple Right-Hand Side Solve SGETRS, DGETRS, CGETRS, and ZGETRS--General Matrix Multiple Right-Hand Side Solve
General Matrix Factorization, Condition Number Reciprocal, and Determinant	SGEFCD	DGEFCD	SGEFCD and DGEFCD--General Matrix Factorization, Condition Number Reciprocal, and Determinant
Positive Definite Real Symmetric or Complex Hermitian Matrix Factorization	SPPF SPOF CPOF SPOTRF^° CPOTRF^°	DPPF DPOF ZPOF DPOTRF^° ZPOTRF^° DPPFP^§	SPPF, DPPF, SPOF, DPOF, CPOF, ZPOF, SPOTRF, DPOTRF, CPOTRF, and ZPOTRF--Positive Definite Real Symmetric or Complex Hermitian Matrix Factorization
Positive Definite Real Symmetric Matrix Solve	SPPS	DPPS	SPPS and DPPS--Positive Definite Real Symmetric Matrix Solve
Positive Definite Real Symmetric or Complex Hermitian Matrix Multiple Right-Hand Side Solve	SPOSM CPOSM SPOTRS^° CPOTRS^°	DPOSM ZPOSM DPOTRS^° ZPOTRS^°	SPOSM, DPOSM, CPOSM, ZPOSM, SPOTRS, DPOTRS, CPOTRS, and ZPOTRS--Positive Definite Real Symmetric or Complex Hermitian Matrix Multiple Right-Hand Side Solve
Positive Definite Real Symmetric Matrix Factorization, Condition Number Reciprocal, and Determinant	SPPFCD SPOFCD	DPPFCD DPOFCD	SPPFCD, DPPFCD, SPOFCD, and DPOFCD--Positive Definite Real Symmetric Matrix Factorization, Condition Number Reciprocal, and Determinant
Symmetric Indefinite Matrix Factorization and Multiple Right-Hand Side Solve		DBSSV	DBSSV--Symmetric Indefinite Matrix Factorization and Multiple Right-Hand Side Solve
Symmetric Indefinite Matrix Factorization		DBSTRF	DBSTRF--Symmetric Indefinite Matrix Factorization
Symmetric Indefinite Matrix Multiple Right-Hand Side Solve		DBSTRS	DBSTRS--Symmetric Indefinite Matrix Multiple Right-Hand Side Solve
General Matrix Inverse, Condition Number Reciprocal, and Determinant	SGEICD SGETRI^°	DGEICD DGETRI^°	SGEICD, DGEICD, SGETRI and DGETRI--General Matrix Inverse
Positive Definite Real Symmetric Matrix Inverse, Condition Number Reciprocal, and Determinant	SPPICD SPOICD SPOTRI^°	DPPICD DPOICD DPOTRI^°	SPPICD, DPPICD, SPOICD, DPOICD, SPOTRI and DPOTRI--Positive Definite Real Symmetric Matrix Inverse
Solution of a Triangular System of Equations with a Single Right-Hand Side	STRSV^ø CTRSV^ø STPSV^ø CTPSV^ø	DTRSV^ø ZTRSV^ø DTPSV^ø ZTPSV^ø	STRSV, DTRSV, CTRSV, ZTRSV, STPSV, DTPSV, CTPSV, and ZTPSV--Solution of a Triangular System of Equations with a Single Right-Hand Side
Solution of Triangular Systems of Equations with Multiple Right-Hand Sides	STRSM^¢ CTRSM^¢	DTRSM^¢ ZTRSM^¢	STRSM, DTRSM, CTRSM, and ZTRSM--Solution of Triangular Systems of Equations with Multiple Right-Hand Sides
Triangular Matrix Inverse	STRI STPI STRTRI^° STPTRI^°	DTRI DTPI DTRTRI^° DTPTRI^°	STRI, DTRI, STPI, DTPI, STRTRI, DTRTRI, STPTRI, and DTPTRI--Triangular Matrix Inverse
^ø Level 2 BLAS ^¢ Level 3 BLAS ^° LAPACK ^§ This subroutine is provided only for migration from earlier releases of ESSL and is not intended for use is new programs. Documentation for this subroutine is no longer provided.