Guide and Reference

List of ESSL Subroutines

This section provides an overview of the subroutines in each of the areas of ESSL.

Appendix A. "Basic Linear Algebra Subprograms (BLAS)" contains a list of Level 1, 2, and 3 Basic Linear Algebra Subprograms (BLAS) included in ESSL.

Appendix B. "LAPACK" contains a list of Linear Algebra Package (LAPACK) subroutines included in ESSL.

Linear Algebra Subprograms

The linear algebra subprograms consist of:

• Vector-scalar linear algebra subprograms (Table 3)
• Sparse vector-scalar linear algebra subprograms (Table 4)
• Matrix-vector linear algebra subprograms (Table 5)
• Sparse matrix-vector linear algebra subprograms (Table 6)

Notes:

1. The term subprograms is used to be consistent with the Basic Linear Algebra Subprograms (BLAS), because many of these subprograms correspond to the BLAS.

2. Some of the linear algebra subprograms were designed in accordance with the Level 1 and Level 2 BLAS 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.

Vector-Scalar Linear Algebra Subprograms

The vector-scalar linear algebra subprograms include a subset of the standard set of Level 1 BLAS. For details on the BLAS, see reference [73]. The remainder of the vector-scalar linear algebra subprograms are commonly used computations provided for your applications. Both real and complex versions of the subprograms are provided.

Table 3. List of Vector-Scalar Linear Algebra Subprograms

Descriptive Name	Short- Precision Subprogram	Long- Precision Subprogram	Page
Position of the First or Last Occurrence of the Vector Element Having the Largest Magnitude	ISAMAX&¤ ICAMAX&¤	IDAMAX&¤ IZAMAX&¤	ISAMAX, IDAMAX, ICAMAX, and IZAMAX--Position of the First or Last Occurrence of the Vector Element Having the Largest Magnitude
Position of the First or Last Occurrence of the Vector Element Having Minimum Absolute Value	ISAMIN&	IDAMIN&	ISAMIN and IDAMIN--Position of the First or Last Occurrence of the Vector Element Having Minimum Absolute Value
Position of the First or Last Occurrence of the Vector Element Having Maximum Value	ISMAX&	IDMAX&	ISMAX and IDMAX--Position of the First or Last Occurrence of the Vector Element Having the Maximum Value
Position of the First or Last Occurrence of the Vector Element Having Minimum Value	ISMIN&	IDMIN&	ISMIN and IDMIN--Position of the First or Last Occurrence of the Vector Element Having Minimum Value
Sum of the Magnitudes of the Elements in a Vector	SASUM&¤ SCASUM&¤	DASUM&¤ DZASUM&¤	SASUM, DASUM, SCASUM, and DZASUM--Sum of the Magnitudes of the Elements in a Vector
Multiply a Vector X by a Scalar, Add to a Vector Y, and Store in the Vector Y	SAXPY¤ CAXPY¤	DAXPY¤ ZAXPY¤	SAXPY, DAXPY, CAXPY, and ZAXPY--Multiply a Vector X by a Scalar, Add to a Vector Y, and Store in the Vector Y
Copy a Vector	SCOPY¤ CCOPY¤	DCOPY¤ ZCOPY¤	SCOPY, DCOPY, CCOPY, and ZCOPY--Copy a Vector
Dot Product of Two Vectors	SDOT&¤ CDOTU&¤ CDOTC&¤	DDOT&¤ ZDOTU&¤ ZDOTC&¤	SDOT, DDOT, CDOTU, ZDOTU, CDOTC, and ZDOTC--Dot Product of Two Vectors
Compute SAXPY or DAXPY N Times	SNAXPY	DNAXPY	SNAXPY and DNAXPY--Compute SAXPY or DAXPY N Times
Compute Special Dot Products N Times	SNDOT	DNDOT	SNDOT and DNDOT--Compute Special Dot Products N Times
Euclidean Length of a Vector with Scaling of Input to Avoid Destructive Underflow and Overflow	SNRM2&¤ SCNRM2&¤	DNRM2&¤ DZNRM2&¤	SNRM2, DNRM2, SCNRM2, and DZNRM2--Euclidean Length of a Vector with Scaling of Input to Avoid Destructive Underflow and Overflow
Euclidean Length of a Vector with No Scaling of Input	SNORM2& CNORM2&	DNORM2& ZNORM2&	SNORM2, DNORM2, CNORM2, and ZNORM2--Euclidean Length of a Vector with No Scaling of Input
Construct a Givens Plane Rotation	SROTG¤ CROTG¤	DROTG¤ ZROTG¤	SROTG, DROTG, CROTG, and ZROTG--Construct a Givens Plane Rotation
Apply a Plane Rotation	SROT¤ CROT¤ CSROT¤	DROT¤ ZROT¤ ZDROT¤	SROT, DROT, CROT, ZROT, CSROT, and ZDROT--Apply a Plane Rotation
Multiply a Vector X by a Scalar and Store in the Vector X	SSCAL¤ CSCAL¤ CSSCAL¤	DSCAL¤ ZSCAL¤ ZDSCAL¤	SSCAL, DSCAL, CSCAL, ZSCAL, CSSCAL, and ZDSCAL--Multiply a Vector X by a Scalar and Store in the Vector X
Interchange the Elements of Two Vectors	SSWAP¤ CSWAP¤	DSWAP¤ ZSWAP¤	SSWAP, DSWAP, CSWAP, and ZSWAP--Interchange the Elements of Two Vectors
Add a Vector X to a Vector Y and Store in a Vector Z	SVEA CVEA	DVEA ZVEA	SVEA, DVEA, CVEA, and ZVEA--Add a Vector X to a Vector Y and Store in a Vector Z
Subtract a Vector Y from a Vector X and Store in a Vector Z	SVES CVES	DVES ZVES	SVES, DVES, CVES, and ZVES--Subtract a Vector Y from a Vector X and Store in a Vector Z
Multiply a Vector X by a Vector Y and Store in a Vector Z	SVEM CVEM	DVEM ZVEM	SVEM, DVEM, CVEM, and ZVEM--Multiply a Vector X by a Vector Y and Store in a Vector Z
Multiply a Vector X by a Scalar and Store in a Vector Y	SYAX CYAX CSYAX	DYAX ZYAX ZDYAX	SYAX, DYAX, CYAX, ZYAX, CSYAX, and ZDYAX--Multiply a Vector X by a Scalar and Store in a Vector Y
Multiply a Vector X by a Scalar, Add to a Vector Y, and Store in a Vector Z	SZAXPY CZAXPY	DZAXPY ZZAXPY	SZAXPY, DZAXPY, CZAXPY, and ZZAXPY--Multiply a Vector X by a Scalar, Add to a Vector Y, and Store in a Vector Z
& This subprogram is invoked as a function in a Fortran program. ¤ Level 1 BLAS

Sparse Vector-Scalar Linear Algebra Subprograms

The sparse vector-scalar linear algebra subprograms operate on sparse vectors; that is, only the nonzero elements of the vector are stored. These subprograms provide similar functions to the vector-scalar subprograms. These subprograms represent a subset of the sparse extensions to the Level 1 BLAS described in reference [29]. Both real and complex versions of the subprograms are provided.

Table 4. List of Sparse Vector-Scalar Linear Algebra Subprograms

Descriptive Name	Short- Precision Subprogram	Long- Precision Subprogram	Page
Scatter the Elements of a Sparse Vector X in Compressed-Vector Storage Mode into Specified Elements of a Sparse Vector Y in Full-Vector Storage Mode	SSCTR CSCTR	DSCTR ZSCTR	SSCTR, DSCTR, CSCTR, ZSCTR--Scatter the Elements of a Sparse Vector X in Compressed-Vector Storage Mode into Specified Elements of a Sparse Vector Y in Full-Vector Storage Mode
Gather Specified Elements of a Sparse Vector Y in Full-Vector Storage Mode into a Sparse Vector X in Compressed-Vector Storage Mode	SGTHR CGTHR	DGTHR ZGTHR	SGTHR, DGTHR, CGTHR, and ZGTHR--Gather Specified Elements of a Sparse Vector Y in Full-Vector Storage Mode into a Sparse Vector X in Compressed-Vector Storage Mode
Gather Specified Elements of a Sparse Vector Y in Full-Vector Mode into a Sparse Vector X in Compressed-Vector Mode, and Zero the Same Specified Elements of Y	SGTHRZ CGTHRZ	DGTHRZ ZGTHRZ	SGTHRZ, DGTHRZ, CGTHRZ, and ZGTHRZ--Gather Specified Elements of a Sparse Vector Y in Full-Vector Mode into a Sparse Vector X in Compressed-Vector Mode, and Zero the Same Specified Elements of Y
Multiply a Sparse Vector X in Compressed-Vector Storage Mode by a Scalar, Add to a Sparse Vector Y in Full-Vector Storage Mode, and Store in the Vector Y	SAXPYI CAXPYI	DAXPYI ZAXPYI	SAXPYI, DAXPYI, CAXPYI, and ZAXPYI--Multiply a Sparse Vector X in Compressed-Vector Storage Mode by a Scalar, Add to a Sparse Vector Y in Full-Vector Storage Mode, and Store in the Vector Y
Dot Product of a Sparse Vector X in Compressed-Vector Storage Mode and a Sparse Vector Y in Full-Vector Storage Mode	SDOTI& CDOTCI& CDOTUI&	DDOTI& ZDOTCI& ZDOTUI&	SDOTI, DDOTI, CDOTUI, ZDOTUI, CDOTCI, and ZDOTCI--Dot Product of a Sparse Vector X in Compressed-Vector Storage Mode and a Sparse Vector Y in Full-Vector Storage Mode
& This subprogram is invoked as a function in a Fortran program.

Matrix-Vector Linear Algebra Subprograms

The matrix-vector linear algebra subprograms operate on a higher-level data structure--matrix-vector rather than vector-scalar--using optimized algorithms to improve performance. These subprograms include a subset of the standard set of Level 2 BLAS. For details on the Level 2 BLAS, see [34] and [35]. Both real and complex versions of the subprograms are provided.

Table 5. List of Matrix-Vector Linear Algebra Subprograms

Descriptive Name	Short- Precision Subprogram	Long- Precision Subprogram	Page
Matrix-Vector Product for a General Matrix, Its Transpose, or Its Conjugate Transpose	SGEMVø CGEMVø SGEMX§ SGEMTX§	DGEMVø ZGEMVø DGEMX§ DGEMTX§	SGEMV, DGEMV, CGEMV, ZGEMV, SGEMX, DGEMX, SGEMTX, and DGEMTX--Matrix-Vector Product for a General Matrix, Its Transpose, or Its Conjugate Transpose
Rank-One Update of a General Matrix	SGERø CGERUø CGERCø	DGERø ZGERUø ZGERCø	SGER, DGER, CGERU, ZGERU, CGERC, and ZGERC--Rank-One Update of a General Matrix
Matrix-Vector Product for a Real Symmetric or Complex Hermitian Matrix	SSPMVø CHPMVø SSYMVø CHEMVø SSLMX§	DSPMVø ZHPMVø DSYMVø ZHEMVø DSLMX§	SSPMV, DSPMV, CHPMV, ZHPMV, SSYMV, DSYMV, CHEMV, ZHEMV, SSLMX, and DSLMX--Matrix-Vector Product for a Real Symmetric or Complex Hermitian Matrix
Rank-One Update of a Real Symmetric or Complex Hermitian Matrix	SSPRø CHPRø SSYRø CHERø SSLR1§	DSPRø ZHPRø DSYRø ZHERø DSLR1§	SSPR, DSPR, CHPR, ZHPR, SSYR, DSYR, CHER, ZHER, SSLR1, and DSLR1 --Rank-One Update of a Real Symmetric or Complex Hermitian Matrix
Rank-Two Update of a Real Symmetric or Complex Hermitian Matrix	SSPR2ø CHPR2ø SSYR2ø CHER2ø SSLR2§	DSPR2ø ZHPR2ø DSYR2ø ZHER2ø DSLR2§	SSPR2, DSPR2, CHPR2, ZHPR2, SSYR2, DSYR2, CHER2, ZHER2, SSLR2, and DSLR2--Rank-Two Update of a Real Symmetric or Complex Hermitian Matrix
Matrix-Vector Product for a General Band Matrix, Its Transpose, or Its Conjugate Transpose	SGBMVø CGBMVø	DGBMVø ZGBMVø	SGBMV, DGBMV, CGBMV, and ZGBMV--Matrix-Vector Product for a General Band Matrix, Its Transpose, or Its Conjugate Transpose
Matrix-Vector Product for a Real Symmetric or Complex Hermitian Band Matrix	SSBMVø CHBMVø	DSBMVø ZHBMVø	SSBMV, DSBMV, CHBMV, and ZHBMV--Matrix-Vector Product for a Real Symmetric or Complex Hermitian Band Matrix
Matrix-Vector Product for a Triangular Matrix, Its Transpose, or Its Conjugate Transpose	STRMVø CTRMVø STPMVø CTPMVø	DTRMVø ZTRMVø DTPMVø ZTPMVø	STRMV, DTRMV, CTRMV, ZTRMV, STPMV, DTPMV, CTPMV, and ZTPMV--Matrix-Vector Product for a Triangular Matrix, Its Transpose, or Its Conjugate Transpose
Matrix-Vector Product for a Triangular Band Matrix, Its Transpose, or Its Conjugate Transpose	STBMVø CTBMVø	DTBMVø ZTBMVø	STBMV, DTBMV, CTBMV, and ZTBMV--Matrix-Vector Product for a Triangular Band Matrix, Its Transpose, or Its Conjugate Transpose
ø Level 2 BLAS § This subroutine is provided only for migration from earlier releases of ESSL and is not intended for use in new programs.

Sparse Matrix-Vector Linear Algebra Subprograms

The sparse matrix-vector linear algebra subprograms operate on sparse matrices; that is, only the nonzero elements of the matrix are stored. These subprograms provide similar functions to the matrix-vector subprograms.

Table 6. List of Sparse Matrix-Vector Linear Algebra Subprograms

Descriptive Name	Long- Precision Subprogram	Page
Matrix-Vector Product for a Sparse Matrix in Compressed-Matrix Storage Mode	DSMMX	DSMMX--Matrix-Vector Product for a Sparse Matrix in Compressed-Matrix Storage Mode
Transpose a Sparse Matrix in Compressed-Matrix Storage Mode	DSMTM	DSMTM--Transpose a Sparse Matrix in Compressed-Matrix Storage Mode
Matrix-Vector Product for a Sparse Matrix or Its Transpose in Compressed-Diagonal Storage Mode	DSDMX	DSDMX--Matrix-Vector Product for a Sparse Matrix or Its Transpose in Compressed-Diagonal Storage Mode

Matrix Operations

Some of the matrix operation subroutines were designed in accordance with the Level 3 BLAS de facto standard. If these subroutines do not comply with the standard as approved, IBM will consider updating them to do so. If IBM updates these subroutines, the updates could require modifications of the calling application program. For details on the Level 3 BLAS, see reference [32]. The matrix operation subroutines also include the commonly used matrix operations: addition, subtraction, multiplication, and transposition.

Table 7. List of Matrix Operation Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
Matrix Addition for General Matrices or Their Transposes	SGEADD CGEADD	DGEADD ZGEADD	SGEADD, DGEADD, CGEADD, and ZGEADD--Matrix Addition for General Matrices or Their Transposes
Matrix Subtraction for General Matrices or Their Transposes	SGESUB CGESUB	DGESUB ZGESUB	SGESUB, DGESUB, CGESUB, and ZGESUB--Matrix Subtraction for General Matrices or Their Transposes
Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes	SGEMUL CGEMUL	DGEMUL ZGEMUL DGEMLP§	SGEMUL, DGEMUL, CGEMUL, and ZGEMUL--Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes
Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes Using Winograd's Variation of Strassen's Algorithm	SGEMMS CGEMMS	DGEMMS ZGEMMS	SGEMMS, DGEMMS, CGEMMS, and ZGEMMS--Matrix Multiplication for General Matrices, Their Transposes, or Conjugate Transposes Using Winograd's Variation of Strassen's Algorithm
Combined Matrix Multiplication and Addition for General Matrices, Their Transposes, or Conjugate Transposes	SGEMM¢ CGEMM¢	DGEMM¢ ZGEMM¢	SGEMM, DGEMM, CGEMM, and ZGEMM--Combined Matrix Multiplication and Addition for General Matrices, Their Transposes, or Conjugate Transposes
Matrix-Matrix Product Where One Matrix is Real or Complex Symmetric or Complex Hermitian	SSYMM¢ CSYMM¢ CHEMM¢	DSYMM¢ ZSYMM¢ ZHEMM¢	SSYMM, DSYMM, CSYMM, ZSYMM, CHEMM, and ZHEMM--Matrix-Matrix Product Where One Matrix is Real or Complex Symmetric or Complex Hermitian
Triangular Matrix-Matrix Product	STRMM¢ CTRMM¢	DTRMM¢ ZTRMM¢	STRMM, DTRMM, CTRMM, and ZTRMM--Triangular Matrix-Matrix Product
Rank-K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix	SSYRK¢ CSYRK¢ CHERK¢	DSYRK¢ ZSYRK¢ ZHERK¢	SSYRK, DSYRK, CSYRK, ZSYRK, CHERK, and ZHERK--Rank-K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix
Rank-2K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix	SSYR2K¢ CSYR2K¢ CHER2K¢	DSYR2K¢ ZSYR2K¢ ZHER2K¢	SSYR2K, DSYR2K, CSYR2K, ZSYR2K, CHER2K, and ZHER2K--Rank-2K Update of a Real or Complex Symmetric or a Complex Hermitian Matrix
General Matrix Transpose (In-Place)	SGETMI CGETMI	DGETMI ZGETMI	SGETMI, DGETMI, CGETMI, and ZGETMI--General Matrix Transpose (In-Place)
General Matrix Transpose (Out-of-Place)	SGETMO CGETMO	DGETMO ZGETMO	SGETMO, DGETMO, CGETMO, and ZGETMO--General Matrix Transpose (Out-of-Place)
¢ Level 3 BLAS § This subroutine is provided only for migration from earlier release of ESSL and is not intended for use in new programs.

Linear Algebraic Equations

The linear algebraic equations consist of:

• Dense linear algebraic equations (Table 8)
• Banded linear algebraic equations (Table 9)
• Sparse linear algebraic equations (Table 10)
• Linear least squares (Table 11)

Notes:

1. 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 [32] and [34]. For details on LAPACK, see [8].

Dense Linear Algebraic Equations

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, 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 8. List of Dense Linear Algebraic Equation Subroutines

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	DPPF DPOF ZPOF DPPFP§	SPPF, DPPF, SPOF, DPOF, CPOF, and ZPOF--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	DPOSM ZPOSM	SPOSM, DPOSM, CPOSM, and ZPOSM--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
General Matrix Inverse, Condition Number Reciprocal, and Determinant	SGEICD	DGEICD	SGEICD and DGEICD--General Matrix Inverse, Condition Number Reciprocal, and Determinant
Positive Definite Real Symmetric Matrix Inverse, Condition Number Reciprocal, and Determinant	SPPICD SPOICD	DPPICD DPOICD	SPPICD, DPPICD, SPOICD, and DPOICD--Positive Definite Real Symmetric Matrix Inverse, Condition Number Reciprocal, and Determinant
Solution of a Triangular System of Equations with a Single Right-Hand Side	STRSVø CTRSVø STPSVø CTPSVø	DTRSVø ZTRSVø DTPSVø ZTPSVø	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	DTRI DTPI	STRI, DTRI, STPI, and DTPI--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 in new programs. Documentation for this subroutine is no longer provided.

Banded Linear Algebraic Equations

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 9. 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	STBSVø CTBSVø	DTBSVø ZTBSVø	STBSV, DTBSV, CTBSV, and ZTBSV--Triangular Band Equation Solve
ø Level 2 BLAS

Sparse Linear Algebraic Equations

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 10. 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 DGKFSP§	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 DSKFSP§	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
§ This subroutine is provided only for migration from earlier releases of ESSL and is not intended for use in new programs. Documentation for this subroutine is no longer provided. £ This subroutine is provided only for migration from earlier releases of ESSL and is not intended for use in new programs. Use DSRIS instead.

Linear Least Squares

The linear least squares subroutines provide least squares solutions to linear systems of equations for real general matrices. Two methods are provided: one that uses the singular value decomposition and another that uses a QR decomposition with column pivoting.

Table 11. 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
Linear Least Squares Solution for a General Matrix Using a QR Decomposition with Column Pivoting	SGELLS	DGELLS	SGELLS and DGELLS--Linear Least Squares Solution for a General Matrix Using a QR Decomposition with Column Pivoting

Eigensystem Analysis

The eigensystem analysis subroutines provide solutions to the algebraic eigensystem analysis problem Az = wz and the generalized eigensystem analysis problem Az = wBz (Table 12). Many of the eigensystem analysis subroutines use the algorithms presented in Linear Algebra by Wilkinson and Reinsch [93] or use adaptations of EISPACK routines, as described in the EISPACK Guide Lecture Notes in Computer Science in reference [81] or in the EISPACK Guide Extension Lecture Notes in Computer Science in reference [55]. (EISPACK is available from the sources listed in reference [49].)

Table 12. List of Eigensystem Analysis Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
Eigenvalues and, Optionally, All or Selected Eigenvectors of a General Matrix	SGEEV CGEEV	DGEEV ZGEEV	SGEEV, DGEEV, CGEEV, and ZGEEV--Eigenvalues and, Optionally, All or Selected Eigenvectors of a General Matrix
Eigenvalues and, Optionally, the Eigenvectors of a Real Symmetric Matrix or a Complex Hermitian Matrix	SSPEV CHPEV	DSPEV ZHPEV	SSPEV, DSPEV, CHPEV, and ZHPEV--Eigenvalues and, Optionally, the Eigenvectors of a Real Symmetric Matrix or a Complex Hermitian Matrix
Extreme Eigenvalues and, Optionally, the Eigenvectors of a Real Symmetric Matrix or a Complex Hermitian Matrix	SSPSV CHPSV	DSPSV ZHPSV	SSPSV, DSPSV, CHPSV, and ZHPSV--Extreme Eigenvalues and, Optionally, the Eigenvectors of a Real Symmetric Matrix or a Complex Hermitian Matrix
Eigenvalues and, Optionally, the Eigenvectors of a Generalized Real Eigensystem, Az=wBz, where A and B Are Real General Matrices	SGEGV	DGEGV	SGEGV and DGEGV--Eigenvalues and, Optionally, the Eigenvectors of a Generalized Real Eigensystem, Az=wBz, where A and B Are Real General Matrices
Eigenvalues and, Optionally, the Eigenvectors of a Generalized Real Symmetric Eigensystem, Az=wBz, where A Is Real Symmetric and B Is Real Symmetric Positive Definite	SSYGV	DSYGV	SSYGV and DSYGV--Eigenvalues and, Optionally, the Eigenvectors of a Generalized Real Symmetric Eigensystem, Az=wBz, where A Is Real Symmetric and B Is Real Symmetric Positive Definite

Fourier Transforms, Convolutions and Correlations, and Related Computations

This signal processing area provides:

• Fourier transform subroutines (Table 13)
• Convolution and correlation subroutines (Table 14)
• Related-computation subroutines (Table 15)

Fourier Transforms

The Fourier transform subroutines perform mixed-radix transforms in one, two, and three dimensions.

Table 13. List of Fourier Transform Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
Complex Fourier Transform	SCFT SCFTP§	DCFT	SCFT and DCFT--Complex Fourier Transform
Real-to-Complex Fourier Transform	SRCFT	DRCFT	SRCFT and DRCFT--Real-to-Complex Fourier Transform
Complex-to-Real Fourier Transform	SCRFT	DCRFT	SCRFT and DCRFT--Complex-to-Real Fourier Transform
Cosine Transform	SCOSF SCOSFT§	DCOSF	SCOSF and DCOSF--Cosine Transform
Sine Transform	SSINF	DSINF	SSINF and DSINF--Sine Transform
Complex Fourier Transform in Two Dimensions	SCFT2 SCFT2P§	DCFT2	SCFT2 and DCFT2--Complex Fourier Transform in Two Dimensions
Real-to-Complex Fourier Transform in Two Dimensions	SRCFT2	DRCFT2	SRCFT2 and DRCFT2--Real-to-Complex Fourier Transform in Two Dimensions
Complex-to-Real Fourier Transform in Two Dimensions	SCRFT2	DCRFT2	SCRFT2 and DCRFT2--Complex-to-Real Fourier Transform in Two Dimensions
Complex Fourier Transform in Three Dimensions	SCFT3 SCFT3P§	DCFT3	SCFT3 and DCFT3--Complex Fourier Transform in Three Dimensions
Real-to-Complex Fourier Transform in Three Dimensions	SRCFT3	DRCFT3	SRCFT3 and DRCFT3--Real-to-Complex Fourier Transform in Three Dimensions
Complex-to-Real Fourier Transform in Three Dimensions	SCRFT3	DCRFT3	SCRFT3 and DCRFT3--Complex-to-Real Fourier Transform in Three Dimensions
§ This subroutine is provided only for migration from earlier releases of ESSL and is not intended for use in new programs. Documentation for this subroutine is no longer provided.

Convolutions and Correlations

The convolution and correlation subroutines provide the choice of using Fourier methods or direct methods. The Fourier-method subroutines contain a high-performance mixed-radix capability. There are also several direct-method subroutines that provide decimated output.

Table 14. List of Convolution and Correlation Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
Convolution or Correlation of One Sequence with One or More Sequences	SCON§ SCOR§		SCON and SCOR--Convolution or Correlation of One Sequence with One or More Sequences
Convolution or Correlation of One Sequence with Another Sequence Using a Direct Method	SCOND SCORD		SCOND and SCORD--Convolution or Correlation of One Sequence with Another Sequence Using a Direct Method
Convolution or Correlation of One Sequence with One or More Sequences Using the Mixed-Radix Fourier Method	SCONF SCORF		SCONF and SCORF--Convolution or Correlation of One Sequence with One or More Sequences Using the Mixed-Radix Fourier Method
Convolution or Correlation with Decimated Output Using a Direct Method	SDCON SDCOR	DDCON DDCOR	SDCON, DDCON, SDCOR, and DDCOR--Convolution or Correlation with Decimated Output Using a Direct Method
Autocorrelation of One or More Sequences	SACOR§		SACOR--Autocorrelation of One or More Sequences
Autocorrelation of One or More Sequences Using the Mixed-Radix Fourier Method	SACORF		SACORF--Autocorrelation of One or More Sequences Using the Mixed-Radix Fourier Method
§ These subroutines are provided only for migration from earlier releases of ESSL and are not intended for use in new programs.

Related Computations

The related-computation subroutines consist of a group of computations that can be used in general signal processing applications. They are similar to those provided on the IBM 3838 Array Processor; however, the ESSL subroutines generally solve a wider range of problems.

Table 15. List of Related-Computation Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
Polynomial Evaluation	SPOLY	DPOLY	SPOLY and DPOLY--Polynomial Evaluation
I-th Zero Crossing	SIZC	DIZC	SIZC and DIZC--I-th Zero Crossing
Time-Varying Recursive Filter	STREC	DTREC	STREC and DTREC--Time-Varying Recursive Filter
Quadratic Interpolation	SQINT	DQINT	SQINT and DQINT--Quadratic Interpolation
Wiener-Levinson Filter Coefficients	SWLEV CWLEV	DWLEV ZWLEV	SWLEV, DWLEV, CWLEV, and ZWLEV--Wiener-Levinson Filter Coefficients

Sorting and Searching

The sorting and searching subroutines operate on three types of data: integer, short-precision real, and long-precision real (Table 16). The sorting subroutines perform sorts with or without index designations. The searching subroutines perform either a binary or sequential search.

Table 16. List of Sorting and Searching Subroutines

Descriptive Name	Integer Subroutine	Short- Precision Subroutine	Long- Precision Subroutine	Page
Sort the Elements of a Sequence	ISORT	SSORT	DSORT	ISORT, SSORT, and DSORT--Sort the Elements of a Sequence
Sort the Elements of a Sequence and Note the Original Element Positions	ISORTX	SSORTX	DSORTX	ISORTX, SSORTX, and DSORTX--Sort the Elements of a Sequence and Note the Original Element Positions
Sort the Elements of a Sequence Using a Stable Sort and Note the Original Element Positions	ISORTS	SSORTS	DSORTS	ISORTS, SSORTS, and DSORTS--Sort the Elements of a Sequence Using a Stable Sort and Note the Original Element Positions
Binary Search for Elements of a Sequence X in a Sorted Sequence Y	IBSRCH	SBSRCH	DBSRCH	IBSRCH, SBSRCH, and DBSRCH--Binary Search for Elements of a Sequence X in a Sorted Sequence Y
Sequential Search for Elements of a Sequence X in the Sequence Y	ISSRCH	SSSRCH	DSSRCH	ISSRCH, SSSRCH, and DSSRCH--Sequential Search for Elements of a Sequence X in the Sequence Y

Interpolation

The interpolation subroutines provide the capabilities of doing polynomial interpolation, local polynomial interpolation, and both one- and two-dimensional cubic spline interpolation (Table 17).

Table 17. List of Interpolation Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
Polynomial Interpolation	SPINT	DPINT	SPINT and DPINT--Polynomial Interpolation
Local Polynomial Interpolation	STPINT	DTPINT	STPINT and DTPINT--Local Polynomial Interpolation
Cubic Spline Interpolation	SCSINT	DCSINT	SCSINT and DCSINT--Cubic Spline Interpolation
Two-Dimensional Cubic Spline Interpolation	SCSIN2	DCSIN2	SCSIN2 and DCSIN2--Two-Dimensional Cubic Spline Interpolation

Numerical Quadrature

The numerical quadrature subroutines provide Gaussian quadrature methods for integrating a tabulated function and a user-supplied function over a finite, semi-infinite, or infinite region of integration (Table 18).

Table 18. List of Numerical Quadrature Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
Numerical Quadrature Performed on a Set of Points	SPTNQ	DPTNQ	SPTNQ and DPTNQ--Numerical Quadrature Performed on a Set of Points
Numerical Quadrature Performed on a Function Using Gauss-Legendre Quadrature	SGLNQ&	DGLNQ&	SGLNQ and DGLNQ--Numerical Quadrature Performed on a Function Using Gauss-Legendre Quadrature
Numerical Quadrature Performed on a Function Over a Rectangle Using Two-Dimensional Gauss-Legendre Quadrature	SGLNQ2&	DGLNQ2&	SGLNQ2 and DGLNQ2--Numerical Quadrature Performed on a Function Over a Rectangle Using Two-Dimensional Gauss-Legendre Quadrature
Numerical Quadrature Performed on a Function Using Gauss-Laguerre Quadrature	SGLGQ&	DGLGQ&	SGLGQ and DGLGQ--Numerical Quadrature Performed on a Function Using Gauss-Laguerre Quadrature
Numerical Quadrature Performed on a Function Using Gauss-Rational Quadrature	SGRAQ&	DGRAQ&	SGRAQ and DGRAQ--Numerical Quadrature Performed on a Function Using Gauss-Rational Quadrature
Numerical Quadrature Performed on a Function Using Gauss-Hermite Quadrature	SGHMQ&	DGHMQ&	SGHMQ and DGHMQ--Numerical Quadrature Performed on a Function Using Gauss-Hermite Quadrature
& This subprogram is invoked as a function in a Fortran program.

Random Number Generation

Random number generation subroutines generate uniformly distributed random numbers or normally distributed random numbers (Table 19).

Table 19. List of Random Number Generation Subroutines

Descriptive Name	Short- Precision Subroutine	Long- Precision Subroutine	Page
Generate a Vector of Uniformly Distributed Random Numbers	SURAND	DURAND	SURAND and DURAND--Generate a Vector of Uniformly Distributed Random Numbers
Generate a Vector of Normally Distributed Random Numbers	SNRAND	DNRAND	SNRAND and DNRAND--Generate a Vector of Normally Distributed Random Numbers
Generate a Vector of Long Period Uniformly Distributed Random Numbers	SURXOR	DURXOR	SURXOR and DURXOR--Generate a Vector of Long Period Uniformly Distributed Random Numbers

Utilities

The utility subroutines perform general service functions that support ESSL, rather than mathematical computations (Table 20).

Table 20. List of Utility Subroutines

Descriptive Name	Subroutine	Page
ESSL Error Information-Handler Subroutine	EINFO	EINFO--ESSL Error Information-Handler Subroutine
ESSL ERRSAV Subroutine for ESSL	ERRSAV	ERRSAV--ESSL ERRSAV Subroutine for ESSL
ESSL ERRSET Subroutine for ESSL	ERRSET	ERRSET--ESSL ERRSET Subroutine for ESSL
ESSL ERRSTR Subroutine for ESSL	ERRSTR	ERRSTR--ESSL ERRSTR Subroutine for ESSL
Set the Vector Section Size (VSS) for the ESSL/370 Scalar Library	IVSSET§
Set the Extended Vector Operations Indicator for the ESSL/370 Scalar Library	IEVOPS§
Determine the Level of ESSL Installed	IESSL	IESSL--Determine the Level of ESSL Installed
Determine the Stride Value for Optimal Performance in Specified Fourier Transform Subroutines	STRIDE	STRIDE--Determine the Stride Value for Optimal Performance in Specified Fourier Transform Subroutines
Convert a Sparse Matrix from Storage-by-Rows to Compressed-Matrix Storage Mode	DSRSM	DSRSM--Convert a Sparse Matrix from Storage-by-Rows to Compressed-Matrix Storage Mode
For a General Sparse Matrix, Convert Between Diagonal-Out and Profile-In Skyline Storage Mode	DGKTRN	DGKTRN--For a General Sparse Matrix, Convert Between Diagonal-Out and Profile-In Skyline Storage Mode
For a Symmetric Sparse Matrix, Convert Between Diagonal-Out and Profile-In Skyline Storage Mode	DSKTRN	DSKTRN--For a Symmetric Sparse Matrix, Convert Between Diagonal-Out and Profile-In Skyline Storage Mode
§ This subroutine is provided for migration from earlier releases of ESSL and is not intended for use in new programs. Documentation for this subroutine is no longer provided.