HP HP-UX Performance Tools User Manual

HP MLIB User’s Guide

Seventh Edition

HP Part No. B6061-96027 and B6061-96028

HP MLIB

December 2004

Printed in USA

Edition: Seventh

Document Numbers: B6061-96027 and B6061-96028 Remarks: Released Dcember 2004 with HP MLIB software version9.0.

User’s Guide split into two volumes with this release.

Edition: Sixth

Document Number: B6061-96023 Remarks: Released September 2003 with HP MLIB software version 8.50.

Edition: Fifth

Document Number: B6061-96020 Remarks: Released September 2002 with HP MLIB software version 8.30.

Edition: Fourth

Document Number: B6061-96017 Remarks: Released June 2002 with HP MLIB software version 8.20.

Edition: Third

Document Number: B6061-96015 Remarks: Released September 2001 with HP MLIB software version 8.10.

Edition: Second

Document Number: B6061-96012 Remarks: Released June 2001 with HP MLIB software version 8.00.

Edition: First

Document Number: B6061-96010 Remarks: Released December 1999 with HP MLIB software version

B.07.00. This HP MLIB User’s Guide contains both HP VECLIB and HP LAPACK information. This document supercedes both the HP MLIB VECLIB User’s Guide, ﬁfth edition (B6061-96006) and the HP MLIB LAPACK User’s Guide, sixth edition (B6061-96005).

Notice

 Copyright 1979-2004 Hewlett-Packard Development Company. All Rights Reserved. Reproduction, adaptation, or translation without prior written permission is prohibited, except as allowed under the copyright laws.

The information contained in this document is subject to change without notice. Hewlett-Packard makes no warranty of any kind with regard to this material,

including, but not limited to, the implied warranties of merchantability and ﬁtness for a particular purpose. Hewlett-Packard shall not be liable for errors contained herein or for incidental or consequential damages in connection with the furnishing, performance or use of this material.

Table of Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

VECLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xv

LAPACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

ScaLAPACK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

SuperLU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

SOLVERS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii

VMATH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

Purpose and audience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

Notational conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii

Documentation resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

Part 1

1 Introduction to VECLIB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

Standardization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Accessing VECLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Compiling and linking (VECLIB) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Problem with +ppu compatibility and duplicated symbols . . . . . . . . . . . . . . . . . . . . 14

Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Parallel processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .17

Linking for parallel or non parallel processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Controlling VECLIB parallelism at runtime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Performance beneﬁts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

OpenMP-based nested parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Message passing-based nested parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Default CPS library stack is too small for MLIB . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

Default Pthread library stack is too small for MLIB . . . . . . . . . . . . . . . . . . . . . . . . . 22

Roundoff effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Data types and precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

VECLIB naming convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Data type and byte length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Operator arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Error handling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Low-level subprograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

High-level subprograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Troubleshooting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .29

HP MLIB man pages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2 Basic Vector Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32

Associated documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

What you need to know to use vector subprograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

BLAS storage conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

BLAS indexing conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Operator arguments in the BLAS Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

Representation of a permutation matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Representation of a Householder matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

Subprograms for basic vector operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

ISAMAX/IDAMAX/IIAMAX/ICAMAX/IZAMAX Index of maximum of magnitudes 40

ISAMIN/IDAMIN/IIAMIN/ICAMIN/IZAMIN Index of minimum of magnitudes . . 43

ISCTxx/IDCTxx/IICTxx/ICCTxx/IZCTxx Count selected vector elements. . . . . . . . 46

ISMAX/IDMAX/IIMAX Index of maximum element of vector . . . . . . . . . . . . . . . . . 49

ISMIN/IDMIN/IIMIN Index of minimum element of vector. . . . . . . . . . . . . . . . . . . 51

ISSVxx/IDSVxx/IISVxx/ICSVxx/IZSVxx Search vector for element . . . . . . . . . . . . 53

SAMAX/DAMAX/IAMAX/SCAMAX/DZAMAX Maximum of magnitudes . . . . . . . . 56

SAMIN/DAMIN/IAMIN/SCAMIN/DZAMIN Minimum of magnitudes . . . . . . . . . . 59

vi Table of Contents

SASUM/DASUM/IASUM/SCASUM/DZASUM Sum of magnitudes . . . . . . . . . . . . 62

SAXPY/DAXPY/CAXPY/CAXPYC/ZAXPY/ZAXPYC Elementary vector operation 65

SAXPYI/DAXPYI/CAXPYI/ZAXPYI Sparse elementary vector operation . . . . . . . 68

SCLIP/DCLIP/ICLIP Two sided vector clip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

SCLIPL/DCLIPL/ICLIPL Left sided vector clip . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

SCLIPR/DCLIPR/ICLIPR Right sided vector clip . . . . . . . . . . . . . . . . . . . . . . . . . . 77

SCOPY/DCOPY/ICOPY/CCOPY/CCOPYC/ZCOPY/ZCOPYC Copy vector . . . . . . . 80

SDOT/DDOT/CDOTC/CDOTU/ZDOTC/ZDOTU Dot product . . . . . . . . . . . . . . . . . 84

SDOTI/DDOTI/CDOTCI/CDOTUI/ZDOTCI/ZDOTUI Sparse dot product . . . . . . . 88

SFRAC/DFRAC Extract fractional parts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

SGTHR/DGTHR/IGTHR/CGTHR/ZGTHR Gather sparse vector . . . . . . . . . . . . . . 94

SGTHRZ/DGTHRZ/IGTHRZ/CGTHRZ/ZGTHRZ Gather and zero sparse vector . 96

SLSTxx/DLSTxx/ILSTxx/CLSTxx/ZLSTxx List selected vector elements . . . . . . . 99

SMAX/DMAX/IMAX Maximum of vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

SMIN/DMIN/IMIN Minimum of vector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

SNRM2/DNRM2/SCNRM2/DZNRM2 Euclidean norm . . . . . . . . . . . . . . . . . . . . . 107

SNRSQ/DNRSQ/SCNRSQ/DZNRSQ Euclidean norm squared . . . . . . . . . . . . . . . 110

SRAMP/DRAMP/IRAMP Generate linear ramp. . . . . . . . . . . . . . . . . . . . . . . . . . . 112

SROT/DROT/CROT/CSROT/ZROT/ZDROT Apply Givens rotation . . . . . . . . . . . 114

SROTG/DROTG/CROTG/ZROTG Construct Givens rotation . . . . . . . . . . . . . . . . 118

SROTI/DROTI Apply sparse Givens rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

SROTM/DROTM Apply modified Givens rotation . . . . . . . . . . . . . . . . . . . . . . . . . 123

SROTMG/DROTMG Construct modified Givens rotation . . . . . . . . . . . . . . . . . . . 127

SRSCL/DRSCL/CRSCL/CSRSCL/ZRSCL/ZDRSCL Scale vector. . . . . . . . . . . . . . 130

SSCAL/DSCAL/CSCAL/CSSCAL/CSCALC/ZSCAL/ZDSCAL/ZSCALC Scale vector . .

133

SSCTR/DSCTR/ISCTR/CSCTR/ZSCTR Scatter sparse vector. . . . . . . . . . . . . . . . 136

SSUM/DSUM/ISUM/CSUM/ZSUM Vector sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

SSWAP/DSWAP/ISWAP/CSWAP/ZSWAP Swap two vectors . . . . . . . . . . . . . . . . . 141

SWDOT/DWDOT/CWDOTC/CWDOTU/ZWDOTC/ZWDOTU Weighted dot product .

145

SZERO/DZERO/IZERO/CZERO/ZZERO Clear vector . . . . . . . . . . . . . . . . . . . . . . 150

F_SAMAX_VAL/F_DAMAX_VAL/F_CAMAX_VAL/F_ZAMAX_VAL Maximum

absolute value and location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

F_SAMIN_VAL/F_DAMIN_VAL/F_CAMIN_VAL/F_ZAMIN_VAL Minimum absolute

value and location . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

F_SAPPLY_GROT/F_DAPPLY_GROT/F_CAPPLY_GROT/F_ZAPPLY_GROT Apply

Table of Contents v i i

plane rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158

F_SAXPBY/F_DAXPBY/F_CAXPBY/F_ZAXPBY Scaled vector accumulation . . . 161 F_SAXPY_DOT/F_DAXPY_DOT/F_CAXPY_DOT/F_ZAXPY_DOT Combine AXPY

and DOT routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

F_SCOPY/F_DCOPY/F_CCOPY/F_ZCOPY Copy vector. . . . . . . . . . . . . . . . . . . . . 167

F_SDOT/F_DDOT/F_CDOT/F_ZDOT Add scaled dot product . . . . . . . . . . . . . . . . 169

F_SFPINFO/F_DFPINFO Environmental inquiry . . . . . . . . . . . . . . . . . . . . . . . . . 172

F_SGEN_GROT/F_DGEN_GROT/F_CGEN_GROT/F_ZGEN_GROT Generate

Givens rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

F_SGEN_HOUSE/F_DGEN_HOUSE/F_CGEN_HOUSE/F_ZGEN_HOUSE Generate

Householder transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

F_SGEN_JROT/F_DGEN_JROT/F_CGEN_JROT/F_ZGEN_JROT Generate Jacobi

rotation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

F_SMAX_VAL/F_DMAX_VAL Maximum value and location. . . . . . . . . . . . . . . . . 181

F_SMIN_VAL/F_DMIN_VAL Minimum value and location. . . . . . . . . . . . . . . . . . 183

F_SNORM/F_DNORM Norm of a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

F_SPERMUTE/F_DPERMUTE/F_CPERMUTE/F_ZPERMUTE Permute vector 187

F_SRSCALE/F_DRSCALE/F_CRSCALE/F_ZRSCALE Reciprocal Scale . . . . . . . 190

F_SSORT/F_DSORT Sort vector entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

F_SSORTV/F_DSORTV Sort vector and return index vector. . . . . . . . . . . . . . . . . 193

F_SSUM/F_DSUM/F_CSUM/F_ZSUM Sum of entries of a vector. . . . . . . . . . . . . 195

F_SSUMSQ/F_DSUMSQ/F_CSUMSQ/F_ZSUMSQ Sum of squares . . . . . . . . . . . 197

F_SSWAP/F_DSWAP/F_CSWAP/F_ZSWAP Interchange vectors . . . . . . . . . . . . . 200

F_SWAXPBY/F_DWAXPBY/F_CWAXPBY/F_ZWAXPBY Scaled vector addition. 202

3 Basic Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Associated documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206

What you need to know to use these subprograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Subroutine naming convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Operator arguments in the BLAS Standard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

Subprograms for basic matrix operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

SGBMV/DGBMV/CGBMV/ZGBMV Matrix-vector multiply. . . . . . . . . . . . . . . . . . 212

SGECPY/DGECPY/CGECPY/ZGECPY Copy general matrix . . . . . . . . . . . . . . . . 219

SGEMM/DGEMM/CGEMM/ZGEMM Matrix-matrix multiply . . . . . . . . . . . . . . . 222

viii Table of Contents

DGEMMS/ZGEMMS Strassen matrix-matrix multiply. . . . . . . . . . . . . . . . . . . . . 227

SGEMV/DGEMV/CGEMV/ZGEMV Matrix-vector multiply . . . . . . . . . . . . . . . . . 232

SGER/DGER/CGERC/CGERU/ZGERC/ZGERU Rank-1 update . . . . . . . . . . . . . . 237

SGETRA/DGETRA/CGETRA/ZGETRA In-place transpose of a general square matrix

241

SSBMV/DSBMV/CHBMV/ZHBMV Matrix-vector multiply. . . . . . . . . . . . . . . . . . 244

SSPMV/DSPMV/CHPMV/ZHPMV Matrix-vector multiply . . . . . . . . . . . . . . . . . . 249

SSPR/DSPR/CHPR/ZHPR Rank-1 update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

SSPR2/DSPR2/CHPR2/ZHPR2 Rank-2 update . . . . . . . . . . . . . . . . . . . . . . . . . . . 259

SSYMM/DSYMM/CHEMM/CSYMM/ZHEMM/ZSYMM Matrix-matrix multiply 265

SSYMV/DSYMV/CHEMV/ZHEMV Matrix-vector multiply. . . . . . . . . . . . . . . . . . 270

SSYR/DSYR/CHER/ZHER Rank-1 update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275

SSYR2/DSYR2/CHER2/ZHER2 Rank-2 update . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

SSYR2K/DSYR2K/CHER2K/CSYR2K/ZHER2K/ZSYR2K Rank-2k update . . . . . 284

SSYRK/DSYRK/CHERK/CSYRK/ZHERK/ZSYRK Rank-k update . . . . . . . . . . . . 289

STBMV/DTBMV/CTBMV/ZTBMV Matrix-vector multiply . . . . . . . . . . . . . . . . . . 294

STBSV/DTBSV/CTBSV/ZTBSV Solve triangular band system . . . . . . . . . . . . . . . 301

STPMV/DTPMV/CTPMV/ZTPMV Matrix-vector multiply . . . . . . . . . . . . . . . . . . 308

STPSV/DTPSV/CTPSV/ZTPSV Solve triangular system . . . . . . . . . . . . . . . . . . . . 313

STRMM/DTRMM/CTRMM/ZTRMM Triangular matrix-matrix multiply . . . . . . 318

STRMV/DTRMV/CTRMV/ZTRMV Matrix-vector multiply . . . . . . . . . . . . . . . . . . 323

STRSM/DTRSM/CTRSM/ZTRSM Solve triangular systems . . . . . . . . . . . . . . . . . 327

STRSV/DTRSV/CTRSV/ZTRSV Solve triangular system . . . . . . . . . . . . . . . . . . . 332

XERBLA Error handler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

F_CHBMV/F_ZHBMV Hermitian banded matrix-vector multiply . . . . . . . . . . . . 340

F_CHEMV/F_ZHEMV Hermitian matrix-vector multiply. . . . . . . . . . . . . . . . . . . 342

F_CHER/F_ZHER Hermitian rank-1 update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

F_CHER2/F_ZHER2 Hermitian rank-2 update . . . . . . . . . . . . . . . . . . . . . . . . . . . 346

F_CHPMV/F_ZHPMV Hermitian packed matrix-vector multiply. . . . . . . . . . . . . 348

F_CHPR/F_ZHPR Hermitian rank-1 update . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350

F_CHPR2/F_ZHPR2 Hermitian rank-2 update . . . . . . . . . . . . . . . . . . . . . . . . . . . 352

F_SFPINFO/F_DFPINFO Environmental inquiry. . . . . . . . . . . . . . . . . . . . . . . . . 354

F_SGBMV/F_DGBMV/F_CGBMV/F_ZGBMV General band matrix-vector multiply .

355

F_SGE_COPY/F_DGE_COPY/F_CGE_COPY/F_ZGE_COPY Matrix copy. . . . . . 358

F_SGE_TRANS/F_DGE_TRANS/F_CGE_TRANS/F_ZGE_TRANS Matrix

transposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

Table of Contents i x

F_SGEMM/F_DGEMM/F_CGEMM/F_ZGEMM General matrix-matrix multiply 362 F_SGEMV/F_DGEMV/F_CGEMV/F_ZGEMV General matrix-vector multiply . . 365 F_SGEMVER/F_DGEMVER/F_CGEMVER/F_ZGEMVER Multiple matrix-vector

multiply, rank 2 update. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368

F_SGEMVT/F_DGEMVT/F_CGEMVT/F_ZGEMVT Multiple matrix-vector multiply

372

F_SGER/F_DGER/F_CGER/F_ZGER General rank-1 update . . . . . . . . . . . . . . . . 375

F_SSBMV/F_DSBMV/F_CSBMV/F_ZSBMV Symmetric band matrix-vector multiply

378 F_SSPMV/F_DSPMV/F_CSPMV/F_ZSPMV Symmetric packed matrix-vector

multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

F_SSPR/F_DSPR/F_CSPR/F_ZSPR Symmetric packed rank-1 update. . . . . . . . . 384

F_SSPR2/F_DSPR2/F_CSPR2/F_ZSPR2 Symmetric rank-2 update . . . . . . . . . . . 386

F_SSYMV/F_DSYMV/F_CSYMV/F_ZSYMV Symmetric matrix-vector multiply . 389

F_SSYR/F_DSYR/F_CSYR/F_ZSYR Symmetric rank-1 update . . . . . . . . . . . . . . . 392

F_SSYR2/F_DSYR2/F_CSYR2/F_ZSYR2 Symmetric rank-2 update . . . . . . . . . . . 394

F_STBMV/F_DTBMV/F_CTBMV/F_ZTBMV Triangular banded matrix-vector

multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397

F_STBSV/F_DTBSV/F_CTBSV/F_ZTBSV Triangular banded solve . . . . . . . . . . . 400

F_STPMV/F_DTPMV/F_CTPMV/F_ZTPMV Triangular packed matrix-vector

multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

F_STPSV/F_DTPSV/F_CTPSV/F_ZTPSV Triangular packed solve. . . . . . . . . . . . 406

F_STRMV/F_DTRMV/F_CTRMV/F_ZTRMV Triangular matrix-vector multiply 408 F_STRMVT/F_DTRMVT/F_CTRMVT/F_ZTRMVT Multiple triangular matrix-vector

multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

F_STRSM/F_DTRSM/F_CTRSM/F_ZTRSM Triangular solve . . . . . . . . . . . . . . . . 414

F_STRSV/F_DTRSV/F_CTRSV/F_ZTRSV Triangular solve . . . . . . . . . . . . . . . . . 417

4 Sparse BLAS Operations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Chapter objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 421

Associated documentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422

What you need to know to use these subprograms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

Subroutine naming convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423

Sparse matrix storage formats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

Operator arguments in the Sparse BLAS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

x Table of Contents

Common arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

SM arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

Order of arguments for args(A) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 440

SBCOMM/DBCOMM/CBCOMM/ZBCOMM Block coordinate matrix-matrix multiply

441 SBDIMM/DBDIMM/CBDIMM/ZBDIMM Block diagonal matrix-matrix multiply 445 SBDISM/DBDISM/CBDISM/ZBDISM Block diagonal format triangular solve . . 449 SBELMM/DBELMM/CBELMM/ZBELMM Block Ellpack matrix-matrix multiply . . .

453 SBELSM/DBELSM/CBELSM/ZBELSM Block Ellpack format triangular solve . 457 SBSCMM/DBSCMM/CBSCMM/ZBSCMM Block sparse column matrix-matrix

multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461

SBSCSM/DBSCSM/CBSCSM/ZBSCSM Block sparse column format triangular solve

465 SBSRMM/DBSRMM/CBSRMM/ZBSRMM Block sparse row matrix-matrix multiply

469 SBSRSM/DBSRSM/CBSRSM/ZBSRSM Block sparse row format triangular solve . . .

473 SCOOMM/DCOOMM/CCOOMM/ZCOOMM Coordinate matrix-matrix multiply 477 SCSCMM/DCSCMM/CCSCMM/ZCSCMM Compressed sparse column matrix-matrix

multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481

SCSCSM/DCSCSM/CCSCSM/ZCSCSM Compressed sparse column format triangular

solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

SCSRMM/DCSRMM/CCSRMM/ZCSRMM Compressed sparse row matrix-matrix

multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 489

SCSRSM/DCSRSM/CCSRSM/ZCSRSM Compressed sparse row format triangular

solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

SDIAMM/DDIAMM/CDIAMM/ZDIAMM Diagonal matrix-matrix multiply . . . . 497

SDIASM/DDIASM/CDIASM/ZDIASM Diagonal format triangular solve. . . . . . . 501

SELLMM/DELLMM/CELLMM/ZELLMM Ellpack matrix-matrix multiply . . . . 505

SELLSM/DELLSM/CELLSM/ZELLSM Ellpack format triangular solve. . . . . . . 509

SJADMM/DJADMM/CJADMM/ZJADMM Jagged diagonal matrix-matrix multiply .

513 SJADSM/DJADSM/CJADSM/ZJADSM Jagged diagonal format triangular solve 517

SSKYMM/DSKYMM/CSKYMM/ZSKYMM Skyline matrix-matrix multiply . . . . 521

SSKYSM/DSKYSM/CSKYSM/ZSKYSM Skyline format triangular solve . . . . . . 525

SVBRMM/DVBRMM/CVBRMM/ZVBRMM Variable block row matrix-matrix

multiply . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 529

Table of Contents x i

SVBRSM/DVBRSM/CVBRSM/ZVBRSM Variable block row format triangular solve

533

xii Table of Contents

Figures

List of Figures xiii

xiv List of Figures

Tables

Table 1-1 VECLIB and VECLIB8 Libraries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

Table 1-2 Compiler Defaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Table 1-3 VECLIB Naming Convention—Data Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Table 1-4 BLAS Standard Operator Arguments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Table 2-1 FPINFO return values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

Table 3-1 Extended BLAS Naming Convention—Data Type . . . . . . . . . . . . . . . . . . . . . 207

Table 3-2 Extended BLAS Naming Convention—Matrix Form . . . . . . . . . . . . . . . . . . . 208

Table 3-3 Extended BLAS Naming Convention—Computation . . . . . . . . . . . . . . . . . . . 208

Table 3-4 Extended BLAS Naming Convention—Subprogram Names . . . . . . . . . . . . . 209

Table 4-1 Sparse BLAS Naming Convention—Data Type . . . . . . . . . . . . . . . . . . . . . . . . 423

Table 4-2 Sparse BLAS Naming Convention—Matrix Form . . . . . . . . . . . . . . . . . . . . . . 424

Table 4-3 4 x 5 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Table 4-4 COO Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425

Table 4-5 CSC Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Table 4-6 MSC Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426

Table 4-7 CSR Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

Table 4-8 MSR Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 428

Table 4-9 5 x 4 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Table 4-10 DIA Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429

Table 4-11 ELL Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

Table 4-12 JAD Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Table 4-13 JAD Row-Permuted Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Table 4-14 5 x 5 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 431

Table 4-15 SKY Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Table 4-16 4 x 6 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432

Table 4-17 BCO Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Table 4-18 BSC Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

List of Tables xiii

Table 4-19 BSR Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435

Table 4-20 6 x 6 Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

Table 4-21 VBR Format Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

xiv List of Tables

VECLIB

Preface

Hewlett-Packard’s high-performance math libraries (HP MLIB) help you speed development of applications and shorten execution time of long-running technical applications.

HP MLIB is a collection of subprograms optimized for use on HP servers and workstations, providing mathematical software and computational kernels for engineering and scientiﬁc applications. HP MLIB can be used on systems ranging from single-processor workstations to multiprocessor high-end servers. HP MLIB is optimized for HP PA-RISC 2.0, Itanium 2, and Opteron processors. HP MLIB has six components; VECLIB, LAPACK, ScaLAPACK, SuperLU, SOLVERS, and VMATH.

HP VECLIB contains robust callable subprograms. Together with a subset of the BLAS Standard subroutines, HP MLIB supports the legacy BLAS, a collection of routines for the solution of sparse symmetric systems of equations, a collection of commonly used Fast Fourier Transforms (FFTs), and convolutions. Although VECLIB was designed for use with Fortran programs, C programs can call VECLIB subprograms, as described in Appendix A. Refer to Part 1 of this manual for HP VECLIB information.

Throughout this document there are references to legacy BLAS and BLAS Standard routines. Legacy BLAS routines include Basic Linear Algebra Subprograms (BLAS), that is the level 1, 2, and 3 BLAS, as well as the Sparse BLAS.

A BLAS standardization effort began with a BLAS Technical (BLAST) Forum meeting in November 1995 at the University of Tennessee. The efforts of the BLAST Forum resulted in a BLAS Standard speciﬁcation in 1999. BLAS Standard routines refer to routines as deﬁned by this BLAS Standard speciﬁcation. HP MLIB supports a subset of BLAS Standard routines. Refer to Chapter 2, “Basic Vector Operations,” and Chapter 3, “Basic Matrix Operations,” for details about supported subprograms.

Preface xv

LAPACK

HP Linear Algebra Package (LAPACK) is a collection of subprograms that provide mathematical software for applications involving linear equations, least squares, eigenvalue problems, and the singular value decomposition. LAPACK is designed to supersede the linear equation and eigenvalue packages, LINPACK and EISPACK. The National Science Foundation, the Defense Advanced Research Projects Agency, and the Department of Energy supported the development of the public-domain version of LAPACK, from which the HP version was derived.

HP LAPACK fully conforms with public domain version 3.0 of LAPACK in all user-visible usage conventions. Refer to Part 2 of this manual for information speciﬁc to HP LAPACK. To supplement the HP speciﬁc information provided in Part 2 of this document, refer to the standard LAPACK Users’ Guide. You can access the latest edition of the LAPACK Users’ Guide at the Netlib repository at the following URL:

http://www.netlib.org/lapack/lug/index..html

ScaLAPACK

ScaLAPACK is a library of high-performance linear algebra routines capable of solving systems of linear equations, linear least squares problems, eigenvalue problems, and singular value problems. ScaLAPACK can also handle many associated computations such as matrix factorizations or estimating condition numbers.

ScaLAPACK is a public domain software that was developed by Oak Ridge National Laboratory. It is designed for distributed computing and uses the Message Passing Interface (MPI) for parallelism. This implementation provides a version of ScaLAPACK tuned on HP servers and built with HP’s MPI. The ScaLAPACK library routines are written in Fortran 77 and are callable from Fortran 90 and C routines. Unlike other MLIB libraries, there is not a version of ScaLAPACK that assumes all integers are 8 bytes in length.

xvi HP MLIB VECLIB User’s Guide

Refer to Part 3 of this manual for information speciﬁc to HP ScaLAPACK. To supplement the HP speciﬁc information provided in Part 3 of this document, refer to the standard ScaLAPACK Users’ Guide. You can access the latest edition of the ScaLAPACK Users’ Guide at the Netlib repository at the following URL:

http://www.netlib.org/scalapack/slug/index..html

SuperLU

This implementation provides the Distributed SuperLU library designed for distributed memory parallel computers. It was based on the public-domain SuperLU_DIST, which was developed at the Lawrence Berkeley National Lab and the University of California at Berkeley.

The library contains a set of subroutines to solve a sparse linear system. The library is implemented in ANSI C, using HP Message Passing Interface (MPI) for communication. The library includes routines to handle both real and complex matrices using double precision. The parallel routine names for the double-precision real version start with the letters “pd” (e.g. pdgstrf). The parallel routine names for the double-precision complex version start with letters “pz” (e.g. pzgstrf). Unlike other MLIB libraries, there is not a version of SuperLU that assumes all integers are 8 bytes in length.

The routines can be called directly from C applications. They may also be called from Fortran applications, however, “bridge” routines (in C) must be supplied. For details about creating bridge routines, please refer to Section 2.9.2 in the SuperLU User’s Guide available at:

http://www.nersc.gov/~xiaoye/SuperLU

SuperLU

SOLVERS

Solvers is a collection of direct sparse linear system solvers and graph partitioning routines. Symmetric systems can be solved using SMP parallelism and structurally-symmetric systems can be solved using out-of-core functionality. These routines have been optimized for use on Hewlett-Packard servers. Features are:

Preface xvii

VMATH

• Sparse symmetric and structurally-symmetric linear equation solutions.

• Sparse symmetric ordinary and generalized eigensystem solutions.

• Out-of-core symmetric and structurally-symmetric linear equation and eigensystems solutions.

• Full METIS functionality This implementation provides the METIS Version 4.0.1 library. It is based

on the public-domain METIS, which was developed at the University of Minnesota, Department of Computer Science, and the Army HPC Research Center. The library contains a set of subroutines for graph partitioning, mesh partitioning, and sparse matrix reordering, as well as auxiliary routines. HP MLIB contains the full METIS functionality as that in the public domain METIS, however, the routine names are different. HP MLIB METIS routine names have been prepended with mlib_ to avoid name conﬂict on applications and libraries that contain their own local version of METIS.

For more information about METIS, please refer to: http://www-users.cs.umn.edu/~karpis/metis/metis/index.html

VMATH

VMATH is a library of vector math routines corresponding to many of the widely used scalar math routines available with C, C++, and Fortran90.

VMATH is intended for computationally intensive mathematical applications amenable to a vector programming style.

VMATH provides two libraries: VMATH, whose interface uses 4-byte integers; and VMATH8, whose interface uses 8-byte integers and is otherwise equivalent to VMATH. VMATH routines come with both Fortran and C interfaces.

For more detailed information on VMATH as well as subprogram speciﬁcations, please refer to the VMATH chapter in Part 5 of this book. The VMATH man pages provide a man page for each subprogram.

xviii HP MLIB VECLIB User’s Guide

Purpose and audience

This guide describes the MLIB software library and shows how to use it. This library provides mathematical software and computational kernels for applications.

The HP MLIB User’s Guide addresses experienced programmers who:

• Convert, develop, or optimize programs for use on HP servers and workstations

• Optimize existing software to improve performance and increase productivity

Purpose and audience

Preface xix

Organization

The HP MLIB User’s Guide describes HP MLIB VECLIB in Part 1, HP MLIB LAPACK in Part 2, HP MLIB ScaLAPACK in Part 3, and HP MLIB Distributed SuperLU in Part 4.

To learn fundamental information necessary for using the VECLIB library, read Chapter 1 and the introductory sections of the other chapters. These sections of background information will help you efﬁciently use the library subprograms.

To learn more about the subject of any chapter, refer to the literature cited in the “Associated documentation” section of each chapter.

Part 1 of this document is organized as follows:

• Chapter 1 introduces general concepts about VECLIB

• Chapter 2 describes basic vector operations included in VECLIB

• Chapter 3 describes basic matrix operations included in VECLIB

• Chapter 4 describes sparse BLAS operations

• Chapter 5 describes the discrete Fourier transforms in VECLIB

• Chapter 6 describes subprograms to compute convolutions and correlations of data sets

• Chapter 7 describes miscellaneous subprograms to produce random numbers, sort the elements of a vector in ascending or descending order, measure time, allocate dynamic memory, and report errors

Part 2 of this document is organized as follows:

• Chapter 8 describes information speciﬁc to Hewlett-Packard’s implementation of LAPACK

• Chapter 9 describes selected LAPACK auxiliary subprograms

Part 3 of this document is organized as follows:

• Chapter 10 describes ScaLAPACK functionality

Part 4 of this document is organized as follows:

• Chapter 11 describes Distributed SuperLU functionality

Part 5 of this document is organized as follows:

• Chapter 12 describes VMATH functionality

xx HP MLIB VECLIB User’s Guide

Organization

Part 6 of this document is organized as follows:

• Chapter 13 explains sparse symmetric linear equation subprograms

• Chapter 14 describes METIS subprograms

• Chapter 15 describes sparse symmetric eigenvalue subprograms

• Chapter 16 describes BCSLIB-EXT functionality Supplemental material is provided as follows:

• Appendix A describes how to call VECLIB and LAPACK subprograms from within C programs

• Appendix B describes LINPACK subprograms available in HP MLIB

• Appendix C lists parallelized subprograms in VECLIB and LAPACK

• An index is included at the back of the manual

Supplemental information for Part 2 of this HP MLIB User’s Guide, is found in the LAPACK Users’ Guide.

The LAPACK Users’ Guide is a publication from the Society for Industrial and Applied Mathematics that provides an introduction to the design of LAPACK as well as complete speciﬁcations for all the driver and computational routines. You can access the latest edition of the LAPACK Users’ Guide at the Netlib repository at the following URL:

http://www.netlib.org/lapack/lug Supplemental information for Part 3 of this HP MLIB User’s Guide, is found in

the ScaLAPACK User’s Guide. The ScaLAPACK Users’ Guide is a publication from the Society for Industrial

and Applied Mathematics that provides an informal introduction to the design of ScaLAPACK as well as a detailed description of its contents and a reference manual. You can access the latest edition of the ScaLAPACK Users’ Guide at the Netlib repository at the following URL:

http://www.netlib.org/scalapack/slug Supplemental information for Part 4 of this HP MLIB User’s Guide, is found in

the SuperLU User’s Guide, which is only available online at: http://www.nersc.gov/~xiaoye/SuperLU

Preface xxi

Notational conventions

The following conventions are used in this manual:

Italics

Italics within text indicate mathematical entities used

or manipulated by the program: for example, solve the

n-by-n system of linear equations Ax = b. Italics within command lines indicate generic

commands, ﬁle names, or subprogram names. Substitute actual commands, ﬁle names, or subprograms for the italicized words. For example, the command line

f90 prog_name.o

instructs you to type the command f90, followed by the name of a program or subprogram object ﬁle.

UPPERCASE BOLDFACE

Fortran statements indicates Fortran keywords and subprogram names that must be typed just as they appear. For example, CALL DAXPY.

within text and in prototype

lowercase boldface

xxii HP MLIB VECLIB User’s Guide

lowercase boldface

generic variable or array names. You should substitute actual variable or array names. The italicized mathematical entities and the lowercase boldface variable and array names usually correspond. For example, A is a matrix and a is the Fortran array containing the matrix:

CALL DAXPY (n, a, x, incx, y, incy) lowercase boldface

ASCII characters that must be typed just as they appear. For example, the command line

f90 prog_name.o

instructs you to type the command f90, followed by the name of a program or subprogram object ﬁle.

within text indicates Fortran

within command lines indicates

UPPERCASE MONOSPACE

UPPERCASE MONOSPACE indicates Fortran programs.

Brackets ( [ ] )

Square brackets in command examples designate optional entries.

NOTE

A NOTE highlights important supplemental information.

Documentation resources

The HP MLIB User’s Guide, the LAPACK Users’ Guide, and the ScaLAPACK Users’ Guide are available in hardcopy and online formats.

For the HP MLIB User’s Guide, refer to: http:// www.hp.com/go/mlib

Documentation resources

For the latest edition of the LAPACK Users’ Guide, refer to: http://www.netlib.org/lapack/lug

For the latest edition of the ScaLAPACK User’s Guide, refer to: http://www.netlib.org/scalapack/slug

For the latest edition of the SuperLU User’s Guide, refer to: http://www.nersc.gov/~xiaoye/SuperLU

The following documents provide supplemental information:

• LAPACK Users’ Guide Philadelphia, PA: Society for Industrial and Applied Mathematics, 1995. This guide provides information on the subprograms provided with the LAPACK library.

• ScaLAPACK Users’ Guide Philadelphia, PA: Society for Industrial and Applied Mathematics. This guide provides information on the subprograms provided with the ScaLAPACK library.

• Parallel Programming Guide for HP-UX Systems. Describes efﬁcient shared-memory parallel programming techniques using HP compilers.

• Fortran/9000 Programmer’s Guide. Describes features and requirements in terms of the tasks a programmer might perform. These tasks include how to compile, link, run, debug, and optimize programs.

Preface xxiii

Documentation resources

• HP-UX Floating-Point Guide. Describes how ﬂoating-point arithmetic is implemented on HP 9000 systems and discusses how ﬂoating-point behavior affects the programmer.

• HP Fortran 90 Programmer’s Guide. Provides extensive usage information (including how to compile and link), suggestions and tools for migrating to HP Fortran 90, and how to call C and HP-UX routines for HP Fortran 90.

• HP Fortran 90 Programmer’s Reference. Provides complete Fortran 90 language reference information. It also covers compiler options, compiler directives, and library information.

• HP C Programming Guide. Contains detailed discussions of selected C topics.

• HP C/HP-UX Reference Manual. Presents reference information on the C programming language as implemented by HP.

• HP aC++ Online Programmer’s Guide. Presents reference and tutorial information on aC++. (This manual is accessed by specifying aCC with the +help command-line option.)

• HP MPI User’s Guide. Describes how to use HP MPI (Message Passing Interface), a library of C- and Fortran-callable routines used for message-passing programming.

• Software Optimization for High Performance Computing: Creating Faster Applications: Provides state-of-the-art solutions for every key aspect of

software performance; both code-based and algorithm-based.

• HP-UX Assembly Language Reference Manual. Describes the HP-UX assembler for the PA-RISC processor.

• HP PA-RISC 2.0 Architecture Reference. Describes the architecture of the PA-RISC 2.0 processor.

• PA-RISC Procedure Calling Conventions Reference. Describes the conventions for creating PA-RISC assembly language procedure calls.

• IA-64 and Elementary Functions. Describes the architecture of the IA-64 processor and how to implement elementary functions.

• METIS documentation is available online at http://www-users.cs.umn.edu/~karypis/metis/metis/index.html.

• OpenMP documentation is available online at http://www.openmp.org.

xxiv HP MLIB VECLIB User’s Guide

Part 1

HP VECLIB

1 Introduction to VECLIB

Overview

VECLIB, a component of HP MLIB, is a collection of subprograms optimized for use on Hewlett-Packard servers and workstations, providing mathematical software and computational kernels for engineering and scientiﬁc applications. This library contains subprograms for:

• Dense vector operations, including the Level 1 BLAS

• Sparse vector and matrix operations, including the Sparse BLAS

• BLAS 3 routine DGEMM is highly tuned

• Matrix operations, including the Level 2 and Level 3 BLAS

• CXML Blas extenstions Compaq Extended Math Library (CXML) is a collection of routines that

performs numerically intensive operations that occur frequently in engineering and scientiﬁc computing, such as linear algebra and signal processing.

HP MLIB adds support for the unique CXML extensions to the legacy BLAS and for the array math functions in CXML. However, the additional support of XCML is not exhaustive. For more information about CXML see the CXML Reference Manual part number AA-PV6VE-TE.

• Discrete Fourier transforms

• Convolution and correlation

• Miscellaneous tasks, such as sorting and generating random numbers

Overview

VECLIB provides two sets of libraries: VECLIB and VECLIB8. To determine if a subprogram is included in VECLIB8, refer to the VECLIB8 section under each subprogram speciﬁcation in the following chapters.

Although VECLIB was designed for use with Fortran programs, C programs can call VECLIB subprograms. Refer to Appendix A, “Calling MLIB routines from C,” for details. Examples are in Fortran, unless otherwise indicated.

Except for subprograms described in Appendix B, “LINPACK Subprograms,” LINPACK, EISPACK, and SKYLINE subprograms are not included in the

Chapter 1 Introduction to VECLIB 1

Chapter objectives

Hewlett-Packard scientiﬁc libraries. Refer to the Appendix, “Converting from LINPACK or EISPACK” in the LAPACK Users’ Guide, for assistance converting programs that currently call LINPACK or EISPACK routines to call LAPACK or VECLIB routines instead.

This chapter provides information necessary for efﬁcient use of VECLIB and includes discussions of:

• Standardization

• Accessing VECLIB

• Optimization

• Parallel processing

• Roundoff effects

• Data types and precision

• VECLIB naming convention

• Data type and byte length

• Operator arguments

• Error handling

• HP MLIB man pages

• Troubleshooting

Chapter objectives

After reading this chapter you will:

• Know how to access VECLIB library subprograms

• Understand how VECLIB works in a parallel computing environment

• Understand VECLIB naming conventions

• Understand roundoff effects

• Understand how VECLIB handles errors

• Know how to access the online HP MLIB man pages

• Know what to do if you experience trouble using VECLIB subprograms

2 HP MLIB User’s Guide

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