Ⅰ GENERAL INFORMATION 1
1 Essentials 3
1.1 LAPACK95 3
1.2 Problems that LAPACK95 can Solve 3
1.3 Computers for which LAPACK95 is Suitable 4
1.4 LAPACK and the BLAS 4
1.5 Availability and Installation of Software 4
1.5.1 LAPACK95 4
1.5.1.1 Incorporating Machine Dependencies 5
1.5.2 LAPACK 6
1.5.3 BLAS 7
1.5.4 Installation Debugging Hints 8
1.5.5 Mirror Repositories of netlib 8
1.5.6 Availability of Software via CD-ROM 8
1.6 Support 9
1.7 Commercial Use 9
2 Contents of LAPACK95 11
2.1 Structure of LAPACK95 11
2.1.1 Levels of Routines 11
2.1.2 Data Types and Precision 11
2.1.3 Naming Scheme 12
2.2 Driver Routines 13
2.2.1 Linear Equations 13
2.2.2 Linear Least Squares(LLS)Problems 13
2.2.3 Generalized Linear Least Squares(LSE and GLM)Problems 15
2.2.4 Standard Eigenvalue and Singular Value Problems 16
2.2.4.1 Symmetric Eigenproblems(SEP) 16
2.2.4.2 Nonsymmetric Eigenproblems(NEP) 17
2.2.4.3 Singular Value Decomposition(SVD) 18
2.2.5 Generalized Eigenvalue and Singular Value Problems 18
2.2.5.1 Generalized Symmetric Definite Eigenproblems(GSEP) 18
2.2.5.2 Generalized Nonsymmetric Eigenproblems(GNEP) 20
2.2.5.3 Generalized Singular Value Decomposition(GSVD) 21
3 Documentation Design and Program Examples 25
3.1 Design of the LAPACK95 Driver Interface 25
3.2 Design and Documentation of Driver Argument Lists 26
3.2.1 Structure of the Documentation 26
3.2.2 Order of Arguments 27
3.2.3 Argument Descriptions 27
3.2.4 Optional Arguments 28
3.2.5 Array Arguments 28
3.3 Error Handling 28
3.4 Matrix Storage Schemes 30
3.5 Design of Interfaces for Computational Routines 30
3.6 How to call an LAPACK95 Routine 31
3.7 Code for One Version of LA_SYEV 33
3.8 LAPACK and LAPACK95 Interface Module Blocks 35
3.8.1 F77_LAPACK Generic Interface Blocks 35
3.8.1.1 LA_SYEV/LA_HEEV 35
3.8.1.2 LA_GESV Multiple RHS Case 37
3.8.1.3 LA_GESV Single RHS Case 37
3.8.2 F95_LAPACK Generic Interface Blocks 38
3.8.2.1 LA_SYEV/LA_HEEV 38
3.8.2.2 LA_GESV 39
3.8.3 LA_LAMCH Interfaces 40
4 Performance and Troubleshooting 41
4.1 Performance of LAPACK95 41
4.1.1 Performance Issues 41
4.1.2 Performance Tables 41
4.2 Accuracy and Stability 47
4.3 Errors and Poor Performance 47
Ⅱ DRIVER ROUTINES 49
5 Driver Routines for Linear Systems 51
5.1 General Linear Systems 51
5.1.1 LA_GESV 51
5.1.2 LA_GESVX 54
5.1.3 LA_GBSV 57
5.1.4 LA_GBSVX 61
5.1.5 LA_GTSV 65
5.1.6 LA_GTSVX 67
5.2 Symmetric/Hermitian Positive Definite Linear Systems 70
5.2.1 LA_POSV 70
5.2.2 LA_POSVX 73
5.2.3 LA_PPSV 77
5.2.4 LA_PPSVX 79
5.2.5 LA_PBSV 82
5.2.6 LA_PBSVX 85
5.2.7 LA_PTSV 89
5.2.8 LA_PTSVX 91
5.3 Symmetric Indefinite Linear Systems 93
5.3.1 LA_SYSV/LA_HESV 93
5.3.2 LA_SYSVX/LA_HESVX 98
5.3.3 LA_SPSV/LA_HPSV 101
5.3.4 LA_SPSVX/LA_HPSVX 104
6 Driver Routines for Least Squares Problems 107
6.1 Linear Least Squares Problems 107
6.1.1 LA_GELS 107
6.1.2 LA_GELSY 110
6.1.3 LA_GELSS/LA_GELSD 112
6.2 Generalized Linear Least Squares Problems 114
6.2.1 LA_GGLSE 114
6.2.2 LA_GGGLM 116
7 Driver Routines for Standard Eigenvalue Problems 119
7.1 Standard Symmetric Eigenvalue Problems 119
7.1.1 LA_SYEV/LA_HEEV/LA_SYEVD/LA_HEEVD 119
7.1.2 LA_SYEVX/LA_HEEVX 122
7.1.3 LA_SYEVR/LA_HEEVR 124
7.1.4 LA_SPEV/LA_HPEV/LA_SPEVD/LA_HPEVD 126
7.1.5 LA_SPEVX/LA_HPEVX 130
7.1.6 LA_SBEV/LA_HBEV/LA_SBEVD/LA_HBEVD 132
7.1.7 LA_SBEVX/LA_HBEVX 135
7.1.8 LA_STEV/LA_STEVD 138
7.1.9 LA_STEVX 140
7.1.10 LA_STEVR 142
7.2 Standard Nonsymmetric Eigenvalue Problems 145
7.2.1 LA_GEES 145
7.2.2 LA_GEESX 149
7.2.3 LA_GEEV 152
7.2.4 LA_GEEVX 156
8 Driver Routines for Generalized Eigenvalue Problems 159
8.1 Generalized Symmetric Eigenvalue Problems 159
8.1.1 LA_SYGV/LA_SYGVD/LA_HEGV/LA_HEGVD 159
8.1.2 LA_SYGVX/LA_HEGVX 163
8.1.3 LA_SPGV/LA_SPGVD/LA_HPGV/LA_HPGVD 166
8.1.4 LA_SPGVX/LA_HPGVX 171
8.1.5 LA_SBGV/LA_SBGVD/LA_HBGV/LA_HBGVD 174
8.1.6 LA_SBGVX/LA_HBGVX 178
8.2 Generalized Nonsymmetric Eigenvalue Problems 181
8.2.1 LA_GGES 181
8.2.2 LA_GGESX 187
8.2.3 LA_GGEV 190
8.2.4 LA_GGEVX 195
9 Driver Routines for Singular Value Problems 201
9.1 Standard Singular Value Problems 201
9.1.1 LA_GESVD/LA_GESDD 201
9.2 Generalized Singular Value Problems 204
9.2.1 LA_GGSVD 204
Ⅲ COMPUTATIONAL ROUTINES 211
10 Computational Routines 213
10.1 Computational Routines for Linear Equations 213
10.1.1 General Linear Systems 213
10.1.2 Symmetric/Hermitian Positive Definite Linear Systems 216
10.1.3 Symmetric Indefinite Linear Systems 221
10.1.4 Triangular Linear Systems 223
10.2 Computational Routines for Orthogonal Factorizations 226
10.3 Computational Routines for the Symmetric Eigenproblem 229
10.4 Computational Routines for the Nonsymmetric eigenproblem 231
10.5 Computational Routines for the Singular Value Decomposition 234
10.6 Computational Routines for the Generalized Symmetric Definite Eigenproblem 236
10.7 Computational Routines for the Generalized Nonsymmetric Eigenproblem 237
10.8 Computational Routines for the Generalized Singular Value Decomposition 239
Bibliography 239
Index by Keyword 245
Index by Routine Name 256
List of Tables 6
1.1 Machine constants returned by LA_LAMCH 6
2.1 Matrix types in the LAPACK naming scheme 12
2.2 Driver routines for linear equations 14
2.3 Driver routines for linear least squares problems 15
2.4 Driver routines for generalized linear least squares problems 16
2.5 Driver routines for standard eigenvalue and singular value problems 19
2.6 Driver routines for generalized eigenvalue and singular value problems 23
4.1 Computer used for running the performance timing 42
4.2 Floating point coefficient of operation counts for LAPACK drivers for n×n matrices(see also Table 3.13 of[1]).The number of operations is α×n3 43
4.3 Performance of LA_GESV in megaflops;n=100 and 1000 43
4.4 Performance of LA_GEEV in megaflops(eigenvalues only);n=100 and 1000 44
4.5 Performance of LA_GEEV in megaflops(eigenvalues and right eigenvectors);n=100 and 1000 44
4.6 Performance of LA_GESVD in megaflops(singular values and left and right singular vectors);n=100 and 1000 45
4.7 Performance of LA_GESDD in megaflops(singular values only);n=100 and 1000 45
4.8 Performance of LA_GESDD in megaflops(singular values and left and right singular vectors);n=100 and 1000 46
List of Figures 32
3.1 Example program calling an LAPACK95 driver routine 32
3.2 Example program calling an LAPACK95 computational routine 33