1SGSVJ1(1)LAPACK routine (version 3.2) SGSVJ1(1)
2
6 SGSVJ1 - is called from SGESVJ as a pre-processor and that is its main
7 purpose
8
10 SUBROUTINE SGSVJ1( JOBV, M, N, N1, A, LDA, D, SVA, MV, V, LDV,
11 + EPS, SFMIN, TOL, NSWEEP, WORK, LWORK, INFO )
13 IMPLICIT NONE
15 REAL EPS, SFMIN, TOL
17 INTEGER INFO, LDA, LDV, LWORK, M, MV, N, N1, NSWEEP
19 CHARACTER*1 JOBV
21 REAL A( LDA, * ), D( N ), SVA( N ), V( LDV, * ),
23 + WORK( LWORK )
25
27 SGSVJ1 is called from SGESVJ as a pre-processor and that is its main
28 purpose. It applies Jacobi rotations in the same way as SGESVJ does,
29 but it targets only particular pivots and it does not check convergence
30 (stopping criterion). Few tunning parameters (marked by [TP]) are
31 available for the implementer.
32 Further Details
33 SGSVJ1 applies few sweeps of Jacobi rotations in the column space of
34 the input M-by-N matrix A. The pivot pairs are taken from the (1,2)
35 off-diagonal block in the corresponding N-by-N Gram matrix A^T * A. The
36 block-entries (tiles) of the (1,2) off-diagonal block are marked by the
37 [x]'s in the following scheme:
38 | * * * [x] [x] [x]|
39 | * * * [x] [x] [x]| Row-cycling in the nblr-by-nblc [x]
40 blocks.
41 | * * * [x] [x] [x]| Row-cyclic pivoting inside each [x]
42 block.
43 |[x] [x] [x] * * * |
44 |[x] [x] [x] * * * |
45 |[x] [x] [x] * * * |
46 In terms of the columns of A, the first N1 columns are rotated
47 'against' the remaining N-N1 columns, trying to increase the angle
48 between the corresponding subspaces. The off-diagonal block is N1-by(N-
49 N1) and it is tiled using quadratic tiles of side KBL. Here, KBL is a
50 tunning parmeter. The number of sweeps is given in NSWEEP and the
51 orthogonality threshold is given in TOL.
52 Contributors
53 ~~~~~~~~~~~~
54 Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
55
57 JOBV (input) CHARACTER*1
58 Specifies whether the output from this procedure is used to
59 compute the matrix V:
60 = 'V': the product of the Jacobi rotations is accumulated by
61 postmulyiplying the N-by-N array V. (See the description of
62 V.) = 'A': the product of the Jacobi rotations is accumulated
63 by postmulyiplying the MV-by-N array V. (See the descriptions
64 of MV and V.) = 'N': the Jacobi rotations are not accumulated.
65 M (input) INTEGER
67 The number of rows of the input matrix A. M >= 0.
68 N (input) INTEGER
70 The number of columns of the input matrix A. M >= N >= 0.
71 N1 (input) INTEGER
73 N1 specifies the 2 x 2 block partition, the first N1 columns
74 are rotated 'against' the remaining N-N1 columns of A.
75 A (input/output) REAL array, dimension (LDA,N)
77 On entry, M-by-N matrix A, such that A*diag(D) represents the
78 input matrix. On exit, A_onexit * D_onexit represents the
79 input matrix A*diag(D) post-multiplied by a sequence of Jacobi
80 rotations, where the rotation threshold and the total number of
81 sweeps are given in TOL and NSWEEP, respectively. (See the
82 descriptions of N1, D, TOL and NSWEEP.)
83 LDA (input) INTEGER
85 The leading dimension of the array A. LDA >= max(1,M).
86 D (input/workspace/output) REAL array, dimension (N)
88 The array D accumulates the scaling factors from the fast
89 scaled Jacobi rotations. On entry, A*diag(D) represents the
90 input matrix. On exit, A_onexit*diag(D_onexit) represents the
91 input matrix post-multiplied by a sequence of Jacobi rotations,
92 where the rotation threshold and the total number of sweeps are
93 given in TOL and NSWEEP, respectively. (See the descriptions
94 of N1, A, TOL and NSWEEP.)
95 SVA (input/workspace/output) REAL array, dimension (N)
97 On entry, SVA contains the Euclidean norms of the columns of
98 the matrix A*diag(D). On exit, SVA contains the Euclidean
99 norms of the columns of the matrix onexit*diag(D_onexit).
100 MV (input) INTEGER
102 If JOBV .EQ. 'A', then MV rows of V are post-multipled by a
103 sequence of Jacobi rotations. If JOBV = 'N', then MV is not
104 referenced.
105 V (input/output) REAL array, dimension (LDV,N)
107 If JOBV .EQ. 'V' then N rows of V are post-multipled by a
108 sequence of Jacobi rotations. If JOBV .EQ. 'A' then MV rows of
109 V are post-multipled by a sequence of Jacobi rotations. If
110 JOBV = 'N', then V is not referenced.
111 LDV (input) INTEGER
113 The leading dimension of the array V, LDV >= 1. If JOBV =
114 'V', LDV .GE. N. If JOBV = 'A', LDV .GE. MV.
115 EPS (input) INTEGER
117 EPS = SLAMCH('Epsilon')
118 SFMIN (input) INTEGER
120 SFMIN = SLAMCH('Safe Minimum')
121 TOL (input) REAL
123 TOL is the threshold for Jacobi rotations. For a pair A(:,p),
124 A(:,q) of pivot columns, the Jacobi rotation is
125 applied only if ABS(COS(angle(A(:,p),A(:,q)))) .GT. TOL.
126 NSWEEP (input) INTEGER
128 NSWEEP is the number of sweeps of Jacobi rotations to be per‐
129 formed.
130 WORK (workspace) REAL array, dimension LWORK.
132 LWORK (input) INTEGER
134 LWORK is the dimension of WORK. LWORK .GE. M.
135 INFO (output) INTEGER
137 = 0 : successful exit.
138 < 0 : if INFO = -i, then the i-th argument had an illegal value
139 LAPACK routine (version 3.2) November 2008 SGSVJ1(1)