1SLALSD(1) LAPACK routine (version 3.1) SLALSD(1)
2
6 SLALSD - the singular value decomposition of A to solve the least
7 squares problem of finding X to minimize the Euclidean norm of each
8 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-
9 by-NRHS
10
12 SUBROUTINE SLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK,
13 WORK, IWORK, INFO )
14 CHARACTER UPLO
16 INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ
18 REAL RCOND
20 INTEGER IWORK( * )
22 REAL B( LDB, * ), D( * ), E( * ), WORK( * )
24
26 SLALSD uses the singular value decomposition of A to solve the least
27 squares problem of finding X to minimize the Euclidean norm of each
28 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-
29 by-NRHS. The solution X overwrites B.
30 The singular values of A smaller than RCOND times the largest singular
32 value are treated as zero in solving the least squares problem; in this
33 case a minimum norm solution is returned. The actual singular values
34 are returned in D in ascending order.
35 This code makes very mild assumptions about floating point arithmetic.
37 It will work on machines with a guard digit in add/subtract, or on
38 those binary machines without guard digits which subtract like the Cray
39 XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on
40 hexadecimal or decimal machines without guard digits, but we know of
41 none.
42
45 UPLO (input) CHARACTER*1
46 = 'U': D and E define an upper bidiagonal matrix.
47 = 'L': D and E define a lower bidiagonal matrix.
48 SMLSIZ (input) INTEGER The maximum size of the subproblems at
50 the bottom of the computation tree.
51 N (input) INTEGER
53 The dimension of the bidiagonal matrix. N >= 0.
54 NRHS (input) INTEGER
56 The number of columns of B. NRHS must be at least 1.
57 D (input/output) REAL array, dimension (N)
59 On entry D contains the main diagonal of the bidiagonal matrix.
60 On exit, if INFO = 0, D contains its singular values.
61 E (input/output) REAL array, dimension (N-1)
63 Contains the super-diagonal entries of the bidiagonal matrix.
64 On exit, E has been destroyed.
65 B (input/output) REAL array, dimension (LDB,NRHS)
67 On input, B contains the right hand sides of the least squares
68 problem. On output, B contains the solution X.
69 LDB (input) INTEGER
71 The leading dimension of B in the calling subprogram. LDB must
72 be at least max(1,N).
73 RCOND (input) REAL
75 The singular values of A less than or equal to RCOND times the
76 largest singular value are treated as zero in solving the least
77 squares problem. If RCOND is negative, machine precision is used
78 instead. For example, if diag(S)*X=B were the least squares
79 problem, where diag(S) is a diagonal matrix of singular values,
80 the solution would be X(i) = B(i) / S(i) if S(i) is greater than
81 RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
82 RCOND*max(S).
83 RANK (output) INTEGER
85 The number of singular values of A greater than RCOND times the
86 largest singular value.
87 WORK (workspace) REAL array, dimension at least
89 (9*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2), where
90 NLVL = max(0, INT(log_2 (N/(SMLSIZ+1))) + 1).
91 IWORK (workspace) INTEGER array, dimension at least
93 (3*N*NLVL + 11*N)
94 INFO (output) INTEGER
96 = 0: successful exit.
97 < 0: if INFO = -i, the i-th argument had an illegal value.
98 > 0: The algorithm failed to compute an singular value while
99 working on the submatrix lying in rows and columns INFO/(N+1)
100 through MOD(INFO,N+1).
101
103 Based on contributions by
104 Ming Gu and Ren-Cang Li, Computer Science Division, University of
105 California at Berkeley, USA
106 Osni Marques, LBNL/NERSC, USA
107 LAPACK routine (version 3.1) November 2006 SLALSD(1)