1CLALSD(1) LAPACK routine (version 3.1) CLALSD(1)
2
6 CLALSD - 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 CLALSD( UPLO, SMLSIZ, N, NRHS, D, E, B, LDB, RCOND, RANK,
13 WORK, RWORK, IWORK, INFO )
14 CHARACTER UPLO
16 INTEGER INFO, LDB, N, NRHS, RANK, SMLSIZ
18 REAL RCOND
20 INTEGER IWORK( * )
22 REAL D( * ), E( * ), RWORK( * )
24 COMPLEX B( LDB, * ), WORK( * )
26
28 CLALSD uses the singular value decomposition of A to solve the least
29 squares problem of finding X to minimize the Euclidean norm of each
30 column of A*X-B, where A is N-by-N upper bidiagonal, and X and B are N-
31 by-NRHS. The solution X overwrites B.
32 The singular values of A smaller than RCOND times the largest singular
34 value are treated as zero in solving the least squares problem; in this
35 case a minimum norm solution is returned. The actual singular values
36 are returned in D in ascending order.
37 This code makes very mild assumptions about floating point arithmetic.
39 It will work on machines with a guard digit in add/subtract, or on
40 those binary machines without guard digits which subtract like the Cray
41 XMP, Cray YMP, Cray C 90, or Cray 2. It could conceivably fail on
42 hexadecimal or decimal machines without guard digits, but we know of
43 none.
44
47 UPLO (input) CHARACTER*1
48 = 'U': D and E define an upper bidiagonal matrix.
49 = 'L': D and E define a lower bidiagonal matrix.
50 SMLSIZ (input) INTEGER The maximum size of the subproblems at
52 the bottom of the computation tree.
53 N (input) INTEGER
55 The dimension of the bidiagonal matrix. N >= 0.
56 NRHS (input) INTEGER
58 The number of columns of B. NRHS must be at least 1.
59 D (input/output) REAL array, dimension (N)
61 On entry D contains the main diagonal of the bidiagonal matrix.
62 On exit, if INFO = 0, D contains its singular values.
63 E (input/output) REAL array, dimension (N-1)
65 Contains the super-diagonal entries of the bidiagonal matrix.
66 On exit, E has been destroyed.
67 B (input/output) COMPLEX array, dimension (LDB,NRHS)
69 On input, B contains the right hand sides of the least squares
70 problem. On output, B contains the solution X.
71 LDB (input) INTEGER
73 The leading dimension of B in the calling subprogram. LDB must
74 be at least max(1,N).
75 RCOND (input) REAL
77 The singular values of A less than or equal to RCOND times the
78 largest singular value are treated as zero in solving the least
79 squares problem. If RCOND is negative, machine precision is used
80 instead. For example, if diag(S)*X=B were the least squares
81 problem, where diag(S) is a diagonal matrix of singular values,
82 the solution would be X(i) = B(i) / S(i) if S(i) is greater than
83 RCOND*max(S), and X(i) = 0 if S(i) is less than or equal to
84 RCOND*max(S).
85 RANK (output) INTEGER
87 The number of singular values of A greater than RCOND times the
88 largest singular value.
89 WORK (workspace) COMPLEX array, dimension (N * NRHS).
91 RWORK (workspace) REAL array, dimension at least
93 (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2),
94 where NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
95 IWORK (workspace) INTEGER array, dimension (3*N*NLVL + 11*N).
97 INFO (output) INTEGER
99 = 0: successful exit.
100 < 0: if INFO = -i, the i-th argument had an illegal value.
101 > 0: The algorithm failed to compute an singular value while
102 working on the submatrix lying in rows and columns INFO/(N+1)
103 through MOD(INFO,N+1).
104
106 Based on contributions by
107 Ming Gu and Ren-Cang Li, Computer Science Division, University of
108 California at Berkeley, USA
109 Osni Marques, LBNL/NERSC, USA
110 LAPACK routine (version 3.1) November 2006 CLALSD(1)