1ZLALSD(1) LAPACK routine (version 3.1) ZLALSD(1)
2
6 ZLALSD - 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 ZLALSD( 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 DOUBLE PRECISION RCOND
20 INTEGER IWORK( * )
22 DOUBLE PRECISION D( * ), E( * ), RWORK( * )
24 COMPLEX*16 B( LDB, * ), WORK( * )
26
28 ZLALSD 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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*16 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) DOUBLE PRECISION
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*16 array, dimension at least
91 (N * NRHS).
92 RWORK (workspace) DOUBLE PRECISION array, dimension at least
94 (9*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS + (SMLSIZ+1)**2),
95 where NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
96 IWORK (workspace) INTEGER array, dimension at least
98 (3*N*NLVL + 11*N).
99 INFO (output) INTEGER
101 = 0: successful exit.
102 < 0: if INFO = -i, the i-th argument had an illegal value.
103 > 0: The algorithm failed to compute an singular value while
104 working on the submatrix lying in rows and columns INFO/(N+1)
105 through MOD(INFO,N+1).
106
108 Based on contributions by
109 Ming Gu and Ren-Cang Li, Computer Science Division, University of
110 California at Berkeley, USA
111 Osni Marques, LBNL/NERSC, USA
112 LAPACK routine (version 3.1) November 2006 ZLALSD(1)