1SGELSD(1) LAPACK driver routine (version 3.2) SGELSD(1)
2
6 SGELSD - computes the minimum-norm solution to a real linear least
7 squares problem
8
10 SUBROUTINE SGELSD( M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK,
11 LWORK, IWORK, INFO )
12 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
14 REAL RCOND
16 INTEGER IWORK( * )
18 REAL A( LDA, * ), B( LDB, * ), S( * ), WORK( * )
20
22 SGELSD computes the minimum-norm solution to a real linear least
23 squares problem:
24 minimize 2-norm(| b - A*x |)
25 using the singular value decomposition (SVD) of A. A is an M-by-N
26 matrix which may be rank-deficient.
27 Several right hand side vectors b and solution vectors x can be handled
28 in a single call; they are stored as the columns of the M-by-NRHS right
29 hand side matrix B and the N-by-NRHS solution matrix X.
30 The problem is solved in three steps:
31 (1) Reduce the coefficient matrix A to bidiagonal form with
32 Householder transformations, reducing the original problem
33 into a "bidiagonal least squares problem" (BLS)
34 (2) Solve the BLS using a divide and conquer approach.
35 (3) Apply back all the Householder tranformations to solve
36 the original least squares problem.
37 The effective rank of A is determined by treating as zero those singu‐
38 lar values which are less than RCOND times the largest singular value.
39 The divide and conquer algorithm makes very mild assumptions about
40 floating point arithmetic. It will work on machines with a guard digit
41 in add/subtract, or on those binary machines without guard digits which
42 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
43 conceivably fail on hexadecimal or decimal machines without guard dig‐
44 its, but we know of none.
45
47 M (input) INTEGER
48 The number of rows of A. M >= 0.
49 N (input) INTEGER
51 The number of columns of A. N >= 0.
52 NRHS (input) INTEGER
54 The number of right hand sides, i.e., the number of columns of
55 the matrices B and X. NRHS >= 0.
56 A (input) REAL array, dimension (LDA,N)
58 On entry, the M-by-N matrix A. On exit, A has been destroyed.
59 LDA (input) INTEGER
61 The leading dimension of the array A. LDA >= max(1,M).
62 B (input/output) REAL array, dimension (LDB,NRHS)
64 On entry, the M-by-NRHS right hand side matrix B. On exit, B
65 is overwritten by the N-by-NRHS solution matrix X. If m >= n
66 and RANK = n, the residual sum-of-squares for the solution in
67 the i-th column is given by the sum of squares of elements
68 n+1:m in that column.
69 LDB (input) INTEGER
71 The leading dimension of the array B. LDB >= max(1,max(M,N)).
72 S (output) REAL array, dimension (min(M,N))
74 The singular values of A in decreasing order. The condition
75 number of A in the 2-norm = S(1)/S(min(m,n)).
76 RCOND (input) REAL
78 RCOND is used to determine the effective rank of A. Singular
79 values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0,
80 machine precision is used instead.
81 RANK (output) INTEGER
83 The effective rank of A, i.e., the number of singular values
84 which are greater than RCOND*S(1).
85 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
87 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
88 LWORK (input) INTEGER
90 The dimension of the array WORK. LWORK must be at least 1. The
91 exact minimum amount of workspace needed depends on M, N and
92 NRHS. As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL
93 + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or
94 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is
95 less than N, the code will execute correctly. SMLSIZ is
96 returned by ILAENV and is equal to the maximum size of the sub‐
97 problems at the bottom of the computation tree (usually about
98 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) +
99 1 ) For good performance, LWORK should generally be larger. If
100 LWORK = -1, then a workspace query is assumed; the routine only
101 calculates the optimal size of the array WORK and the minimum
102 size of the array IWORK, and returns these values as the first
103 entries of the WORK and IWORK arrays, and no error message
104 related to LWORK is issued by XERBLA.
105 IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))
107 LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN(
108 M,N ). On exit, if INFO = 0, IWORK(1) returns the minimum
109 LIWORK.
110 INFO (output) INTEGER
112 = 0: successful exit
113 < 0: if INFO = -i, the i-th argument had an illegal value.
114 > 0: the algorithm for computing the SVD failed to converge;
115 if INFO = i, i off-diagonal elements of an intermediate bidiag‐
116 onal form did not converge to zero.
117
119 Based on contributions by
120 Ming Gu and Ren-Cang Li, Computer Science Division, University of
121 California at Berkeley, USA
122 Osni Marques, LBNL/NERSC, USA
123 LAPACK driver routine (version 3.N2o)vember 2008 SGELSD(1)