1SGELSD(1) LAPACK driver routine (version 3.1) SGELSD(1)
2
6 SGELSD - the minimum-norm solution to a real linear least squares prob‐
7 lem
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
29 in a single call; they are stored as the columns of the M-by-NRHS right
30 hand side matrix B and the N-by-NRHS solution matrix X.
31 The problem is solved in three steps:
33 (1) Reduce the coefficient matrix A to bidiagonal form with
34 Householder transformations, reducing the original problem
35 into a "bidiagonal least squares problem" (BLS)
36 (2) Solve the BLS using a divide and conquer approach.
37 (3) Apply back all the Householder tranformations to solve
38 the original least squares problem.
39 The effective rank of A is determined by treating as zero those singu‐
41 lar values which are less than RCOND times the largest singular value.
42 The divide and conquer algorithm makes very mild assumptions about
44 floating point arithmetic. It will work on machines with a guard digit
45 in add/subtract, or on those binary machines without guard digits which
46 subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2. It could
47 conceivably fail on hexadecimal or decimal machines without guard dig‐
48 its, but we know of none.
49
52 M (input) INTEGER
53 The number of rows of A. M >= 0.
54 N (input) INTEGER
56 The number of columns of A. N >= 0.
57 NRHS (input) INTEGER
59 The number of right hand sides, i.e., the number of columns of
60 the matrices B and X. NRHS >= 0.
61 A (input) REAL array, dimension (LDA,N)
63 On entry, the M-by-N matrix A. On exit, A has been destroyed.
64 LDA (input) INTEGER
66 The leading dimension of the array A. LDA >= max(1,M).
67 B (input/output) REAL array, dimension (LDB,NRHS)
69 On entry, the M-by-NRHS right hand side matrix B. On exit, B
70 is overwritten by the N-by-NRHS solution matrix X. If m >= n
71 and RANK = n, the residual sum-of-squares for the solution in
72 the i-th column is given by the sum of squares of elements
73 n+1:m in that column.
74 LDB (input) INTEGER
76 The leading dimension of the array B. LDB >= max(1,max(M,N)).
77 S (output) REAL array, dimension (min(M,N))
79 The singular values of A in decreasing order. The condition
80 number of A in the 2-norm = S(1)/S(min(m,n)).
81 RCOND (input) REAL
83 RCOND is used to determine the effective rank of A. Singular
84 values S(i) <= RCOND*S(1) are treated as zero. If RCOND < 0,
85 machine precision is used instead.
86 RANK (output) INTEGER
88 The effective rank of A, i.e., the number of singular values
89 which are greater than RCOND*S(1).
90 WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))
92 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
93 LWORK (input) INTEGER
95 The dimension of the array WORK. LWORK must be at least 1. The
96 exact minimum amount of workspace needed depends on M, N and
97 NRHS. As long as LWORK is at least 12*N + 2*N*SMLSIZ + 8*N*NLVL
98 + N*NRHS + (SMLSIZ+1)**2, if M is greater than or equal to N or
99 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2, if M is
100 less than N, the code will execute correctly. SMLSIZ is
101 returned by ILAENV and is equal to the maximum size of the sub‐
102 problems at the bottom of the computation tree (usually about
103 25), and NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) +
104 1 ) For good performance, LWORK should generally be larger.
105 If LWORK = -1, then a workspace query is assumed; the routine
107 only calculates the optimal size of the array WORK and the min‐
108 imum size of the array IWORK, and returns these values as the
109 first entries of the WORK and IWORK arrays, and no error mes‐
110 sage related to LWORK is issued by XERBLA.
111 IWORK (workspace) INTEGER array, dimension (MAX(1,LIWORK))
113 LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN), where MINMN = MIN(
114 M,N ). On exit, if INFO = 0, IWORK(1) returns the minimum
115 LIWORK.
116 INFO (output) INTEGER
118 = 0: successful exit
119 < 0: if INFO = -i, the i-th argument had an illegal value.
120 > 0: the algorithm for computing the SVD failed to converge;
121 if INFO = i, i off-diagonal elements of an intermediate bidiag‐
122 onal form did not converge to zero.
123
125 Based on contributions by
126 Ming Gu and Ren-Cang Li, Computer Science Division, University of
127 California at Berkeley, USA
128 Osni Marques, LBNL/NERSC, USA
129 LAPACK driver routine (version 3.N1o)vember 2006 SGELSD(1)