1DGELS(1) LAPACK driver routine (version 3.1) DGELS(1)
2
6 DGELS - overdetermined or underdetermined real linear systems involving
7 an M-by-N matrix A, or its transpose, using a QR or LQ factorization of
8 A
9
11 SUBROUTINE DGELS( TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO
12 )
13 CHARACTER TRANS
15 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS
17 DOUBLE PRECISION A( LDA, * ), B( LDB, * ), WORK( * )
19
21 DGELS solves overdetermined or underdetermined real linear systems
22 involving an M-by-N matrix A, or its transpose, using a QR or LQ fac‐
23 torization of A. It is assumed that A has full rank.
24 The following options are provided:
26 1. If TRANS = 'N' and m >= n: find the least squares solution of
28 an overdetermined system, i.e., solve the least squares problem
29 minimize || B - A*X ||.
30 2. If TRANS = 'N' and m < n: find the minimum norm solution of
32 an underdetermined system A * X = B.
33 3. If TRANS = 'T' and m >= n: find the minimum norm solution of
35 an undetermined system A**T * X = B.
36 4. If TRANS = 'T' and m < n: find the least squares solution of
38 an overdetermined system, i.e., solve the least squares problem
39 minimize || B - A**T * X ||.
40 Several right hand side vectors b and solution vectors x can be handled
42 in a single call; they are stored as the columns of the M-by-NRHS right
43 hand side matrix B and the N-by-NRHS solution matrix X.
44
47 TRANS (input) CHARACTER*1
48 = 'N': the linear system involves A;
49 = 'T': the linear system involves A**T.
50 M (input) INTEGER
52 The number of rows of the matrix A. M >= 0.
53 N (input) INTEGER
55 The number of columns of the matrix A. N >= 0.
56 NRHS (input) INTEGER
58 The number of right hand sides, i.e., the number of columns of
59 the matrices B and X. NRHS >=0.
60 A (input/output) DOUBLE PRECISION array, dimension (LDA,N)
62 On entry, the M-by-N matrix A. On exit, if M >= N, A is over‐
63 written by details of its QR factorization as returned by DGE‐
64 QRF; if M < N, A is overwritten by details of its LQ factor‐
65 ization as returned by DGELQF.
66 LDA (input) INTEGER
68 The leading dimension of the array A. LDA >= max(1,M).
69 B (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
71 On entry, the matrix B of right hand side vectors, stored
72 columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if
73 TRANS = 'T'. On exit, if INFO = 0, B is overwritten by the
74 solution vectors, stored columnwise: if TRANS = 'N' and m >= n,
75 rows 1 to n of B contain the least squares solution vectors;
76 the residual sum of squares for the solution in each column is
77 given by the sum of squares of elements N+1 to M in that col‐
78 umn; if TRANS = 'N' and m < n, rows 1 to N of B contain the
79 minimum norm solution vectors; if TRANS = 'T' and m >= n, rows
80 1 to M of B contain the minimum norm solution vectors; if TRANS
81 = 'T' and m < n, rows 1 to M of B contain the least squares
82 solution vectors; the residual sum of squares for the solution
83 in each column is given by the sum of squares of elements M+1
84 to N in that column.
85 LDB (input) INTEGER
87 The leading dimension of the array B. LDB >= MAX(1,M,N).
88 WORK (workspace/output) DOUBLE PRECISION array, dimension
90 (MAX(1,LWORK))
91 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
92 LWORK (input) INTEGER
94 The dimension of the array WORK. LWORK >= max( 1, MN + max(
95 MN, NRHS ) ). For optimal performance, LWORK >= max( 1, MN +
96 max( MN, NRHS )*NB ). where MN = min(M,N) and NB is the opti‐
97 mum block size.
98 If LWORK = -1, then a workspace query is assumed; the routine
100 only calculates the optimal size of the WORK array, returns
101 this value as the first entry of the WORK array, and no error
102 message related to LWORK is issued by XERBLA.
103 INFO (output) INTEGER
105 = 0: successful exit
106 < 0: if INFO = -i, the i-th argument had an illegal value
107 > 0: if INFO = i, the i-th diagonal element of the triangular
108 factor of A is zero, so that A does not have full rank; the
109 least squares solution could not be computed.
110 LAPACK driver routine (version 3.N1o)vember 2006 DGELS(1)