1SGELSX(1) LAPACK driver routine (version 3.1) SGELSX(1)
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6 SGELSX - i deprecated and has been replaced by routine SGELSY
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9 SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
10 INFO )
11 INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
13 REAL RCOND
15 INTEGER JPVT( * )
17 REAL A( LDA, * ), B( LDB, * ), WORK( * )
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21 This routine is deprecated and has been replaced by routine SGELSY.
22 SGELSX computes the minimum-norm solution to a real linear least
24 squares problem:
25 minimize || A * X - B ||
26 using a complete orthogonal factorization of A. A is an M-by-N matrix
27 which may be rank-deficient.
28 Several right hand side vectors b and solution vectors x can be handled
30 in a single call; they are stored as the columns of the M-by-NRHS right
31 hand side matrix B and the N-by-NRHS solution matrix X.
32 The routine first computes a QR factorization with column pivoting:
34 A * P = Q * [ R11 R12 ]
35 [ 0 R22 ]
36 with R11 defined as the largest leading submatrix whose estimated con‐
37 dition number is less than 1/RCOND. The order of R11, RANK, is the
38 effective rank of A.
39 Then, R22 is considered to be negligible, and R12 is annihilated by
41 orthogonal transformations from the right, arriving at the complete
42 orthogonal factorization:
43 A * P = Q * [ T11 0 ] * Z
44 [ 0 0 ]
45 The minimum-norm solution is then
46 X = P * Z' [ inv(T11)*Q1'*B ]
47 [ 0 ]
48 where Q1 consists of the first RANK columns of Q.
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52 M (input) INTEGER
53 The number of rows of the matrix A. M >= 0.
54 N (input) INTEGER
56 The number of columns of the matrix A. N >= 0.
57 NRHS (input) INTEGER
59 The number of right hand sides, i.e., the number of columns of
60 matrices B and X. NRHS >= 0.
61 A (input/output) REAL array, dimension (LDA,N)
63 On entry, the M-by-N matrix A. On exit, A has been overwritten
64 by details of its complete orthogonal factorization.
65 LDA (input) INTEGER
67 The leading dimension of the array A. LDA >= max(1,M).
68 B (input/output) REAL array, dimension (LDB,NRHS)
70 On entry, the M-by-NRHS right hand side matrix B. On exit, the
71 N-by-NRHS solution matrix X. If m >= n and RANK = n, the
72 residual sum-of-squares for the solution in the i-th column is
73 given by the sum of squares of elements N+1:M in that column.
74 LDB (input) INTEGER
76 The leading dimension of the array B. LDB >= max(1,M,N).
77 JPVT (input/output) INTEGER array, dimension (N)
79 On entry, if JPVT(i) .ne. 0, the i-th column of A is an initial
80 column, otherwise it is a free column. Before the QR factor‐
81 ization of A, all initial columns are permuted to the leading
82 positions; only the remaining free columns are moved as a
83 result of column pivoting during the factorization. On exit,
84 if JPVT(i) = k, then the i-th column of A*P was the k-th column
85 of A.
86 RCOND (input) REAL
88 RCOND is used to determine the effective rank of A, which is
89 defined as the order of the largest leading triangular subma‐
90 trix R11 in the QR factorization with pivoting of A, whose
91 estimated condition number < 1/RCOND.
92 RANK (output) INTEGER
94 The effective rank of A, i.e., the order of the submatrix R11.
95 This is the same as the order of the submatrix T11 in the com‐
96 plete orthogonal factorization of A.
97 WORK (workspace) REAL array, dimension
99 (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
100 INFO (output) INTEGER
102 = 0: successful exit
103 < 0: if INFO = -i, the i-th argument had an illegal value
104 LAPACK driver routine (version 3.N1o)vember 2006 SGELSX(1)