1ZGELSX(1) LAPACK driver routine (version 3.2) ZGELSX(1)
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6 ZGELSX - routine i deprecated and has been replaced by routine ZGELSY
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9 SUBROUTINE ZGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
10 RWORK, INFO )
11 INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
13 DOUBLE PRECISION RCOND
15 INTEGER JPVT( * )
17 DOUBLE PRECISION RWORK( * )
19 COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
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23 This routine is deprecated and has been replaced by routine ZGELSY.
24 ZGELSX computes the minimum-norm solution to a complex linear least
25 squares problem:
26 minimize || A * X - B ||
27 using a complete orthogonal factorization of A. A is an M-by-N matrix
28 which may be rank-deficient.
29 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:
33 A * P = Q * [ R11 R12 ]
34 [ 0 R22 ]
35 with R11 defined as the largest leading submatrix whose estimated con‐
36 dition number is less than 1/RCOND. The order of R11, RANK, is the
37 effective rank of A.
38 Then, R22 is considered to be negligible, and R12 is annihilated by
39 unitary transformations from the right, arriving at the complete
40 orthogonal factorization:
41 A * P = Q * [ T11 0 ] * Z
42 [ 0 0 ]
43 The minimum-norm solution is then
44 X = P * Z' [ inv(T11)*Q1'*B ]
45 [ 0 ]
46 where Q1 consists of the first RANK columns of Q.
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49 M (input) INTEGER
50 The number of rows of the matrix A. M >= 0.
51 N (input) INTEGER
53 The number of columns of the matrix A. N >= 0.
54 NRHS (input) INTEGER
56 The number of right hand sides, i.e., the number of columns of
57 matrices B and X. NRHS >= 0.
58 A (input/output) COMPLEX*16 array, dimension (LDA,N)
60 On entry, the M-by-N matrix A. On exit, A has been overwritten
61 by details of its complete orthogonal factorization.
62 LDA (input) INTEGER
64 The leading dimension of the array A. LDA >= max(1,M).
65 B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
67 On entry, the M-by-NRHS right hand side matrix B. On exit, the
68 N-by-NRHS solution matrix X. If m >= n and RANK = n, the
69 residual sum-of-squares for the solution in the i-th column is
70 given by the sum of squares of elements N+1:M in that column.
71 LDB (input) INTEGER
73 The leading dimension of the array B. LDB >= max(1,M,N).
74 JPVT (input/output) INTEGER array, dimension (N)
76 On entry, if JPVT(i) .ne. 0, the i-th column of A is an initial
77 column, otherwise it is a free column. Before the QR factor‐
78 ization of A, all initial columns are permuted to the leading
79 positions; only the remaining free columns are moved as a
80 result of column pivoting during the factorization. On exit,
81 if JPVT(i) = k, then the i-th column of A*P was the k-th column
82 of A.
83 RCOND (input) DOUBLE PRECISION
85 RCOND is used to determine the effective rank of A, which is
86 defined as the order of the largest leading triangular subma‐
87 trix R11 in the QR factorization with pivoting of A, whose
88 estimated condition number < 1/RCOND.
89 RANK (output) INTEGER
91 The effective rank of A, i.e., the order of the submatrix R11.
92 This is the same as the order of the submatrix T11 in the com‐
93 plete orthogonal factorization of A.
94 WORK (workspace) COMPLEX*16 array, dimension
96 (min(M,N) + max( N, 2*min(M,N)+NRHS )),
97 RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
99 INFO (output) INTEGER
101 = 0: successful exit
102 < 0: if INFO = -i, the i-th argument had an illegal value
103 LAPACK driver routine (version 3.N2o)vember 2008 ZGELSX(1)