1ZGELSY(1) LAPACK driver routine (version 3.2) ZGELSY(1)
2
6 ZGELSY - computes the minimum-norm solution to a complex linear least
7 squares problem
8
10 SUBROUTINE ZGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK,
11 LWORK, RWORK, INFO )
12 INTEGER INFO, LDA, LDB, LWORK, M, N, NRHS, RANK
14 DOUBLE PRECISION RCOND
16 INTEGER JPVT( * )
18 DOUBLE PRECISION RWORK( * )
20 COMPLEX*16 A( LDA, * ), B( LDB, * ), WORK( * )
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24 ZGELSY 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.
47 This routine is basically identical to the original xGELSX except three
48 differences:
49 o The permutation of matrix B (the right hand side) is faster and
50 more simple.
51 o The call to the subroutine xGEQPF has been substituted by the
52 the call to the subroutine xGEQP3. This subroutine is a Blas-3
53 version of the QR factorization with column pivoting.
54 o Matrix B (the right hand side) is updated with Blas-3.
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57 M (input) INTEGER
58 The number of rows of the matrix A. M >= 0.
59 N (input) INTEGER
61 The number of columns of the matrix A. N >= 0.
62 NRHS (input) INTEGER
64 The number of right hand sides, i.e., the number of columns of
65 matrices B and X. NRHS >= 0.
66 A (input/output) COMPLEX*16 array, dimension (LDA,N)
68 On entry, the M-by-N matrix A. On exit, A has been overwritten
69 by details of its complete orthogonal factorization.
70 LDA (input) INTEGER
72 The leading dimension of the array A. LDA >= max(1,M).
73 B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
75 On entry, the M-by-NRHS right hand side matrix B. On exit, the
76 N-by-NRHS solution matrix X.
77 LDB (input) INTEGER
79 The leading dimension of the array B. LDB >= max(1,M,N).
80 JPVT (input/output) INTEGER array, dimension (N)
82 On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
83 to the front of AP, otherwise column i is a free column. On
84 exit, if JPVT(i) = k, then the i-th column of A*P was the k-th
85 column of A.
86 RCOND (input) DOUBLE PRECISION
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/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
99 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
100 LWORK (input) INTEGER
102 The dimension of the array WORK. The unblocked strategy
103 requires that: LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS ) where MN
104 = min(M,N). The block algorithm requires that: LWORK >= MN +
105 MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS ) where NB is an
106 upper bound on the blocksize returned by ILAENV for the rou‐
107 tines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR, and ZUNMRZ. If LWORK =
108 -1, then a workspace query is assumed; the routine only calcu‐
109 lates the optimal size of the WORK array, returns this value as
110 the first entry of the WORK array, and no error message related
111 to LWORK is issued by XERBLA.
112 RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
114 INFO (output) INTEGER
116 = 0: successful exit
117 < 0: if INFO = -i, the i-th argument had an illegal value
118
120 Based on contributions by
121 A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
122 E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
123 G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
124 LAPACK driver routine (version 3.N2o)vember 2008 ZGELSY(1)