1ZGELSY(1) LAPACK driver routine (version 3.1) ZGELSY(1)
2
6 ZGELSY - the minimum-norm solution to a complex linear least squares
7 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( * )
22
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
31 in a single call; they are stored as the columns of the M-by-NRHS right
32 hand side matrix B and the N-by-NRHS solution matrix X.
33 The routine first computes a QR factorization with column pivoting:
35 A * P = Q * [ R11 R12 ]
36 [ 0 R22 ]
37 with R11 defined as the largest leading submatrix whose estimated con‐
38 dition number is less than 1/RCOND. The order of R11, RANK, is the
39 effective rank of A.
40 Then, R22 is considered to be negligible, and R12 is annihilated by
42 unitary transformations from the right, arriving at the complete
43 orthogonal factorization:
44 A * P = Q * [ T11 0 ] * Z
45 [ 0 0 ]
46 The minimum-norm solution is then
47 X = P * Z' [ inv(T11)*Q1'*B ]
48 [ 0 ]
49 where Q1 consists of the first RANK columns of Q.
50 This routine is basically identical to the original xGELSX except three
52 differences:
53 o The permutation of matrix B (the right hand side) is faster and
54 more simple.
55 o The call to the subroutine xGEQPF has been substituted by the
56 the call to the subroutine xGEQP3. This subroutine is a Blas-3
57 version of the QR factorization with column pivoting.
58 o Matrix B (the right hand side) is updated with Blas-3.
59
62 M (input) INTEGER
63 The number of rows of the matrix A. M >= 0.
64 N (input) INTEGER
66 The number of columns of the matrix A. N >= 0.
67 NRHS (input) INTEGER
69 The number of right hand sides, i.e., the number of columns of
70 matrices B and X. NRHS >= 0.
71 A (input/output) COMPLEX*16 array, dimension (LDA,N)
73 On entry, the M-by-N matrix A. On exit, A has been overwritten
74 by details of its complete orthogonal factorization.
75 LDA (input) INTEGER
77 The leading dimension of the array A. LDA >= max(1,M).
78 B (input/output) COMPLEX*16 array, dimension (LDB,NRHS)
80 On entry, the M-by-NRHS right hand side matrix B. On exit, the
81 N-by-NRHS solution matrix X.
82 LDB (input) INTEGER
84 The leading dimension of the array B. LDB >= max(1,M,N).
85 JPVT (input/output) INTEGER array, dimension (N)
87 On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
88 to the front of AP, otherwise column i is a free column. On
89 exit, if JPVT(i) = k, then the i-th column of A*P was the k-th
90 column of A.
91 RCOND (input) DOUBLE PRECISION
93 RCOND is used to determine the effective rank of A, which is
94 defined as the order of the largest leading triangular subma‐
95 trix R11 in the QR factorization with pivoting of A, whose
96 estimated condition number < 1/RCOND.
97 RANK (output) INTEGER
99 The effective rank of A, i.e., the order of the submatrix R11.
100 This is the same as the order of the submatrix T11 in the com‐
101 plete orthogonal factorization of A.
102 WORK (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
104 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
105 LWORK (input) INTEGER
107 The dimension of the array WORK. The unblocked strategy
108 requires that: LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS ) where MN
109 = min(M,N). The block algorithm requires that: LWORK >= MN +
110 MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS ) where NB is an
111 upper bound on the blocksize returned by ILAENV for the rou‐
112 tines ZGEQP3, ZTZRZF, CTZRQF, ZUNMQR, and ZUNMRZ.
113 If LWORK = -1, then a workspace query is assumed; the routine
115 only calculates the optimal size of the WORK array, returns
116 this value as the first entry of the WORK array, and no error
117 message related to LWORK is issued by XERBLA.
118 RWORK (workspace) DOUBLE PRECISION array, dimension (2*N)
120 INFO (output) INTEGER
122 = 0: successful exit
123 < 0: if INFO = -i, the i-th argument had an illegal value
124
126 Based on contributions by
127 A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA
128 E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
129 G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain
130 LAPACK driver routine (version 3.N1o)vember 2006 ZGELSY(1)