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cgelsy-f(3)

cgelsy.f –

Synopsis

Functions/Subroutines

subroutine cgelsy (M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, RWORK, INFO)
CGELSY solves overdetermined or underdetermined systems for GE matrices

Function/Subroutine Documentation

subroutine cgelsy (integerM, integerN, integerNRHS, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, integer, dimension( * )JPVT, realRCOND, integerRANK, complex, dimension( * )WORK, integerLWORK, real, dimension( * )RWORK, integerINFO)

CGELSY solves overdetermined or underdetermined systems for GE matrices

Purpose:

 CGELSY computes the minimum-norm solution to a complex linear least
 squares problem:
 minimize || A * X - B ||
 using a complete orthogonal factorization of A. A is an M-by-N
 matrix which may be rank-deficient.
 Several right hand side vectors b and solution vectors x can be
 handled in a single call; they are stored as the columns of the
 M-by-NRHS right hand side matrix B and the N-by-NRHS solution
 matrix X.
 The routine first computes a QR factorization with column pivoting:
 A * P = Q * [ R11 R12 ]
 [ 0 R22 ]
 with R11 defined as the largest leading submatrix whose estimated
 condition number is less than 1/RCOND. The order of R11, RANK,
 is the effective rank of A.
 Then, R22 is considered to be negligible, and R12 is annihilated
 by unitary transformations from the right, arriving at the
 complete orthogonal factorization:
 A * P = Q * [ T11 0 ] * Z
 [ 0 0 ]
 The minimum-norm solution is then
 X = P * Z**H [ inv(T11)*Q1**H*B ]
 [ 0 ]
 where Q1 consists of the first RANK columns of Q.
 This routine is basically identical to the original xGELSX except
 three differences:
 o The permutation of matrix B (the right hand side) is faster and
 more simple.
 o The call to the subroutine xGEQPF has been substituted by the
 the call to the subroutine xGEQP3. This subroutine is a Blas-3
 version of the QR factorization with column pivoting.
 o Matrix B (the right hand side) is updated with Blas-3.

 

Parameters:

M

 M is INTEGER
 The number of rows of the matrix A. M >= 0.

N

 N is INTEGER
 The number of columns of the matrix A. N >= 0.

NRHS

 NRHS is INTEGER
 The number of right hand sides, i.e., the number of
 columns of matrices B and X. NRHS >= 0.

A

 A is COMPLEX array, dimension (LDA,N)
 On entry, the M-by-N matrix A.
 On exit, A has been overwritten by details of its
 complete orthogonal factorization.

LDA

 LDA is INTEGER
 The leading dimension of the array A. LDA >= max(1,M).

B

 B is COMPLEX array, dimension (LDB,NRHS)
 On entry, the M-by-NRHS right hand side matrix B.
 On exit, the N-by-NRHS solution matrix X.

LDB

 LDB is INTEGER
 The leading dimension of the array B. LDB >= max(1,M,N).

JPVT

 JPVT is INTEGER array, dimension (N)
 On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
 to the front of AP, otherwise column i is a free column.
 On exit, if JPVT(i) = k, then the i-th column of A*P
 was the k-th column of A.

RCOND

 RCOND is REAL
 RCOND is used to determine the effective rank of A, which
 is defined as the order of the largest leading triangular
 submatrix R11 in the QR factorization with pivoting of A,
 whose estimated condition number < 1/RCOND.

RANK

 RANK is INTEGER
 The effective rank of A, i.e., the order of the submatrix
 R11. This is the same as the order of the submatrix T11
 in the complete orthogonal factorization of A.

WORK

 WORK is COMPLEX array, dimension (MAX(1,LWORK))
 On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

LWORK

 LWORK is INTEGER
 The dimension of the array WORK.
 The unblocked strategy requires that:
 LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
 where MN = min(M,N).
 The block algorithm requires that:
 LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
 where NB is an upper bound on the blocksize returned
 by ILAENV for the routines CGEQP3, CTZRZF, CTZRQF, CUNMQR,
 and CUNMRZ.
 If LWORK = -1, then a workspace query is assumed; the routine
 only calculates the optimal size of the WORK array, returns
 this value as the first entry of the WORK array, and no error
 message related to LWORK is issued by XERBLA.

RWORK

 RWORK is REAL array, dimension (2*N)

INFO

 INFO is INTEGER
 = 0: successful exit
 < 0: if INFO = -i, the i-th argument had an illegal value

 

Author:

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Date:

November 2011

Contributors:

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA

 E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain 

 G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain 

 

Definition at line 210 of file cgelsy.f.

Author

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