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RedHat 9 (Linux i386) - man page for cgelsy (redhat section l)

CGELSY(l)					)					CGELSY(l)

NAME
       CGELSY - compute the minimum-norm solution to a complex linear least squares problem

SYNOPSIS
       SUBROUTINE CGELSY( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, RWORK, INFO
			  )

	   INTEGER	  INFO, LDA, LDB, LWORK, M, N, NRHS, RANK

	   REAL 	  RCOND

	   INTEGER	  JPVT( * )

	   REAL 	  RWORK( * )

	   COMPLEX	  A( LDA, * ), B( LDB, * ), WORK( * )

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 transforma-
       tions 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' [ inv(T11)*Q1'*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.

ARGUMENTS
       M       (input) INTEGER
	       The number of rows of the matrix A.  M >= 0.

       N       (input) INTEGER
	       The number of columns of the matrix A.  N >= 0.

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

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

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

       JPVT    (input/output) 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  col-
	       umn of A*P was the k-th column of A.

       RCOND   (input) 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  piv-
	       oting of A, whose estimated condition number < 1/RCOND.

       RANK    (output) 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    (workspace/output) COMPLEX array, dimension (LWORK)
	       On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

       LWORK   (input) 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   (workspace) REAL array, dimension (2*N)

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

FURTHER DETAILS
       Based on contributions by
	 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

LAPACK version 3.0			   15 June 2000 				CGELSY(l)


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