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cgelsd.f(3)				      LAPACK				      cgelsd.f(3)

NAME
       cgelsd.f -

SYNOPSIS
   Functions/Subroutines
       subroutine cgelsd (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK, LWORK, RWORK, IWORK,
	   INFO)
	    CGELSD computes the minimum-norm solution to a linear least squares problem for GE
	   matrices

Function/Subroutine Documentation
   subroutine cgelsd (integerM, integerN, integerNRHS, complex, dimension( lda, * )A, integerLDA,
       complex, dimension( ldb, * )B, integerLDB, real, dimension( * )S, realRCOND, integerRANK,
       complex, dimension( * )WORK, integerLWORK, real, dimension( * )RWORK, integer, dimension(
       * )IWORK, integerINFO)
	CGELSD computes the minimum-norm solution to a linear least squares problem for GE
       matrices

       Purpose:

	    CGELSD computes the minimum-norm solution to a real linear least
	    squares problem:
		minimize 2-norm(| b - A*x |)
	    using the singular value decomposition (SVD) 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 problem is solved in three steps:
	    (1) Reduce the coefficient matrix A to bidiagonal form with
		Householder tranformations, reducing the original problem
		into a "bidiagonal least squares problem" (BLS)
	    (2) Solve the BLS using a divide and conquer approach.
	    (3) Apply back all the Householder tranformations to solve
		the original least squares problem.

	    The effective rank of A is determined by treating as zero those
	    singular values which are less than RCOND times the largest singular
	    value.

	    The divide and conquer algorithm makes very mild assumptions about
	    floating point arithmetic. It will work on machines with a guard
	    digit in add/subtract, or on those binary machines without guard
	    digits which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or
	    Cray-2. It could conceivably fail on hexadecimal or decimal machines
	    without guard digits, but we know of none.

       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 the 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 destroyed.

	   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, B is overwritten by the N-by-NRHS solution matrix X.
		     If m >= n and RANK = n, the residual sum-of-squares for
		     the solution in the i-th column is given by the sum of
		     squares of the modulus of elements n+1:m in that column.

	   LDB

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

	   S

		     S is REAL array, dimension (min(M,N))
		     The singular values of A in decreasing order.
		     The condition number of A in the 2-norm = S(1)/S(min(m,n)).

	   RCOND

		     RCOND is REAL
		     RCOND is used to determine the effective rank of A.
		     Singular values S(i) <= RCOND*S(1) are treated as zero.
		     If RCOND < 0, machine precision is used instead.

	   RANK

		     RANK is INTEGER
		     The effective rank of A, i.e., the number of singular values
		     which are greater than RCOND*S(1).

	   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. LWORK must be at least 1.
		     The exact minimum amount of workspace needed depends on M,
		     N and NRHS. As long as LWORK is at least
			 2 * N + N * NRHS
		     if M is greater than or equal to N or
			 2 * M + M * NRHS
		     if M is less than N, the code will execute correctly.
		     For good performance, LWORK should generally be larger.

		     If LWORK = -1, then a workspace query is assumed; the routine
		     only calculates the optimal size of the array WORK and the
		     minimum sizes of the arrays RWORK and IWORK, and returns
		     these values as the first entries of the WORK, RWORK and
		     IWORK arrays, and no error message related to LWORK is issued
		     by XERBLA.

	   RWORK

		     RWORK is REAL array, dimension (MAX(1,LRWORK))
		     LRWORK >=
			10*N + 2*N*SMLSIZ + 8*N*NLVL + 3*SMLSIZ*NRHS +
			MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
		     if M is greater than or equal to N or
			10*M + 2*M*SMLSIZ + 8*M*NLVL + 3*SMLSIZ*NRHS +
			MAX( (SMLSIZ+1)**2, N*(1+NRHS) + 2*NRHS )
		     if M is less than N, the code will execute correctly.
		     SMLSIZ is returned by ILAENV and is equal to the maximum
		     size of the subproblems at the bottom of the computation
		     tree (usually about 25), and
			NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
		     On exit, if INFO = 0, RWORK(1) returns the minimum LRWORK.

	   IWORK

		     IWORK is INTEGER array, dimension (MAX(1,LIWORK))
		     LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
		     where MINMN = MIN( M,N ).
		     On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

	   INFO

		     INFO is INTEGER
		     = 0: successful exit
		     < 0: if INFO = -i, the i-th argument had an illegal value.
		     > 0:  the algorithm for computing the SVD failed to converge;
			   if INFO = i, i off-diagonal elements of an intermediate
			   bidiagonal form did not converge to zero.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   November 2011

       Contributors:
	   Ming Gu and Ren-Cang Li, Computer Science Division, University of California at
	   Berkeley, USA
	    Osni Marques, LBNL/NERSC, USA

       Definition at line 225 of file cgelsd.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      cgelsd.f(3)
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