sgelsd.f(3)							      LAPACK							       sgelsd.f(3)

NAME

       sgelsd.f -

SYNOPSIS

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

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

       Purpose:

	    SGELSD 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 transformations, 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 A. M >= 0.

	   N

		     N is INTEGER
		     The number of columns of 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 REAL 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 REAL 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 elements n+1:m in that column.

	   LDB

		     LDB is INTEGER
		     The leading dimension of the array B. LDB >= max(1,max(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 REAL 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
			 12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
		     if M is greater than or equal to N or
			 12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
		     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 )
		     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 size of the array IWORK, and returns these values as
		     the first entries of the WORK and IWORK arrays, and no error
		     message related to LWORK is issued by XERBLA.

	   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 210 of file sgelsd.f.

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

Version 3.4.1							  Sun May 26 2013						       sgelsd.f(3)