Provided by: liblapack-doc_3.3.1-1_all bug


       LAPACK-3 - computes the minimum-norm solution to a real linear least squares problem



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

           DOUBLE         PRECISION RCOND

           INTEGER        IWORK( * )

           DOUBLE         PRECISION A( LDA, * ), B( LDB, * ), S( * ), WORK( * )


       DGELSD 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
        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.


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

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

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

        A       (input) DOUBLE PRECISION array, dimension (LDA,N)
                On entry, the M-by-N matrix A.
                On exit, A has been destroyed.

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

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

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

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

        LWORK   (input) 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 WORK array, returns
                this value as the first entry of the WORK array, and no error
                message related to LWORK is issued by XERBLA.

        IWORK   (workspace) 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    (output) 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.


        Based on contributions by
           Ming Gu and Ren-Cang Li, Computer Science Division, University of
             California at Berkeley, USA
           Osni Marques, LBNL/NERSC, USA

 LAPACK driver routine (version 3.2.2)      April 2011                            DGELSD(3lapack)