ZGELSD(3)      LAPACK routine of NEC Numeric Library Collection      ZGELSD(3)



NAME
       ZGELSD

SYNOPSIS
       SUBROUTINE ZGELSD (M, N, NRHS, A, LDA, B, LDB, S, RCOND, RANK, WORK,
           LWORK, RWORK, IWORK, INFO)



PURPOSE
            ZGELSD 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.




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

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

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

           A         (input)
                     A is COMPLEX*16 array, dimension (LDA,N)
                     On entry, the M-by-N matrix A.
                     On exit, A has been destroyed.

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

           B         (input/output)
                     B is COMPLEX*16 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       (input)
                     LDB is INTEGER
                     The leading dimension of the array B.  LDB >= max(1,M,N).

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

           WORK      (output)
                     WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
                     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

           LWORK     (input)
                     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     (output)
                     RWORK is DOUBLE PRECISION 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     (output)
                     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      (output)
                     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.



LAPACK routine                  31 October 2017                      ZGELSD(3)