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



NAME
       SBDSQR

SYNOPSIS
       SUBROUTINE SBDSQR (UPLO, N, NCVT, NRU, NCC, D, E, VT, LDVT, U, LDU, C,
           LDC, WORK, INFO)



PURPOSE
            SBDSQR computes the singular values and, optionally, the right and/or
            left singular vectors from the singular value decomposition (SVD) of
            a real N-by-N (upper or lower) bidiagonal matrix B using the implicit
            zero-shift QR algorithm.  The SVD of B has the form

               B = Q * S * P**T

            where S is the diagonal matrix of singular values, Q is an orthogonal
            matrix of left singular vectors, and P is an orthogonal matrix of
            right singular vectors.  If left singular vectors are requested, this
            subroutine actually returns U*Q instead of Q, and, if right singular
            vectors are requested, this subroutine returns P**T*VT instead of
            P**T, for given real input matrices U and VT.  When U and VT are the
            orthogonal matrices that reduce a general matrix A to bidiagonal
            form:  A = U*B*VT, as computed by SGEBRD, then

               A = (U*Q) * S * (P**T*VT)

            is the SVD of A.  Optionally, the subroutine may also compute Q**T*C
            for a given real input matrix C.





ARGUMENTS
           UPLO      (input)
                     UPLO is CHARACTER*1
                     = 'U':  B is upper bidiagonal;
                     = 'L':  B is lower bidiagonal.

           N         (input)
                     N is INTEGER
                     The order of the matrix B.  N >= 0.

           NCVT      (input)
                     NCVT is INTEGER
                     The number of columns of the matrix VT. NCVT >= 0.

           NRU       (input)
                     NRU is INTEGER
                     The number of rows of the matrix U. NRU >= 0.

           NCC       (input)
                     NCC is INTEGER
                     The number of columns of the matrix C. NCC >= 0.

           D         (input/output)
                     D is REAL array, dimension (N)
                     On entry, the n diagonal elements of the bidiagonal matrix B.
                     On exit, if INFO=0, the singular values of B in decreasing
                     order.

           E         (input/output)
                     E is REAL array, dimension (N-1)
                     On entry, the N-1 offdiagonal elements of the bidiagonal
                     matrix B.
                     On exit, if INFO = 0, E is destroyed; if INFO > 0, D and E
                     will contain the diagonal and superdiagonal elements of a
                     bidiagonal matrix orthogonally equivalent to the one given
                     as input.

           VT        (input/output)
                     VT is REAL array, dimension (LDVT, NCVT)
                     On entry, an N-by-NCVT matrix VT.
                     On exit, VT is overwritten by P**T * VT.
                     Not referenced if NCVT = 0.

           LDVT      (input)
                     LDVT is INTEGER
                     The leading dimension of the array VT.
                     LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0.

           U         (input/output)
                     U is REAL array, dimension (LDU, N)
                     On entry, an NRU-by-N matrix U.
                     On exit, U is overwritten by U * Q.
                     Not referenced if NRU = 0.

           LDU       (input)
                     LDU is INTEGER
                     The leading dimension of the array U.  LDU >= max(1,NRU).

           C         (input/output)
                     C is REAL array, dimension (LDC, NCC)
                     On entry, an N-by-NCC matrix C.
                     On exit, C is overwritten by Q**T * C.
                     Not referenced if NCC = 0.

           LDC       (input)
                     LDC is INTEGER
                     The leading dimension of the array C.
                     LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0.

           WORK      (output)
                     WORK is REAL array, dimension (4*N)

           INFO      (output)
                     INFO is INTEGER
                     = 0:  successful exit
                     < 0:  If INFO = -i, the i-th argument had an illegal value
                     > 0:
                        if NCVT = NRU = NCC = 0,
                           = 1, a split was marked by a positive value in E
                           = 2, current block of Z not diagonalized after 30*N
                                iterations (in inner while loop)
                           = 3, termination criterion of outer while loop not met
                                (program created more than N unreduced blocks)
                        else NCVT = NRU = NCC = 0,
                              the algorithm did not converge; D and E contain the
                              elements of a bidiagonal matrix which is orthogonally
                              similar to the input matrix B;  if INFO = i, i
                              elements of E have not converged to zero.



       Internal Parameters:


             TOLMUL  REAL, default = max(10,min(100,EPS**(-1/8)))
                     TOLMUL controls the convergence criterion of the QR loop.
                     If it is positive, TOLMUL*EPS is the desired relative
                        precision in the computed singular values.
                     If it is negative, abs(TOLMUL*EPS*sigma_max) is the
                        desired absolute accuracy in the computed singular
                        values (corresponds to relative accuracy
                        abs(TOLMUL*EPS) in the largest singular value.
                     abs(TOLMUL) should be between 1 and 1/EPS, and preferably
                        between 10 (for fast convergence) and .1/EPS
                        (for there to be some accuracy in the results).
                     Default is to lose at either one eighth or 2 of the
                        available decimal digits in each computed singular value
                        (whichever is smaller).

             MAXITR  INTEGER, default = 6
                     MAXITR controls the maximum number of passes of the
                     algorithm through its inner loop. The algorithms stops
                     (and so fails to converge) if the number of passes
                     through the inner loop exceeds MAXITR*N**2.



LAPACK routine                  31 October 2017                      SBDSQR(3)