sbdsqr.f3

!>
!> 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.
!>
!> See  by J. Demmel and W. Kahan,
!> LAPACK Working Note #3 (or SIAM J. Sci. Statist. Comput. vol. 11,
!> no. 5, pp. 873-912, Sept 1990) and
!>  by
!> B. Parlett and V. Fernando, Technical Report CPAM-554, Mathematics
!> Department, University of California at Berkeley, July 1992
!> for a detailed description of the algorithm.
!> 

UPLO

!>          UPLO is CHARACTER*1
!>          = 'U':  B is upper bidiagonal;
!>          = 'L':  B is lower bidiagonal.
!> 

N

!>          N is INTEGER
!>          The order of the matrix B.  N >= 0.
!> 

NCVT

!>          NCVT is INTEGER
!>          The number of columns of the matrix VT. NCVT >= 0.
!> 

NRU

!>          NRU is INTEGER
!>          The number of rows of the matrix U. NRU >= 0.
!> 

NCC

!>          NCC is INTEGER
!>          The number of columns of the matrix C. NCC >= 0.
!> 

D

!>          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

!>          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

!>          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

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

U

!>          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

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

C

!>          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

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

WORK

!>          WORK is REAL array, dimension (LWORK)
!>          LWORK = 4*N, if NCVT = NRU = NCC = 0, and
!>          LWORK = 4*(N-1), otherwise
!> 

INFO

!>          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.
!> 

!>  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.
!> 

!>  Bug report from Cezary Dendek.
!>  On March 23rd 2017, the INTEGER variable MAXIT = MAXITR*N**2 is
!>  removed since it can overflow pretty easily (for N larger or equal
!>  than 18,919). We instead use MAXITDIVN = MAXITR*N.
!> 

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