zbdsqr()

subroutine zbdsqr	(	character	uplo,
		integer	n,
		integer	ncvt,
		integer	nru,
		integer	ncc,
		double precision, dimension( * )	d,
		double precision, dimension( * )	e,
		complex*16, dimension( ldvt, * )	vt,
		integer	ldvt,
		complex*16, dimension( ldu, * )	u,
		integer	ldu,
		complex*16, dimension( ldc, * )	c,
		integer	ldc,
		double precision, dimension( * )	rwork,
		integer	info )

ZBDSQR

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Purpose:

!>
!> ZBDSQR 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**H
!>
!> 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**H*VT instead of
!> P**H, for given complex input matrices U and VT.  When U and VT are
!> the unitary matrices that reduce a general matrix A to bidiagonal
!> form: A = U*B*VT, as computed by ZGEBRD, then
!>
!>    A = (U*Q) * S * (P**H*VT)
!>
!> is the SVD of A.  Optionally, the subroutine may also compute Q**H*C
!> for a given complex 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.
!> 

Parameters

[in]	UPLO	!> UPLO is CHARACTER*1 !> = 'U': B is upper bidiagonal; !> = 'L': B is lower bidiagonal. !>
[in]	N	!> N is INTEGER !> The order of the matrix B. N >= 0. !>
[in]	NCVT	!> NCVT is INTEGER !> The number of columns of the matrix VT. NCVT >= 0. !>
[in]	NRU	!> NRU is INTEGER !> The number of rows of the matrix U. NRU >= 0. !>
[in]	NCC	!> NCC is INTEGER !> The number of columns of the matrix C. NCC >= 0. !>
[in,out]	D	!> D is DOUBLE PRECISION 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. !>
[in,out]	E	!> E is DOUBLE PRECISION 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. !>
[in,out]	VT	!> VT is COMPLEX*16 array, dimension (LDVT, NCVT) !> On entry, an N-by-NCVT matrix VT. !> On exit, VT is overwritten by P**H * VT. !> Not referenced if NCVT = 0. !>
[in]	LDVT	!> LDVT is INTEGER !> The leading dimension of the array VT. !> LDVT >= max(1,N) if NCVT > 0; LDVT >= 1 if NCVT = 0. !>
[in,out]	U	!> U is COMPLEX*16 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. !>
[in]	LDU	!> LDU is INTEGER !> The leading dimension of the array U. LDU >= max(1,NRU). !>
[in,out]	C	!> C is COMPLEX*16 array, dimension (LDC, NCC) !> On entry, an N-by-NCC matrix C. !> On exit, C is overwritten by Q**H * C. !> Not referenced if NCC = 0. !>
[in]	LDC	!> LDC is INTEGER !> The leading dimension of the array C. !> LDC >= max(1,N) if NCC > 0; LDC >=1 if NCC = 0. !>
[out]	RWORK	!> RWORK is DOUBLE PRECISION array, dimension (LRWORK) !> LRWORK = 4*N, if NCVT = NRU = NCC = 0, and !> LRWORK = 4*(N-1), otherwise !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: If INFO = -i, the i-th argument had an illegal value !> > 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  DOUBLE PRECISION, 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.
!>
!> 

Note:

!>  Bug report from Cezary Dendek.
!>  On November 3rd 2023, 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.
!> 

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

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