slasd2()

subroutine slasd2	(	integer	nl,
		integer	nr,
		integer	sqre,
		integer	k,
		real, dimension( * )	d,
		real, dimension( * )	z,
		real	alpha,
		real	beta,
		real, dimension( ldu, * )	u,
		integer	ldu,
		real, dimension( ldvt, * )	vt,
		integer	ldvt,
		real, dimension( * )	dsigma,
		real, dimension( ldu2, * )	u2,
		integer	ldu2,
		real, dimension( ldvt2, * )	vt2,
		integer	ldvt2,
		integer, dimension( * )	idxp,
		integer, dimension( * )	idx,
		integer, dimension( * )	idxc,
		integer, dimension( * )	idxq,
		integer, dimension( * )	coltyp,
		integer	info )

SLASD2 merges the two sets of singular values together into a single sorted set. Used by sbdsdc.

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Purpose:

!>
!> SLASD2 merges the two sets of singular values together into a single
!> sorted set.  Then it tries to deflate the size of the problem.
!> There are two ways in which deflation can occur:  when two or more
!> singular values are close together or if there is a tiny entry in the
!> Z vector.  For each such occurrence the order of the related secular
!> equation problem is reduced by one.
!>
!> SLASD2 is called from SLASD1.
!> 

Parameters

[in]	NL	!> NL is INTEGER !> The row dimension of the upper block. NL >= 1. !>
[in]	NR	!> NR is INTEGER !> The row dimension of the lower block. NR >= 1. !>
[in]	SQRE	!> SQRE is INTEGER !> = 0: the lower block is an NR-by-NR square matrix. !> = 1: the lower block is an NR-by-(NR+1) rectangular matrix. !> !> The bidiagonal matrix has N = NL + NR + 1 rows and !> M = N + SQRE >= N columns. !>
[out]	K	!> K is INTEGER !> Contains the dimension of the non-deflated matrix, !> This is the order of the related secular equation. 1 <= K <=N. !>
[in,out]	D	!> D is REAL array, dimension (N) !> On entry D contains the singular values of the two submatrices !> to be combined. On exit D contains the trailing (N-K) updated !> singular values (those which were deflated) sorted into !> increasing order. !>
[out]	Z	!> Z is REAL array, dimension (N) !> On exit Z contains the updating row vector in the secular !> equation. !>
[in]	ALPHA	!> ALPHA is REAL !> Contains the diagonal element associated with the added row. !>
[in]	BETA	!> BETA is REAL !> Contains the off-diagonal element associated with the added !> row. !>
[in,out]	U	!> U is REAL array, dimension (LDU,N) !> On entry U contains the left singular vectors of two !> submatrices in the two square blocks with corners at (1,1), !> (NL, NL), and (NL+2, NL+2), (N,N). !> On exit U contains the trailing (N-K) updated left singular !> vectors (those which were deflated) in its last N-K columns. !>
[in]	LDU	!> LDU is INTEGER !> The leading dimension of the array U. LDU >= N. !>
[in,out]	VT	!> VT is REAL array, dimension (LDVT,M) !> On entry VT**T contains the right singular vectors of two !> submatrices in the two square blocks with corners at (1,1), !> (NL+1, NL+1), and (NL+2, NL+2), (M,M). !> On exit VT**T contains the trailing (N-K) updated right singular !> vectors (those which were deflated) in its last N-K columns. !> In case SQRE =1, the last row of VT spans the right null !> space. !>
[in]	LDVT	!> LDVT is INTEGER !> The leading dimension of the array VT. LDVT >= M. !>
[out]	DSIGMA	!> DSIGMA is REAL array, dimension (N) !> Contains a copy of the diagonal elements (K-1 singular values !> and one zero) in the secular equation. !>
[out]	U2	!> U2 is REAL array, dimension (LDU2,N) !> Contains a copy of the first K-1 left singular vectors which !> will be used by SLASD3 in a matrix multiply (SGEMM) to solve !> for the new left singular vectors. U2 is arranged into four !> blocks. The first block contains a column with 1 at NL+1 and !> zero everywhere else; the second block contains non-zero !> entries only at and above NL; the third contains non-zero !> entries only below NL+1; and the fourth is dense. !>
[in]	LDU2	!> LDU2 is INTEGER !> The leading dimension of the array U2. LDU2 >= N. !>
[out]	VT2	!> VT2 is REAL array, dimension (LDVT2,N) !> VT2**T contains a copy of the first K right singular vectors !> which will be used by SLASD3 in a matrix multiply (SGEMM) to !> solve for the new right singular vectors. VT2 is arranged into !> three blocks. The first block contains a row that corresponds !> to the special 0 diagonal element in SIGMA; the second block !> contains non-zeros only at and before NL +1; the third block !> contains non-zeros only at and after NL +2. !>
[in]	LDVT2	!> LDVT2 is INTEGER !> The leading dimension of the array VT2. LDVT2 >= M. !>
[out]	IDXP	!> IDXP is INTEGER array, dimension (N) !> This will contain the permutation used to place deflated !> values of D at the end of the array. On output IDXP(2:K) !> points to the nondeflated D-values and IDXP(K+1:N) !> points to the deflated singular values. !>
[out]	IDX	!> IDX is INTEGER array, dimension (N) !> This will contain the permutation used to sort the contents of !> D into ascending order. !>
[out]	IDXC	!> IDXC is INTEGER array, dimension (N) !> This will contain the permutation used to arrange the columns !> of the deflated U matrix into three groups: the first group !> contains non-zero entries only at and above NL, the second !> contains non-zero entries only below NL+2, and the third is !> dense. !>
[in,out]	IDXQ	!> IDXQ is INTEGER array, dimension (N) !> This contains the permutation which separately sorts the two !> sub-problems in D into ascending order. Note that entries in !> the first hlaf of this permutation must first be moved one !> position backward; and entries in the second half !> must first have NL+1 added to their values. !>
[out]	COLTYP	!> COLTYP is INTEGER array, dimension (N) !> As workspace, this will contain a label which will indicate !> which of the following types a column in the U2 matrix or a !> row in the VT2 matrix is: !> 1 : non-zero in the upper half only !> 2 : non-zero in the lower half only !> 3 : dense !> 4 : deflated !> !> On exit, it is an array of dimension 4, with COLTYP(I) being !> the dimension of the I-th type columns. !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit. !> < 0: if INFO = -i, the i-th argument had an illegal value. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA