DLASD6 man page on IRIX

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DLASD6(3S)							    DLASD6(3S)

NAME
     DLASD6 - compute the SVD of an updated upper bidiagonal matrix B obtained
     by merging two smaller ones by appending a row

SYNOPSIS
     SUBROUTINE DLASD6( ICOMPQ, NL, NR, SQRE, D, VF, VL, ALPHA, BETA, IDXQ,
			PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM, POLES,
			DIFL, DIFR, Z, K, C, S, WORK, IWORK, INFO )

	 INTEGER	GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL, NR, SQRE

	 DOUBLE		PRECISION ALPHA, BETA, C, S

	 INTEGER	GIVCOL( LDGCOL, * ), IDXQ( * ), IWORK( * ), PERM( * )

	 DOUBLE		PRECISION D( * ), DIFL( * ), DIFR( * ), GIVNUM(
			LDGNUM, * ), POLES( LDGNUM, * ), VF( * ), VL( * ),
			WORK( * ), Z( * )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     DLASD6 computes the SVD of an updated upper bidiagonal matrix B obtained
     by merging two smaller ones by appending a row. This routine is used only
     for the problem which requires all singular values and optionally
     singular vector matrices in factored form.	 B is an N-by-M matrix with N
     = NL + NR + 1 and M = N + SQRE.  A related subroutine, DLASD1, handles
     the case in which all singular values and singular vectors of the
     bidiagonal matrix are desired.

     DLASD6 computes the SVD as follows:

		   ( D1(in)  0	  0	0 )
       B = U(in) * (   Z1'   a	 Z2'	b ) * VT(in)
		   (   0     0	 D2(in) 0 )

	 = U(out) * ( D(out) 0) * VT(out)

     where Z' = (Z1' a Z2' b) = u' VT', and u is a vector of dimension M with
     ALPHA and BETA in the NL+1 and NL+2 th entries and zeros elsewhere; and
     the entry b is empty if SQRE = 0.

									Page 1

DLASD6(3S)							    DLASD6(3S)

     The singular values of B can be computed using D1, D2, the first
     components of all the right singular vectors of the lower block, and the
     last components of all the right singular vectors of the upper block.
     These components are stored and updated in VF and VL, respectively, in
     DLASD6. Hence U and VT are not explicitly referenced.

     The singular values are stored in D. The algorithm consists of two
     stages:

	   The first stage consists of deflating the size of the problem
	   when there are multiple singular values or if there is a zero
	   in the Z vector. For each such occurence the dimension of the
	   secular equation problem is reduced by one. This stage is
	   performed by the routine DLASD7.

	   The second stage consists of calculating the updated
	   singular values. This is done by finding the roots of the
	   secular equation via the routine DLASD4 (as called by DLASD8).
	   This routine also updates VF and VL and computes the distances
	   between the updated singular values and the old singular
	   values.

     DLASD6 is called from DLASDA.

ARGUMENTS
     ICOMPQ (input) INTEGER Specifies whether singular vectors are to be
     computed in factored form:
     = 0: Compute singular values only.
     = 1: Compute singular vectors in factored form as well.

     NL	    (input) INTEGER
	    The row dimension of the upper block.  NL >= 1.

     NR	    (input) INTEGER
	    The row dimension of the lower block.  NR >= 1.

     SQRE   (input) 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 row dimension N = NL + NR + 1, and
	    column dimension M = N + SQRE.

     D	    (input/output) DOUBLE PRECISION array, dimension ( NL+NR+1 ).
	    On entry D(1:NL,1:NL) contains the singular values of the
	    upper block, and D(NL+2:N) contains the singular values
	    of the lower block. On exit D(1:N) contains the singular values of
	    the modified matrix.

									Page 2

DLASD6(3S)							    DLASD6(3S)

     VF	    (input/output) DOUBLE PRECISION array, dimension ( M )
	    On entry, VF(1:NL+1) contains the first components of all
	    right singular vectors of the upper block; and VF(NL+2:M) contains
	    the first components of all right singular vectors of the lower
	    block. On exit, VF contains the first components of all right
	    singular vectors of the bidiagonal matrix.

     VL	    (input/output) DOUBLE PRECISION array, dimension ( M )
	    On entry, VL(1:NL+1) contains the  last components of all
	    right singular vectors of the upper block; and VL(NL+2:M) contains
	    the last components of all right singular vectors of the lower
	    block. On exit, VL contains the last components of all right
	    singular vectors of the bidiagonal matrix.

     ALPHA  (input) DOUBLE PRECISION
	    Contains the diagonal element associated with the added row.

     BETA   (input) DOUBLE PRECISION
	    Contains the off-diagonal element associated with the added row.

     IDXQ   (output) INTEGER array, dimension ( N )
	    This contains the permutation which will reintegrate the
	    subproblem just solved back into sorted order, i.e.	 D( IDXQ( I =
	    1, N ) ) will be in ascending order.

     PERM   (output) INTEGER array, dimension ( N )
	    The permutations (from deflation and sorting) to be applied to
	    each block. Not referenced if ICOMPQ = 0.

	    GIVPTR (output) INTEGER The number of Givens rotations which took
	    place in this subproblem. Not referenced if ICOMPQ = 0.

	    GIVCOL (output) INTEGER array, dimension ( LDGCOL, 2 ) Each pair
	    of numbers indicates a pair of columns to take place in a Givens
	    rotation. Not referenced if ICOMPQ = 0.

	    LDGCOL (input) INTEGER leading dimension of GIVCOL, must be at
	    least N.

	    GIVNUM (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
	    Each number indicates the C or S value to be used in the
	    corresponding Givens rotation. Not referenced if ICOMPQ = 0.

	    LDGNUM (input) INTEGER The leading dimension of GIVNUM and POLES,
	    must be at least N.

     POLES  (output) DOUBLE PRECISION array, dimension ( LDGNUM, 2 )
	    On exit, POLES(1,*) is an array containing the new singular values
	    obtained from solving the secular equation, and POLES(2,*) is an
	    array containing the poles in the secular equation. Not referenced
	    if ICOMPQ = 0.

									Page 3

DLASD6(3S)							    DLASD6(3S)

     DIFL   (output) DOUBLE PRECISION array, dimension ( N )
	    On exit, DIFL(I) is the distance between I-th updated (undeflated)
	    singular value and the I-th (undeflated) old singular value.

     DIFR   (output) DOUBLE PRECISION array,
	    dimension ( LDGNUM, 2 ) if ICOMPQ = 1 and dimension ( N ) if
	    ICOMPQ = 0.	 On exit, DIFR(I, 1) is the distance between I-th
	    updated (undeflated) singular value and the I+1-th (undeflated)
	    old singular value.

	    If ICOMPQ = 1, DIFR(1:K,2) is an array containing the normalizing
	    factors for the right singular vector matrix.

	    See DLASD8 for details on DIFL and DIFR.

     Z	    (output) DOUBLE PRECISION array, dimension ( M )
	    The first elements of this array contain the components of the
	    deflation-adjusted updating row vector.

     K	    (output) INTEGER
	    Contains the dimension of the non-deflated matrix, This is the
	    order of the related secular equation. 1 <= K <=N.

     C	    (output) DOUBLE PRECISION
	    C contains garbage if SQRE =0 and the C-value of a Givens rotation
	    related to the right null space if SQRE = 1.

     S	    (output) DOUBLE PRECISION
	    S contains garbage if SQRE =0 and the S-value of a Givens rotation
	    related to the right null space if SQRE = 1.

     WORK   (workspace) DOUBLE PRECISION array, dimension ( 4 * M )

     IWORK  (workspace) INTEGER array, dimension ( 3 * N )

     INFO   (output) INTEGER
	    = 0:  successful exit.
	    < 0:  if INFO = -i, the i-th argument had an illegal value.
	    > 0:  if INFO = 1, an singular value did not converge

FURTHER DETAILS
     Based on contributions by
	Ming Gu and Huan Ren, Computer Science Division, University of
	California at Berkeley, USA

SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.

									Page 4

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