SUBROUTINE SLASD7( ICOMPQ, NL, NR, SQRE, K, D, Z, ZW, VF, VFW, VL,
$ VLW, ALPHA, BETA, DSIGMA, IDX, IDXP, IDXQ,
$ PERM, GIVPTR, GIVCOL, LDGCOL, GIVNUM, LDGNUM,
$ C, S, INFO )
*
* -- LAPACK auxiliary routine (version 3.1) --
* Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
* November 2006
*
* .. Scalar Arguments ..
INTEGER GIVPTR, ICOMPQ, INFO, K, LDGCOL, LDGNUM, NL,
$ NR, SQRE
REAL ALPHA, BETA, C, S
* ..
* .. Array Arguments ..
INTEGER GIVCOL( LDGCOL, * ), IDX( * ), IDXP( * ),
$ IDXQ( * ), PERM( * )
REAL D( * ), DSIGMA( * ), GIVNUM( LDGNUM, * ),
$ VF( * ), VFW( * ), VL( * ), VLW( * ), Z( * ),
$ ZW( * )
* ..
*
* Purpose
* =======
*
* SLASD7 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.
*
* SLASD7 is called from SLASD6.
*
* Arguments
* =========
*
* ICOMPQ (input) INTEGER
* Specifies whether singular vectors are to be computed
* in compact form, as follows:
* = 0: Compute singular values only.
* = 1: Compute singular vectors of upper
* bidiagonal matrix in compact form.
*
* 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
* N = NL + NR + 1 rows and
* M = N + SQRE >= N columns.
*
* K (output) INTEGER
* Contains the dimension of the non-deflated matrix, this is
* the order of the related secular equation. 1 <= K <=N.
*
* D (input/output) 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.
*
* Z (output) REAL array, dimension ( M )
* On exit Z contains the updating row vector in the secular
* equation.
*
* ZW (workspace) REAL array, dimension ( M )
* Workspace for Z.
*
* VF (input/output) REAL 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.
*
* VFW (workspace) REAL array, dimension ( M )
* Workspace for VF.
*
* VL (input/output) REAL 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.
*
* VLW (workspace) REAL array, dimension ( M )
* Workspace for VL.
*
* ALPHA (input) REAL
* Contains the diagonal element associated with the added row.
*
* BETA (input) REAL
* Contains the off-diagonal element associated with the added
* row.
*
* DSIGMA (output) REAL array, dimension ( N )
* Contains a copy of the diagonal elements (K-1 singular values
* and one zero) in the secular equation.
*
* IDX (workspace) INTEGER array, dimension ( N )
* This will contain the permutation used to sort the contents of
* D into ascending order.
*
* IDXP (workspace) 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.
*
* IDXQ (input) 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 half 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.
*
* PERM (output) INTEGER array, dimension ( N )
* The permutations (from deflation and sorting) to be applied
* to each singular 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
* The leading dimension of GIVCOL, must be at least N.
*
* GIVNUM (output) REAL 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, must be at least N.
*
* C (output) REAL
* 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) REAL
* S contains garbage if SQRE =0 and the S-value of a Givens
* rotation related to the right null space if SQRE = 1.
*
* INFO (output) INTEGER
* = 0: successful exit.
* < 0: if INFO = -i, the i-th argument had an illegal value.
*
* Further Details
* ===============
*
* Based on contributions by
* Ming Gu and Huan Ren, Computer Science Division, University of
* California at Berkeley, USA
*
* =====================================================================
*
* .. Parameters ..
REAL ZERO, ONE, TWO, EIGHT
PARAMETER ( ZERO = 0.0E+0, ONE = 1.0E+0, TWO = 2.0E+0,
$ EIGHT = 8.0E+0 )
* ..
* .. Local Scalars ..
*
INTEGER I, IDXI, IDXJ, IDXJP, J, JP, JPREV, K2, M, N,
$ NLP1, NLP2
REAL EPS, HLFTOL, TAU, TOL, Z1
* ..
* .. External Subroutines ..
EXTERNAL SCOPY, SLAMRG, SROT, XERBLA
* ..
* .. External Functions ..
REAL SLAMCH, SLAPY2
EXTERNAL SLAMCH, SLAPY2
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, MAX
* ..
* .. Executable Statements ..
*
* Test the input parameters.
*
INFO = 0
N = NL + NR + 1
M = N + SQRE
*
IF( ( ICOMPQ.LT.0 ) .OR. ( ICOMPQ.GT.1 ) ) THEN
INFO = -1
ELSE IF( NL.LT.1 ) THEN
INFO = -2
ELSE IF( NR.LT.1 ) THEN
INFO = -3
ELSE IF( ( SQRE.LT.0 ) .OR. ( SQRE.GT.1 ) ) THEN
INFO = -4
ELSE IF( LDGCOL.LT.N ) THEN
INFO = -22
ELSE IF( LDGNUM.LT.N ) THEN
INFO = -24
END IF
IF( INFO.NE.0 ) THEN
CALL XERBLA( 'SLASD7', -INFO )
RETURN
END IF
*
NLP1 = NL + 1
NLP2 = NL + 2
IF( ICOMPQ.EQ.1 ) THEN
GIVPTR = 0
END IF
*
* Generate the first part of the vector Z and move the singular
* values in the first part of D one position backward.
*
Z1 = ALPHA*VL( NLP1 )
VL( NLP1 ) = ZERO
TAU = VF( NLP1 )
DO 10 I = NL, 1, -1
Z( I+1 ) = ALPHA*VL( I )
VL( I ) = ZERO
VF( I+1 ) = VF( I )
D( I+1 ) = D( I )
IDXQ( I+1 ) = IDXQ( I ) + 1
10 CONTINUE
VF( 1 ) = TAU
*
* Generate the second part of the vector Z.
*
DO 20 I = NLP2, M
Z( I ) = BETA*VF( I )
VF( I ) = ZERO
20 CONTINUE
*
* Sort the singular values into increasing order
*
DO 30 I = NLP2, N
IDXQ( I ) = IDXQ( I ) + NLP1
30 CONTINUE
*
* DSIGMA, IDXC, IDXC, and ZW are used as storage space.
*
DO 40 I = 2, N
DSIGMA( I ) = D( IDXQ( I ) )
ZW( I ) = Z( IDXQ( I ) )
VFW( I ) = VF( IDXQ( I ) )
VLW( I ) = VL( IDXQ( I ) )
40 CONTINUE
*
CALL SLAMRG( NL, NR, DSIGMA( 2 ), 1, 1, IDX( 2 ) )
*
DO 50 I = 2, N
IDXI = 1 + IDX( I )
D( I ) = DSIGMA( IDXI )
Z( I ) = ZW( IDXI )
VF( I ) = VFW( IDXI )
VL( I ) = VLW( IDXI )
50 CONTINUE
*
* Calculate the allowable deflation tolerence
*
EPS = SLAMCH( 'Epsilon' )
TOL = MAX( ABS( ALPHA ), ABS( BETA ) )
TOL = EIGHT*EIGHT*EPS*MAX( ABS( D( N ) ), TOL )
*
* There are 2 kinds of deflation -- first a value in the z-vector
* is small, second two (or more) singular values are very close
* together (their difference is small).
*
* If the value in the z-vector is small, we simply permute the
* array so that the corresponding singular value is moved to the
* end.
*
* If two values in the D-vector are close, we perform a two-sided
* rotation designed to make one of the corresponding z-vector
* entries zero, and then permute the array so that the deflated
* singular value is moved to the end.
*
* If there are multiple singular values then the problem deflates.
* Here the number of equal singular values are found. As each equal
* singular value is found, an elementary reflector is computed to
* rotate the corresponding singular subspace so that the
* corresponding components of Z are zero in this new basis.
*
K = 1
K2 = N + 1
DO 60 J = 2, N
IF( ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
IDXP( K2 ) = J
IF( J.EQ.N )
$ GO TO 100
ELSE
JPREV = J
GO TO 70
END IF
60 CONTINUE
70 CONTINUE
J = JPREV
80 CONTINUE
J = J + 1
IF( J.GT.N )
$ GO TO 90
IF( ABS( Z( J ) ).LE.TOL ) THEN
*
* Deflate due to small z component.
*
K2 = K2 - 1
IDXP( K2 ) = J
ELSE
*
* Check if singular values are close enough to allow deflation.
*
IF( ABS( D( J )-D( JPREV ) ).LE.TOL ) THEN
*
* Deflation is possible.
*
S = Z( JPREV )
C = Z( J )
*
* Find sqrt(a**2+b**2) without overflow or
* destructive underflow.
*
TAU = SLAPY2( C, S )
Z( J ) = TAU
Z( JPREV ) = ZERO
C = C / TAU
S = -S / TAU
*
* Record the appropriate Givens rotation
*
IF( ICOMPQ.EQ.1 ) THEN
GIVPTR = GIVPTR + 1
IDXJP = IDXQ( IDX( JPREV )+1 )
IDXJ = IDXQ( IDX( J )+1 )
IF( IDXJP.LE.NLP1 ) THEN
IDXJP = IDXJP - 1
END IF
IF( IDXJ.LE.NLP1 ) THEN
IDXJ = IDXJ - 1
END IF
GIVCOL( GIVPTR, 2 ) = IDXJP
GIVCOL( GIVPTR, 1 ) = IDXJ
GIVNUM( GIVPTR, 2 ) = C
GIVNUM( GIVPTR, 1 ) = S
END IF
CALL SROT( 1, VF( JPREV ), 1, VF( J ), 1, C, S )
CALL SROT( 1, VL( JPREV ), 1, VL( J ), 1, C, S )
K2 = K2 - 1
IDXP( K2 ) = JPREV
JPREV = J
ELSE
K = K + 1
ZW( K ) = Z( JPREV )
DSIGMA( K ) = D( JPREV )
IDXP( K ) = JPREV
JPREV = J
END IF
END IF
GO TO 80
90 CONTINUE
*
* Record the last singular value.
*
K = K + 1
ZW( K ) = Z( JPREV )
DSIGMA( K ) = D( JPREV )
IDXP( K ) = JPREV
*
100 CONTINUE
*
* Sort the singular values into DSIGMA. The singular values which
* were not deflated go into the first K slots of DSIGMA, except
* that DSIGMA(1) is treated separately.
*
DO 110 J = 2, N
JP = IDXP( J )
DSIGMA( J ) = D( JP )
VFW( J ) = VF( JP )
VLW( J ) = VL( JP )
110 CONTINUE
IF( ICOMPQ.EQ.1 ) THEN
DO 120 J = 2, N
JP = IDXP( J )
PERM( J ) = IDXQ( IDX( JP )+1 )
IF( PERM( J ).LE.NLP1 ) THEN
PERM( J ) = PERM( J ) - 1
END IF
120 CONTINUE
END IF
*
* The deflated singular values go back into the last N - K slots of
* D.
*
CALL SCOPY( N-K, DSIGMA( K+1 ), 1, D( K+1 ), 1 )
*
* Determine DSIGMA(1), DSIGMA(2), Z(1), VF(1), VL(1), VF(M), and
* VL(M).
*
DSIGMA( 1 ) = ZERO
HLFTOL = TOL / TWO
IF( ABS( DSIGMA( 2 ) ).LE.HLFTOL )
$ DSIGMA( 2 ) = HLFTOL
IF( M.GT.N ) THEN
Z( 1 ) = SLAPY2( Z1, Z( M ) )
IF( Z( 1 ).LE.TOL ) THEN
C = ONE
S = ZERO
Z( 1 ) = TOL
ELSE
C = Z1 / Z( 1 )
S = -Z( M ) / Z( 1 )
END IF
CALL SROT( 1, VF( M ), 1, VF( 1 ), 1, C, S )
CALL SROT( 1, VL( M ), 1, VL( 1 ), 1, C, S )
ELSE
IF( ABS( Z1 ).LE.TOL ) THEN
Z( 1 ) = TOL
ELSE
Z( 1 ) = Z1
END IF
END IF
*
* Restore Z, VF, and VL.
*
CALL SCOPY( K-1, ZW( 2 ), 1, Z( 2 ), 1 )
CALL SCOPY( N-1, VFW( 2 ), 1, VF( 2 ), 1 )
CALL SCOPY( N-1, VLW( 2 ), 1, VL( 2 ), 1 )
*
RETURN
*
* End of SLASD7
*
END