*> \brief \b SLASD7 merges the two sets of singular values together into a single sorted set. Then it tries to deflate the size of the problem. Used by sbdsdc. * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download SLASD7 + dependencies *> *> [TGZ] *> *> [ZIP] *> *> [TXT] *> \endhtmlonly * * Definition: * =========== * * 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 ) * * .. 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( * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> 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. *> \endverbatim * * Arguments: * ========== * *> \param[in] ICOMPQ *> \verbatim *> ICOMPQ is 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. *> \endverbatim *> *> \param[in] NL *> \verbatim *> NL is INTEGER *> The row dimension of the upper block. NL >= 1. *> \endverbatim *> *> \param[in] NR *> \verbatim *> NR is INTEGER *> The row dimension of the lower block. NR >= 1. *> \endverbatim *> *> \param[in] SQRE *> \verbatim *> 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. *> \endverbatim *> *> \param[out] K *> \verbatim *> K is INTEGER *> Contains the dimension of the non-deflated matrix, this is *> the order of the related secular equation. 1 <= K <=N. *> \endverbatim *> *> \param[in,out] D *> \verbatim *> 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. *> \endverbatim *> *> \param[out] Z *> \verbatim *> Z is REAL array, dimension ( M ) *> On exit Z contains the updating row vector in the secular *> equation. *> \endverbatim *> *> \param[out] ZW *> \verbatim *> ZW is REAL array, dimension ( M ) *> Workspace for Z. *> \endverbatim *> *> \param[in,out] VF *> \verbatim *> VF is 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. *> \endverbatim *> *> \param[out] VFW *> \verbatim *> VFW is REAL array, dimension ( M ) *> Workspace for VF. *> \endverbatim *> *> \param[in,out] VL *> \verbatim *> VL is 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. *> \endverbatim *> *> \param[out] VLW *> \verbatim *> VLW is REAL array, dimension ( M ) *> Workspace for VL. *> \endverbatim *> *> \param[in] ALPHA *> \verbatim *> ALPHA is REAL *> Contains the diagonal element associated with the added row. *> \endverbatim *> *> \param[in] BETA *> \verbatim *> BETA is REAL *> Contains the off-diagonal element associated with the added *> row. *> \endverbatim *> *> \param[out] DSIGMA *> \verbatim *> DSIGMA is REAL array, dimension ( N ) *> Contains a copy of the diagonal elements (K-1 singular values *> and one zero) in the secular equation. *> \endverbatim *> *> \param[out] IDX *> \verbatim *> IDX is INTEGER array, dimension ( N ) *> This will contain the permutation used to sort the contents of *> D into ascending order. *> \endverbatim *> *> \param[out] IDXP *> \verbatim *> 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. *> \endverbatim *> *> \param[in] IDXQ *> \verbatim *> 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 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. *> \endverbatim *> *> \param[out] PERM *> \verbatim *> PERM is INTEGER array, dimension ( N ) *> The permutations (from deflation and sorting) to be applied *> to each singular block. Not referenced if ICOMPQ = 0. *> \endverbatim *> *> \param[out] GIVPTR *> \verbatim *> GIVPTR is INTEGER *> The number of Givens rotations which took place in this *> subproblem. Not referenced if ICOMPQ = 0. *> \endverbatim *> *> \param[out] GIVCOL *> \verbatim *> GIVCOL is 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. *> \endverbatim *> *> \param[in] LDGCOL *> \verbatim *> LDGCOL is INTEGER *> The leading dimension of GIVCOL, must be at least N. *> \endverbatim *> *> \param[out] GIVNUM *> \verbatim *> GIVNUM is 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. *> \endverbatim *> *> \param[in] LDGNUM *> \verbatim *> LDGNUM is INTEGER *> The leading dimension of GIVNUM, must be at least N. *> \endverbatim *> *> \param[out] C *> \verbatim *> C is REAL *> C contains garbage if SQRE =0 and the C-value of a Givens *> rotation related to the right null space if SQRE = 1. *> \endverbatim *> *> \param[out] S *> \verbatim *> S is REAL *> S contains garbage if SQRE =0 and the S-value of a Givens *> rotation related to the right null space if SQRE = 1. *> \endverbatim *> *> \param[out] INFO *> \verbatim *> INFO is INTEGER *> = 0: successful exit. *> < 0: if INFO = -i, the i-th argument had an illegal value. *> \endverbatim * * Authors: * ======== * *> \author Univ. of Tennessee *> \author Univ. of California Berkeley *> \author Univ. of Colorado Denver *> \author NAG Ltd. * *> \date December 2016 * *> \ingroup OTHERauxiliary * *> \par Contributors: * ================== *> *> Ming Gu and Huan Ren, Computer Science Division, University of *> California at Berkeley, USA *> * ===================================================================== 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 3.7.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * December 2016 * * .. 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( * ) * .. * * ===================================================================== * * .. 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