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dbbcsd.f
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*> \brief \b DBBCSD
*
* =========== DOCUMENTATION ===========
*
* Online html documentation available at
* http://www.netlib.org/lapack/explore-html/
*
*> \htmlonly
*> Download DBBCSD + dependencies
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/dbbcsd.f">
*> [TGZ]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dbbcsd.f">
*> [ZIP]</a>
*> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dbbcsd.f">
*> [TXT]</a>
*> \endhtmlonly
*
* Definition:
* ===========
*
* SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
* THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T,
* V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E,
* B22D, B22E, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS
* INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
* ..
* .. Array Arguments ..
* DOUBLE PRECISION B11D( * ), B11E( * ), B12D( * ), B12E( * ),
* $ B21D( * ), B21E( * ), B22D( * ), B22E( * ),
* $ PHI( * ), THETA( * ), WORK( * )
* DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
* $ V2T( LDV2T, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> DBBCSD computes the CS decomposition of an orthogonal matrix in
*> bidiagonal-block form,
*>
*>
*> [ B11 | B12 0 0 ]
*> [ 0 | 0 -I 0 ]
*> X = [----------------]
*> [ B21 | B22 0 0 ]
*> [ 0 | 0 0 I ]
*>
*> [ C | -S 0 0 ]
*> [ U1 | ] [ 0 | 0 -I 0 ] [ V1 | ]**T
*> = [---------] [---------------] [---------] .
*> [ | U2 ] [ S | C 0 0 ] [ | V2 ]
*> [ 0 | 0 0 I ]
*>
*> X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
*> than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
*> transposed and/or permuted. This can be done in constant time using
*> the TRANS and SIGNS options. See DORCSD for details.)
*>
*> The bidiagonal matrices B11, B12, B21, and B22 are represented
*> implicitly by angles THETA(1:Q) and PHI(1:Q-1).
*>
*> The orthogonal matrices U1, U2, V1T, and V2T are input/output.
*> The input matrices are pre- or post-multiplied by the appropriate
*> singular vector matrices.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOBU1
*> \verbatim
*> JOBU1 is CHARACTER
*> = 'Y': U1 is updated;
*> otherwise: U1 is not updated.
*> \endverbatim
*>
*> \param[in] JOBU2
*> \verbatim
*> JOBU2 is CHARACTER
*> = 'Y': U2 is updated;
*> otherwise: U2 is not updated.
*> \endverbatim
*>
*> \param[in] JOBV1T
*> \verbatim
*> JOBV1T is CHARACTER
*> = 'Y': V1T is updated;
*> otherwise: V1T is not updated.
*> \endverbatim
*>
*> \param[in] JOBV2T
*> \verbatim
*> JOBV2T is CHARACTER
*> = 'Y': V2T is updated;
*> otherwise: V2T is not updated.
*> \endverbatim
*>
*> \param[in] TRANS
*> \verbatim
*> TRANS is CHARACTER
*> = 'T': X, U1, U2, V1T, and V2T are stored in row-major
*> order;
*> otherwise: X, U1, U2, V1T, and V2T are stored in column-
*> major order.
*> \endverbatim
*>
*> \param[in] M
*> \verbatim
*> M is INTEGER
*> The number of rows and columns in X, the orthogonal matrix in
*> bidiagonal-block form.
*> \endverbatim
*>
*> \param[in] P
*> \verbatim
*> P is INTEGER
*> The number of rows in the top-left block of X. 0 <= P <= M.
*> \endverbatim
*>
*> \param[in] Q
*> \verbatim
*> Q is INTEGER
*> The number of columns in the top-left block of X.
*> 0 <= Q <= MIN(P,M-P,M-Q).
*> \endverbatim
*>
*> \param[in,out] THETA
*> \verbatim
*> THETA is DOUBLE PRECISION array, dimension (Q)
*> On entry, the angles THETA(1),...,THETA(Q) that, along with
*> PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
*> form. On exit, the angles whose cosines and sines define the
*> diagonal blocks in the CS decomposition.
*> \endverbatim
*>
*> \param[in,out] PHI
*> \verbatim
*> PHI is DOUBLE PRECISION array, dimension (Q-1)
*> The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
*> THETA(Q), define the matrix in bidiagonal-block form.
*> \endverbatim
*>
*> \param[in,out] U1
*> \verbatim
*> U1 is DOUBLE PRECISION array, dimension (LDU1,P)
*> On entry, a P-by-P matrix. On exit, U1 is postmultiplied
*> by the left singular vector matrix common to [ B11 ; 0 ] and
*> [ B12 0 0 ; 0 -I 0 0 ].
*> \endverbatim
*>
*> \param[in] LDU1
*> \verbatim
*> LDU1 is INTEGER
*> The leading dimension of the array U1, LDU1 >= MAX(1,P).
*> \endverbatim
*>
*> \param[in,out] U2
*> \verbatim
*> U2 is DOUBLE PRECISION array, dimension (LDU2,M-P)
*> On entry, an (M-P)-by-(M-P) matrix. On exit, U2 is
*> postmultiplied by the left singular vector matrix common to
*> [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].
*> \endverbatim
*>
*> \param[in] LDU2
*> \verbatim
*> LDU2 is INTEGER
*> The leading dimension of the array U2, LDU2 >= MAX(1,M-P).
*> \endverbatim
*>
*> \param[in,out] V1T
*> \verbatim
*> V1T is DOUBLE PRECISION array, dimension (LDV1T,Q)
*> On entry, a Q-by-Q matrix. On exit, V1T is premultiplied
*> by the transpose of the right singular vector
*> matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].
*> \endverbatim
*>
*> \param[in] LDV1T
*> \verbatim
*> LDV1T is INTEGER
*> The leading dimension of the array V1T, LDV1T >= MAX(1,Q).
*> \endverbatim
*>
*> \param[in,out] V2T
*> \verbatim
*> V2T is DOUBLE PRECISION array, dimension (LDV2T,M-Q)
*> On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is
*> premultiplied by the transpose of the right
*> singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
*> [ B22 0 0 ; 0 0 I ].
*> \endverbatim
*>
*> \param[in] LDV2T
*> \verbatim
*> LDV2T is INTEGER
*> The leading dimension of the array V2T, LDV2T >= MAX(1,M-Q).
*> \endverbatim
*>
*> \param[out] B11D
*> \verbatim
*> B11D is DOUBLE PRECISION array, dimension (Q)
*> When DBBCSD converges, B11D contains the cosines of THETA(1),
*> ..., THETA(Q). If DBBCSD fails to converge, then B11D
*> contains the diagonal of the partially reduced top-left
*> block.
*> \endverbatim
*>
*> \param[out] B11E
*> \verbatim
*> B11E is DOUBLE PRECISION array, dimension (Q-1)
*> When DBBCSD converges, B11E contains zeros. If DBBCSD fails
*> to converge, then B11E contains the superdiagonal of the
*> partially reduced top-left block.
*> \endverbatim
*>
*> \param[out] B12D
*> \verbatim
*> B12D is DOUBLE PRECISION array, dimension (Q)
*> When DBBCSD converges, B12D contains the negative sines of
*> THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
*> B12D contains the diagonal of the partially reduced top-right
*> block.
*> \endverbatim
*>
*> \param[out] B12E
*> \verbatim
*> B12E is DOUBLE PRECISION array, dimension (Q-1)
*> When DBBCSD converges, B12E contains zeros. If DBBCSD fails
*> to converge, then B12E contains the subdiagonal of the
*> partially reduced top-right block.
*> \endverbatim
*>
*> \param[out] B21D
*> \verbatim
*> B21D is DOUBLE PRECISION array, dimension (Q)
*> When DBBCSD converges, B21D contains the negative sines of
*> THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
*> B21D contains the diagonal of the partially reduced bottom-left
*> block.
*> \endverbatim
*>
*> \param[out] B21E
*> \verbatim
*> B21E is DOUBLE PRECISION array, dimension (Q-1)
*> When DBBCSD converges, B21E contains zeros. If DBBCSD fails
*> to converge, then B21E contains the subdiagonal of the
*> partially reduced bottom-left block.
*> \endverbatim
*>
*> \param[out] B22D
*> \verbatim
*> B22D is DOUBLE PRECISION array, dimension (Q)
*> When DBBCSD converges, B22D contains the negative sines of
*> THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
*> B22D contains the diagonal of the partially reduced bottom-right
*> block.
*> \endverbatim
*>
*> \param[out] B22E
*> \verbatim
*> B22E is DOUBLE PRECISION array, dimension (Q-1)
*> When DBBCSD converges, B22E contains zeros. If DBBCSD fails
*> to converge, then B22E contains the subdiagonal of the
*> partially reduced bottom-right block.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK))
*> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> LWORK is INTEGER
*> The dimension of the array WORK. LWORK >= MAX(1,8*Q).
*>
*> If LWORK = -1, then a workspace query is assumed; the
*> routine only calculates the optimal size of the WORK array,
*> returns this value as the first entry of the work array, and
*> no error message related to LWORK is issued by XERBLA.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> INFO is INTEGER
*> = 0: successful exit.
*> < 0: if INFO = -i, the i-th argument had an illegal value.
*> > 0: if DBBCSD did not converge, INFO specifies the number
*> of nonzero entries in PHI, and B11D, B11E, etc.,
*> contain the partially reduced matrix.
*> \endverbatim
*
*> \par Internal Parameters:
* =========================
*>
*> \verbatim
*> TOLMUL DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(-1/8)))
*> TOLMUL controls the convergence criterion of the QR loop.
*> Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
*> are within TOLMUL*EPS of either bound.
*> \endverbatim
*
*> \par References:
* ================
*>
*> [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
*> Algorithms, 50(1):33-65, 2009.
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date June 2016
*
*> \ingroup doubleOTHERcomputational
*
* =====================================================================
SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
$ THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T,
$ V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E,
$ B22D, B22E, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.7.1) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* June 2016
*
* .. Scalar Arguments ..
CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS
INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
* ..
* .. Array Arguments ..
DOUBLE PRECISION B11D( * ), B11E( * ), B12D( * ), B12E( * ),
$ B21D( * ), B21E( * ), B22D( * ), B22E( * ),
$ PHI( * ), THETA( * ), WORK( * )
DOUBLE PRECISION U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
$ V2T( LDV2T, * )
* ..
*
* ===================================================================
*
* .. Parameters ..
INTEGER MAXITR
PARAMETER ( MAXITR = 6 )
DOUBLE PRECISION HUNDRED, MEIGHTH, ONE, PIOVER2, TEN, ZERO
PARAMETER ( HUNDRED = 100.0D0, MEIGHTH = -0.125D0,
$ ONE = 1.0D0, PIOVER2 = 1.57079632679489662D0,
$ TEN = 10.0D0, ZERO = 0.0D0 )
DOUBLE PRECISION NEGONE
PARAMETER ( NEGONE = -1.0D0 )
* ..
* .. Local Scalars ..
LOGICAL COLMAJOR, LQUERY, RESTART11, RESTART12,
$ RESTART21, RESTART22, WANTU1, WANTU2, WANTV1T,
$ WANTV2T
INTEGER I, IMIN, IMAX, ITER, IU1CS, IU1SN, IU2CS,
$ IU2SN, IV1TCS, IV1TSN, IV2TCS, IV2TSN, J,
$ LWORKMIN, LWORKOPT, MAXIT, MINI
DOUBLE PRECISION B11BULGE, B12BULGE, B21BULGE, B22BULGE, DUMMY,
$ EPS, MU, NU, R, SIGMA11, SIGMA21,
$ TEMP, THETAMAX, THETAMIN, THRESH, TOL, TOLMUL,
$ UNFL, X1, X2, Y1, Y2
*
* .. External Subroutines ..
EXTERNAL DLASR, DSCAL, DSWAP, DLARTGP, DLARTGS, DLAS2,
$ XERBLA
* ..
* .. External Functions ..
DOUBLE PRECISION DLAMCH
LOGICAL LSAME
EXTERNAL LSAME, DLAMCH
* ..
* .. Intrinsic Functions ..
INTRINSIC ABS, ATAN2, COS, MAX, MIN, SIN, SQRT
* ..
* .. Executable Statements ..
*
* Test input arguments
*
INFO = 0
LQUERY = LWORK .EQ. -1
WANTU1 = LSAME( JOBU1, 'Y' )
WANTU2 = LSAME( JOBU2, 'Y' )
WANTV1T = LSAME( JOBV1T, 'Y' )
WANTV2T = LSAME( JOBV2T, 'Y' )
COLMAJOR = .NOT. LSAME( TRANS, 'T' )
*
IF( M .LT. 0 ) THEN
INFO = -6
ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
INFO = -7
ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
INFO = -8
ELSE IF( Q .GT. P .OR. Q .GT. M-P .OR. Q .GT. M-Q ) THEN
INFO = -8
ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
INFO = -12
ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
INFO = -14
ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
INFO = -16
ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
INFO = -18
END IF
*
* Quick return if Q = 0
*
IF( INFO .EQ. 0 .AND. Q .EQ. 0 ) THEN
LWORKMIN = 1
WORK(1) = LWORKMIN
RETURN
END IF
*
* Compute workspace
*
IF( INFO .EQ. 0 ) THEN
IU1CS = 1
IU1SN = IU1CS + Q
IU2CS = IU1SN + Q
IU2SN = IU2CS + Q
IV1TCS = IU2SN + Q
IV1TSN = IV1TCS + Q
IV2TCS = IV1TSN + Q
IV2TSN = IV2TCS + Q
LWORKOPT = IV2TSN + Q - 1
LWORKMIN = LWORKOPT
WORK(1) = LWORKOPT
IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
INFO = -28
END IF
END IF
*
IF( INFO .NE. 0 ) THEN
CALL XERBLA( 'DBBCSD', -INFO )
RETURN
ELSE IF( LQUERY ) THEN
RETURN
END IF
*
* Get machine constants
*
EPS = DLAMCH( 'Epsilon' )
UNFL = DLAMCH( 'Safe minimum' )
TOLMUL = MAX( TEN, MIN( HUNDRED, EPS**MEIGHTH ) )
TOL = TOLMUL*EPS
THRESH = MAX( TOL, MAXITR*Q*Q*UNFL )
*
* Test for negligible sines or cosines
*
DO I = 1, Q
IF( THETA(I) .LT. THRESH ) THEN
THETA(I) = ZERO
ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN
THETA(I) = PIOVER2
END IF
END DO
DO I = 1, Q-1
IF( PHI(I) .LT. THRESH ) THEN
PHI(I) = ZERO
ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN
PHI(I) = PIOVER2
END IF
END DO
*
* Initial deflation
*
IMAX = Q
DO WHILE( IMAX .GT. 1 )
IF( PHI(IMAX-1) .NE. ZERO ) THEN
EXIT
END IF
IMAX = IMAX - 1
END DO
IMIN = IMAX - 1
IF ( IMIN .GT. 1 ) THEN
DO WHILE( PHI(IMIN-1) .NE. ZERO )
IMIN = IMIN - 1
IF ( IMIN .LE. 1 ) EXIT
END DO
END IF
*
* Initialize iteration counter
*
MAXIT = MAXITR*Q*Q
ITER = 0
*
* Begin main iteration loop
*
DO WHILE( IMAX .GT. 1 )
*
* Compute the matrix entries
*
B11D(IMIN) = COS( THETA(IMIN) )
B21D(IMIN) = -SIN( THETA(IMIN) )
DO I = IMIN, IMAX - 1
B11E(I) = -SIN( THETA(I) ) * SIN( PHI(I) )
B11D(I+1) = COS( THETA(I+1) ) * COS( PHI(I) )
B12D(I) = SIN( THETA(I) ) * COS( PHI(I) )
B12E(I) = COS( THETA(I+1) ) * SIN( PHI(I) )
B21E(I) = -COS( THETA(I) ) * SIN( PHI(I) )
B21D(I+1) = -SIN( THETA(I+1) ) * COS( PHI(I) )
B22D(I) = COS( THETA(I) ) * COS( PHI(I) )
B22E(I) = -SIN( THETA(I+1) ) * SIN( PHI(I) )
END DO
B12D(IMAX) = SIN( THETA(IMAX) )
B22D(IMAX) = COS( THETA(IMAX) )
*
* Abort if not converging; otherwise, increment ITER
*
IF( ITER .GT. MAXIT ) THEN
INFO = 0
DO I = 1, Q
IF( PHI(I) .NE. ZERO )
$ INFO = INFO + 1
END DO
RETURN
END IF
*
ITER = ITER + IMAX - IMIN
*
* Compute shifts
*
THETAMAX = THETA(IMIN)
THETAMIN = THETA(IMIN)
DO I = IMIN+1, IMAX
IF( THETA(I) > THETAMAX )
$ THETAMAX = THETA(I)
IF( THETA(I) < THETAMIN )
$ THETAMIN = THETA(I)
END DO
*
IF( THETAMAX .GT. PIOVER2 - THRESH ) THEN
*
* Zero on diagonals of B11 and B22; induce deflation with a
* zero shift
*
MU = ZERO
NU = ONE
*
ELSE IF( THETAMIN .LT. THRESH ) THEN
*
* Zero on diagonals of B12 and B22; induce deflation with a
* zero shift
*
MU = ONE
NU = ZERO
*
ELSE
*
* Compute shifts for B11 and B21 and use the lesser
*
CALL DLAS2( B11D(IMAX-1), B11E(IMAX-1), B11D(IMAX), SIGMA11,
$ DUMMY )
CALL DLAS2( B21D(IMAX-1), B21E(IMAX-1), B21D(IMAX), SIGMA21,
$ DUMMY )
*
IF( SIGMA11 .LE. SIGMA21 ) THEN
MU = SIGMA11
NU = SQRT( ONE - MU**2 )
IF( MU .LT. THRESH ) THEN
MU = ZERO
NU = ONE
END IF
ELSE
NU = SIGMA21
MU = SQRT( 1.0 - NU**2 )
IF( NU .LT. THRESH ) THEN
MU = ONE
NU = ZERO
END IF
END IF
END IF
*
* Rotate to produce bulges in B11 and B21
*
IF( MU .LE. NU ) THEN
CALL DLARTGS( B11D(IMIN), B11E(IMIN), MU,
$ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
ELSE
CALL DLARTGS( B21D(IMIN), B21E(IMIN), NU,
$ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
END IF
*
TEMP = WORK(IV1TCS+IMIN-1)*B11D(IMIN) +
$ WORK(IV1TSN+IMIN-1)*B11E(IMIN)
B11E(IMIN) = WORK(IV1TCS+IMIN-1)*B11E(IMIN) -
$ WORK(IV1TSN+IMIN-1)*B11D(IMIN)
B11D(IMIN) = TEMP
B11BULGE = WORK(IV1TSN+IMIN-1)*B11D(IMIN+1)
B11D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B11D(IMIN+1)
TEMP = WORK(IV1TCS+IMIN-1)*B21D(IMIN) +
$ WORK(IV1TSN+IMIN-1)*B21E(IMIN)
B21E(IMIN) = WORK(IV1TCS+IMIN-1)*B21E(IMIN) -
$ WORK(IV1TSN+IMIN-1)*B21D(IMIN)
B21D(IMIN) = TEMP
B21BULGE = WORK(IV1TSN+IMIN-1)*B21D(IMIN+1)
B21D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B21D(IMIN+1)
*
* Compute THETA(IMIN)
*
THETA( IMIN ) = ATAN2( SQRT( B21D(IMIN)**2+B21BULGE**2 ),
$ SQRT( B11D(IMIN)**2+B11BULGE**2 ) )
*
* Chase the bulges in B11(IMIN+1,IMIN) and B21(IMIN+1,IMIN)
*
IF( B11D(IMIN)**2+B11BULGE**2 .GT. THRESH**2 ) THEN
CALL DLARTGP( B11BULGE, B11D(IMIN), WORK(IU1SN+IMIN-1),
$ WORK(IU1CS+IMIN-1), R )
ELSE IF( MU .LE. NU ) THEN
CALL DLARTGS( B11E( IMIN ), B11D( IMIN + 1 ), MU,
$ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
ELSE
CALL DLARTGS( B12D( IMIN ), B12E( IMIN ), NU,
$ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
END IF
IF( B21D(IMIN)**2+B21BULGE**2 .GT. THRESH**2 ) THEN
CALL DLARTGP( B21BULGE, B21D(IMIN), WORK(IU2SN+IMIN-1),
$ WORK(IU2CS+IMIN-1), R )
ELSE IF( NU .LT. MU ) THEN
CALL DLARTGS( B21E( IMIN ), B21D( IMIN + 1 ), NU,
$ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
ELSE
CALL DLARTGS( B22D(IMIN), B22E(IMIN), MU,
$ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
END IF
WORK(IU2CS+IMIN-1) = -WORK(IU2CS+IMIN-1)
WORK(IU2SN+IMIN-1) = -WORK(IU2SN+IMIN-1)
*
TEMP = WORK(IU1CS+IMIN-1)*B11E(IMIN) +
$ WORK(IU1SN+IMIN-1)*B11D(IMIN+1)
B11D(IMIN+1) = WORK(IU1CS+IMIN-1)*B11D(IMIN+1) -
$ WORK(IU1SN+IMIN-1)*B11E(IMIN)
B11E(IMIN) = TEMP
IF( IMAX .GT. IMIN+1 ) THEN
B11BULGE = WORK(IU1SN+IMIN-1)*B11E(IMIN+1)
B11E(IMIN+1) = WORK(IU1CS+IMIN-1)*B11E(IMIN+1)
END IF
TEMP = WORK(IU1CS+IMIN-1)*B12D(IMIN) +
$ WORK(IU1SN+IMIN-1)*B12E(IMIN)
B12E(IMIN) = WORK(IU1CS+IMIN-1)*B12E(IMIN) -
$ WORK(IU1SN+IMIN-1)*B12D(IMIN)
B12D(IMIN) = TEMP
B12BULGE = WORK(IU1SN+IMIN-1)*B12D(IMIN+1)
B12D(IMIN+1) = WORK(IU1CS+IMIN-1)*B12D(IMIN+1)
TEMP = WORK(IU2CS+IMIN-1)*B21E(IMIN) +
$ WORK(IU2SN+IMIN-1)*B21D(IMIN+1)
B21D(IMIN+1) = WORK(IU2CS+IMIN-1)*B21D(IMIN+1) -
$ WORK(IU2SN+IMIN-1)*B21E(IMIN)
B21E(IMIN) = TEMP
IF( IMAX .GT. IMIN+1 ) THEN
B21BULGE = WORK(IU2SN+IMIN-1)*B21E(IMIN+1)
B21E(IMIN+1) = WORK(IU2CS+IMIN-1)*B21E(IMIN+1)
END IF
TEMP = WORK(IU2CS+IMIN-1)*B22D(IMIN) +
$ WORK(IU2SN+IMIN-1)*B22E(IMIN)
B22E(IMIN) = WORK(IU2CS+IMIN-1)*B22E(IMIN) -
$ WORK(IU2SN+IMIN-1)*B22D(IMIN)
B22D(IMIN) = TEMP
B22BULGE = WORK(IU2SN+IMIN-1)*B22D(IMIN+1)
B22D(IMIN+1) = WORK(IU2CS+IMIN-1)*B22D(IMIN+1)
*
* Inner loop: chase bulges from B11(IMIN,IMIN+2),
* B12(IMIN,IMIN+1), B21(IMIN,IMIN+2), and B22(IMIN,IMIN+1) to
* bottom-right
*
DO I = IMIN+1, IMAX-1
*
* Compute PHI(I-1)
*
X1 = SIN(THETA(I-1))*B11E(I-1) + COS(THETA(I-1))*B21E(I-1)
X2 = SIN(THETA(I-1))*B11BULGE + COS(THETA(I-1))*B21BULGE
Y1 = SIN(THETA(I-1))*B12D(I-1) + COS(THETA(I-1))*B22D(I-1)
Y2 = SIN(THETA(I-1))*B12BULGE + COS(THETA(I-1))*B22BULGE
*
PHI(I-1) = ATAN2( SQRT(X1**2+X2**2), SQRT(Y1**2+Y2**2) )
*
* Determine if there are bulges to chase or if a new direct
* summand has been reached
*
RESTART11 = B11E(I-1)**2 + B11BULGE**2 .LE. THRESH**2
RESTART21 = B21E(I-1)**2 + B21BULGE**2 .LE. THRESH**2
RESTART12 = B12D(I-1)**2 + B12BULGE**2 .LE. THRESH**2
RESTART22 = B22D(I-1)**2 + B22BULGE**2 .LE. THRESH**2
*
* If possible, chase bulges from B11(I-1,I+1), B12(I-1,I),
* B21(I-1,I+1), and B22(I-1,I). If necessary, restart bulge-
* chasing by applying the original shift again.
*
IF( .NOT. RESTART11 .AND. .NOT. RESTART21 ) THEN
CALL DLARTGP( X2, X1, WORK(IV1TSN+I-1), WORK(IV1TCS+I-1),
$ R )
ELSE IF( .NOT. RESTART11 .AND. RESTART21 ) THEN
CALL DLARTGP( B11BULGE, B11E(I-1), WORK(IV1TSN+I-1),
$ WORK(IV1TCS+I-1), R )
ELSE IF( RESTART11 .AND. .NOT. RESTART21 ) THEN
CALL DLARTGP( B21BULGE, B21E(I-1), WORK(IV1TSN+I-1),
$ WORK(IV1TCS+I-1), R )
ELSE IF( MU .LE. NU ) THEN
CALL DLARTGS( B11D(I), B11E(I), MU, WORK(IV1TCS+I-1),
$ WORK(IV1TSN+I-1) )
ELSE
CALL DLARTGS( B21D(I), B21E(I), NU, WORK(IV1TCS+I-1),
$ WORK(IV1TSN+I-1) )
END IF
WORK(IV1TCS+I-1) = -WORK(IV1TCS+I-1)
WORK(IV1TSN+I-1) = -WORK(IV1TSN+I-1)
IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( Y2, Y1, WORK(IV2TSN+I-1-1),
$ WORK(IV2TCS+I-1-1), R )
ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
CALL DLARTGP( B12BULGE, B12D(I-1), WORK(IV2TSN+I-1-1),
$ WORK(IV2TCS+I-1-1), R )
ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( B22BULGE, B22D(I-1), WORK(IV2TSN+I-1-1),
$ WORK(IV2TCS+I-1-1), R )
ELSE IF( NU .LT. MU ) THEN
CALL DLARTGS( B12E(I-1), B12D(I), NU, WORK(IV2TCS+I-1-1),
$ WORK(IV2TSN+I-1-1) )
ELSE
CALL DLARTGS( B22E(I-1), B22D(I), MU, WORK(IV2TCS+I-1-1),
$ WORK(IV2TSN+I-1-1) )
END IF
*
TEMP = WORK(IV1TCS+I-1)*B11D(I) + WORK(IV1TSN+I-1)*B11E(I)
B11E(I) = WORK(IV1TCS+I-1)*B11E(I) -
$ WORK(IV1TSN+I-1)*B11D(I)
B11D(I) = TEMP
B11BULGE = WORK(IV1TSN+I-1)*B11D(I+1)
B11D(I+1) = WORK(IV1TCS+I-1)*B11D(I+1)
TEMP = WORK(IV1TCS+I-1)*B21D(I) + WORK(IV1TSN+I-1)*B21E(I)
B21E(I) = WORK(IV1TCS+I-1)*B21E(I) -
$ WORK(IV1TSN+I-1)*B21D(I)
B21D(I) = TEMP
B21BULGE = WORK(IV1TSN+I-1)*B21D(I+1)
B21D(I+1) = WORK(IV1TCS+I-1)*B21D(I+1)
TEMP = WORK(IV2TCS+I-1-1)*B12E(I-1) +
$ WORK(IV2TSN+I-1-1)*B12D(I)
B12D(I) = WORK(IV2TCS+I-1-1)*B12D(I) -
$ WORK(IV2TSN+I-1-1)*B12E(I-1)
B12E(I-1) = TEMP
B12BULGE = WORK(IV2TSN+I-1-1)*B12E(I)
B12E(I) = WORK(IV2TCS+I-1-1)*B12E(I)
TEMP = WORK(IV2TCS+I-1-1)*B22E(I-1) +
$ WORK(IV2TSN+I-1-1)*B22D(I)
B22D(I) = WORK(IV2TCS+I-1-1)*B22D(I) -
$ WORK(IV2TSN+I-1-1)*B22E(I-1)
B22E(I-1) = TEMP
B22BULGE = WORK(IV2TSN+I-1-1)*B22E(I)
B22E(I) = WORK(IV2TCS+I-1-1)*B22E(I)
*
* Compute THETA(I)
*
X1 = COS(PHI(I-1))*B11D(I) + SIN(PHI(I-1))*B12E(I-1)
X2 = COS(PHI(I-1))*B11BULGE + SIN(PHI(I-1))*B12BULGE
Y1 = COS(PHI(I-1))*B21D(I) + SIN(PHI(I-1))*B22E(I-1)
Y2 = COS(PHI(I-1))*B21BULGE + SIN(PHI(I-1))*B22BULGE
*
THETA(I) = ATAN2( SQRT(Y1**2+Y2**2), SQRT(X1**2+X2**2) )
*
* Determine if there are bulges to chase or if a new direct
* summand has been reached
*
RESTART11 = B11D(I)**2 + B11BULGE**2 .LE. THRESH**2
RESTART12 = B12E(I-1)**2 + B12BULGE**2 .LE. THRESH**2
RESTART21 = B21D(I)**2 + B21BULGE**2 .LE. THRESH**2
RESTART22 = B22E(I-1)**2 + B22BULGE**2 .LE. THRESH**2
*
* If possible, chase bulges from B11(I+1,I), B12(I+1,I-1),
* B21(I+1,I), and B22(I+1,I-1). If necessary, restart bulge-
* chasing by applying the original shift again.
*
IF( .NOT. RESTART11 .AND. .NOT. RESTART12 ) THEN
CALL DLARTGP( X2, X1, WORK(IU1SN+I-1), WORK(IU1CS+I-1),
$ R )
ELSE IF( .NOT. RESTART11 .AND. RESTART12 ) THEN
CALL DLARTGP( B11BULGE, B11D(I), WORK(IU1SN+I-1),
$ WORK(IU1CS+I-1), R )
ELSE IF( RESTART11 .AND. .NOT. RESTART12 ) THEN
CALL DLARTGP( B12BULGE, B12E(I-1), WORK(IU1SN+I-1),
$ WORK(IU1CS+I-1), R )
ELSE IF( MU .LE. NU ) THEN
CALL DLARTGS( B11E(I), B11D(I+1), MU, WORK(IU1CS+I-1),
$ WORK(IU1SN+I-1) )
ELSE
CALL DLARTGS( B12D(I), B12E(I), NU, WORK(IU1CS+I-1),
$ WORK(IU1SN+I-1) )
END IF
IF( .NOT. RESTART21 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( Y2, Y1, WORK(IU2SN+I-1), WORK(IU2CS+I-1),
$ R )
ELSE IF( .NOT. RESTART21 .AND. RESTART22 ) THEN
CALL DLARTGP( B21BULGE, B21D(I), WORK(IU2SN+I-1),
$ WORK(IU2CS+I-1), R )
ELSE IF( RESTART21 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( B22BULGE, B22E(I-1), WORK(IU2SN+I-1),
$ WORK(IU2CS+I-1), R )
ELSE IF( NU .LT. MU ) THEN
CALL DLARTGS( B21E(I), B21E(I+1), NU, WORK(IU2CS+I-1),
$ WORK(IU2SN+I-1) )
ELSE
CALL DLARTGS( B22D(I), B22E(I), MU, WORK(IU2CS+I-1),
$ WORK(IU2SN+I-1) )
END IF
WORK(IU2CS+I-1) = -WORK(IU2CS+I-1)
WORK(IU2SN+I-1) = -WORK(IU2SN+I-1)
*
TEMP = WORK(IU1CS+I-1)*B11E(I) + WORK(IU1SN+I-1)*B11D(I+1)
B11D(I+1) = WORK(IU1CS+I-1)*B11D(I+1) -
$ WORK(IU1SN+I-1)*B11E(I)
B11E(I) = TEMP
IF( I .LT. IMAX - 1 ) THEN
B11BULGE = WORK(IU1SN+I-1)*B11E(I+1)
B11E(I+1) = WORK(IU1CS+I-1)*B11E(I+1)
END IF
TEMP = WORK(IU2CS+I-1)*B21E(I) + WORK(IU2SN+I-1)*B21D(I+1)
B21D(I+1) = WORK(IU2CS+I-1)*B21D(I+1) -
$ WORK(IU2SN+I-1)*B21E(I)
B21E(I) = TEMP
IF( I .LT. IMAX - 1 ) THEN
B21BULGE = WORK(IU2SN+I-1)*B21E(I+1)
B21E(I+1) = WORK(IU2CS+I-1)*B21E(I+1)
END IF
TEMP = WORK(IU1CS+I-1)*B12D(I) + WORK(IU1SN+I-1)*B12E(I)
B12E(I) = WORK(IU1CS+I-1)*B12E(I) - WORK(IU1SN+I-1)*B12D(I)
B12D(I) = TEMP
B12BULGE = WORK(IU1SN+I-1)*B12D(I+1)
B12D(I+1) = WORK(IU1CS+I-1)*B12D(I+1)
TEMP = WORK(IU2CS+I-1)*B22D(I) + WORK(IU2SN+I-1)*B22E(I)
B22E(I) = WORK(IU2CS+I-1)*B22E(I) - WORK(IU2SN+I-1)*B22D(I)
B22D(I) = TEMP
B22BULGE = WORK(IU2SN+I-1)*B22D(I+1)
B22D(I+1) = WORK(IU2CS+I-1)*B22D(I+1)
*
END DO
*
* Compute PHI(IMAX-1)
*
X1 = SIN(THETA(IMAX-1))*B11E(IMAX-1) +
$ COS(THETA(IMAX-1))*B21E(IMAX-1)
Y1 = SIN(THETA(IMAX-1))*B12D(IMAX-1) +
$ COS(THETA(IMAX-1))*B22D(IMAX-1)
Y2 = SIN(THETA(IMAX-1))*B12BULGE + COS(THETA(IMAX-1))*B22BULGE
*
PHI(IMAX-1) = ATAN2( ABS(X1), SQRT(Y1**2+Y2**2) )
*
* Chase bulges from B12(IMAX-1,IMAX) and B22(IMAX-1,IMAX)
*
RESTART12 = B12D(IMAX-1)**2 + B12BULGE**2 .LE. THRESH**2
RESTART22 = B22D(IMAX-1)**2 + B22BULGE**2 .LE. THRESH**2
*
IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( Y2, Y1, WORK(IV2TSN+IMAX-1-1),
$ WORK(IV2TCS+IMAX-1-1), R )
ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
CALL DLARTGP( B12BULGE, B12D(IMAX-1), WORK(IV2TSN+IMAX-1-1),
$ WORK(IV2TCS+IMAX-1-1), R )
ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
CALL DLARTGP( B22BULGE, B22D(IMAX-1), WORK(IV2TSN+IMAX-1-1),
$ WORK(IV2TCS+IMAX-1-1), R )
ELSE IF( NU .LT. MU ) THEN
CALL DLARTGS( B12E(IMAX-1), B12D(IMAX), NU,
$ WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
ELSE
CALL DLARTGS( B22E(IMAX-1), B22D(IMAX), MU,
$ WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
END IF
*
TEMP = WORK(IV2TCS+IMAX-1-1)*B12E(IMAX-1) +
$ WORK(IV2TSN+IMAX-1-1)*B12D(IMAX)
B12D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B12D(IMAX) -
$ WORK(IV2TSN+IMAX-1-1)*B12E(IMAX-1)
B12E(IMAX-1) = TEMP
TEMP = WORK(IV2TCS+IMAX-1-1)*B22E(IMAX-1) +
$ WORK(IV2TSN+IMAX-1-1)*B22D(IMAX)
B22D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B22D(IMAX) -
$ WORK(IV2TSN+IMAX-1-1)*B22E(IMAX-1)
B22E(IMAX-1) = TEMP
*
* Update singular vectors
*
IF( WANTU1 ) THEN
IF( COLMAJOR ) THEN
CALL DLASR( 'R', 'V', 'F', P, IMAX-IMIN+1,
$ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
$ U1(1,IMIN), LDU1 )
ELSE
CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, P,
$ WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
$ U1(IMIN,1), LDU1 )
END IF
END IF
IF( WANTU2 ) THEN
IF( COLMAJOR ) THEN
CALL DLASR( 'R', 'V', 'F', M-P, IMAX-IMIN+1,
$ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
$ U2(1,IMIN), LDU2 )
ELSE
CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-P,
$ WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
$ U2(IMIN,1), LDU2 )
END IF
END IF
IF( WANTV1T ) THEN
IF( COLMAJOR ) THEN
CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, Q,
$ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
$ V1T(IMIN,1), LDV1T )
ELSE
CALL DLASR( 'R', 'V', 'F', Q, IMAX-IMIN+1,
$ WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
$ V1T(1,IMIN), LDV1T )
END IF
END IF
IF( WANTV2T ) THEN
IF( COLMAJOR ) THEN
CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-Q,
$ WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
$ V2T(IMIN,1), LDV2T )
ELSE
CALL DLASR( 'R', 'V', 'F', M-Q, IMAX-IMIN+1,
$ WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
$ V2T(1,IMIN), LDV2T )
END IF
END IF
*
* Fix signs on B11(IMAX-1,IMAX) and B21(IMAX-1,IMAX)
*
IF( B11E(IMAX-1)+B21E(IMAX-1) .GT. 0 ) THEN
B11D(IMAX) = -B11D(IMAX)
B21D(IMAX) = -B21D(IMAX)
IF( WANTV1T ) THEN
IF( COLMAJOR ) THEN
CALL DSCAL( Q, NEGONE, V1T(IMAX,1), LDV1T )
ELSE
CALL DSCAL( Q, NEGONE, V1T(1,IMAX), 1 )
END IF
END IF
END IF
*
* Compute THETA(IMAX)
*
X1 = COS(PHI(IMAX-1))*B11D(IMAX) +
$ SIN(PHI(IMAX-1))*B12E(IMAX-1)
Y1 = COS(PHI(IMAX-1))*B21D(IMAX) +
$ SIN(PHI(IMAX-1))*B22E(IMAX-1)
*
THETA(IMAX) = ATAN2( ABS(Y1), ABS(X1) )
*
* Fix signs on B11(IMAX,IMAX), B12(IMAX,IMAX-1), B21(IMAX,IMAX),
* and B22(IMAX,IMAX-1)
*
IF( B11D(IMAX)+B12E(IMAX-1) .LT. 0 ) THEN
B12D(IMAX) = -B12D(IMAX)
IF( WANTU1 ) THEN
IF( COLMAJOR ) THEN
CALL DSCAL( P, NEGONE, U1(1,IMAX), 1 )
ELSE
CALL DSCAL( P, NEGONE, U1(IMAX,1), LDU1 )
END IF
END IF
END IF
IF( B21D(IMAX)+B22E(IMAX-1) .GT. 0 ) THEN
B22D(IMAX) = -B22D(IMAX)
IF( WANTU2 ) THEN
IF( COLMAJOR ) THEN
CALL DSCAL( M-P, NEGONE, U2(1,IMAX), 1 )
ELSE
CALL DSCAL( M-P, NEGONE, U2(IMAX,1), LDU2 )
END IF
END IF
END IF
*
* Fix signs on B12(IMAX,IMAX) and B22(IMAX,IMAX)
*
IF( B12D(IMAX)+B22D(IMAX) .LT. 0 ) THEN
IF( WANTV2T ) THEN
IF( COLMAJOR ) THEN
CALL DSCAL( M-Q, NEGONE, V2T(IMAX,1), LDV2T )
ELSE
CALL DSCAL( M-Q, NEGONE, V2T(1,IMAX), 1 )
END IF
END IF
END IF
*
* Test for negligible sines or cosines
*
DO I = IMIN, IMAX
IF( THETA(I) .LT. THRESH ) THEN