numeric-linalg
Educational material on the SciPy implementation of numerical linear algebra algorithms
| Name | Size | Mode | |
| .. | |||
| lapack/SRC/cbbcsd.f | 39328B | -rw-r--r-- |
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*> \brief \b CBBCSD * * =========== DOCUMENTATION =========== * * Online html documentation available at * http://www.netlib.org/lapack/explore-html/ * *> \htmlonly *> Download CBBCSD + dependencies *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/cbbcsd.f"> *> [TGZ]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/cbbcsd.f"> *> [ZIP]</a> *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/cbbcsd.f"> *> [TXT]</a> *> \endhtmlonly * * Definition: * =========== * * SUBROUTINE CBBCSD( 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, RWORK, LRWORK, INFO ) * * .. Scalar Arguments .. * CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS * INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LRWORK, M, P, Q * .. * .. Array Arguments .. * REAL B11D( * ), B11E( * ), B12D( * ), B12E( * ), * $ B21D( * ), B21E( * ), B22D( * ), B22E( * ), * $ PHI( * ), THETA( * ), RWORK( * ) * COMPLEX U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ), * $ V2T( LDV2T, * ) * .. * * *> \par Purpose: * ============= *> *> \verbatim *> *> CBBCSD computes the CS decomposition of a unitary 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 | ]**H *> = [---------] [---------------] [---------] . *> [ | 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 CUNCSD for details.) *> *> The bidiagonal matrices B11, B12, B21, and B22 are represented *> implicitly by angles THETA(1:Q) and PHI(1:Q-1). *> *> The unitary 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 unitary 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 REAL 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 REAL 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 COMPLEX 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 COMPLEX 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 COMPLEX array, dimension (LDV1T,Q) *> On entry, a Q-by-Q matrix. On exit, V1T is premultiplied *> by the conjugate 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 COMPLEX array, dimension (LDV2T,M-Q) *> On entry, an (M-Q)-by-(M-Q) matrix. On exit, V2T is *> premultiplied by the conjugate 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 REAL array, dimension (Q) *> When CBBCSD converges, B11D contains the cosines of THETA(1), *> ..., THETA(Q). If CBBCSD fails to converge, then B11D *> contains the diagonal of the partially reduced top-left *> block. *> \endverbatim *> *> \param[out] B11E *> \verbatim *> B11E is REAL array, dimension (Q-1) *> When CBBCSD converges, B11E contains zeros. If CBBCSD fails *> to converge, then B11E contains the superdiagonal of the *> partially reduced top-left block. *> \endverbatim *> *> \param[out] B12D *> \verbatim *> B12D is REAL array, dimension (Q) *> When CBBCSD converges, B12D contains the negative sines of *> THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then *> B12D contains the diagonal of the partially reduced top-right *> block. *> \endverbatim *> *> \param[out] B12E *> \verbatim *> B12E is REAL array, dimension (Q-1) *> When CBBCSD converges, B12E contains zeros. If CBBCSD fails *> to converge, then B12E contains the subdiagonal of the *> partially reduced top-right block. *> \endverbatim *> *> \param[out] B21D *> \verbatim *> B21D is REAL array, dimension (Q) *> When CBBCSD converges, B21D contains the negative sines of *> THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then *> B21D contains the diagonal of the partially reduced bottom-left *> block. *> \endverbatim *> *> \param[out] B21E *> \verbatim *> B21E is REAL array, dimension (Q-1) *> When CBBCSD converges, B21E contains zeros. If CBBCSD fails *> to converge, then B21E contains the subdiagonal of the *> partially reduced bottom-left block. *> \endverbatim *> *> \param[out] B22D *> \verbatim *> B22D is REAL array, dimension (Q) *> When CBBCSD converges, B22D contains the negative sines of *> THETA(1), ..., THETA(Q). If CBBCSD fails to converge, then *> B22D contains the diagonal of the partially reduced bottom-right *> block. *> \endverbatim *> *> \param[out] B22E *> \verbatim *> B22E is REAL array, dimension (Q-1) *> When CBBCSD converges, B22E contains zeros. If CBBCSD fails *> to converge, then B22E contains the subdiagonal of the *> partially reduced bottom-right block. *> \endverbatim *> *> \param[out] RWORK *> \verbatim *> RWORK is REAL array, dimension (MAX(1,LRWORK)) *> On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK. *> \endverbatim *> *> \param[in] LRWORK *> \verbatim *> LRWORK is INTEGER *> The dimension of the array RWORK. LRWORK >= MAX(1,8*Q). *> *> If LRWORK = -1, then a workspace query is assumed; the *> routine only calculates the optimal size of the RWORK array, *> returns this value as the first entry of the work array, and *> no error message related to LRWORK 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 CBBCSD 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 REAL, 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. * *> \ingroup bbcsd * * ===================================================================== SUBROUTINE CBBCSD( 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, RWORK, LRWORK, INFO ) * * -- LAPACK computational routine -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- * * .. Scalar Arguments .. CHARACTER JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS INTEGER INFO, LDU1, LDU2, LDV1T, LDV2T, LRWORK, M, P, Q * .. * .. Array Arguments .. REAL B11D( * ), B11E( * ), B12D( * ), B12E( * ), $ B21D( * ), B21E( * ), B22D( * ), B22E( * ), $ PHI( * ), THETA( * ), RWORK( * ) COMPLEX U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ), $ V2T( LDV2T, * ) * .. * * =================================================================== * * .. Parameters .. INTEGER MAXITR PARAMETER ( MAXITR = 6 ) REAL HUNDRED, MEIGHTH, ONE, TEN, ZERO PARAMETER ( HUNDRED = 100.0E0, MEIGHTH = -0.125E0, $ ONE = 1.0E0, TEN = 10.0E0, ZERO = 0.0E0 ) COMPLEX NEGONECOMPLEX PARAMETER ( NEGONECOMPLEX = (-1.0E0,0.0E0) ) REAL PIOVER2 PARAMETER ( PIOVER2 = 1.57079632679489661923132169163975144210E0 ) * .. * .. 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, $ LRWORKMIN, LRWORKOPT, MAXIT, MINI REAL 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 CLASR, CSCAL, CSWAP, SLARTGP, SLARTGS, $ SLAS2, $ XERBLA * .. * .. External Functions .. REAL SLAMCH LOGICAL LSAME EXTERNAL LSAME, SLAMCH * .. * .. Intrinsic Functions .. INTRINSIC ABS, ATAN2, COS, MAX, MIN, SIN, SQRT * .. * .. Executable Statements .. * * Test input arguments * INFO = 0 LQUERY = LRWORK .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 LRWORKMIN = 1 RWORK(1) = REAL( LRWORKMIN ) 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 LRWORKOPT = IV2TSN + Q - 1 LRWORKMIN = LRWORKOPT RWORK(1) = REAL( LRWORKOPT ) IF( LRWORK .LT. LRWORKMIN .AND. .NOT. LQUERY ) THEN INFO = -28 END IF END IF * IF( INFO .NE. 0 ) THEN CALL XERBLA( 'CBBCSD', -INFO ) RETURN ELSE IF( LQUERY ) THEN RETURN END IF * * Get machine constants * EPS = SLAMCH( 'Epsilon' ) UNFL = SLAMCH( 'Safe minimum' ) TOLMUL = MAX( TEN, MIN( HUNDRED, EPS**MEIGHTH ) ) TOL = TOLMUL*EPS THRESH = MAX( TOL, REAL( 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 SLAS2( B11D(IMAX-1), B11E(IMAX-1), B11D(IMAX), $ SIGMA11, $ DUMMY ) CALL SLAS2( 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 SLARTGS( B11D(IMIN), B11E(IMIN), MU, $ RWORK(IV1TCS+IMIN-1), RWORK(IV1TSN+IMIN-1) ) ELSE CALL SLARTGS( B21D(IMIN), B21E(IMIN), NU, $ RWORK(IV1TCS+IMIN-1), RWORK(IV1TSN+IMIN-1) ) END IF * TEMP = RWORK(IV1TCS+IMIN-1)*B11D(IMIN) + $ RWORK(IV1TSN+IMIN-1)*B11E(IMIN) B11E(IMIN) = RWORK(IV1TCS+IMIN-1)*B11E(IMIN) - $ RWORK(IV1TSN+IMIN-1)*B11D(IMIN) B11D(IMIN) = TEMP B11BULGE = RWORK(IV1TSN+IMIN-1)*B11D(IMIN+1) B11D(IMIN+1) = RWORK(IV1TCS+IMIN-1)*B11D(IMIN+1) TEMP = RWORK(IV1TCS+IMIN-1)*B21D(IMIN) + $ RWORK(IV1TSN+IMIN-1)*B21E(IMIN) B21E(IMIN) = RWORK(IV1TCS+IMIN-1)*B21E(IMIN) - $ RWORK(IV1TSN+IMIN-1)*B21D(IMIN) B21D(IMIN) = TEMP B21BULGE = RWORK(IV1TSN+IMIN-1)*B21D(IMIN+1) B21D(IMIN+1) = RWORK(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 SLARTGP( B11BULGE, B11D(IMIN), RWORK(IU1SN+IMIN-1), $ RWORK(IU1CS+IMIN-1), R ) ELSE IF( MU .LE. NU ) THEN CALL SLARTGS( B11E( IMIN ), B11D( IMIN + 1 ), MU, $ RWORK(IU1CS+IMIN-1), RWORK(IU1SN+IMIN-1) ) ELSE CALL SLARTGS( B12D( IMIN ), B12E( IMIN ), NU, $ RWORK(IU1CS+IMIN-1), RWORK(IU1SN+IMIN-1) ) END IF IF( B21D(IMIN)**2+B21BULGE**2 .GT. THRESH**2 ) THEN CALL SLARTGP( B21BULGE, B21D(IMIN), RWORK(IU2SN+IMIN-1), $ RWORK(IU2CS+IMIN-1), R ) ELSE IF( NU .LT. MU ) THEN CALL SLARTGS( B21E( IMIN ), B21D( IMIN + 1 ), NU, $ RWORK(IU2CS+IMIN-1), RWORK(IU2SN+IMIN-1) ) ELSE CALL SLARTGS( B22D(IMIN), B22E(IMIN), MU, $ RWORK(IU2CS+IMIN-1), RWORK(IU2SN+IMIN-1) ) END IF RWORK(IU2CS+IMIN-1) = -RWORK(IU2CS+IMIN-1) RWORK(IU2SN+IMIN-1) = -RWORK(IU2SN+IMIN-1) * TEMP = RWORK(IU1CS+IMIN-1)*B11E(IMIN) + $ RWORK(IU1SN+IMIN-1)*B11D(IMIN+1) B11D(IMIN+1) = RWORK(IU1CS+IMIN-1)*B11D(IMIN+1) - $ RWORK(IU1SN+IMIN-1)*B11E(IMIN) B11E(IMIN) = TEMP IF( IMAX .GT. IMIN+1 ) THEN B11BULGE = RWORK(IU1SN+IMIN-1)*B11E(IMIN+1) B11E(IMIN+1) = RWORK(IU1CS+IMIN-1)*B11E(IMIN+1) END IF TEMP = RWORK(IU1CS+IMIN-1)*B12D(IMIN) + $ RWORK(IU1SN+IMIN-1)*B12E(IMIN) B12E(IMIN) = RWORK(IU1CS+IMIN-1)*B12E(IMIN) - $ RWORK(IU1SN+IMIN-1)*B12D(IMIN) B12D(IMIN) = TEMP B12BULGE = RWORK(IU1SN+IMIN-1)*B12D(IMIN+1) B12D(IMIN+1) = RWORK(IU1CS+IMIN-1)*B12D(IMIN+1) TEMP = RWORK(IU2CS+IMIN-1)*B21E(IMIN) + $ RWORK(IU2SN+IMIN-1)*B21D(IMIN+1) B21D(IMIN+1) = RWORK(IU2CS+IMIN-1)*B21D(IMIN+1) - $ RWORK(IU2SN+IMIN-1)*B21E(IMIN) B21E(IMIN) = TEMP IF( IMAX .GT. IMIN+1 ) THEN B21BULGE = RWORK(IU2SN+IMIN-1)*B21E(IMIN+1) B21E(IMIN+1) = RWORK(IU2CS+IMIN-1)*B21E(IMIN+1) END IF TEMP = RWORK(IU2CS+IMIN-1)*B22D(IMIN) + $ RWORK(IU2SN+IMIN-1)*B22E(IMIN) B22E(IMIN) = RWORK(IU2CS+IMIN-1)*B22E(IMIN) - $ RWORK(IU2SN+IMIN-1)*B22D(IMIN) B22D(IMIN) = TEMP B22BULGE = RWORK(IU2SN+IMIN-1)*B22D(IMIN+1) B22D(IMIN+1) = RWORK(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 SLARTGP( X2, X1, RWORK(IV1TSN+I-1), $ RWORK(IV1TCS+I-1), R ) ELSE IF( .NOT. RESTART11 .AND. RESTART21 ) THEN CALL SLARTGP( B11BULGE, B11E(I-1), RWORK(IV1TSN+I-1), $ RWORK(IV1TCS+I-1), R ) ELSE IF( RESTART11 .AND. .NOT. RESTART21 ) THEN CALL SLARTGP( B21BULGE, B21E(I-1), RWORK(IV1TSN+I-1), $ RWORK(IV1TCS+I-1), R ) ELSE IF( MU .LE. NU ) THEN CALL SLARTGS( B11D(I), B11E(I), MU, RWORK(IV1TCS+I-1), $ RWORK(IV1TSN+I-1) ) ELSE CALL SLARTGS( B21D(I), B21E(I), NU, RWORK(IV1TCS+I-1), $ RWORK(IV1TSN+I-1) ) END IF RWORK(IV1TCS+I-1) = -RWORK(IV1TCS+I-1) RWORK(IV1TSN+I-1) = -RWORK(IV1TSN+I-1) IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN CALL SLARTGP( Y2, Y1, RWORK(IV2TSN+I-1-1), $ RWORK(IV2TCS+I-1-1), R ) ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN CALL SLARTGP( B12BULGE, B12D(I-1), $ RWORK(IV2TSN+I-1-1), $ RWORK(IV2TCS+I-1-1), R ) ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN CALL SLARTGP( B22BULGE, B22D(I-1), $ RWORK(IV2TSN+I-1-1), $ RWORK(IV2TCS+I-1-1), R ) ELSE IF( NU .LT. MU ) THEN CALL SLARTGS( B12E(I-1), B12D(I), NU, $ RWORK(IV2TCS+I-1-1), RWORK(IV2TSN+I-1-1) ) ELSE CALL SLARTGS( B22E(I-1), B22D(I), MU, $ RWORK(IV2TCS+I-1-1), RWORK(IV2TSN+I-1-1) ) END IF * TEMP = RWORK(IV1TCS+I-1)*B11D(I) + RWORK(IV1TSN+I-1)*B11E(I) B11E(I) = RWORK(IV1TCS+I-1)*B11E(I) - $ RWORK(IV1TSN+I-1)*B11D(I) B11D(I) = TEMP B11BULGE = RWORK(IV1TSN+I-1)*B11D(I+1) B11D(I+1) = RWORK(IV1TCS+I-1)*B11D(I+1) TEMP = RWORK(IV1TCS+I-1)*B21D(I) + RWORK(IV1TSN+I-1)*B21E(I) B21E(I) = RWORK(IV1TCS+I-1)*B21E(I) - $ RWORK(IV1TSN+I-1)*B21D(I) B21D(I) = TEMP B21BULGE = RWORK(IV1TSN+I-1)*B21D(I+1) B21D(I+1) = RWORK(IV1TCS+I-1)*B21D(I+1) TEMP = RWORK(IV2TCS+I-1-1)*B12E(I-1) + $ RWORK(IV2TSN+I-1-1)*B12D(I) B12D(I) = RWORK(IV2TCS+I-1-1)*B12D(I) - $ RWORK(IV2TSN+I-1-1)*B12E(I-1) B12E(I-1) = TEMP B12BULGE = RWORK(IV2TSN+I-1-1)*B12E(I) B12E(I) = RWORK(IV2TCS+I-1-1)*B12E(I) TEMP = RWORK(IV2TCS+I-1-1)*B22E(I-1) + $ RWORK(IV2TSN+I-1-1)*B22D(I) B22D(I) = RWORK(IV2TCS+I-1-1)*B22D(I) - $ RWORK(IV2TSN+I-1-1)*B22E(I-1) B22E(I-1) = TEMP B22BULGE = RWORK(IV2TSN+I-1-1)*B22E(I) B22E(I) = RWORK(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 SLARTGP( X2, X1, RWORK(IU1SN+I-1), $ RWORK(IU1CS+I-1), $ R ) ELSE IF( .NOT. RESTART11 .AND. RESTART12 ) THEN CALL SLARTGP( B11BULGE, B11D(I), RWORK(IU1SN+I-1), $ RWORK(IU1CS+I-1), R ) ELSE IF( RESTART11 .AND. .NOT. RESTART12 ) THEN CALL SLARTGP( B12BULGE, B12E(I-1), RWORK(IU1SN+I-1), $ RWORK(IU1CS+I-1), R ) ELSE IF( MU .LE. NU ) THEN CALL SLARTGS( B11E(I), B11D(I+1), MU, $ RWORK(IU1CS+I-1), $ RWORK(IU1SN+I-1) ) ELSE CALL SLARTGS( B12D(I), B12E(I), NU, RWORK(IU1CS+I-1), $ RWORK(IU1SN+I-1) ) END IF IF( .NOT. RESTART21 .AND. .NOT. RESTART22 ) THEN CALL SLARTGP( Y2, Y1, RWORK(IU2SN+I-1), $ RWORK(IU2CS+I-1), $ R ) ELSE IF( .NOT. RESTART21 .AND. RESTART22 ) THEN CALL SLARTGP( B21BULGE, B21D(I), RWORK(IU2SN+I-1), $ RWORK(IU2CS+I-1), R ) ELSE IF( RESTART21 .AND. .NOT. RESTART22 ) THEN CALL SLARTGP( B22BULGE, B22E(I-1), RWORK(IU2SN+I-1), $ RWORK(IU2CS+I-1), R ) ELSE IF( NU .LT. MU ) THEN CALL SLARTGS( B21E(I), B21E(I+1), NU, $ RWORK(IU2CS+I-1), $ RWORK(IU2SN+I-1) ) ELSE CALL SLARTGS( B22D(I), B22E(I), MU, RWORK(IU2CS+I-1), $ RWORK(IU2SN+I-1) ) END IF RWORK(IU2CS+I-1) = -RWORK(IU2CS+I-1) RWORK(IU2SN+I-1) = -RWORK(IU2SN+I-1) * TEMP = RWORK(IU1CS+I-1)*B11E(I) + RWORK(IU1SN+I-1)*B11D(I+1) B11D(I+1) = RWORK(IU1CS+I-1)*B11D(I+1) - $ RWORK(IU1SN+I-1)*B11E(I) B11E(I) = TEMP IF( I .LT. IMAX - 1 ) THEN B11BULGE = RWORK(IU1SN+I-1)*B11E(I+1) B11E(I+1) = RWORK(IU1CS+I-1)*B11E(I+1) END IF TEMP = RWORK(IU2CS+I-1)*B21E(I) + RWORK(IU2SN+I-1)*B21D(I+1) B21D(I+1) = RWORK(IU2CS+I-1)*B21D(I+1) - $ RWORK(IU2SN+I-1)*B21E(I) B21E(I) = TEMP IF( I .LT. IMAX - 1 ) THEN B21BULGE = RWORK(IU2SN+I-1)*B21E(I+1) B21E(I+1) = RWORK(IU2CS+I-1)*B21E(I+1) END IF TEMP = RWORK(IU1CS+I-1)*B12D(I) + RWORK(IU1SN+I-1)*B12E(I) B12E(I) = RWORK(IU1CS+I-1)*B12E(I) - $ RWORK(IU1SN+I-1)*B12D(I) B12D(I) = TEMP B12BULGE = RWORK(IU1SN+I-1)*B12D(I+1) B12D(I+1) = RWORK(IU1CS+I-1)*B12D(I+1) TEMP = RWORK(IU2CS+I-1)*B22D(I) + RWORK(IU2SN+I-1)*B22E(I) B22E(I) = RWORK(IU2CS+I-1)*B22E(I) - $ RWORK(IU2SN+I-1)*B22D(I) B22D(I) = TEMP B22BULGE = RWORK(IU2SN+I-1)*B22D(I+1) B22D(I+1) = RWORK(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 SLARTGP( Y2, Y1, RWORK(IV2TSN+IMAX-1-1), $ RWORK(IV2TCS+IMAX-1-1), R ) ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN CALL SLARTGP( B12BULGE, B12D(IMAX-1), $ RWORK(IV2TSN+IMAX-1-1), $ RWORK(IV2TCS+IMAX-1-1), R ) ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN CALL SLARTGP( B22BULGE, B22D(IMAX-1), $ RWORK(IV2TSN+IMAX-1-1), $ RWORK(IV2TCS+IMAX-1-1), R ) ELSE IF( NU .LT. MU ) THEN CALL SLARTGS( B12E(IMAX-1), B12D(IMAX), NU, $ RWORK(IV2TCS+IMAX-1-1), $ RWORK(IV2TSN+IMAX-1-1) ) ELSE CALL SLARTGS( B22E(IMAX-1), B22D(IMAX), MU, $ RWORK(IV2TCS+IMAX-1-1), $ RWORK(IV2TSN+IMAX-1-1) ) END IF * TEMP = RWORK(IV2TCS+IMAX-1-1)*B12E(IMAX-1) + $ RWORK(IV2TSN+IMAX-1-1)*B12D(IMAX) B12D(IMAX) = RWORK(IV2TCS+IMAX-1-1)*B12D(IMAX) - $ RWORK(IV2TSN+IMAX-1-1)*B12E(IMAX-1) B12E(IMAX-1) = TEMP TEMP = RWORK(IV2TCS+IMAX-1-1)*B22E(IMAX-1) + $ RWORK(IV2TSN+IMAX-1-1)*B22D(IMAX) B22D(IMAX) = RWORK(IV2TCS+IMAX-1-1)*B22D(IMAX) - $ RWORK(IV2TSN+IMAX-1-1)*B22E(IMAX-1) B22E(IMAX-1) = TEMP * * Update singular vectors * IF( WANTU1 ) THEN IF( COLMAJOR ) THEN CALL CLASR( 'R', 'V', 'F', P, IMAX-IMIN+1, $ RWORK(IU1CS+IMIN-1), RWORK(IU1SN+IMIN-1), $ U1(1,IMIN), LDU1 ) ELSE CALL CLASR( 'L', 'V', 'F', IMAX-IMIN+1, P, $ RWORK(IU1CS+IMIN-1), RWORK(IU1SN+IMIN-1), $ U1(IMIN,1), LDU1 ) END IF END IF IF( WANTU2 ) THEN IF( COLMAJOR ) THEN CALL CLASR( 'R', 'V', 'F', M-P, IMAX-IMIN+1, $ RWORK(IU2CS+IMIN-1), RWORK(IU2SN+IMIN-1), $ U2(1,IMIN), LDU2 ) ELSE CALL CLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-P, $ RWORK(IU2CS+IMIN-1), RWORK(IU2SN+IMIN-1), $ U2(IMIN,1), LDU2 ) END IF END IF IF( WANTV1T ) THEN IF( COLMAJOR ) THEN CALL CLASR( 'L', 'V', 'F', IMAX-IMIN+1, Q, $ RWORK(IV1TCS+IMIN-1), RWORK(IV1TSN+IMIN-1), $ V1T(IMIN,1), LDV1T ) ELSE CALL CLASR( 'R', 'V', 'F', Q, IMAX-IMIN+1, $ RWORK(IV1TCS+IMIN-1), RWORK(IV1TSN+IMIN-1), $ V1T(1,IMIN), LDV1T ) END IF END IF IF( WANTV2T ) THEN IF( COLMAJOR ) THEN CALL CLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-Q, $ RWORK(IV2TCS+IMIN-1), RWORK(IV2TSN+IMIN-1), $ V2T(IMIN,1), LDV2T ) ELSE CALL CLASR( 'R', 'V', 'F', M-Q, IMAX-IMIN+1, $ RWORK(IV2TCS+IMIN-1), RWORK(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 CSCAL( Q, NEGONECOMPLEX, V1T(IMAX,1), LDV1T ) ELSE CALL CSCAL( Q, NEGONECOMPLEX, 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 CSCAL( P, NEGONECOMPLEX, U1(1,IMAX), 1 ) ELSE CALL CSCAL( P, NEGONECOMPLEX, 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 CSCAL( M-P, NEGONECOMPLEX, U2(1,IMAX), 1 ) ELSE CALL CSCAL( M-P, NEGONECOMPLEX, 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 CSCAL( M-Q, NEGONECOMPLEX, V2T(IMAX,1), $ LDV2T ) ELSE CALL CSCAL( M-Q, NEGONECOMPLEX, 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 THETA(I) = ZERO ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN THETA(I) = PIOVER2 END IF END DO DO I = IMIN, IMAX-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 * * Deflate * IF (IMAX .GT. 1) THEN DO WHILE( PHI(IMAX-1) .EQ. ZERO ) IMAX = IMAX - 1 IF (IMAX .LE. 1) EXIT END DO END IF IF( IMIN .GT. IMAX - 1 ) $ 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 * * Repeat main iteration loop * END DO * * Postprocessing: order THETA from least to greatest * DO I = 1, Q * MINI = I THETAMIN = THETA(I) DO J = I+1, Q IF( THETA(J) .LT. THETAMIN ) THEN MINI = J THETAMIN = THETA(J) END IF END DO * IF( MINI .NE. I ) THEN THETA(MINI) = THETA(I) THETA(I) = THETAMIN IF( COLMAJOR ) THEN IF( WANTU1 ) $ CALL CSWAP( P, U1(1,I), 1, U1(1,MINI), 1 ) IF( WANTU2 ) $ CALL CSWAP( M-P, U2(1,I), 1, U2(1,MINI), 1 ) IF( WANTV1T ) $ CALL CSWAP( Q, V1T(I,1), LDV1T, V1T(MINI,1), $ LDV1T ) IF( WANTV2T ) $ CALL CSWAP( M-Q, V2T(I,1), LDV2T, V2T(MINI,1), $ LDV2T ) ELSE IF( WANTU1 ) $ CALL CSWAP( P, U1(I,1), LDU1, U1(MINI,1), LDU1 ) IF( WANTU2 ) $ CALL CSWAP( M-P, U2(I,1), LDU2, U2(MINI,1), LDU2 ) IF( WANTV1T ) $ CALL CSWAP( Q, V1T(1,I), 1, V1T(1,MINI), 1 ) IF( WANTV2T ) $ CALL CSWAP( M-Q, V2T(1,I), 1, V2T(1,MINI), 1 ) END IF END IF * END DO * RETURN * * End of CBBCSD * END