LAPACK: SRC/dbbcsd.f Source File

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00001       SUBROUTINE DBBCSD( JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS, M, P, Q,
00002      $                   THETA, PHI, U1, LDU1, U2, LDU2, V1T, LDV1T,
00003      $                   V2T, LDV2T, B11D, B11E, B12D, B12E, B21D, B21E,
00004      $                   B22D, B22E, WORK, LWORK, INFO )
00005       IMPLICIT NONE
00006 *
00007 *  -- LAPACK routine (version 3.3.0) --
00008 *
00009 *  -- Contributed by Brian Sutton of the Randolph-Macon College --
00010 *  -- November 2010
00011 *
00012 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00013 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--     
00014 *
00015 *     .. Scalar Arguments ..
00016       CHARACTER          JOBU1, JOBU2, JOBV1T, JOBV2T, TRANS
00017       INTEGER            INFO, LDU1, LDU2, LDV1T, LDV2T, LWORK, M, P, Q
00018 *     ..
00019 *     .. Array Arguments ..
00020       DOUBLE PRECISION   B11D( * ), B11E( * ), B12D( * ), B12E( * ),
00021      $                   B21D( * ), B21E( * ), B22D( * ), B22E( * ),
00022      $                   PHI( * ), THETA( * ), WORK( * )
00023       DOUBLE PRECISION   U1( LDU1, * ), U2( LDU2, * ), V1T( LDV1T, * ),
00024      $                   V2T( LDV2T, * )
00025 *     ..
00026 *
00027 *  Purpose
00028 *  =======
00029 *
00030 *  DBBCSD computes the CS decomposition of an orthogonal matrix in
00031 *  bidiagonal-block form,
00032 *
00033 *
00034 *      [ B11 | B12 0  0 ]
00035 *      [  0  |  0 -I  0 ]
00036 *  X = [----------------]
00037 *      [ B21 | B22 0  0 ]
00038 *      [  0  |  0  0  I ]
00039 *
00040 *                                [  C | -S  0  0 ]
00041 *                    [ U1 |    ] [  0 |  0 -I  0 ] [ V1 |    ]**T
00042 *                  = [---------] [---------------] [---------]   .
00043 *                    [    | U2 ] [  S |  C  0  0 ] [    | V2 ]
00044 *                                [  0 |  0  0  I ]
00045 *
00046 *  X is M-by-M, its top-left block is P-by-Q, and Q must be no larger
00047 *  than P, M-P, or M-Q. (If Q is not the smallest index, then X must be
00048 *  transposed and/or permuted. This can be done in constant time using
00049 *  the TRANS and SIGNS options. See DORCSD for details.)
00050 *
00051 *  The bidiagonal matrices B11, B12, B21, and B22 are represented
00052 *  implicitly by angles THETA(1:Q) and PHI(1:Q-1).
00053 *
00054 *  The orthogonal matrices U1, U2, V1T, and V2T are input/output.
00055 *  The input matrices are pre- or post-multiplied by the appropriate
00056 *  singular vector matrices.
00057 *
00058 *  Arguments
00059 *  =========
00060 *
00061 *  JOBU1   (input) CHARACTER
00062 *          = 'Y':      U1 is updated;
00063 *          otherwise:  U1 is not updated.
00064 *
00065 *  JOBU2   (input) CHARACTER
00066 *          = 'Y':      U2 is updated;
00067 *          otherwise:  U2 is not updated.
00068 *
00069 *  JOBV1T  (input) CHARACTER
00070 *          = 'Y':      V1T is updated;
00071 *          otherwise:  V1T is not updated.
00072 *
00073 *  JOBV2T  (input) CHARACTER
00074 *          = 'Y':      V2T is updated;
00075 *          otherwise:  V2T is not updated.
00076 *
00077 *  TRANS   (input) CHARACTER
00078 *          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
00079 *                      order;
00080 *          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
00081 *                      major order.
00082 *
00083 *  M       (input) INTEGER
00084 *          The number of rows and columns in X, the orthogonal matrix in
00085 *          bidiagonal-block form.
00086 *
00087 *  P       (input) INTEGER
00088 *          The number of rows in the top-left block of X. 0 <= P <= M.
00089 *
00090 *  Q       (input) INTEGER
00091 *          The number of columns in the top-left block of X.
00092 *          0 <= Q <= MIN(P,M-P,M-Q).
00093 *
00094 *  THETA   (input/output) DOUBLE PRECISION array, dimension (Q)
00095 *          On entry, the angles THETA(1),...,THETA(Q) that, along with
00096 *          PHI(1), ...,PHI(Q-1), define the matrix in bidiagonal-block
00097 *          form. On exit, the angles whose cosines and sines define the
00098 *          diagonal blocks in the CS decomposition.
00099 *
00100 *  PHI     (input/workspace) DOUBLE PRECISION array, dimension (Q-1)
00101 *          The angles PHI(1),...,PHI(Q-1) that, along with THETA(1),...,
00102 *          THETA(Q), define the matrix in bidiagonal-block form.
00103 *
00104 *  U1      (input/output) DOUBLE PRECISION array, dimension (LDU1,P)
00105 *          On entry, an LDU1-by-P matrix. On exit, U1 is postmultiplied
00106 *          by the left singular vector matrix common to [ B11 ; 0 ] and
00107 *          [ B12 0 0 ; 0 -I 0 0 ].
00108 *
00109 *  LDU1    (input) INTEGER
00110 *          The leading dimension of the array U1.
00111 *
00112 *  U2      (input/output) DOUBLE PRECISION array, dimension (LDU2,M-P)
00113 *          On entry, an LDU2-by-(M-P) matrix. On exit, U2 is
00114 *          postmultiplied by the left singular vector matrix common to
00115 *          [ B21 ; 0 ] and [ B22 0 0 ; 0 0 I ].
00116 *
00117 *  LDU2    (input) INTEGER
00118 *          The leading dimension of the array U2.
00119 *
00120 *  V1T     (input/output) DOUBLE PRECISION array, dimension (LDV1T,Q)
00121 *          On entry, a LDV1T-by-Q matrix. On exit, V1T is premultiplied
00122 *          by the transpose of the right singular vector
00123 *          matrix common to [ B11 ; 0 ] and [ B21 ; 0 ].
00124 *
00125 *  LDV1T   (input) INTEGER
00126 *          The leading dimension of the array V1T.
00127 *
00128 *  V2T     (input/output) DOUBLE PRECISION array, dimenison (LDV2T,M-Q)
00129 *          On entry, a LDV2T-by-(M-Q) matrix. On exit, V2T is
00130 *          premultiplied by the transpose of the right
00131 *          singular vector matrix common to [ B12 0 0 ; 0 -I 0 ] and
00132 *          [ B22 0 0 ; 0 0 I ].
00133 *
00134 *  LDV2T   (input) INTEGER
00135 *          The leading dimension of the array V2T.
00136 *
00137 *  B11D    (output) DOUBLE PRECISION array, dimension (Q)
00138 *          When DBBCSD converges, B11D contains the cosines of THETA(1),
00139 *          ..., THETA(Q). If DBBCSD fails to converge, then B11D
00140 *          contains the diagonal of the partially reduced top-left
00141 *          block.
00142 *
00143 *  B11E    (output) DOUBLE PRECISION array, dimension (Q-1)
00144 *          When DBBCSD converges, B11E contains zeros. If DBBCSD fails
00145 *          to converge, then B11E contains the superdiagonal of the
00146 *          partially reduced top-left block.
00147 *
00148 *  B12D    (output) DOUBLE PRECISION array, dimension (Q)
00149 *          When DBBCSD converges, B12D contains the negative sines of
00150 *          THETA(1), ..., THETA(Q). If DBBCSD fails to converge, then
00151 *          B12D contains the diagonal of the partially reduced top-right
00152 *          block.
00153 *
00154 *  B12E    (output) DOUBLE PRECISION array, dimension (Q-1)
00155 *          When DBBCSD converges, B12E contains zeros. If DBBCSD fails
00156 *          to converge, then B12E contains the subdiagonal of the
00157 *          partially reduced top-right block.
00158 *
00159 *  WORK    (workspace) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
00160 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00161 *
00162 *  LWORK   (input) INTEGER
00163 *          The dimension of the array WORK. LWORK >= MAX(1,8*Q).
00164 *
00165 *          If LWORK = -1, then a workspace query is assumed; the
00166 *          routine only calculates the optimal size of the WORK array,
00167 *          returns this value as the first entry of the work array, and
00168 *          no error message related to LWORK is issued by XERBLA.
00169 *
00170 *  INFO    (output) INTEGER
00171 *          = 0:  successful exit.
00172 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00173 *          > 0:  if DBBCSD did not converge, INFO specifies the number
00174 *                of nonzero entries in PHI, and B11D, B11E, etc.,
00175 *                contain the partially reduced matrix.
00176 *
00177 *  Reference
00178 *  =========
00179 *
00180 *  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
00181 *      Algorithms, 50(1):33-65, 2009.
00182 *
00183 *  Internal Parameters
00184 *  ===================
00185 *
00186 *  TOLMUL  DOUBLE PRECISION, default = MAX(10,MIN(100,EPS**(-1/8)))
00187 *          TOLMUL controls the convergence criterion of the QR loop.
00188 *          Angles THETA(i), PHI(i) are rounded to 0 or PI/2 when they
00189 *          are within TOLMUL*EPS of either bound.
00190 *
00191 *  ===================================================================
00192 *
00193 *     .. Parameters ..
00194       INTEGER            MAXITR
00195       PARAMETER          ( MAXITR = 6 )
00196       DOUBLE PRECISION   HUNDRED, MEIGHTH, ONE, PIOVER2, TEN, ZERO
00197       PARAMETER          ( HUNDRED = 100.0D0, MEIGHTH = -0.125D0,
00198      $                     ONE = 1.0D0, PIOVER2 = 1.57079632679489662D0,
00199      $                     TEN = 10.0D0, ZERO = 0.0D0 )
00200       DOUBLE PRECISION   NEGONECOMPLEX
00201       PARAMETER          ( NEGONECOMPLEX = -1.0D0 )
00202 *     ..
00203 *     .. Local Scalars ..
00204       LOGICAL            COLMAJOR, LQUERY, RESTART11, RESTART12,
00205      $                   RESTART21, RESTART22, WANTU1, WANTU2, WANTV1T,
00206      $                   WANTV2T
00207       INTEGER            I, IMIN, IMAX, ITER, IU1CS, IU1SN, IU2CS,
00208      $                   IU2SN, IV1TCS, IV1TSN, IV2TCS, IV2TSN, J,
00209      $                   LWORKMIN, LWORKOPT, MAXIT, MINI
00210       DOUBLE PRECISION   B11BULGE, B12BULGE, B21BULGE, B22BULGE, DUMMY,
00211      $                   EPS, MU, NU, R, SIGMA11, SIGMA21,
00212      $                   TEMP, THETAMAX, THETAMIN, THRESH, TOL, TOLMUL,
00213      $                   UNFL, X1, X2, Y1, Y2
00214 *
00215 *     .. External Subroutines ..
00216       EXTERNAL           DLASR, DSCAL, DSWAP, DLARTGP, DLARTGS, DLAS2,
00217      $                   XERBLA
00218 *     ..
00219 *     .. External Functions ..
00220       DOUBLE PRECISION   DLAMCH
00221       LOGICAL            LSAME
00222       EXTERNAL           LSAME, DLAMCH
00223 *     ..
00224 *     .. Intrinsic Functions ..
00225       INTRINSIC          ABS, ATAN2, COS, MAX, MIN, SIN, SQRT
00226 *     ..
00227 *     .. Executable Statements ..
00228 *
00229 *     Test input arguments
00230 *
00231       INFO = 0
00232       LQUERY = LWORK .EQ. -1
00233       WANTU1 = LSAME( JOBU1, 'Y' )
00234       WANTU2 = LSAME( JOBU2, 'Y' )
00235       WANTV1T = LSAME( JOBV1T, 'Y' )
00236       WANTV2T = LSAME( JOBV2T, 'Y' )
00237       COLMAJOR = .NOT. LSAME( TRANS, 'T' )
00238 *
00239       IF( M .LT. 0 ) THEN
00240          INFO = -6
00241       ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
00242          INFO = -7
00243       ELSE IF( Q .LT. 0 .OR. Q .GT. M ) THEN
00244          INFO = -8
00245       ELSE IF( Q .GT. P .OR. Q .GT. M-P .OR. Q .GT. M-Q ) THEN
00246          INFO = -8
00247       ELSE IF( WANTU1 .AND. LDU1 .LT. P ) THEN
00248          INFO = -12
00249       ELSE IF( WANTU2 .AND. LDU2 .LT. M-P ) THEN
00250          INFO = -14
00251       ELSE IF( WANTV1T .AND. LDV1T .LT. Q ) THEN
00252          INFO = -16
00253       ELSE IF( WANTV2T .AND. LDV2T .LT. M-Q ) THEN
00254          INFO = -18
00255       END IF
00256 *
00257 *     Quick return if Q = 0
00258 *
00259       IF( INFO .EQ. 0 .AND. Q .EQ. 0 ) THEN
00260          LWORKMIN = 1
00261          WORK(1) = LWORKMIN
00262          RETURN
00263       END IF
00264 *
00265 *     Compute workspace
00266 *
00267       IF( INFO .EQ. 0 ) THEN
00268          IU1CS = 1
00269          IU1SN = IU1CS + Q
00270          IU2CS = IU1SN + Q
00271          IU2SN = IU2CS + Q
00272          IV1TCS = IU2SN + Q
00273          IV1TSN = IV1TCS + Q
00274          IV2TCS = IV1TSN + Q
00275          IV2TSN = IV2TCS + Q
00276          LWORKOPT = IV2TSN + Q - 1
00277          LWORKMIN = LWORKOPT
00278          WORK(1) = LWORKOPT
00279          IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
00280             INFO = -28
00281          END IF
00282       END IF
00283 *
00284       IF( INFO .NE. 0 ) THEN
00285          CALL XERBLA( 'DBBCSD', -INFO )
00286          RETURN
00287       ELSE IF( LQUERY ) THEN
00288          RETURN
00289       END IF
00290 *
00291 *     Get machine constants
00292 *
00293       EPS = DLAMCH( 'Epsilon' )
00294       UNFL = DLAMCH( 'Safe minimum' )
00295       TOLMUL = MAX( TEN, MIN( HUNDRED, EPS**MEIGHTH ) )
00296       TOL = TOLMUL*EPS
00297       THRESH = MAX( TOL, MAXITR*Q*Q*UNFL )
00298 *
00299 *     Test for negligible sines or cosines
00300 *
00301       DO I = 1, Q
00302          IF( THETA(I) .LT. THRESH ) THEN
00303             THETA(I) = ZERO
00304          ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN
00305             THETA(I) = PIOVER2
00306          END IF
00307       END DO
00308       DO I = 1, Q-1
00309          IF( PHI(I) .LT. THRESH ) THEN
00310             PHI(I) = ZERO
00311          ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN
00312             PHI(I) = PIOVER2
00313          END IF
00314       END DO
00315 *
00316 *     Initial deflation
00317 *
00318       IMAX = Q
00319       DO WHILE( ( IMAX .GT. 1 ) .AND. ( PHI(IMAX-1) .EQ. ZERO ) )
00320          IMAX = IMAX - 1
00321       END DO
00322       IMIN = IMAX - 1
00323       IF  ( IMIN .GT. 1 ) THEN
00324          DO WHILE( PHI(IMIN-1) .NE. ZERO )
00325             IMIN = IMIN - 1
00326             IF  ( IMIN .LE. 1 ) EXIT
00327          END DO
00328       END IF
00329 *
00330 *     Initialize iteration counter
00331 *
00332       MAXIT = MAXITR*Q*Q
00333       ITER = 0
00334 *
00335 *     Begin main iteration loop
00336 *
00337       DO WHILE( IMAX .GT. 1 )
00338 *
00339 *        Compute the matrix entries
00340 *
00341          B11D(IMIN) = COS( THETA(IMIN) )
00342          B21D(IMIN) = -SIN( THETA(IMIN) )
00343          DO I = IMIN, IMAX - 1
00344             B11E(I) = -SIN( THETA(I) ) * SIN( PHI(I) )
00345             B11D(I+1) = COS( THETA(I+1) ) * COS( PHI(I) )
00346             B12D(I) = SIN( THETA(I) ) * COS( PHI(I) )
00347             B12E(I) = COS( THETA(I+1) ) * SIN( PHI(I) )
00348             B21E(I) = -COS( THETA(I) ) * SIN( PHI(I) )
00349             B21D(I+1) = -SIN( THETA(I+1) ) * COS( PHI(I) )
00350             B22D(I) = COS( THETA(I) ) * COS( PHI(I) )
00351             B22E(I) = -SIN( THETA(I+1) ) * SIN( PHI(I) )
00352          END DO
00353          B12D(IMAX) = SIN( THETA(IMAX) )
00354          B22D(IMAX) = COS( THETA(IMAX) )
00355 *
00356 *        Abort if not converging; otherwise, increment ITER
00357 *
00358          IF( ITER .GT. MAXIT ) THEN
00359             INFO = 0
00360             DO I = 1, Q
00361                IF( PHI(I) .NE. ZERO )
00362      $            INFO = INFO + 1
00363             END DO
00364             RETURN
00365          END IF
00366 *
00367          ITER = ITER + IMAX - IMIN
00368 *
00369 *        Compute shifts
00370 *
00371          THETAMAX = THETA(IMIN)
00372          THETAMIN = THETA(IMIN)
00373          DO I = IMIN+1, IMAX
00374             IF( THETA(I) > THETAMAX )
00375      $         THETAMAX = THETA(I)
00376             IF( THETA(I) < THETAMIN )
00377      $         THETAMIN = THETA(I)
00378          END DO
00379 *
00380          IF( THETAMAX .GT. PIOVER2 - THRESH ) THEN
00381 *
00382 *           Zero on diagonals of B11 and B22; induce deflation with a
00383 *           zero shift
00384 *
00385             MU = ZERO
00386             NU = ONE
00387 *
00388          ELSE IF( THETAMIN .LT. THRESH ) THEN
00389 *
00390 *           Zero on diagonals of B12 and B22; induce deflation with a
00391 *           zero shift
00392 *
00393             MU = ONE
00394             NU = ZERO
00395 *
00396          ELSE
00397 *
00398 *           Compute shifts for B11 and B21 and use the lesser
00399 *
00400             CALL DLAS2( B11D(IMAX-1), B11E(IMAX-1), B11D(IMAX), SIGMA11,
00401      $                  DUMMY )
00402             CALL DLAS2( B21D(IMAX-1), B21E(IMAX-1), B21D(IMAX), SIGMA21,
00403      $                  DUMMY )
00404 *
00405             IF( SIGMA11 .LE. SIGMA21 ) THEN
00406                MU = SIGMA11
00407                NU = SQRT( ONE - MU**2 )
00408                IF( MU .LT. THRESH ) THEN
00409                   MU = ZERO
00410                   NU = ONE
00411                END IF
00412             ELSE
00413                NU = SIGMA21
00414                MU = SQRT( 1.0 - NU**2 )
00415                IF( NU .LT. THRESH ) THEN
00416                   MU = ONE
00417                   NU = ZERO
00418                END IF
00419             END IF
00420          END IF
00421 *
00422 *        Rotate to produce bulges in B11 and B21
00423 *
00424          IF( MU .LE. NU ) THEN
00425             CALL DLARTGS( B11D(IMIN), B11E(IMIN), MU,
00426      $                    WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
00427          ELSE
00428             CALL DLARTGS( B21D(IMIN), B21E(IMIN), NU,
00429      $                    WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1) )
00430          END IF
00431 *
00432          TEMP = WORK(IV1TCS+IMIN-1)*B11D(IMIN) +
00433      $          WORK(IV1TSN+IMIN-1)*B11E(IMIN)
00434          B11E(IMIN) = WORK(IV1TCS+IMIN-1)*B11E(IMIN) -
00435      $                WORK(IV1TSN+IMIN-1)*B11D(IMIN)
00436          B11D(IMIN) = TEMP
00437          B11BULGE = WORK(IV1TSN+IMIN-1)*B11D(IMIN+1)
00438          B11D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B11D(IMIN+1)
00439          TEMP = WORK(IV1TCS+IMIN-1)*B21D(IMIN) +
00440      $          WORK(IV1TSN+IMIN-1)*B21E(IMIN)
00441          B21E(IMIN) = WORK(IV1TCS+IMIN-1)*B21E(IMIN) -
00442      $                WORK(IV1TSN+IMIN-1)*B21D(IMIN)
00443          B21D(IMIN) = TEMP
00444          B21BULGE = WORK(IV1TSN+IMIN-1)*B21D(IMIN+1)
00445          B21D(IMIN+1) = WORK(IV1TCS+IMIN-1)*B21D(IMIN+1)
00446 *
00447 *        Compute THETA(IMIN)
00448 *
00449          THETA( IMIN ) = ATAN2( SQRT( B21D(IMIN)**2+B21BULGE**2 ),
00450      $                   SQRT( B11D(IMIN)**2+B11BULGE**2 ) )
00451 *
00452 *        Chase the bulges in B11(IMIN+1,IMIN) and B21(IMIN+1,IMIN)
00453 *
00454          IF( B11D(IMIN)**2+B11BULGE**2 .GT. THRESH**2 ) THEN
00455             CALL DLARTGP( B11BULGE, B11D(IMIN), WORK(IU1SN+IMIN-1),
00456      $                    WORK(IU1CS+IMIN-1), R )
00457          ELSE IF( MU .LE. NU ) THEN
00458             CALL DLARTGS( B11E( IMIN ), B11D( IMIN + 1 ), MU,
00459      $                    WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
00460          ELSE
00461             CALL DLARTGS( B12D( IMIN ), B12E( IMIN ), NU,
00462      $                    WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1) )
00463          END IF
00464          IF( B21D(IMIN)**2+B21BULGE**2 .GT. THRESH**2 ) THEN
00465             CALL DLARTGP( B21BULGE, B21D(IMIN), WORK(IU2SN+IMIN-1),
00466      $                    WORK(IU2CS+IMIN-1), R )
00467          ELSE IF( NU .LT. MU ) THEN
00468             CALL DLARTGS( B21E( IMIN ), B21D( IMIN + 1 ), NU,
00469      $                    WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
00470          ELSE
00471             CALL DLARTGS( B22D(IMIN), B22E(IMIN), MU,
00472      $                    WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1) )
00473          END IF
00474          WORK(IU2CS+IMIN-1) = -WORK(IU2CS+IMIN-1)
00475          WORK(IU2SN+IMIN-1) = -WORK(IU2SN+IMIN-1)
00476 *
00477          TEMP = WORK(IU1CS+IMIN-1)*B11E(IMIN) +
00478      $          WORK(IU1SN+IMIN-1)*B11D(IMIN+1)
00479          B11D(IMIN+1) = WORK(IU1CS+IMIN-1)*B11D(IMIN+1) -
00480      $                  WORK(IU1SN+IMIN-1)*B11E(IMIN)
00481          B11E(IMIN) = TEMP
00482          IF( IMAX .GT. IMIN+1 ) THEN
00483             B11BULGE = WORK(IU1SN+IMIN-1)*B11E(IMIN+1)
00484             B11E(IMIN+1) = WORK(IU1CS+IMIN-1)*B11E(IMIN+1)
00485          END IF
00486          TEMP = WORK(IU1CS+IMIN-1)*B12D(IMIN) +
00487      $          WORK(IU1SN+IMIN-1)*B12E(IMIN)
00488          B12E(IMIN) = WORK(IU1CS+IMIN-1)*B12E(IMIN) -
00489      $                WORK(IU1SN+IMIN-1)*B12D(IMIN)
00490          B12D(IMIN) = TEMP
00491          B12BULGE = WORK(IU1SN+IMIN-1)*B12D(IMIN+1)
00492          B12D(IMIN+1) = WORK(IU1CS+IMIN-1)*B12D(IMIN+1)
00493          TEMP = WORK(IU2CS+IMIN-1)*B21E(IMIN) +
00494      $          WORK(IU2SN+IMIN-1)*B21D(IMIN+1)
00495          B21D(IMIN+1) = WORK(IU2CS+IMIN-1)*B21D(IMIN+1) -
00496      $                  WORK(IU2SN+IMIN-1)*B21E(IMIN)
00497          B21E(IMIN) = TEMP
00498          IF( IMAX .GT. IMIN+1 ) THEN
00499             B21BULGE = WORK(IU2SN+IMIN-1)*B21E(IMIN+1)
00500             B21E(IMIN+1) = WORK(IU2CS+IMIN-1)*B21E(IMIN+1)
00501          END IF
00502          TEMP = WORK(IU2CS+IMIN-1)*B22D(IMIN) +
00503      $          WORK(IU2SN+IMIN-1)*B22E(IMIN)
00504          B22E(IMIN) = WORK(IU2CS+IMIN-1)*B22E(IMIN) -
00505      $                WORK(IU2SN+IMIN-1)*B22D(IMIN)
00506          B22D(IMIN) = TEMP
00507          B22BULGE = WORK(IU2SN+IMIN-1)*B22D(IMIN+1)
00508          B22D(IMIN+1) = WORK(IU2CS+IMIN-1)*B22D(IMIN+1)
00509 *
00510 *        Inner loop: chase bulges from B11(IMIN,IMIN+2),
00511 *        B12(IMIN,IMIN+1), B21(IMIN,IMIN+2), and B22(IMIN,IMIN+1) to
00512 *        bottom-right
00513 *
00514          DO I = IMIN+1, IMAX-1
00515 *
00516 *           Compute PHI(I-1)
00517 *
00518             X1 = SIN(THETA(I-1))*B11E(I-1) + COS(THETA(I-1))*B21E(I-1)
00519             X2 = SIN(THETA(I-1))*B11BULGE + COS(THETA(I-1))*B21BULGE
00520             Y1 = SIN(THETA(I-1))*B12D(I-1) + COS(THETA(I-1))*B22D(I-1)
00521             Y2 = SIN(THETA(I-1))*B12BULGE + COS(THETA(I-1))*B22BULGE
00522 *
00523             PHI(I-1) = ATAN2( SQRT(X1**2+X2**2), SQRT(Y1**2+Y2**2) )
00524 *
00525 *           Determine if there are bulges to chase or if a new direct
00526 *           summand has been reached
00527 *
00528             RESTART11 = B11E(I-1)**2 + B11BULGE**2 .LE. THRESH**2
00529             RESTART21 = B21E(I-1)**2 + B21BULGE**2 .LE. THRESH**2
00530             RESTART12 = B12D(I-1)**2 + B12BULGE**2 .LE. THRESH**2
00531             RESTART22 = B22D(I-1)**2 + B22BULGE**2 .LE. THRESH**2
00532 *
00533 *           If possible, chase bulges from B11(I-1,I+1), B12(I-1,I),
00534 *           B21(I-1,I+1), and B22(I-1,I). If necessary, restart bulge-
00535 *           chasing by applying the original shift again.
00536 *
00537             IF( .NOT. RESTART11 .AND. .NOT. RESTART21 ) THEN
00538                CALL DLARTGP( X2, X1, WORK(IV1TSN+I-1), WORK(IV1TCS+I-1),
00539      $                       R )
00540             ELSE IF( .NOT. RESTART11 .AND. RESTART21 ) THEN
00541                CALL DLARTGP( B11BULGE, B11E(I-1), WORK(IV1TSN+I-1),
00542      $                       WORK(IV1TCS+I-1), R )
00543             ELSE IF( RESTART11 .AND. .NOT. RESTART21 ) THEN
00544                CALL DLARTGP( B21BULGE, B21E(I-1), WORK(IV1TSN+I-1),
00545      $                       WORK(IV1TCS+I-1), R )
00546             ELSE IF( MU .LE. NU ) THEN
00547                CALL DLARTGS( B11D(I), B11E(I), MU, WORK(IV1TCS+I-1),
00548      $                       WORK(IV1TSN+I-1) )
00549             ELSE
00550                CALL DLARTGS( B21D(I), B21E(I), NU, WORK(IV1TCS+I-1),
00551      $                       WORK(IV1TSN+I-1) )
00552             END IF
00553             WORK(IV1TCS+I-1) = -WORK(IV1TCS+I-1)
00554             WORK(IV1TSN+I-1) = -WORK(IV1TSN+I-1)
00555             IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
00556                CALL DLARTGP( Y2, Y1, WORK(IV2TSN+I-1-1),
00557      $                       WORK(IV2TCS+I-1-1), R )
00558             ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
00559                CALL DLARTGP( B12BULGE, B12D(I-1), WORK(IV2TSN+I-1-1),
00560      $                       WORK(IV2TCS+I-1-1), R )
00561             ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
00562                CALL DLARTGP( B22BULGE, B22D(I-1), WORK(IV2TSN+I-1-1),
00563      $                       WORK(IV2TCS+I-1-1), R )
00564             ELSE IF( NU .LT. MU ) THEN
00565                CALL DLARTGS( B12E(I-1), B12D(I), NU, WORK(IV2TCS+I-1-1),
00566      $                       WORK(IV2TSN+I-1-1) )
00567             ELSE
00568                CALL DLARTGS( B22E(I-1), B22D(I), MU, WORK(IV2TCS+I-1-1),
00569      $                       WORK(IV2TSN+I-1-1) )
00570             END IF
00571 *
00572             TEMP = WORK(IV1TCS+I-1)*B11D(I) + WORK(IV1TSN+I-1)*B11E(I)
00573             B11E(I) = WORK(IV1TCS+I-1)*B11E(I) -
00574      $                WORK(IV1TSN+I-1)*B11D(I)
00575             B11D(I) = TEMP
00576             B11BULGE = WORK(IV1TSN+I-1)*B11D(I+1)
00577             B11D(I+1) = WORK(IV1TCS+I-1)*B11D(I+1)
00578             TEMP = WORK(IV1TCS+I-1)*B21D(I) + WORK(IV1TSN+I-1)*B21E(I)
00579             B21E(I) = WORK(IV1TCS+I-1)*B21E(I) -
00580      $                WORK(IV1TSN+I-1)*B21D(I)
00581             B21D(I) = TEMP
00582             B21BULGE = WORK(IV1TSN+I-1)*B21D(I+1)
00583             B21D(I+1) = WORK(IV1TCS+I-1)*B21D(I+1)
00584             TEMP = WORK(IV2TCS+I-1-1)*B12E(I-1) +
00585      $             WORK(IV2TSN+I-1-1)*B12D(I)
00586             B12D(I) = WORK(IV2TCS+I-1-1)*B12D(I) -
00587      $                WORK(IV2TSN+I-1-1)*B12E(I-1)
00588             B12E(I-1) = TEMP
00589             B12BULGE = WORK(IV2TSN+I-1-1)*B12E(I)
00590             B12E(I) = WORK(IV2TCS+I-1-1)*B12E(I)
00591             TEMP = WORK(IV2TCS+I-1-1)*B22E(I-1) +
00592      $             WORK(IV2TSN+I-1-1)*B22D(I)
00593             B22D(I) = WORK(IV2TCS+I-1-1)*B22D(I) -
00594      $                WORK(IV2TSN+I-1-1)*B22E(I-1)
00595             B22E(I-1) = TEMP
00596             B22BULGE = WORK(IV2TSN+I-1-1)*B22E(I)
00597             B22E(I) = WORK(IV2TCS+I-1-1)*B22E(I)
00598 *
00599 *           Compute THETA(I)
00600 *
00601             X1 = COS(PHI(I-1))*B11D(I) + SIN(PHI(I-1))*B12E(I-1)
00602             X2 = COS(PHI(I-1))*B11BULGE + SIN(PHI(I-1))*B12BULGE
00603             Y1 = COS(PHI(I-1))*B21D(I) + SIN(PHI(I-1))*B22E(I-1)
00604             Y2 = COS(PHI(I-1))*B21BULGE + SIN(PHI(I-1))*B22BULGE
00605 *
00606             THETA(I) = ATAN2( SQRT(Y1**2+Y2**2), SQRT(X1**2+X2**2) )
00607 *
00608 *           Determine if there are bulges to chase or if a new direct
00609 *           summand has been reached
00610 *
00611             RESTART11 =   B11D(I)**2 + B11BULGE**2 .LE. THRESH**2
00612             RESTART12 = B12E(I-1)**2 + B12BULGE**2 .LE. THRESH**2
00613             RESTART21 =   B21D(I)**2 + B21BULGE**2 .LE. THRESH**2
00614             RESTART22 = B22E(I-1)**2 + B22BULGE**2 .LE. THRESH**2
00615 *
00616 *           If possible, chase bulges from B11(I+1,I), B12(I+1,I-1),
00617 *           B21(I+1,I), and B22(I+1,I-1). If necessary, restart bulge-
00618 *           chasing by applying the original shift again.
00619 *
00620             IF( .NOT. RESTART11 .AND. .NOT. RESTART12 ) THEN
00621                CALL DLARTGP( X2, X1, WORK(IU1SN+I-1), WORK(IU1CS+I-1),
00622      $                       R )
00623             ELSE IF( .NOT. RESTART11 .AND. RESTART12 ) THEN
00624                CALL DLARTGP( B11BULGE, B11D(I), WORK(IU1SN+I-1),
00625      $                       WORK(IU1CS+I-1), R )
00626             ELSE IF( RESTART11 .AND. .NOT. RESTART12 ) THEN
00627                CALL DLARTGP( B12BULGE, B12E(I-1), WORK(IU1SN+I-1),
00628      $                       WORK(IU1CS+I-1), R )
00629             ELSE IF( MU .LE. NU ) THEN
00630                CALL DLARTGS( B11E(I), B11D(I+1), MU, WORK(IU1CS+I-1),
00631      $                       WORK(IU1SN+I-1) )
00632             ELSE
00633                CALL DLARTGS( B12D(I), B12E(I), NU, WORK(IU1CS+I-1),
00634      $                       WORK(IU1SN+I-1) )
00635             END IF
00636             IF( .NOT. RESTART21 .AND. .NOT. RESTART22 ) THEN
00637                CALL DLARTGP( Y2, Y1, WORK(IU2SN+I-1), WORK(IU2CS+I-1),
00638      $                       R )
00639             ELSE IF( .NOT. RESTART21 .AND. RESTART22 ) THEN
00640                CALL DLARTGP( B21BULGE, B21D(I), WORK(IU2SN+I-1),
00641      $                       WORK(IU2CS+I-1), R )
00642             ELSE IF( RESTART21 .AND. .NOT. RESTART22 ) THEN
00643                CALL DLARTGP( B22BULGE, B22E(I-1), WORK(IU2SN+I-1),
00644      $                       WORK(IU2CS+I-1), R )
00645             ELSE IF( NU .LT. MU ) THEN
00646                CALL DLARTGS( B21E(I), B21E(I+1), NU, WORK(IU2CS+I-1),
00647      $                       WORK(IU2SN+I-1) )
00648             ELSE
00649                CALL DLARTGS( B22D(I), B22E(I), MU, WORK(IU2CS+I-1),
00650      $                       WORK(IU2SN+I-1) )
00651             END IF
00652             WORK(IU2CS+I-1) = -WORK(IU2CS+I-1)
00653             WORK(IU2SN+I-1) = -WORK(IU2SN+I-1)
00654 *
00655             TEMP = WORK(IU1CS+I-1)*B11E(I) + WORK(IU1SN+I-1)*B11D(I+1)
00656             B11D(I+1) = WORK(IU1CS+I-1)*B11D(I+1) -
00657      $                  WORK(IU1SN+I-1)*B11E(I)
00658             B11E(I) = TEMP
00659             IF( I .LT. IMAX - 1 ) THEN
00660                B11BULGE = WORK(IU1SN+I-1)*B11E(I+1)
00661                B11E(I+1) = WORK(IU1CS+I-1)*B11E(I+1)
00662             END IF
00663             TEMP = WORK(IU2CS+I-1)*B21E(I) + WORK(IU2SN+I-1)*B21D(I+1)
00664             B21D(I+1) = WORK(IU2CS+I-1)*B21D(I+1) -
00665      $                  WORK(IU2SN+I-1)*B21E(I)
00666             B21E(I) = TEMP
00667             IF( I .LT. IMAX - 1 ) THEN
00668                B21BULGE = WORK(IU2SN+I-1)*B21E(I+1)
00669                B21E(I+1) = WORK(IU2CS+I-1)*B21E(I+1)
00670             END IF
00671             TEMP = WORK(IU1CS+I-1)*B12D(I) + WORK(IU1SN+I-1)*B12E(I)
00672             B12E(I) = WORK(IU1CS+I-1)*B12E(I) - WORK(IU1SN+I-1)*B12D(I)
00673             B12D(I) = TEMP
00674             B12BULGE = WORK(IU1SN+I-1)*B12D(I+1)
00675             B12D(I+1) = WORK(IU1CS+I-1)*B12D(I+1)
00676             TEMP = WORK(IU2CS+I-1)*B22D(I) + WORK(IU2SN+I-1)*B22E(I)
00677             B22E(I) = WORK(IU2CS+I-1)*B22E(I) - WORK(IU2SN+I-1)*B22D(I)
00678             B22D(I) = TEMP
00679             B22BULGE = WORK(IU2SN+I-1)*B22D(I+1)
00680             B22D(I+1) = WORK(IU2CS+I-1)*B22D(I+1)
00681 *
00682          END DO
00683 *
00684 *        Compute PHI(IMAX-1)
00685 *
00686          X1 = SIN(THETA(IMAX-1))*B11E(IMAX-1) +
00687      $        COS(THETA(IMAX-1))*B21E(IMAX-1)
00688          Y1 = SIN(THETA(IMAX-1))*B12D(IMAX-1) +
00689      $        COS(THETA(IMAX-1))*B22D(IMAX-1)
00690          Y2 = SIN(THETA(IMAX-1))*B12BULGE + COS(THETA(IMAX-1))*B22BULGE
00691 *
00692          PHI(IMAX-1) = ATAN2( ABS(X1), SQRT(Y1**2+Y2**2) )
00693 *
00694 *        Chase bulges from B12(IMAX-1,IMAX) and B22(IMAX-1,IMAX)
00695 *
00696          RESTART12 = B12D(IMAX-1)**2 + B12BULGE**2 .LE. THRESH**2
00697          RESTART22 = B22D(IMAX-1)**2 + B22BULGE**2 .LE. THRESH**2
00698 *
00699          IF( .NOT. RESTART12 .AND. .NOT. RESTART22 ) THEN
00700             CALL DLARTGP( Y2, Y1, WORK(IV2TSN+IMAX-1-1),
00701      $                    WORK(IV2TCS+IMAX-1-1), R )
00702          ELSE IF( .NOT. RESTART12 .AND. RESTART22 ) THEN
00703             CALL DLARTGP( B12BULGE, B12D(IMAX-1), WORK(IV2TSN+IMAX-1-1),
00704      $                    WORK(IV2TCS+IMAX-1-1), R )
00705          ELSE IF( RESTART12 .AND. .NOT. RESTART22 ) THEN
00706             CALL DLARTGP( B22BULGE, B22D(IMAX-1), WORK(IV2TSN+IMAX-1-1),
00707      $                    WORK(IV2TCS+IMAX-1-1), R )
00708          ELSE IF( NU .LT. MU ) THEN
00709             CALL DLARTGS( B12E(IMAX-1), B12D(IMAX), NU,
00710      $                    WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
00711          ELSE
00712             CALL DLARTGS( B22E(IMAX-1), B22D(IMAX), MU,
00713      $                    WORK(IV2TCS+IMAX-1-1), WORK(IV2TSN+IMAX-1-1) )
00714          END IF
00715 *
00716          TEMP = WORK(IV2TCS+IMAX-1-1)*B12E(IMAX-1) +
00717      $          WORK(IV2TSN+IMAX-1-1)*B12D(IMAX)
00718          B12D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B12D(IMAX) -
00719      $                WORK(IV2TSN+IMAX-1-1)*B12E(IMAX-1)
00720          B12E(IMAX-1) = TEMP
00721          TEMP = WORK(IV2TCS+IMAX-1-1)*B22E(IMAX-1) +
00722      $          WORK(IV2TSN+IMAX-1-1)*B22D(IMAX)
00723          B22D(IMAX) = WORK(IV2TCS+IMAX-1-1)*B22D(IMAX) -
00724      $                WORK(IV2TSN+IMAX-1-1)*B22E(IMAX-1)
00725          B22E(IMAX-1) = TEMP
00726 *
00727 *        Update singular vectors
00728 *
00729          IF( WANTU1 ) THEN
00730             IF( COLMAJOR ) THEN
00731                CALL DLASR( 'R', 'V', 'F', P, IMAX-IMIN+1,
00732      $                     WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
00733      $                     U1(1,IMIN), LDU1 )
00734             ELSE
00735                CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, P,
00736      $                     WORK(IU1CS+IMIN-1), WORK(IU1SN+IMIN-1),
00737      $                     U1(IMIN,1), LDU1 )
00738             END IF
00739          END IF
00740          IF( WANTU2 ) THEN
00741             IF( COLMAJOR ) THEN
00742                CALL DLASR( 'R', 'V', 'F', M-P, IMAX-IMIN+1,
00743      $                     WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
00744      $                     U2(1,IMIN), LDU2 )
00745             ELSE
00746                CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-P,
00747      $                     WORK(IU2CS+IMIN-1), WORK(IU2SN+IMIN-1),
00748      $                     U2(IMIN,1), LDU2 )
00749             END IF
00750          END IF
00751          IF( WANTV1T ) THEN
00752             IF( COLMAJOR ) THEN
00753                CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, Q,
00754      $                     WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
00755      $                     V1T(IMIN,1), LDV1T )
00756             ELSE
00757                CALL DLASR( 'R', 'V', 'F', Q, IMAX-IMIN+1,
00758      $                     WORK(IV1TCS+IMIN-1), WORK(IV1TSN+IMIN-1),
00759      $                     V1T(1,IMIN), LDV1T )
00760             END IF
00761          END IF
00762          IF( WANTV2T ) THEN
00763             IF( COLMAJOR ) THEN
00764                CALL DLASR( 'L', 'V', 'F', IMAX-IMIN+1, M-Q,
00765      $                     WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
00766      $                     V2T(IMIN,1), LDV2T )
00767             ELSE
00768                CALL DLASR( 'R', 'V', 'F', M-Q, IMAX-IMIN+1,
00769      $                     WORK(IV2TCS+IMIN-1), WORK(IV2TSN+IMIN-1),
00770      $                     V2T(1,IMIN), LDV2T )
00771             END IF
00772          END IF
00773 *
00774 *        Fix signs on B11(IMAX-1,IMAX) and B21(IMAX-1,IMAX)
00775 *
00776          IF( B11E(IMAX-1)+B21E(IMAX-1) .GT. 0 ) THEN
00777             B11D(IMAX) = -B11D(IMAX)
00778             B21D(IMAX) = -B21D(IMAX)
00779             IF( WANTV1T ) THEN
00780                IF( COLMAJOR ) THEN
00781                   CALL DSCAL( Q, NEGONECOMPLEX, V1T(IMAX,1), LDV1T )
00782                ELSE
00783                   CALL DSCAL( Q, NEGONECOMPLEX, V1T(1,IMAX), 1 )
00784                END IF
00785             END IF
00786          END IF
00787 *
00788 *        Compute THETA(IMAX)
00789 *
00790          X1 = COS(PHI(IMAX-1))*B11D(IMAX) +
00791      $        SIN(PHI(IMAX-1))*B12E(IMAX-1)
00792          Y1 = COS(PHI(IMAX-1))*B21D(IMAX) +
00793      $        SIN(PHI(IMAX-1))*B22E(IMAX-1)
00794 *
00795          THETA(IMAX) = ATAN2( ABS(Y1), ABS(X1) )
00796 *
00797 *        Fix signs on B11(IMAX,IMAX), B12(IMAX,IMAX-1), B21(IMAX,IMAX),
00798 *        and B22(IMAX,IMAX-1)
00799 *
00800          IF( B11D(IMAX)+B12E(IMAX-1) .LT. 0 ) THEN
00801             B12D(IMAX) = -B12D(IMAX)
00802             IF( WANTU1 ) THEN
00803                IF( COLMAJOR ) THEN
00804                   CALL DSCAL( P, NEGONECOMPLEX, U1(1,IMAX), 1 )
00805                ELSE
00806                   CALL DSCAL( P, NEGONECOMPLEX, U1(IMAX,1), LDU1 )
00807                END IF
00808             END IF
00809          END IF
00810          IF( B21D(IMAX)+B22E(IMAX-1) .GT. 0 ) THEN
00811             B22D(IMAX) = -B22D(IMAX)
00812             IF( WANTU2 ) THEN
00813                IF( COLMAJOR ) THEN
00814                   CALL DSCAL( M-P, NEGONECOMPLEX, U2(1,IMAX), 1 )
00815                ELSE
00816                   CALL DSCAL( M-P, NEGONECOMPLEX, U2(IMAX,1), LDU2 )
00817                END IF
00818             END IF
00819          END IF
00820 *
00821 *        Fix signs on B12(IMAX,IMAX) and B22(IMAX,IMAX)
00822 *
00823          IF( B12D(IMAX)+B22D(IMAX) .LT. 0 ) THEN
00824             IF( WANTV2T ) THEN
00825                IF( COLMAJOR ) THEN
00826                   CALL DSCAL( M-Q, NEGONECOMPLEX, V2T(IMAX,1), LDV2T )
00827                ELSE
00828                   CALL DSCAL( M-Q, NEGONECOMPLEX, V2T(1,IMAX), 1 )
00829                END IF
00830             END IF
00831          END IF
00832 *
00833 *        Test for negligible sines or cosines
00834 *
00835          DO I = IMIN, IMAX
00836             IF( THETA(I) .LT. THRESH ) THEN
00837                THETA(I) = ZERO
00838             ELSE IF( THETA(I) .GT. PIOVER2-THRESH ) THEN
00839                THETA(I) = PIOVER2
00840             END IF
00841          END DO
00842          DO I = IMIN, IMAX-1
00843             IF( PHI(I) .LT. THRESH ) THEN
00844                PHI(I) = ZERO
00845             ELSE IF( PHI(I) .GT. PIOVER2-THRESH ) THEN
00846                PHI(I) = PIOVER2
00847             END IF
00848          END DO
00849 *
00850 *        Deflate
00851 *
00852          IF (IMAX .GT. 1) THEN
00853             DO WHILE( PHI(IMAX-1) .EQ. ZERO )
00854                IMAX = IMAX - 1
00855                IF (IMAX .LE. 1) EXIT
00856             END DO
00857          END IF
00858          IF( IMIN .GT. IMAX - 1 )
00859      $      IMIN = IMAX - 1
00860          IF (IMIN .GT. 1) THEN
00861             DO WHILE (PHI(IMIN-1) .NE. ZERO)
00862                 IMIN = IMIN - 1
00863                 IF (IMIN .LE. 1) EXIT
00864             END DO
00865          END IF
00866 *
00867 *        Repeat main iteration loop
00868 *
00869       END DO
00870 *
00871 *     Postprocessing: order THETA from least to greatest
00872 *
00873       DO I = 1, Q
00874 *
00875          MINI = I
00876          THETAMIN = THETA(I)
00877          DO J = I+1, Q
00878             IF( THETA(J) .LT. THETAMIN ) THEN
00879                MINI = J
00880                THETAMIN = THETA(J)
00881             END IF
00882          END DO
00883 *
00884          IF( MINI .NE. I ) THEN
00885             THETA(MINI) = THETA(I)
00886             THETA(I) = THETAMIN
00887             IF( COLMAJOR ) THEN
00888                IF( WANTU1 )
00889      $            CALL DSWAP( P, U1(1,I), 1, U1(1,MINI), 1 )
00890                IF( WANTU2 )
00891      $            CALL DSWAP( M-P, U2(1,I), 1, U2(1,MINI), 1 )
00892                IF( WANTV1T )
00893      $            CALL DSWAP( Q, V1T(I,1), LDV1T, V1T(MINI,1), LDV1T )
00894                IF( WANTV2T )
00895      $            CALL DSWAP( M-Q, V2T(I,1), LDV2T, V2T(MINI,1),
00896      $               LDV2T )
00897             ELSE
00898                IF( WANTU1 )
00899      $            CALL DSWAP( P, U1(I,1), LDU1, U1(MINI,1), LDU1 )
00900                IF( WANTU2 )
00901      $            CALL DSWAP( M-P, U2(I,1), LDU2, U2(MINI,1), LDU2 )
00902                IF( WANTV1T )
00903      $            CALL DSWAP( Q, V1T(1,I), 1, V1T(1,MINI), 1 )
00904                IF( WANTV2T )
00905      $            CALL DSWAP( M-Q, V2T(1,I), 1, V2T(1,MINI), 1 )
00906             END IF
00907          END IF
00908 *
00909       END DO
00910 *
00911       RETURN
00912 *
00913 *     End of DBBCSD
00914 *
00915       END
00916