LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dorbdb.f
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00001 *> \brief \b DORBDB
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download DORBDB + dependencies 
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00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/dorbdb.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/dorbdb.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
00022 *                          X21, LDX21, X22, LDX22, THETA, PHI, TAUP1,
00023 *                          TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO )
00024 * 
00025 *       .. Scalar Arguments ..
00026 *       CHARACTER          SIGNS, TRANS
00027 *       INTEGER            INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P,
00028 *      $                   Q
00029 *       ..
00030 *       .. Array Arguments ..
00031 *       DOUBLE PRECISION   PHI( * ), THETA( * )
00032 *       DOUBLE PRECISION   TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
00033 *      $                   WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
00034 *      $                   X21( LDX21, * ), X22( LDX22, * )
00035 *       ..
00036 *  
00037 *
00038 *> \par Purpose:
00039 *  =============
00040 *>
00041 *> \verbatim
00042 *>
00043 *> DORBDB simultaneously bidiagonalizes the blocks of an M-by-M
00044 *> partitioned orthogonal matrix X:
00045 *>
00046 *>                                 [ B11 | B12 0  0 ]
00047 *>     [ X11 | X12 ]   [ P1 |    ] [  0  |  0 -I  0 ] [ Q1 |    ]**T
00048 *> X = [-----------] = [---------] [----------------] [---------]   .
00049 *>     [ X21 | X22 ]   [    | P2 ] [ B21 | B22 0  0 ] [    | Q2 ]
00050 *>                                 [  0  |  0  0  I ]
00051 *>
00052 *> X11 is P-by-Q. Q must be no larger than P, M-P, or M-Q. (If this is
00053 *> not the case, then X must be transposed and/or permuted. This can be
00054 *> done in constant time using the TRANS and SIGNS options. See DORCSD
00055 *> for details.)
00056 *>
00057 *> The orthogonal matrices P1, P2, Q1, and Q2 are P-by-P, (M-P)-by-
00058 *> (M-P), Q-by-Q, and (M-Q)-by-(M-Q), respectively. They are
00059 *> represented implicitly by Householder vectors.
00060 *>
00061 *> B11, B12, B21, and B22 are Q-by-Q bidiagonal matrices represented
00062 *> implicitly by angles THETA, PHI.
00063 *> \endverbatim
00064 *
00065 *  Arguments:
00066 *  ==========
00067 *
00068 *> \param[in] TRANS
00069 *> \verbatim
00070 *>          TRANS is CHARACTER
00071 *>          = 'T':      X, U1, U2, V1T, and V2T are stored in row-major
00072 *>                      order;
00073 *>          otherwise:  X, U1, U2, V1T, and V2T are stored in column-
00074 *>                      major order.
00075 *> \endverbatim
00076 *>
00077 *> \param[in] SIGNS
00078 *> \verbatim
00079 *>          SIGNS is CHARACTER
00080 *>          = 'O':      The lower-left block is made nonpositive (the
00081 *>                      "other" convention);
00082 *>          otherwise:  The upper-right block is made nonpositive (the
00083 *>                      "default" convention).
00084 *> \endverbatim
00085 *>
00086 *> \param[in] M
00087 *> \verbatim
00088 *>          M is INTEGER
00089 *>          The number of rows and columns in X.
00090 *> \endverbatim
00091 *>
00092 *> \param[in] P
00093 *> \verbatim
00094 *>          P is INTEGER
00095 *>          The number of rows in X11 and X12. 0 <= P <= M.
00096 *> \endverbatim
00097 *>
00098 *> \param[in] Q
00099 *> \verbatim
00100 *>          Q is INTEGER
00101 *>          The number of columns in X11 and X21. 0 <= Q <=
00102 *>          MIN(P,M-P,M-Q).
00103 *> \endverbatim
00104 *>
00105 *> \param[in,out] X11
00106 *> \verbatim
00107 *>          X11 is DOUBLE PRECISION array, dimension (LDX11,Q)
00108 *>          On entry, the top-left block of the orthogonal matrix to be
00109 *>          reduced. On exit, the form depends on TRANS:
00110 *>          If TRANS = 'N', then
00111 *>             the columns of tril(X11) specify reflectors for P1,
00112 *>             the rows of triu(X11,1) specify reflectors for Q1;
00113 *>          else TRANS = 'T', and
00114 *>             the rows of triu(X11) specify reflectors for P1,
00115 *>             the columns of tril(X11,-1) specify reflectors for Q1.
00116 *> \endverbatim
00117 *>
00118 *> \param[in] LDX11
00119 *> \verbatim
00120 *>          LDX11 is INTEGER
00121 *>          The leading dimension of X11. If TRANS = 'N', then LDX11 >=
00122 *>          P; else LDX11 >= Q.
00123 *> \endverbatim
00124 *>
00125 *> \param[in,out] X12
00126 *> \verbatim
00127 *>          X12 is DOUBLE PRECISION array, dimension (LDX12,M-Q)
00128 *>          On entry, the top-right block of the orthogonal matrix to
00129 *>          be reduced. On exit, the form depends on TRANS:
00130 *>          If TRANS = 'N', then
00131 *>             the rows of triu(X12) specify the first P reflectors for
00132 *>             Q2;
00133 *>          else TRANS = 'T', and
00134 *>             the columns of tril(X12) specify the first P reflectors
00135 *>             for Q2.
00136 *> \endverbatim
00137 *>
00138 *> \param[in] LDX12
00139 *> \verbatim
00140 *>          LDX12 is INTEGER
00141 *>          The leading dimension of X12. If TRANS = 'N', then LDX12 >=
00142 *>          P; else LDX11 >= M-Q.
00143 *> \endverbatim
00144 *>
00145 *> \param[in,out] X21
00146 *> \verbatim
00147 *>          X21 is DOUBLE PRECISION array, dimension (LDX21,Q)
00148 *>          On entry, the bottom-left block of the orthogonal matrix to
00149 *>          be reduced. On exit, the form depends on TRANS:
00150 *>          If TRANS = 'N', then
00151 *>             the columns of tril(X21) specify reflectors for P2;
00152 *>          else TRANS = 'T', and
00153 *>             the rows of triu(X21) specify reflectors for P2.
00154 *> \endverbatim
00155 *>
00156 *> \param[in] LDX21
00157 *> \verbatim
00158 *>          LDX21 is INTEGER
00159 *>          The leading dimension of X21. If TRANS = 'N', then LDX21 >=
00160 *>          M-P; else LDX21 >= Q.
00161 *> \endverbatim
00162 *>
00163 *> \param[in,out] X22
00164 *> \verbatim
00165 *>          X22 is DOUBLE PRECISION array, dimension (LDX22,M-Q)
00166 *>          On entry, the bottom-right block of the orthogonal matrix to
00167 *>          be reduced. On exit, the form depends on TRANS:
00168 *>          If TRANS = 'N', then
00169 *>             the rows of triu(X22(Q+1:M-P,P+1:M-Q)) specify the last
00170 *>             M-P-Q reflectors for Q2,
00171 *>          else TRANS = 'T', and
00172 *>             the columns of tril(X22(P+1:M-Q,Q+1:M-P)) specify the last
00173 *>             M-P-Q reflectors for P2.
00174 *> \endverbatim
00175 *>
00176 *> \param[in] LDX22
00177 *> \verbatim
00178 *>          LDX22 is INTEGER
00179 *>          The leading dimension of X22. If TRANS = 'N', then LDX22 >=
00180 *>          M-P; else LDX22 >= M-Q.
00181 *> \endverbatim
00182 *>
00183 *> \param[out] THETA
00184 *> \verbatim
00185 *>          THETA is DOUBLE PRECISION array, dimension (Q)
00186 *>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
00187 *>          be computed from the angles THETA and PHI. See Further
00188 *>          Details.
00189 *> \endverbatim
00190 *>
00191 *> \param[out] PHI
00192 *> \verbatim
00193 *>          PHI is DOUBLE PRECISION array, dimension (Q-1)
00194 *>          The entries of the bidiagonal blocks B11, B12, B21, B22 can
00195 *>          be computed from the angles THETA and PHI. See Further
00196 *>          Details.
00197 *> \endverbatim
00198 *>
00199 *> \param[out] TAUP1
00200 *> \verbatim
00201 *>          TAUP1 is DOUBLE PRECISION array, dimension (P)
00202 *>          The scalar factors of the elementary reflectors that define
00203 *>          P1.
00204 *> \endverbatim
00205 *>
00206 *> \param[out] TAUP2
00207 *> \verbatim
00208 *>          TAUP2 is DOUBLE PRECISION array, dimension (M-P)
00209 *>          The scalar factors of the elementary reflectors that define
00210 *>          P2.
00211 *> \endverbatim
00212 *>
00213 *> \param[out] TAUQ1
00214 *> \verbatim
00215 *>          TAUQ1 is DOUBLE PRECISION array, dimension (Q)
00216 *>          The scalar factors of the elementary reflectors that define
00217 *>          Q1.
00218 *> \endverbatim
00219 *>
00220 *> \param[out] TAUQ2
00221 *> \verbatim
00222 *>          TAUQ2 is DOUBLE PRECISION array, dimension (M-Q)
00223 *>          The scalar factors of the elementary reflectors that define
00224 *>          Q2.
00225 *> \endverbatim
00226 *>
00227 *> \param[out] WORK
00228 *> \verbatim
00229 *>          WORK is DOUBLE PRECISION array, dimension (LWORK)
00230 *> \endverbatim
00231 *>
00232 *> \param[in] LWORK
00233 *> \verbatim
00234 *>          LWORK is INTEGER
00235 *>          The dimension of the array WORK. LWORK >= M-Q.
00236 *>
00237 *>          If LWORK = -1, then a workspace query is assumed; the routine
00238 *>          only calculates the optimal size of the WORK array, returns
00239 *>          this value as the first entry of the WORK array, and no error
00240 *>          message related to LWORK is issued by XERBLA.
00241 *> \endverbatim
00242 *>
00243 *> \param[out] INFO
00244 *> \verbatim
00245 *>          INFO is INTEGER
00246 *>          = 0:  successful exit.
00247 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
00248 *> \endverbatim
00249 *
00250 *  Authors:
00251 *  ========
00252 *
00253 *> \author Univ. of Tennessee 
00254 *> \author Univ. of California Berkeley 
00255 *> \author Univ. of Colorado Denver 
00256 *> \author NAG Ltd. 
00257 *
00258 *> \date November 2011
00259 *
00260 *> \ingroup doubleOTHERcomputational
00261 *
00262 *> \par Further Details:
00263 *  =====================
00264 *>
00265 *> \verbatim
00266 *>
00267 *>  The bidiagonal blocks B11, B12, B21, and B22 are represented
00268 *>  implicitly by angles THETA(1), ..., THETA(Q) and PHI(1), ...,
00269 *>  PHI(Q-1). B11 and B21 are upper bidiagonal, while B21 and B22 are
00270 *>  lower bidiagonal. Every entry in each bidiagonal band is a product
00271 *>  of a sine or cosine of a THETA with a sine or cosine of a PHI. See
00272 *>  [1] or DORCSD for details.
00273 *>
00274 *>  P1, P2, Q1, and Q2 are represented as products of elementary
00275 *>  reflectors. See DORCSD for details on generating P1, P2, Q1, and Q2
00276 *>  using DORGQR and DORGLQ.
00277 *> \endverbatim
00278 *
00279 *> \par References:
00280 *  ================
00281 *>
00282 *>  [1] Brian D. Sutton. Computing the complete CS decomposition. Numer.
00283 *>      Algorithms, 50(1):33-65, 2009.
00284 *>
00285 *  =====================================================================
00286       SUBROUTINE DORBDB( TRANS, SIGNS, M, P, Q, X11, LDX11, X12, LDX12,
00287      $                   X21, LDX21, X22, LDX22, THETA, PHI, TAUP1,
00288      $                   TAUP2, TAUQ1, TAUQ2, WORK, LWORK, INFO )
00289 *
00290 *  -- LAPACK computational routine (version 3.4.0) --
00291 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00292 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00293 *     November 2011
00294 *
00295 *     .. Scalar Arguments ..
00296       CHARACTER          SIGNS, TRANS
00297       INTEGER            INFO, LDX11, LDX12, LDX21, LDX22, LWORK, M, P,
00298      $                   Q
00299 *     ..
00300 *     .. Array Arguments ..
00301       DOUBLE PRECISION   PHI( * ), THETA( * )
00302       DOUBLE PRECISION   TAUP1( * ), TAUP2( * ), TAUQ1( * ), TAUQ2( * ),
00303      $                   WORK( * ), X11( LDX11, * ), X12( LDX12, * ),
00304      $                   X21( LDX21, * ), X22( LDX22, * )
00305 *     ..
00306 *
00307 *  ====================================================================
00308 *
00309 *     .. Parameters ..
00310       DOUBLE PRECISION   REALONE
00311       PARAMETER          ( REALONE = 1.0D0 )
00312       DOUBLE PRECISION   ONE
00313       PARAMETER          ( ONE = 1.0D0 )
00314 *     ..
00315 *     .. Local Scalars ..
00316       LOGICAL            COLMAJOR, LQUERY
00317       INTEGER            I, LWORKMIN, LWORKOPT
00318       DOUBLE PRECISION   Z1, Z2, Z3, Z4
00319 *     ..
00320 *     .. External Subroutines ..
00321       EXTERNAL           DAXPY, DLARF, DLARFGP, DSCAL, XERBLA
00322 *     ..
00323 *     .. External Functions ..
00324       DOUBLE PRECISION   DNRM2
00325       LOGICAL            LSAME
00326       EXTERNAL           DNRM2, LSAME
00327 *     ..
00328 *     .. Intrinsic Functions
00329       INTRINSIC          ATAN2, COS, MAX, SIN
00330 *     ..
00331 *     .. Executable Statements ..
00332 *
00333 *     Test input arguments
00334 *
00335       INFO = 0
00336       COLMAJOR = .NOT. LSAME( TRANS, 'T' )
00337       IF( .NOT. LSAME( SIGNS, 'O' ) ) THEN
00338          Z1 = REALONE
00339          Z2 = REALONE
00340          Z3 = REALONE
00341          Z4 = REALONE
00342       ELSE
00343          Z1 = REALONE
00344          Z2 = -REALONE
00345          Z3 = REALONE
00346          Z4 = -REALONE
00347       END IF
00348       LQUERY = LWORK .EQ. -1
00349 *
00350       IF( M .LT. 0 ) THEN
00351          INFO = -3
00352       ELSE IF( P .LT. 0 .OR. P .GT. M ) THEN
00353          INFO = -4
00354       ELSE IF( Q .LT. 0 .OR. Q .GT. P .OR. Q .GT. M-P .OR.
00355      $         Q .GT. M-Q ) THEN
00356          INFO = -5
00357       ELSE IF( COLMAJOR .AND. LDX11 .LT. MAX( 1, P ) ) THEN
00358          INFO = -7
00359       ELSE IF( .NOT.COLMAJOR .AND. LDX11 .LT. MAX( 1, Q ) ) THEN
00360          INFO = -7
00361       ELSE IF( COLMAJOR .AND. LDX12 .LT. MAX( 1, P ) ) THEN
00362          INFO = -9
00363       ELSE IF( .NOT.COLMAJOR .AND. LDX12 .LT. MAX( 1, M-Q ) ) THEN
00364          INFO = -9
00365       ELSE IF( COLMAJOR .AND. LDX21 .LT. MAX( 1, M-P ) ) THEN
00366          INFO = -11
00367       ELSE IF( .NOT.COLMAJOR .AND. LDX21 .LT. MAX( 1, Q ) ) THEN
00368          INFO = -11
00369       ELSE IF( COLMAJOR .AND. LDX22 .LT. MAX( 1, M-P ) ) THEN
00370          INFO = -13
00371       ELSE IF( .NOT.COLMAJOR .AND. LDX22 .LT. MAX( 1, M-Q ) ) THEN
00372          INFO = -13
00373       END IF
00374 *
00375 *     Compute workspace
00376 *
00377       IF( INFO .EQ. 0 ) THEN
00378          LWORKOPT = M - Q
00379          LWORKMIN = M - Q
00380          WORK(1) = LWORKOPT
00381          IF( LWORK .LT. LWORKMIN .AND. .NOT. LQUERY ) THEN
00382             INFO = -21
00383          END IF
00384       END IF
00385       IF( INFO .NE. 0 ) THEN
00386          CALL XERBLA( 'xORBDB', -INFO )
00387          RETURN
00388       ELSE IF( LQUERY ) THEN
00389          RETURN
00390       END IF
00391 *
00392 *     Handle column-major and row-major separately
00393 *
00394       IF( COLMAJOR ) THEN
00395 *
00396 *        Reduce columns 1, ..., Q of X11, X12, X21, and X22 
00397 *
00398          DO I = 1, Q
00399 *
00400             IF( I .EQ. 1 ) THEN
00401                CALL DSCAL( P-I+1, Z1, X11(I,I), 1 )
00402             ELSE
00403                CALL DSCAL( P-I+1, Z1*COS(PHI(I-1)), X11(I,I), 1 )
00404                CALL DAXPY( P-I+1, -Z1*Z3*Z4*SIN(PHI(I-1)), X12(I,I-1),
00405      $                     1, X11(I,I), 1 )
00406             END IF
00407             IF( I .EQ. 1 ) THEN
00408                CALL DSCAL( M-P-I+1, Z2, X21(I,I), 1 )
00409             ELSE
00410                CALL DSCAL( M-P-I+1, Z2*COS(PHI(I-1)), X21(I,I), 1 )
00411                CALL DAXPY( M-P-I+1, -Z2*Z3*Z4*SIN(PHI(I-1)), X22(I,I-1),
00412      $                     1, X21(I,I), 1 )
00413             END IF
00414 *
00415             THETA(I) = ATAN2( DNRM2( M-P-I+1, X21(I,I), 1 ),
00416      $                 DNRM2( P-I+1, X11(I,I), 1 ) )
00417 *
00418             CALL DLARFGP( P-I+1, X11(I,I), X11(I+1,I), 1, TAUP1(I) )
00419             X11(I,I) = ONE
00420             CALL DLARFGP( M-P-I+1, X21(I,I), X21(I+1,I), 1, TAUP2(I) )
00421             X21(I,I) = ONE
00422 *
00423             CALL DLARF( 'L', P-I+1, Q-I, X11(I,I), 1, TAUP1(I),
00424      $                  X11(I,I+1), LDX11, WORK )
00425             CALL DLARF( 'L', P-I+1, M-Q-I+1, X11(I,I), 1, TAUP1(I),
00426      $                  X12(I,I), LDX12, WORK )
00427             CALL DLARF( 'L', M-P-I+1, Q-I, X21(I,I), 1, TAUP2(I),
00428      $                  X21(I,I+1), LDX21, WORK )
00429             CALL DLARF( 'L', M-P-I+1, M-Q-I+1, X21(I,I), 1, TAUP2(I),
00430      $                  X22(I,I), LDX22, WORK )
00431 *
00432             IF( I .LT. Q ) THEN
00433                CALL DSCAL( Q-I, -Z1*Z3*SIN(THETA(I)), X11(I,I+1),
00434      $                     LDX11 )
00435                CALL DAXPY( Q-I, Z2*Z3*COS(THETA(I)), X21(I,I+1), LDX21,
00436      $                     X11(I,I+1), LDX11 )
00437             END IF
00438             CALL DSCAL( M-Q-I+1, -Z1*Z4*SIN(THETA(I)), X12(I,I), LDX12 )
00439             CALL DAXPY( M-Q-I+1, Z2*Z4*COS(THETA(I)), X22(I,I), LDX22,
00440      $                  X12(I,I), LDX12 )
00441 *
00442             IF( I .LT. Q )
00443      $         PHI(I) = ATAN2( DNRM2( Q-I, X11(I,I+1), LDX11 ),
00444      $                  DNRM2( M-Q-I+1, X12(I,I), LDX12 ) )
00445 *
00446             IF( I .LT. Q ) THEN
00447                CALL DLARFGP( Q-I, X11(I,I+1), X11(I,I+2), LDX11,
00448      $                       TAUQ1(I) )
00449                X11(I,I+1) = ONE
00450             END IF
00451             CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
00452      $                    TAUQ2(I) )
00453             X12(I,I) = ONE
00454 *
00455             IF( I .LT. Q ) THEN
00456                CALL DLARF( 'R', P-I, Q-I, X11(I,I+1), LDX11, TAUQ1(I),
00457      $                     X11(I+1,I+1), LDX11, WORK )
00458                CALL DLARF( 'R', M-P-I, Q-I, X11(I,I+1), LDX11, TAUQ1(I),
00459      $                     X21(I+1,I+1), LDX21, WORK )
00460             END IF
00461             CALL DLARF( 'R', P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I),
00462      $                  X12(I+1,I), LDX12, WORK )
00463             CALL DLARF( 'R', M-P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I),
00464      $                  X22(I+1,I), LDX22, WORK )
00465 *
00466          END DO
00467 *
00468 *        Reduce columns Q + 1, ..., P of X12, X22
00469 *
00470          DO I = Q + 1, P
00471 *
00472             CALL DSCAL( M-Q-I+1, -Z1*Z4, X12(I,I), LDX12 )
00473             CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I,I+1), LDX12,
00474      $                    TAUQ2(I) )
00475             X12(I,I) = ONE
00476 *
00477             CALL DLARF( 'R', P-I, M-Q-I+1, X12(I,I), LDX12, TAUQ2(I),
00478      $                  X12(I+1,I), LDX12, WORK )
00479             IF( M-P-Q .GE. 1 )
00480      $         CALL DLARF( 'R', M-P-Q, M-Q-I+1, X12(I,I), LDX12,
00481      $                     TAUQ2(I), X22(Q+1,I), LDX22, WORK )
00482 *
00483          END DO
00484 *
00485 *        Reduce columns P + 1, ..., M - Q of X12, X22
00486 *
00487          DO I = 1, M - P - Q
00488 *
00489             CALL DSCAL( M-P-Q-I+1, Z2*Z4, X22(Q+I,P+I), LDX22 )
00490             CALL DLARFGP( M-P-Q-I+1, X22(Q+I,P+I), X22(Q+I,P+I+1),
00491      $                    LDX22, TAUQ2(P+I) )
00492             X22(Q+I,P+I) = ONE
00493             CALL DLARF( 'R', M-P-Q-I, M-P-Q-I+1, X22(Q+I,P+I), LDX22,
00494      $                  TAUQ2(P+I), X22(Q+I+1,P+I), LDX22, WORK )
00495 *
00496          END DO
00497 *
00498       ELSE
00499 *
00500 *        Reduce columns 1, ..., Q of X11, X12, X21, X22
00501 *
00502          DO I = 1, Q
00503 *
00504             IF( I .EQ. 1 ) THEN
00505                CALL DSCAL( P-I+1, Z1, X11(I,I), LDX11 )
00506             ELSE
00507                CALL DSCAL( P-I+1, Z1*COS(PHI(I-1)), X11(I,I), LDX11 )
00508                CALL DAXPY( P-I+1, -Z1*Z3*Z4*SIN(PHI(I-1)), X12(I-1,I),
00509      $                     LDX12, X11(I,I), LDX11 )
00510             END IF
00511             IF( I .EQ. 1 ) THEN
00512                CALL DSCAL( M-P-I+1, Z2, X21(I,I), LDX21 )
00513             ELSE
00514                CALL DSCAL( M-P-I+1, Z2*COS(PHI(I-1)), X21(I,I), LDX21 )
00515                CALL DAXPY( M-P-I+1, -Z2*Z3*Z4*SIN(PHI(I-1)), X22(I-1,I),
00516      $                     LDX22, X21(I,I), LDX21 )
00517             END IF
00518 *
00519             THETA(I) = ATAN2( DNRM2( M-P-I+1, X21(I,I), LDX21 ),
00520      $                 DNRM2( P-I+1, X11(I,I), LDX11 ) )
00521 *
00522             CALL DLARFGP( P-I+1, X11(I,I), X11(I,I+1), LDX11, TAUP1(I) )
00523             X11(I,I) = ONE
00524             CALL DLARFGP( M-P-I+1, X21(I,I), X21(I,I+1), LDX21,
00525      $                    TAUP2(I) )
00526             X21(I,I) = ONE
00527 *
00528             CALL DLARF( 'R', Q-I, P-I+1, X11(I,I), LDX11, TAUP1(I),
00529      $                  X11(I+1,I), LDX11, WORK )
00530             CALL DLARF( 'R', M-Q-I+1, P-I+1, X11(I,I), LDX11, TAUP1(I),
00531      $                  X12(I,I), LDX12, WORK )
00532             CALL DLARF( 'R', Q-I, M-P-I+1, X21(I,I), LDX21, TAUP2(I),
00533      $                  X21(I+1,I), LDX21, WORK )
00534             CALL DLARF( 'R', M-Q-I+1, M-P-I+1, X21(I,I), LDX21,
00535      $                  TAUP2(I), X22(I,I), LDX22, WORK )
00536 *
00537             IF( I .LT. Q ) THEN
00538                CALL DSCAL( Q-I, -Z1*Z3*SIN(THETA(I)), X11(I+1,I), 1 )
00539                CALL DAXPY( Q-I, Z2*Z3*COS(THETA(I)), X21(I+1,I), 1,
00540      $                     X11(I+1,I), 1 )
00541             END IF
00542             CALL DSCAL( M-Q-I+1, -Z1*Z4*SIN(THETA(I)), X12(I,I), 1 )
00543             CALL DAXPY( M-Q-I+1, Z2*Z4*COS(THETA(I)), X22(I,I), 1,
00544      $                  X12(I,I), 1 )
00545 *
00546             IF( I .LT. Q )
00547      $         PHI(I) = ATAN2( DNRM2( Q-I, X11(I+1,I), 1 ),
00548      $                  DNRM2( M-Q-I+1, X12(I,I), 1 ) )
00549 *
00550             IF( I .LT. Q ) THEN
00551                CALL DLARFGP( Q-I, X11(I+1,I), X11(I+2,I), 1, TAUQ1(I) )
00552                X11(I+1,I) = ONE
00553             END IF
00554             CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1, TAUQ2(I) )
00555             X12(I,I) = ONE
00556 *
00557             IF( I .LT. Q ) THEN
00558                CALL DLARF( 'L', Q-I, P-I, X11(I+1,I), 1, TAUQ1(I),
00559      $                     X11(I+1,I+1), LDX11, WORK )
00560                CALL DLARF( 'L', Q-I, M-P-I, X11(I+1,I), 1, TAUQ1(I),
00561      $                     X21(I+1,I+1), LDX21, WORK )
00562             END IF
00563             CALL DLARF( 'L', M-Q-I+1, P-I, X12(I,I), 1, TAUQ2(I),
00564      $                  X12(I,I+1), LDX12, WORK )
00565             CALL DLARF( 'L', M-Q-I+1, M-P-I, X12(I,I), 1, TAUQ2(I),
00566      $                  X22(I,I+1), LDX22, WORK )
00567 *
00568          END DO
00569 *
00570 *        Reduce columns Q + 1, ..., P of X12, X22
00571 *
00572          DO I = Q + 1, P
00573 *
00574             CALL DSCAL( M-Q-I+1, -Z1*Z4, X12(I,I), 1 )
00575             CALL DLARFGP( M-Q-I+1, X12(I,I), X12(I+1,I), 1, TAUQ2(I) )
00576             X12(I,I) = ONE
00577 *
00578             CALL DLARF( 'L', M-Q-I+1, P-I, X12(I,I), 1, TAUQ2(I),
00579      $                  X12(I,I+1), LDX12, WORK )
00580             IF( M-P-Q .GE. 1 )
00581      $         CALL DLARF( 'L', M-Q-I+1, M-P-Q, X12(I,I), 1, TAUQ2(I),
00582      $                     X22(I,Q+1), LDX22, WORK )
00583 *
00584          END DO
00585 *
00586 *        Reduce columns P + 1, ..., M - Q of X12, X22
00587 *
00588          DO I = 1, M - P - Q
00589 *
00590             CALL DSCAL( M-P-Q-I+1, Z2*Z4, X22(P+I,Q+I), 1 )
00591             CALL DLARFGP( M-P-Q-I+1, X22(P+I,Q+I), X22(P+I+1,Q+I), 1,
00592      $                    TAUQ2(P+I) )
00593             X22(P+I,Q+I) = ONE
00594 *
00595             CALL DLARF( 'L', M-P-Q-I+1, M-P-Q-I, X22(P+I,Q+I), 1,
00596      $                  TAUQ2(P+I), X22(P+I,Q+I+1), LDX22, WORK )
00597 *
00598          END DO
00599 *
00600       END IF
00601 *
00602       RETURN
00603 *
00604 *     End of DORBDB
00605 *
00606       END
00607 
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