LAPACK 3.3.0

ztgsja.f

Go to the documentation of this file.
00001       SUBROUTINE ZTGSJA( JOBU, JOBV, JOBQ, M, P, N, K, L, A, LDA, B,
00002      $                   LDB, TOLA, TOLB, ALPHA, BETA, U, LDU, V, LDV,
00003      $                   Q, LDQ, WORK, NCYCLE, INFO )
00004 *
00005 *  -- LAPACK routine (version 3.2.1)                                 --
00006 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00007 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00008 *  -- April 2009                                                      --
00009 *
00010 *     .. Scalar Arguments ..
00011       CHARACTER          JOBQ, JOBU, JOBV
00012       INTEGER            INFO, K, L, LDA, LDB, LDQ, LDU, LDV, M, N,
00013      $                   NCYCLE, P
00014       DOUBLE PRECISION   TOLA, TOLB
00015 *     ..
00016 *     .. Array Arguments ..
00017       DOUBLE PRECISION   ALPHA( * ), BETA( * )
00018       COMPLEX*16         A( LDA, * ), B( LDB, * ), Q( LDQ, * ),
00019      $                   U( LDU, * ), V( LDV, * ), WORK( * )
00020 *     ..
00021 *
00022 *  Purpose
00023 *  =======
00024 *
00025 *  ZTGSJA computes the generalized singular value decomposition (GSVD)
00026 *  of two complex upper triangular (or trapezoidal) matrices A and B.
00027 *
00028 *  On entry, it is assumed that matrices A and B have the following
00029 *  forms, which may be obtained by the preprocessing subroutine ZGGSVP
00030 *  from a general M-by-N matrix A and P-by-N matrix B:
00031 *
00032 *               N-K-L  K    L
00033 *     A =    K ( 0    A12  A13 ) if M-K-L >= 0;
00034 *            L ( 0     0   A23 )
00035 *        M-K-L ( 0     0    0  )
00036 *
00037 *             N-K-L  K    L
00038 *     A =  K ( 0    A12  A13 ) if M-K-L < 0;
00039 *        M-K ( 0     0   A23 )
00040 *
00041 *             N-K-L  K    L
00042 *     B =  L ( 0     0   B13 )
00043 *        P-L ( 0     0    0  )
00044 *
00045 *  where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
00046 *  upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
00047 *  otherwise A23 is (M-K)-by-L upper trapezoidal.
00048 *
00049 *  On exit,
00050 *
00051 *         U'*A*Q = D1*( 0 R ),    V'*B*Q = D2*( 0 R ),
00052 *
00053 *  where U, V and Q are unitary matrices, Z' denotes the conjugate
00054 *  transpose of Z, R is a nonsingular upper triangular matrix, and D1
00055 *  and D2 are ``diagonal'' matrices, which are of the following
00056 *  structures:
00057 *
00058 *  If M-K-L >= 0,
00059 *
00060 *                      K  L
00061 *         D1 =     K ( I  0 )
00062 *                  L ( 0  C )
00063 *              M-K-L ( 0  0 )
00064 *
00065 *                     K  L
00066 *         D2 = L   ( 0  S )
00067 *              P-L ( 0  0 )
00068 *
00069 *                 N-K-L  K    L
00070 *    ( 0 R ) = K (  0   R11  R12 ) K
00071 *              L (  0    0   R22 ) L
00072 *
00073 *  where
00074 *
00075 *    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
00076 *    S = diag( BETA(K+1),  ... , BETA(K+L) ),
00077 *    C**2 + S**2 = I.
00078 *
00079 *    R is stored in A(1:K+L,N-K-L+1:N) on exit.
00080 *
00081 *  If M-K-L < 0,
00082 *
00083 *                 K M-K K+L-M
00084 *      D1 =   K ( I  0    0   )
00085 *           M-K ( 0  C    0   )
00086 *
00087 *                   K M-K K+L-M
00088 *      D2 =   M-K ( 0  S    0   )
00089 *           K+L-M ( 0  0    I   )
00090 *             P-L ( 0  0    0   )
00091 *
00092 *                 N-K-L  K   M-K  K+L-M
00093 * ( 0 R ) =    K ( 0    R11  R12  R13  )
00094 *            M-K ( 0     0   R22  R23  )
00095 *          K+L-M ( 0     0    0   R33  )
00096 *
00097 *  where
00098 *  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
00099 *  S = diag( BETA(K+1),  ... , BETA(M) ),
00100 *  C**2 + S**2 = I.
00101 *
00102 *  R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
00103 *      (  0  R22 R23 )
00104 *  in B(M-K+1:L,N+M-K-L+1:N) on exit.
00105 *
00106 *  The computation of the unitary transformation matrices U, V or Q
00107 *  is optional.  These matrices may either be formed explicitly, or they
00108 *  may be postmultiplied into input matrices U1, V1, or Q1.
00109 *
00110 *  Arguments
00111 *  =========
00112 *
00113 *  JOBU    (input) CHARACTER*1
00114 *          = 'U':  U must contain a unitary matrix U1 on entry, and
00115 *                  the product U1*U is returned;
00116 *          = 'I':  U is initialized to the unit matrix, and the
00117 *                  unitary matrix U is returned;
00118 *          = 'N':  U is not computed.
00119 *
00120 *  JOBV    (input) CHARACTER*1
00121 *          = 'V':  V must contain a unitary matrix V1 on entry, and
00122 *                  the product V1*V is returned;
00123 *          = 'I':  V is initialized to the unit matrix, and the
00124 *                  unitary matrix V is returned;
00125 *          = 'N':  V is not computed.
00126 *
00127 *  JOBQ    (input) CHARACTER*1
00128 *          = 'Q':  Q must contain a unitary matrix Q1 on entry, and
00129 *                  the product Q1*Q is returned;
00130 *          = 'I':  Q is initialized to the unit matrix, and the
00131 *                  unitary matrix Q is returned;
00132 *          = 'N':  Q is not computed.
00133 *
00134 *  M       (input) INTEGER
00135 *          The number of rows of the matrix A.  M >= 0.
00136 *
00137 *  P       (input) INTEGER
00138 *          The number of rows of the matrix B.  P >= 0.
00139 *
00140 *  N       (input) INTEGER
00141 *          The number of columns of the matrices A and B.  N >= 0.
00142 *
00143 *  K       (input) INTEGER
00144 *  L       (input) INTEGER
00145 *          K and L specify the subblocks in the input matrices A and B:
00146 *          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,,N-L+1:N)
00147 *          of A and B, whose GSVD is going to be computed by ZTGSJA.
00148 *          See Further Details.
00149 *
00150 *  A       (input/output) COMPLEX*16 array, dimension (LDA,N)
00151 *          On entry, the M-by-N matrix A.
00152 *          On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
00153 *          matrix R or part of R.  See Purpose for details.
00154 *
00155 *  LDA     (input) INTEGER
00156 *          The leading dimension of the array A. LDA >= max(1,M).
00157 *
00158 *  B       (input/output) COMPLEX*16 array, dimension (LDB,N)
00159 *          On entry, the P-by-N matrix B.
00160 *          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
00161 *          a part of R.  See Purpose for details.
00162 *
00163 *  LDB     (input) INTEGER
00164 *          The leading dimension of the array B. LDB >= max(1,P).
00165 *
00166 *  TOLA    (input) DOUBLE PRECISION
00167 *  TOLB    (input) DOUBLE PRECISION
00168 *          TOLA and TOLB are the convergence criteria for the Jacobi-
00169 *          Kogbetliantz iteration procedure. Generally, they are the
00170 *          same as used in the preprocessing step, say
00171 *              TOLA = MAX(M,N)*norm(A)*MAZHEPS,
00172 *              TOLB = MAX(P,N)*norm(B)*MAZHEPS.
00173 *
00174 *  ALPHA   (output) DOUBLE PRECISION array, dimension (N)
00175 *  BETA    (output) DOUBLE PRECISION array, dimension (N)
00176 *          On exit, ALPHA and BETA contain the generalized singular
00177 *          value pairs of A and B;
00178 *            ALPHA(1:K) = 1,
00179 *            BETA(1:K)  = 0,
00180 *          and if M-K-L >= 0,
00181 *            ALPHA(K+1:K+L) = diag(C),
00182 *            BETA(K+1:K+L)  = diag(S),
00183 *          or if M-K-L < 0,
00184 *            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
00185 *            BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
00186 *          Furthermore, if K+L < N,
00187 *            ALPHA(K+L+1:N) = 0
00188 *            BETA(K+L+1:N)  = 0.
00189 *
00190 *  U       (input/output) COMPLEX*16 array, dimension (LDU,M)
00191 *          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
00192 *          the unitary matrix returned by ZGGSVP).
00193 *          On exit,
00194 *          if JOBU = 'I', U contains the unitary matrix U;
00195 *          if JOBU = 'U', U contains the product U1*U.
00196 *          If JOBU = 'N', U is not referenced.
00197 *
00198 *  LDU     (input) INTEGER
00199 *          The leading dimension of the array U. LDU >= max(1,M) if
00200 *          JOBU = 'U'; LDU >= 1 otherwise.
00201 *
00202 *  V       (input/output) COMPLEX*16 array, dimension (LDV,P)
00203 *          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
00204 *          the unitary matrix returned by ZGGSVP).
00205 *          On exit,
00206 *          if JOBV = 'I', V contains the unitary matrix V;
00207 *          if JOBV = 'V', V contains the product V1*V.
00208 *          If JOBV = 'N', V is not referenced.
00209 *
00210 *  LDV     (input) INTEGER
00211 *          The leading dimension of the array V. LDV >= max(1,P) if
00212 *          JOBV = 'V'; LDV >= 1 otherwise.
00213 *
00214 *  Q       (input/output) COMPLEX*16 array, dimension (LDQ,N)
00215 *          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
00216 *          the unitary matrix returned by ZGGSVP).
00217 *          On exit,
00218 *          if JOBQ = 'I', Q contains the unitary matrix Q;
00219 *          if JOBQ = 'Q', Q contains the product Q1*Q.
00220 *          If JOBQ = 'N', Q is not referenced.
00221 *
00222 *  LDQ     (input) INTEGER
00223 *          The leading dimension of the array Q. LDQ >= max(1,N) if
00224 *          JOBQ = 'Q'; LDQ >= 1 otherwise.
00225 *
00226 *  WORK    (workspace) COMPLEX*16 array, dimension (2*N)
00227 *
00228 *  NCYCLE  (output) INTEGER
00229 *          The number of cycles required for convergence.
00230 *
00231 *  INFO    (output) INTEGER
00232 *          = 0:  successful exit
00233 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00234 *          = 1:  the procedure does not converge after MAXIT cycles.
00235 *
00236 *  Internal Parameters
00237 *  ===================
00238 *
00239 *  MAXIT   INTEGER
00240 *          MAXIT specifies the total loops that the iterative procedure
00241 *          may take. If after MAXIT cycles, the routine fails to
00242 *          converge, we return INFO = 1.
00243 *
00244 *  Further Details
00245 *  ===============
00246 *
00247 *  ZTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
00248 *  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
00249 *  matrix B13 to the form:
00250 *
00251 *           U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
00252 *
00253 *  where U1, V1 and Q1 are unitary matrix, and Z' is the conjugate
00254 *  transpose of Z.  C1 and S1 are diagonal matrices satisfying
00255 *
00256 *                C1**2 + S1**2 = I,
00257 *
00258 *  and R1 is an L-by-L nonsingular upper triangular matrix.
00259 *
00260 *  =====================================================================
00261 *
00262 *     .. Parameters ..
00263       INTEGER            MAXIT
00264       PARAMETER          ( MAXIT = 40 )
00265       DOUBLE PRECISION   ZERO, ONE
00266       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
00267       COMPLEX*16         CZERO, CONE
00268       PARAMETER          ( CZERO = ( 0.0D+0, 0.0D+0 ),
00269      $                   CONE = ( 1.0D+0, 0.0D+0 ) )
00270 *     ..
00271 *     .. Local Scalars ..
00272 *
00273       LOGICAL            INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
00274       INTEGER            I, J, KCYCLE
00275       DOUBLE PRECISION   A1, A3, B1, B3, CSQ, CSU, CSV, ERROR, GAMMA,
00276      $                   RWK, SSMIN
00277       COMPLEX*16         A2, B2, SNQ, SNU, SNV
00278 *     ..
00279 *     .. External Functions ..
00280       LOGICAL            LSAME
00281       EXTERNAL           LSAME
00282 *     ..
00283 *     .. External Subroutines ..
00284       EXTERNAL           DLARTG, XERBLA, ZCOPY, ZDSCAL, ZLAGS2, ZLAPLL,
00285      $                   ZLASET, ZROT
00286 *     ..
00287 *     .. Intrinsic Functions ..
00288       INTRINSIC          ABS, DBLE, DCONJG, MAX, MIN
00289 *     ..
00290 *     .. Executable Statements ..
00291 *
00292 *     Decode and test the input parameters
00293 *
00294       INITU = LSAME( JOBU, 'I' )
00295       WANTU = INITU .OR. LSAME( JOBU, 'U' )
00296 *
00297       INITV = LSAME( JOBV, 'I' )
00298       WANTV = INITV .OR. LSAME( JOBV, 'V' )
00299 *
00300       INITQ = LSAME( JOBQ, 'I' )
00301       WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
00302 *
00303       INFO = 0
00304       IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
00305          INFO = -1
00306       ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
00307          INFO = -2
00308       ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
00309          INFO = -3
00310       ELSE IF( M.LT.0 ) THEN
00311          INFO = -4
00312       ELSE IF( P.LT.0 ) THEN
00313          INFO = -5
00314       ELSE IF( N.LT.0 ) THEN
00315          INFO = -6
00316       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00317          INFO = -10
00318       ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
00319          INFO = -12
00320       ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
00321          INFO = -18
00322       ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
00323          INFO = -20
00324       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
00325          INFO = -22
00326       END IF
00327       IF( INFO.NE.0 ) THEN
00328          CALL XERBLA( 'ZTGSJA', -INFO )
00329          RETURN
00330       END IF
00331 *
00332 *     Initialize U, V and Q, if necessary
00333 *
00334       IF( INITU )
00335      $   CALL ZLASET( 'Full', M, M, CZERO, CONE, U, LDU )
00336       IF( INITV )
00337      $   CALL ZLASET( 'Full', P, P, CZERO, CONE, V, LDV )
00338       IF( INITQ )
00339      $   CALL ZLASET( 'Full', N, N, CZERO, CONE, Q, LDQ )
00340 *
00341 *     Loop until convergence
00342 *
00343       UPPER = .FALSE.
00344       DO 40 KCYCLE = 1, MAXIT
00345 *
00346          UPPER = .NOT.UPPER
00347 *
00348          DO 20 I = 1, L - 1
00349             DO 10 J = I + 1, L
00350 *
00351                A1 = ZERO
00352                A2 = CZERO
00353                A3 = ZERO
00354                IF( K+I.LE.M )
00355      $            A1 = DBLE( A( K+I, N-L+I ) )
00356                IF( K+J.LE.M )
00357      $            A3 = DBLE( A( K+J, N-L+J ) )
00358 *
00359                B1 = DBLE( B( I, N-L+I ) )
00360                B3 = DBLE( B( J, N-L+J ) )
00361 *
00362                IF( UPPER ) THEN
00363                   IF( K+I.LE.M )
00364      $               A2 = A( K+I, N-L+J )
00365                   B2 = B( I, N-L+J )
00366                ELSE
00367                   IF( K+J.LE.M )
00368      $               A2 = A( K+J, N-L+I )
00369                   B2 = B( J, N-L+I )
00370                END IF
00371 *
00372                CALL ZLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
00373      $                      CSV, SNV, CSQ, SNQ )
00374 *
00375 *              Update (K+I)-th and (K+J)-th rows of matrix A: U'*A
00376 *
00377                IF( K+J.LE.M )
00378      $            CALL ZROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
00379      $                       LDA, CSU, DCONJG( SNU ) )
00380 *
00381 *              Update I-th and J-th rows of matrix B: V'*B
00382 *
00383                CALL ZROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
00384      $                    CSV, DCONJG( SNV ) )
00385 *
00386 *              Update (N-L+I)-th and (N-L+J)-th columns of matrices
00387 *              A and B: A*Q and B*Q
00388 *
00389                CALL ZROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
00390      $                    A( 1, N-L+I ), 1, CSQ, SNQ )
00391 *
00392                CALL ZROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
00393      $                    SNQ )
00394 *
00395                IF( UPPER ) THEN
00396                   IF( K+I.LE.M )
00397      $               A( K+I, N-L+J ) = CZERO
00398                   B( I, N-L+J ) = CZERO
00399                ELSE
00400                   IF( K+J.LE.M )
00401      $               A( K+J, N-L+I ) = CZERO
00402                   B( J, N-L+I ) = CZERO
00403                END IF
00404 *
00405 *              Ensure that the diagonal elements of A and B are real.
00406 *
00407                IF( K+I.LE.M )
00408      $            A( K+I, N-L+I ) = DBLE( A( K+I, N-L+I ) )
00409                IF( K+J.LE.M )
00410      $            A( K+J, N-L+J ) = DBLE( A( K+J, N-L+J ) )
00411                B( I, N-L+I ) = DBLE( B( I, N-L+I ) )
00412                B( J, N-L+J ) = DBLE( B( J, N-L+J ) )
00413 *
00414 *              Update unitary matrices U, V, Q, if desired.
00415 *
00416                IF( WANTU .AND. K+J.LE.M )
00417      $            CALL ZROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
00418      $                       SNU )
00419 *
00420                IF( WANTV )
00421      $            CALL ZROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
00422 *
00423                IF( WANTQ )
00424      $            CALL ZROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
00425      $                       SNQ )
00426 *
00427    10       CONTINUE
00428    20    CONTINUE
00429 *
00430          IF( .NOT.UPPER ) THEN
00431 *
00432 *           The matrices A13 and B13 were lower triangular at the start
00433 *           of the cycle, and are now upper triangular.
00434 *
00435 *           Convergence test: test the parallelism of the corresponding
00436 *           rows of A and B.
00437 *
00438             ERROR = ZERO
00439             DO 30 I = 1, MIN( L, M-K )
00440                CALL ZCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
00441                CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
00442                CALL ZLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
00443                ERROR = MAX( ERROR, SSMIN )
00444    30       CONTINUE
00445 *
00446             IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
00447      $         GO TO 50
00448          END IF
00449 *
00450 *        End of cycle loop
00451 *
00452    40 CONTINUE
00453 *
00454 *     The algorithm has not converged after MAXIT cycles.
00455 *
00456       INFO = 1
00457       GO TO 100
00458 *
00459    50 CONTINUE
00460 *
00461 *     If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
00462 *     Compute the generalized singular value pairs (ALPHA, BETA), and
00463 *     set the triangular matrix R to array A.
00464 *
00465       DO 60 I = 1, K
00466          ALPHA( I ) = ONE
00467          BETA( I ) = ZERO
00468    60 CONTINUE
00469 *
00470       DO 70 I = 1, MIN( L, M-K )
00471 *
00472          A1 = DBLE( A( K+I, N-L+I ) )
00473          B1 = DBLE( B( I, N-L+I ) )
00474 *
00475          IF( A1.NE.ZERO ) THEN
00476             GAMMA = B1 / A1
00477 *
00478             IF( GAMMA.LT.ZERO ) THEN
00479                CALL ZDSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
00480                IF( WANTV )
00481      $            CALL ZDSCAL( P, -ONE, V( 1, I ), 1 )
00482             END IF
00483 *
00484             CALL DLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
00485      $                   RWK )
00486 *
00487             IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
00488                CALL ZDSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
00489      $                      LDA )
00490             ELSE
00491                CALL ZDSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
00492      $                      LDB )
00493                CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
00494      $                     LDA )
00495             END IF
00496 *
00497          ELSE
00498             ALPHA( K+I ) = ZERO
00499             BETA( K+I ) = ONE
00500             CALL ZCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
00501      $                  LDA )
00502          END IF
00503    70 CONTINUE
00504 *
00505 *     Post-assignment
00506 *
00507       DO 80 I = M + 1, K + L
00508          ALPHA( I ) = ZERO
00509          BETA( I ) = ONE
00510    80 CONTINUE
00511 *
00512       IF( K+L.LT.N ) THEN
00513          DO 90 I = K + L + 1, N
00514             ALPHA( I ) = ZERO
00515             BETA( I ) = ZERO
00516    90    CONTINUE
00517       END IF
00518 *
00519   100 CONTINUE
00520       NCYCLE = KCYCLE
00521 *
00522       RETURN
00523 *
00524 *     End of ZTGSJA
00525 *
00526       END
 All Files Functions