LAPACK 3.3.0

stgsja.f

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00001       SUBROUTINE STGSJA( 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       REAL               TOLA, TOLB
00015 *     ..
00016 *     .. Array Arguments ..
00017       REAL               A( LDA, * ), ALPHA( * ), B( LDB, * ),
00018      $                   BETA( * ), Q( LDQ, * ), U( LDU, * ),
00019      $                   V( LDV, * ), WORK( * )
00020 *     ..
00021 *
00022 *  Purpose
00023 *  =======
00024 *
00025 *  STGSJA computes the generalized singular value decomposition (GSVD)
00026 *  of two real 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 SGGSVP
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 orthogonal matrices, Z' denotes the transpose
00054 *  of Z, R is a nonsingular upper triangular matrix, and D1 and D2 are
00055 *  ``diagonal'' matrices, which are of the following structures:
00056 *
00057 *  If M-K-L >= 0,
00058 *
00059 *                      K  L
00060 *         D1 =     K ( I  0 )
00061 *                  L ( 0  C )
00062 *              M-K-L ( 0  0 )
00063 *
00064 *                    K  L
00065 *         D2 = L   ( 0  S )
00066 *              P-L ( 0  0 )
00067 *
00068 *                 N-K-L  K    L
00069 *    ( 0 R ) = K (  0   R11  R12 ) K
00070 *              L (  0    0   R22 ) L
00071 *
00072 *  where
00073 *
00074 *    C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
00075 *    S = diag( BETA(K+1),  ... , BETA(K+L) ),
00076 *    C**2 + S**2 = I.
00077 *
00078 *    R is stored in A(1:K+L,N-K-L+1:N) on exit.
00079 *
00080 *  If M-K-L < 0,
00081 *
00082 *                 K M-K K+L-M
00083 *      D1 =   K ( I  0    0   )
00084 *           M-K ( 0  C    0   )
00085 *
00086 *                   K M-K K+L-M
00087 *      D2 =   M-K ( 0  S    0   )
00088 *           K+L-M ( 0  0    I   )
00089 *             P-L ( 0  0    0   )
00090 *
00091 *                 N-K-L  K   M-K  K+L-M
00092 * ( 0 R ) =    K ( 0    R11  R12  R13  )
00093 *            M-K ( 0     0   R22  R23  )
00094 *          K+L-M ( 0     0    0   R33  )
00095 *
00096 *  where
00097 *  C = diag( ALPHA(K+1), ... , ALPHA(M) ),
00098 *  S = diag( BETA(K+1),  ... , BETA(M) ),
00099 *  C**2 + S**2 = I.
00100 *
00101 *  R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
00102 *      (  0  R22 R23 )
00103 *  in B(M-K+1:L,N+M-K-L+1:N) on exit.
00104 *
00105 *  The computation of the orthogonal transformation matrices U, V or Q
00106 *  is optional.  These matrices may either be formed explicitly, or they
00107 *  may be postmultiplied into input matrices U1, V1, or Q1.
00108 *
00109 *  Arguments
00110 *  =========
00111 *
00112 *  JOBU    (input) CHARACTER*1
00113 *          = 'U':  U must contain an orthogonal matrix U1 on entry, and
00114 *                  the product U1*U is returned;
00115 *          = 'I':  U is initialized to the unit matrix, and the
00116 *                  orthogonal matrix U is returned;
00117 *          = 'N':  U is not computed.
00118 *
00119 *  JOBV    (input) CHARACTER*1
00120 *          = 'V':  V must contain an orthogonal matrix V1 on entry, and
00121 *                  the product V1*V is returned;
00122 *          = 'I':  V is initialized to the unit matrix, and the
00123 *                  orthogonal matrix V is returned;
00124 *          = 'N':  V is not computed.
00125 *
00126 *  JOBQ    (input) CHARACTER*1
00127 *          = 'Q':  Q must contain an orthogonal matrix Q1 on entry, and
00128 *                  the product Q1*Q is returned;
00129 *          = 'I':  Q is initialized to the unit matrix, and the
00130 *                  orthogonal matrix Q is returned;
00131 *          = 'N':  Q is not computed.
00132 *
00133 *  M       (input) INTEGER
00134 *          The number of rows of the matrix A.  M >= 0.
00135 *
00136 *  P       (input) INTEGER
00137 *          The number of rows of the matrix B.  P >= 0.
00138 *
00139 *  N       (input) INTEGER
00140 *          The number of columns of the matrices A and B.  N >= 0.
00141 *
00142 *  K       (input) INTEGER
00143 *  L       (input) INTEGER
00144 *          K and L specify the subblocks in the input matrices A and B:
00145 *          A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N)
00146 *          of A and B, whose GSVD is going to be computed by STGSJA.
00147 *          See Further Details.
00148 *
00149 *  A       (input/output) REAL array, dimension (LDA,N)
00150 *          On entry, the M-by-N matrix A.
00151 *          On exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular
00152 *          matrix R or part of R.  See Purpose for details.
00153 *
00154 *  LDA     (input) INTEGER
00155 *          The leading dimension of the array A. LDA >= max(1,M).
00156 *
00157 *  B       (input/output) REAL array, dimension (LDB,N)
00158 *          On entry, the P-by-N matrix B.
00159 *          On exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains
00160 *          a part of R.  See Purpose for details.
00161 *
00162 *  LDB     (input) INTEGER
00163 *          The leading dimension of the array B. LDB >= max(1,P).
00164 *
00165 *  TOLA    (input) REAL
00166 *  TOLB    (input) REAL
00167 *          TOLA and TOLB are the convergence criteria for the Jacobi-
00168 *          Kogbetliantz iteration procedure. Generally, they are the
00169 *          same as used in the preprocessing step, say
00170 *              TOLA = max(M,N)*norm(A)*MACHEPS,
00171 *              TOLB = max(P,N)*norm(B)*MACHEPS.
00172 *
00173 *  ALPHA   (output) REAL array, dimension (N)
00174 *  BETA    (output) REAL array, dimension (N)
00175 *          On exit, ALPHA and BETA contain the generalized singular
00176 *          value pairs of A and B;
00177 *            ALPHA(1:K) = 1,
00178 *            BETA(1:K)  = 0,
00179 *          and if M-K-L >= 0,
00180 *            ALPHA(K+1:K+L) = diag(C),
00181 *            BETA(K+1:K+L)  = diag(S),
00182 *          or if M-K-L < 0,
00183 *            ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
00184 *            BETA(K+1:M) = S, BETA(M+1:K+L) = 1.
00185 *          Furthermore, if K+L < N,
00186 *            ALPHA(K+L+1:N) = 0 and
00187 *            BETA(K+L+1:N)  = 0.
00188 *
00189 *  U       (input/output) REAL array, dimension (LDU,M)
00190 *          On entry, if JOBU = 'U', U must contain a matrix U1 (usually
00191 *          the orthogonal matrix returned by SGGSVP).
00192 *          On exit,
00193 *          if JOBU = 'I', U contains the orthogonal matrix U;
00194 *          if JOBU = 'U', U contains the product U1*U.
00195 *          If JOBU = 'N', U is not referenced.
00196 *
00197 *  LDU     (input) INTEGER
00198 *          The leading dimension of the array U. LDU >= max(1,M) if
00199 *          JOBU = 'U'; LDU >= 1 otherwise.
00200 *
00201 *  V       (input/output) REAL array, dimension (LDV,P)
00202 *          On entry, if JOBV = 'V', V must contain a matrix V1 (usually
00203 *          the orthogonal matrix returned by SGGSVP).
00204 *          On exit,
00205 *          if JOBV = 'I', V contains the orthogonal matrix V;
00206 *          if JOBV = 'V', V contains the product V1*V.
00207 *          If JOBV = 'N', V is not referenced.
00208 *
00209 *  LDV     (input) INTEGER
00210 *          The leading dimension of the array V. LDV >= max(1,P) if
00211 *          JOBV = 'V'; LDV >= 1 otherwise.
00212 *
00213 *  Q       (input/output) REAL array, dimension (LDQ,N)
00214 *          On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually
00215 *          the orthogonal matrix returned by SGGSVP).
00216 *          On exit,
00217 *          if JOBQ = 'I', Q contains the orthogonal matrix Q;
00218 *          if JOBQ = 'Q', Q contains the product Q1*Q.
00219 *          If JOBQ = 'N', Q is not referenced.
00220 *
00221 *  LDQ     (input) INTEGER
00222 *          The leading dimension of the array Q. LDQ >= max(1,N) if
00223 *          JOBQ = 'Q'; LDQ >= 1 otherwise.
00224 *
00225 *  WORK    (workspace) REAL array, dimension (2*N)
00226 *
00227 *  NCYCLE  (output) INTEGER
00228 *          The number of cycles required for convergence.
00229 *
00230 *  INFO    (output) INTEGER
00231 *          = 0:  successful exit
00232 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00233 *          = 1:  the procedure does not converge after MAXIT cycles.
00234 *
00235 *  Internal Parameters
00236 *  ===================
00237 *
00238 *  MAXIT   INTEGER
00239 *          MAXIT specifies the total loops that the iterative procedure
00240 *          may take. If after MAXIT cycles, the routine fails to
00241 *          converge, we return INFO = 1.
00242 *
00243 *  Further Details
00244 *  ===============
00245 *
00246 *  STGSJA essentially uses a variant of Kogbetliantz algorithm to reduce
00247 *  min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L
00248 *  matrix B13 to the form:
00249 *
00250 *           U1'*A13*Q1 = C1*R1; V1'*B13*Q1 = S1*R1,
00251 *
00252 *  where U1, V1 and Q1 are orthogonal matrix, and Z' is the transpose
00253 *  of Z.  C1 and S1 are diagonal matrices satisfying
00254 *
00255 *                C1**2 + S1**2 = I,
00256 *
00257 *  and R1 is an L-by-L nonsingular upper triangular matrix.
00258 *
00259 *  =====================================================================
00260 *
00261 *     .. Parameters ..
00262       INTEGER            MAXIT
00263       PARAMETER          ( MAXIT = 40 )
00264       REAL               ZERO, ONE
00265       PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0 )
00266 *     ..
00267 *     .. Local Scalars ..
00268 *
00269       LOGICAL            INITQ, INITU, INITV, UPPER, WANTQ, WANTU, WANTV
00270       INTEGER            I, J, KCYCLE
00271       REAL               A1, A2, A3, B1, B2, B3, CSQ, CSU, CSV, ERROR,
00272      $                   GAMMA, RWK, SNQ, SNU, SNV, SSMIN
00273 *     ..
00274 *     .. External Functions ..
00275       LOGICAL            LSAME
00276       EXTERNAL           LSAME
00277 *     ..
00278 *     .. External Subroutines ..
00279       EXTERNAL           SCOPY, SLAGS2, SLAPLL, SLARTG, SLASET, SROT,
00280      $                   SSCAL, XERBLA
00281 *     ..
00282 *     .. Intrinsic Functions ..
00283       INTRINSIC          ABS, MAX, MIN
00284 *     ..
00285 *     .. Executable Statements ..
00286 *
00287 *     Decode and test the input parameters
00288 *
00289       INITU = LSAME( JOBU, 'I' )
00290       WANTU = INITU .OR. LSAME( JOBU, 'U' )
00291 *
00292       INITV = LSAME( JOBV, 'I' )
00293       WANTV = INITV .OR. LSAME( JOBV, 'V' )
00294 *
00295       INITQ = LSAME( JOBQ, 'I' )
00296       WANTQ = INITQ .OR. LSAME( JOBQ, 'Q' )
00297 *
00298       INFO = 0
00299       IF( .NOT.( INITU .OR. WANTU .OR. LSAME( JOBU, 'N' ) ) ) THEN
00300          INFO = -1
00301       ELSE IF( .NOT.( INITV .OR. WANTV .OR. LSAME( JOBV, 'N' ) ) ) THEN
00302          INFO = -2
00303       ELSE IF( .NOT.( INITQ .OR. WANTQ .OR. LSAME( JOBQ, 'N' ) ) ) THEN
00304          INFO = -3
00305       ELSE IF( M.LT.0 ) THEN
00306          INFO = -4
00307       ELSE IF( P.LT.0 ) THEN
00308          INFO = -5
00309       ELSE IF( N.LT.0 ) THEN
00310          INFO = -6
00311       ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
00312          INFO = -10
00313       ELSE IF( LDB.LT.MAX( 1, P ) ) THEN
00314          INFO = -12
00315       ELSE IF( LDU.LT.1 .OR. ( WANTU .AND. LDU.LT.M ) ) THEN
00316          INFO = -18
00317       ELSE IF( LDV.LT.1 .OR. ( WANTV .AND. LDV.LT.P ) ) THEN
00318          INFO = -20
00319       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
00320          INFO = -22
00321       END IF
00322       IF( INFO.NE.0 ) THEN
00323          CALL XERBLA( 'STGSJA', -INFO )
00324          RETURN
00325       END IF
00326 *
00327 *     Initialize U, V and Q, if necessary
00328 *
00329       IF( INITU )
00330      $   CALL SLASET( 'Full', M, M, ZERO, ONE, U, LDU )
00331       IF( INITV )
00332      $   CALL SLASET( 'Full', P, P, ZERO, ONE, V, LDV )
00333       IF( INITQ )
00334      $   CALL SLASET( 'Full', N, N, ZERO, ONE, Q, LDQ )
00335 *
00336 *     Loop until convergence
00337 *
00338       UPPER = .FALSE.
00339       DO 40 KCYCLE = 1, MAXIT
00340 *
00341          UPPER = .NOT.UPPER
00342 *
00343          DO 20 I = 1, L - 1
00344             DO 10 J = I + 1, L
00345 *
00346                A1 = ZERO
00347                A2 = ZERO
00348                A3 = ZERO
00349                IF( K+I.LE.M )
00350      $            A1 = A( K+I, N-L+I )
00351                IF( K+J.LE.M )
00352      $            A3 = A( K+J, N-L+J )
00353 *
00354                B1 = B( I, N-L+I )
00355                B3 = B( J, N-L+J )
00356 *
00357                IF( UPPER ) THEN
00358                   IF( K+I.LE.M )
00359      $               A2 = A( K+I, N-L+J )
00360                   B2 = B( I, N-L+J )
00361                ELSE
00362                   IF( K+J.LE.M )
00363      $               A2 = A( K+J, N-L+I )
00364                   B2 = B( J, N-L+I )
00365                END IF
00366 *
00367                CALL SLAGS2( UPPER, A1, A2, A3, B1, B2, B3, CSU, SNU,
00368      $                      CSV, SNV, CSQ, SNQ )
00369 *
00370 *              Update (K+I)-th and (K+J)-th rows of matrix A: U'*A
00371 *
00372                IF( K+J.LE.M )
00373      $            CALL SROT( L, A( K+J, N-L+1 ), LDA, A( K+I, N-L+1 ),
00374      $                       LDA, CSU, SNU )
00375 *
00376 *              Update I-th and J-th rows of matrix B: V'*B
00377 *
00378                CALL SROT( L, B( J, N-L+1 ), LDB, B( I, N-L+1 ), LDB,
00379      $                    CSV, SNV )
00380 *
00381 *              Update (N-L+I)-th and (N-L+J)-th columns of matrices
00382 *              A and B: A*Q and B*Q
00383 *
00384                CALL SROT( MIN( K+L, M ), A( 1, N-L+J ), 1,
00385      $                    A( 1, N-L+I ), 1, CSQ, SNQ )
00386 *
00387                CALL SROT( L, B( 1, N-L+J ), 1, B( 1, N-L+I ), 1, CSQ,
00388      $                    SNQ )
00389 *
00390                IF( UPPER ) THEN
00391                   IF( K+I.LE.M )
00392      $               A( K+I, N-L+J ) = ZERO
00393                   B( I, N-L+J ) = ZERO
00394                ELSE
00395                   IF( K+J.LE.M )
00396      $               A( K+J, N-L+I ) = ZERO
00397                   B( J, N-L+I ) = ZERO
00398                END IF
00399 *
00400 *              Update orthogonal matrices U, V, Q, if desired.
00401 *
00402                IF( WANTU .AND. K+J.LE.M )
00403      $            CALL SROT( M, U( 1, K+J ), 1, U( 1, K+I ), 1, CSU,
00404      $                       SNU )
00405 *
00406                IF( WANTV )
00407      $            CALL SROT( P, V( 1, J ), 1, V( 1, I ), 1, CSV, SNV )
00408 *
00409                IF( WANTQ )
00410      $            CALL SROT( N, Q( 1, N-L+J ), 1, Q( 1, N-L+I ), 1, CSQ,
00411      $                       SNQ )
00412 *
00413    10       CONTINUE
00414    20    CONTINUE
00415 *
00416          IF( .NOT.UPPER ) THEN
00417 *
00418 *           The matrices A13 and B13 were lower triangular at the start
00419 *           of the cycle, and are now upper triangular.
00420 *
00421 *           Convergence test: test the parallelism of the corresponding
00422 *           rows of A and B.
00423 *
00424             ERROR = ZERO
00425             DO 30 I = 1, MIN( L, M-K )
00426                CALL SCOPY( L-I+1, A( K+I, N-L+I ), LDA, WORK, 1 )
00427                CALL SCOPY( L-I+1, B( I, N-L+I ), LDB, WORK( L+1 ), 1 )
00428                CALL SLAPLL( L-I+1, WORK, 1, WORK( L+1 ), 1, SSMIN )
00429                ERROR = MAX( ERROR, SSMIN )
00430    30       CONTINUE
00431 *
00432             IF( ABS( ERROR ).LE.MIN( TOLA, TOLB ) )
00433      $         GO TO 50
00434          END IF
00435 *
00436 *        End of cycle loop
00437 *
00438    40 CONTINUE
00439 *
00440 *     The algorithm has not converged after MAXIT cycles.
00441 *
00442       INFO = 1
00443       GO TO 100
00444 *
00445    50 CONTINUE
00446 *
00447 *     If ERROR <= MIN(TOLA,TOLB), then the algorithm has converged.
00448 *     Compute the generalized singular value pairs (ALPHA, BETA), and
00449 *     set the triangular matrix R to array A.
00450 *
00451       DO 60 I = 1, K
00452          ALPHA( I ) = ONE
00453          BETA( I ) = ZERO
00454    60 CONTINUE
00455 *
00456       DO 70 I = 1, MIN( L, M-K )
00457 *
00458          A1 = A( K+I, N-L+I )
00459          B1 = B( I, N-L+I )
00460 *
00461          IF( A1.NE.ZERO ) THEN
00462             GAMMA = B1 / A1
00463 *
00464 *           change sign if necessary
00465 *
00466             IF( GAMMA.LT.ZERO ) THEN
00467                CALL SSCAL( L-I+1, -ONE, B( I, N-L+I ), LDB )
00468                IF( WANTV )
00469      $            CALL SSCAL( P, -ONE, V( 1, I ), 1 )
00470             END IF
00471 *
00472             CALL SLARTG( ABS( GAMMA ), ONE, BETA( K+I ), ALPHA( K+I ),
00473      $                   RWK )
00474 *
00475             IF( ALPHA( K+I ).GE.BETA( K+I ) ) THEN
00476                CALL SSCAL( L-I+1, ONE / ALPHA( K+I ), A( K+I, N-L+I ),
00477      $                     LDA )
00478             ELSE
00479                CALL SSCAL( L-I+1, ONE / BETA( K+I ), B( I, N-L+I ),
00480      $                     LDB )
00481                CALL SCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
00482      $                     LDA )
00483             END IF
00484 *
00485          ELSE
00486 *
00487             ALPHA( K+I ) = ZERO
00488             BETA( K+I ) = ONE
00489             CALL SCOPY( L-I+1, B( I, N-L+I ), LDB, A( K+I, N-L+I ),
00490      $                  LDA )
00491 *
00492          END IF
00493 *
00494    70 CONTINUE
00495 *
00496 *     Post-assignment
00497 *
00498       DO 80 I = M + 1, K + L
00499          ALPHA( I ) = ZERO
00500          BETA( I ) = ONE
00501    80 CONTINUE
00502 *
00503       IF( K+L.LT.N ) THEN
00504          DO 90 I = K + L + 1, N
00505             ALPHA( I ) = ZERO
00506             BETA( I ) = ZERO
00507    90    CONTINUE
00508       END IF
00509 *
00510   100 CONTINUE
00511       NCYCLE = KCYCLE
00512       RETURN
00513 *
00514 *     End of STGSJA
00515 *
00516       END
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