Annotation of rpl/lapack/lapack/ztrsen.f, revision 1.6

1.1       bertrand    1:       SUBROUTINE ZTRSEN( JOB, COMPQ, SELECT, N, T, LDT, Q, LDQ, W, M, S,
                      2:      $                   SEP, WORK, LWORK, INFO )
                      3: *
                      4: *  -- LAPACK routine (version 3.2) --
                      5: *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
                      6: *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
                      7: *     November 2006
                      8: *
                      9: *     Modified to call ZLACN2 in place of ZLACON, 10 Feb 03, SJH.
                     10: *
                     11: *     .. Scalar Arguments ..
                     12:       CHARACTER          COMPQ, JOB
                     13:       INTEGER            INFO, LDQ, LDT, LWORK, M, N
                     14:       DOUBLE PRECISION   S, SEP
                     15: *     ..
                     16: *     .. Array Arguments ..
                     17:       LOGICAL            SELECT( * )
                     18:       COMPLEX*16         Q( LDQ, * ), T( LDT, * ), W( * ), WORK( * )
                     19: *     ..
                     20: *
                     21: *  Purpose
                     22: *  =======
                     23: *
                     24: *  ZTRSEN reorders the Schur factorization of a complex matrix
                     25: *  A = Q*T*Q**H, so that a selected cluster of eigenvalues appears in
                     26: *  the leading positions on the diagonal of the upper triangular matrix
                     27: *  T, and the leading columns of Q form an orthonormal basis of the
                     28: *  corresponding right invariant subspace.
                     29: *
                     30: *  Optionally the routine computes the reciprocal condition numbers of
                     31: *  the cluster of eigenvalues and/or the invariant subspace.
                     32: *
                     33: *  Arguments
                     34: *  =========
                     35: *
                     36: *  JOB     (input) CHARACTER*1
                     37: *          Specifies whether condition numbers are required for the
                     38: *          cluster of eigenvalues (S) or the invariant subspace (SEP):
                     39: *          = 'N': none;
                     40: *          = 'E': for eigenvalues only (S);
                     41: *          = 'V': for invariant subspace only (SEP);
                     42: *          = 'B': for both eigenvalues and invariant subspace (S and
                     43: *                 SEP).
                     44: *
                     45: *  COMPQ   (input) CHARACTER*1
                     46: *          = 'V': update the matrix Q of Schur vectors;
                     47: *          = 'N': do not update Q.
                     48: *
                     49: *  SELECT  (input) LOGICAL array, dimension (N)
                     50: *          SELECT specifies the eigenvalues in the selected cluster. To
                     51: *          select the j-th eigenvalue, SELECT(j) must be set to .TRUE..
                     52: *
                     53: *  N       (input) INTEGER
                     54: *          The order of the matrix T. N >= 0.
                     55: *
                     56: *  T       (input/output) COMPLEX*16 array, dimension (LDT,N)
                     57: *          On entry, the upper triangular matrix T.
                     58: *          On exit, T is overwritten by the reordered matrix T, with the
                     59: *          selected eigenvalues as the leading diagonal elements.
                     60: *
                     61: *  LDT     (input) INTEGER
                     62: *          The leading dimension of the array T. LDT >= max(1,N).
                     63: *
                     64: *  Q       (input/output) COMPLEX*16 array, dimension (LDQ,N)
                     65: *          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
                     66: *          On exit, if COMPQ = 'V', Q has been postmultiplied by the
                     67: *          unitary transformation matrix which reorders T; the leading M
                     68: *          columns of Q form an orthonormal basis for the specified
                     69: *          invariant subspace.
                     70: *          If COMPQ = 'N', Q is not referenced.
                     71: *
                     72: *  LDQ     (input) INTEGER
                     73: *          The leading dimension of the array Q.
                     74: *          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
                     75: *
                     76: *  W       (output) COMPLEX*16 array, dimension (N)
                     77: *          The reordered eigenvalues of T, in the same order as they
                     78: *          appear on the diagonal of T.
                     79: *
                     80: *  M       (output) INTEGER
                     81: *          The dimension of the specified invariant subspace.
                     82: *          0 <= M <= N.
                     83: *
                     84: *  S       (output) DOUBLE PRECISION
                     85: *          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
                     86: *          condition number for the selected cluster of eigenvalues.
                     87: *          S cannot underestimate the true reciprocal condition number
                     88: *          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
                     89: *          If JOB = 'N' or 'V', S is not referenced.
                     90: *
                     91: *  SEP     (output) DOUBLE PRECISION
                     92: *          If JOB = 'V' or 'B', SEP is the estimated reciprocal
                     93: *          condition number of the specified invariant subspace. If
                     94: *          M = 0 or N, SEP = norm(T).
                     95: *          If JOB = 'N' or 'E', SEP is not referenced.
                     96: *
                     97: *  WORK    (workspace/output) COMPLEX*16 array, dimension (MAX(1,LWORK))
                     98: *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
                     99: *
                    100: *  LWORK   (input) INTEGER
                    101: *          The dimension of the array WORK.
                    102: *          If JOB = 'N', LWORK >= 1;
                    103: *          if JOB = 'E', LWORK = max(1,M*(N-M));
                    104: *          if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).
                    105: *
                    106: *          If LWORK = -1, then a workspace query is assumed; the routine
                    107: *          only calculates the optimal size of the WORK array, returns
                    108: *          this value as the first entry of the WORK array, and no error
                    109: *          message related to LWORK is issued by XERBLA.
                    110: *
                    111: *  INFO    (output) INTEGER
                    112: *          = 0:  successful exit
                    113: *          < 0:  if INFO = -i, the i-th argument had an illegal value
                    114: *
                    115: *  Further Details
                    116: *  ===============
                    117: *
                    118: *  ZTRSEN first collects the selected eigenvalues by computing a unitary
                    119: *  transformation Z to move them to the top left corner of T. In other
                    120: *  words, the selected eigenvalues are the eigenvalues of T11 in:
                    121: *
                    122: *                Z'*T*Z = ( T11 T12 ) n1
                    123: *                         (  0  T22 ) n2
                    124: *                            n1  n2
                    125: *
                    126: *  where N = n1+n2 and Z' means the conjugate transpose of Z. The first
                    127: *  n1 columns of Z span the specified invariant subspace of T.
                    128: *
                    129: *  If T has been obtained from the Schur factorization of a matrix
                    130: *  A = Q*T*Q', then the reordered Schur factorization of A is given by
                    131: *  A = (Q*Z)*(Z'*T*Z)*(Q*Z)', and the first n1 columns of Q*Z span the
                    132: *  corresponding invariant subspace of A.
                    133: *
                    134: *  The reciprocal condition number of the average of the eigenvalues of
                    135: *  T11 may be returned in S. S lies between 0 (very badly conditioned)
                    136: *  and 1 (very well conditioned). It is computed as follows. First we
                    137: *  compute R so that
                    138: *
                    139: *                         P = ( I  R ) n1
                    140: *                             ( 0  0 ) n2
                    141: *                               n1 n2
                    142: *
                    143: *  is the projector on the invariant subspace associated with T11.
                    144: *  R is the solution of the Sylvester equation:
                    145: *
                    146: *                        T11*R - R*T22 = T12.
                    147: *
                    148: *  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
                    149: *  the two-norm of M. Then S is computed as the lower bound
                    150: *
                    151: *                      (1 + F-norm(R)**2)**(-1/2)
                    152: *
                    153: *  on the reciprocal of 2-norm(P), the true reciprocal condition number.
                    154: *  S cannot underestimate 1 / 2-norm(P) by more than a factor of
                    155: *  sqrt(N).
                    156: *
                    157: *  An approximate error bound for the computed average of the
                    158: *  eigenvalues of T11 is
                    159: *
                    160: *                         EPS * norm(T) / S
                    161: *
                    162: *  where EPS is the machine precision.
                    163: *
                    164: *  The reciprocal condition number of the right invariant subspace
                    165: *  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
                    166: *  SEP is defined as the separation of T11 and T22:
                    167: *
                    168: *                     sep( T11, T22 ) = sigma-min( C )
                    169: *
                    170: *  where sigma-min(C) is the smallest singular value of the
                    171: *  n1*n2-by-n1*n2 matrix
                    172: *
                    173: *     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )
                    174: *
                    175: *  I(m) is an m by m identity matrix, and kprod denotes the Kronecker
                    176: *  product. We estimate sigma-min(C) by the reciprocal of an estimate of
                    177: *  the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
                    178: *  cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).
                    179: *
                    180: *  When SEP is small, small changes in T can cause large changes in
                    181: *  the invariant subspace. An approximate bound on the maximum angular
                    182: *  error in the computed right invariant subspace is
                    183: *
                    184: *                      EPS * norm(T) / SEP
                    185: *
                    186: *  =====================================================================
                    187: *
                    188: *     .. Parameters ..
                    189:       DOUBLE PRECISION   ZERO, ONE
                    190:       PARAMETER          ( ZERO = 0.0D+0, ONE = 1.0D+0 )
                    191: *     ..
                    192: *     .. Local Scalars ..
                    193:       LOGICAL            LQUERY, WANTBH, WANTQ, WANTS, WANTSP
                    194:       INTEGER            IERR, K, KASE, KS, LWMIN, N1, N2, NN
                    195:       DOUBLE PRECISION   EST, RNORM, SCALE
                    196: *     ..
                    197: *     .. Local Arrays ..
                    198:       INTEGER            ISAVE( 3 )
                    199:       DOUBLE PRECISION   RWORK( 1 )
                    200: *     ..
                    201: *     .. External Functions ..
                    202:       LOGICAL            LSAME
                    203:       DOUBLE PRECISION   ZLANGE
                    204:       EXTERNAL           LSAME, ZLANGE
                    205: *     ..
                    206: *     .. External Subroutines ..
                    207:       EXTERNAL           XERBLA, ZLACN2, ZLACPY, ZTREXC, ZTRSYL
                    208: *     ..
                    209: *     .. Intrinsic Functions ..
                    210:       INTRINSIC          MAX, SQRT
                    211: *     ..
                    212: *     .. Executable Statements ..
                    213: *
                    214: *     Decode and test the input parameters.
                    215: *
                    216:       WANTBH = LSAME( JOB, 'B' )
                    217:       WANTS = LSAME( JOB, 'E' ) .OR. WANTBH
                    218:       WANTSP = LSAME( JOB, 'V' ) .OR. WANTBH
                    219:       WANTQ = LSAME( COMPQ, 'V' )
                    220: *
                    221: *     Set M to the number of selected eigenvalues.
                    222: *
                    223:       M = 0
                    224:       DO 10 K = 1, N
                    225:          IF( SELECT( K ) )
                    226:      $      M = M + 1
                    227:    10 CONTINUE
                    228: *
                    229:       N1 = M
                    230:       N2 = N - M
                    231:       NN = N1*N2
                    232: *
                    233:       INFO = 0
                    234:       LQUERY = ( LWORK.EQ.-1 )
                    235: *
                    236:       IF( WANTSP ) THEN
                    237:          LWMIN = MAX( 1, 2*NN )
                    238:       ELSE IF( LSAME( JOB, 'N' ) ) THEN
                    239:          LWMIN = 1
                    240:       ELSE IF( LSAME( JOB, 'E' ) ) THEN
                    241:          LWMIN = MAX( 1, NN )
                    242:       END IF
                    243: *
                    244:       IF( .NOT.LSAME( JOB, 'N' ) .AND. .NOT.WANTS .AND. .NOT.WANTSP )
                    245:      $     THEN
                    246:          INFO = -1
                    247:       ELSE IF( .NOT.LSAME( COMPQ, 'N' ) .AND. .NOT.WANTQ ) THEN
                    248:          INFO = -2
                    249:       ELSE IF( N.LT.0 ) THEN
                    250:          INFO = -4
                    251:       ELSE IF( LDT.LT.MAX( 1, N ) ) THEN
                    252:          INFO = -6
                    253:       ELSE IF( LDQ.LT.1 .OR. ( WANTQ .AND. LDQ.LT.N ) ) THEN
                    254:          INFO = -8
                    255:       ELSE IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
                    256:          INFO = -14
                    257:       END IF
                    258: *
                    259:       IF( INFO.EQ.0 ) THEN
                    260:          WORK( 1 ) = LWMIN
                    261:       END IF
                    262: *
                    263:       IF( INFO.NE.0 ) THEN
                    264:          CALL XERBLA( 'ZTRSEN', -INFO )
                    265:          RETURN
                    266:       ELSE IF( LQUERY ) THEN
                    267:          RETURN
                    268:       END IF
                    269: *
                    270: *     Quick return if possible
                    271: *
                    272:       IF( M.EQ.N .OR. M.EQ.0 ) THEN
                    273:          IF( WANTS )
                    274:      $      S = ONE
                    275:          IF( WANTSP )
                    276:      $      SEP = ZLANGE( '1', N, N, T, LDT, RWORK )
                    277:          GO TO 40
                    278:       END IF
                    279: *
                    280: *     Collect the selected eigenvalues at the top left corner of T.
                    281: *
                    282:       KS = 0
                    283:       DO 20 K = 1, N
                    284:          IF( SELECT( K ) ) THEN
                    285:             KS = KS + 1
                    286: *
                    287: *           Swap the K-th eigenvalue to position KS.
                    288: *
                    289:             IF( K.NE.KS )
                    290:      $         CALL ZTREXC( COMPQ, N, T, LDT, Q, LDQ, K, KS, IERR )
                    291:          END IF
                    292:    20 CONTINUE
                    293: *
                    294:       IF( WANTS ) THEN
                    295: *
                    296: *        Solve the Sylvester equation for R:
                    297: *
                    298: *           T11*R - R*T22 = scale*T12
                    299: *
                    300:          CALL ZLACPY( 'F', N1, N2, T( 1, N1+1 ), LDT, WORK, N1 )
                    301:          CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT, T( N1+1, N1+1 ),
                    302:      $                LDT, WORK, N1, SCALE, IERR )
                    303: *
                    304: *        Estimate the reciprocal of the condition number of the cluster
                    305: *        of eigenvalues.
                    306: *
                    307:          RNORM = ZLANGE( 'F', N1, N2, WORK, N1, RWORK )
                    308:          IF( RNORM.EQ.ZERO ) THEN
                    309:             S = ONE
                    310:          ELSE
                    311:             S = SCALE / ( SQRT( SCALE*SCALE / RNORM+RNORM )*
                    312:      $          SQRT( RNORM ) )
                    313:          END IF
                    314:       END IF
                    315: *
                    316:       IF( WANTSP ) THEN
                    317: *
                    318: *        Estimate sep(T11,T22).
                    319: *
                    320:          EST = ZERO
                    321:          KASE = 0
                    322:    30    CONTINUE
                    323:          CALL ZLACN2( NN, WORK( NN+1 ), WORK, EST, KASE, ISAVE )
                    324:          IF( KASE.NE.0 ) THEN
                    325:             IF( KASE.EQ.1 ) THEN
                    326: *
                    327: *              Solve T11*R - R*T22 = scale*X.
                    328: *
                    329:                CALL ZTRSYL( 'N', 'N', -1, N1, N2, T, LDT,
                    330:      $                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
                    331:      $                      IERR )
                    332:             ELSE
                    333: *
                    334: *              Solve T11'*R - R*T22' = scale*X.
                    335: *
                    336:                CALL ZTRSYL( 'C', 'C', -1, N1, N2, T, LDT,
                    337:      $                      T( N1+1, N1+1 ), LDT, WORK, N1, SCALE,
                    338:      $                      IERR )
                    339:             END IF
                    340:             GO TO 30
                    341:          END IF
                    342: *
                    343:          SEP = SCALE / EST
                    344:       END IF
                    345: *
                    346:    40 CONTINUE
                    347: *
                    348: *     Copy reordered eigenvalues to W.
                    349: *
                    350:       DO 50 K = 1, N
                    351:          W( K ) = T( K, K )
                    352:    50 CONTINUE
                    353: *
                    354:       WORK( 1 ) = LWMIN
                    355: *
                    356:       RETURN
                    357: *
                    358: *     End of ZTRSEN
                    359: *
                    360:       END

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