LAPACK 3.11.0
LAPACK: Linear Algebra PACKage
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◆ strsen()

subroutine strsen ( character  JOB,
character  COMPQ,
logical, dimension( * )  SELECT,
integer  N,
real, dimension( ldt, * )  T,
integer  LDT,
real, dimension( ldq, * )  Q,
integer  LDQ,
real, dimension( * )  WR,
real, dimension( * )  WI,
integer  M,
real  S,
real  SEP,
real, dimension( * )  WORK,
integer  LWORK,
integer, dimension( * )  IWORK,
integer  LIWORK,
integer  INFO 
)

STRSEN

Download STRSEN + dependencies [TGZ] [ZIP] [TXT]

Purpose:
 STRSEN reorders the real Schur factorization of a real matrix
 A = Q*T*Q**T, so that a selected cluster of eigenvalues appears in
 the leading diagonal blocks of the upper quasi-triangular matrix T,
 and the leading columns of Q form an orthonormal basis of the
 corresponding right invariant subspace.

 Optionally the routine computes the reciprocal condition numbers of
 the cluster of eigenvalues and/or the invariant subspace.

 T must be in Schur canonical form (as returned by SHSEQR), that is,
 block upper triangular with 1-by-1 and 2-by-2 diagonal blocks; each
 2-by-2 diagonal block has its diagonal elements equal and its
 off-diagonal elements of opposite sign.
Parameters
[in]JOB
          JOB is CHARACTER*1
          Specifies whether condition numbers are required for the
          cluster of eigenvalues (S) or the invariant subspace (SEP):
          = 'N': none;
          = 'E': for eigenvalues only (S);
          = 'V': for invariant subspace only (SEP);
          = 'B': for both eigenvalues and invariant subspace (S and
                 SEP).
[in]COMPQ
          COMPQ is CHARACTER*1
          = 'V': update the matrix Q of Schur vectors;
          = 'N': do not update Q.
[in]SELECT
          SELECT is LOGICAL array, dimension (N)
          SELECT specifies the eigenvalues in the selected cluster. To
          select a real eigenvalue w(j), SELECT(j) must be set to
          .TRUE.. To select a complex conjugate pair of eigenvalues
          w(j) and w(j+1), corresponding to a 2-by-2 diagonal block,
          either SELECT(j) or SELECT(j+1) or both must be set to
          .TRUE.; a complex conjugate pair of eigenvalues must be
          either both included in the cluster or both excluded.
[in]N
          N is INTEGER
          The order of the matrix T. N >= 0.
[in,out]T
          T is REAL array, dimension (LDT,N)
          On entry, the upper quasi-triangular matrix T, in Schur
          canonical form.
          On exit, T is overwritten by the reordered matrix T, again in
          Schur canonical form, with the selected eigenvalues in the
          leading diagonal blocks.
[in]LDT
          LDT is INTEGER
          The leading dimension of the array T. LDT >= max(1,N).
[in,out]Q
          Q is REAL array, dimension (LDQ,N)
          On entry, if COMPQ = 'V', the matrix Q of Schur vectors.
          On exit, if COMPQ = 'V', Q has been postmultiplied by the
          orthogonal transformation matrix which reorders T; the
          leading M columns of Q form an orthonormal basis for the
          specified invariant subspace.
          If COMPQ = 'N', Q is not referenced.
[in]LDQ
          LDQ is INTEGER
          The leading dimension of the array Q.
          LDQ >= 1; and if COMPQ = 'V', LDQ >= N.
[out]WR
          WR is REAL array, dimension (N)
[out]WI
          WI is REAL array, dimension (N)

          The real and imaginary parts, respectively, of the reordered
          eigenvalues of T. The eigenvalues are stored in the same
          order as on the diagonal of T, with WR(i) = T(i,i) and, if
          T(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) > 0 and
          WI(i+1) = -WI(i). Note that if a complex eigenvalue is
          sufficiently ill-conditioned, then its value may differ
          significantly from its value before reordering.
[out]M
          M is INTEGER
          The dimension of the specified invariant subspace.
          0 < = M <= N.
[out]S
          S is REAL
          If JOB = 'E' or 'B', S is a lower bound on the reciprocal
          condition number for the selected cluster of eigenvalues.
          S cannot underestimate the true reciprocal condition number
          by more than a factor of sqrt(N). If M = 0 or N, S = 1.
          If JOB = 'N' or 'V', S is not referenced.
[out]SEP
          SEP is REAL
          If JOB = 'V' or 'B', SEP is the estimated reciprocal
          condition number of the specified invariant subspace. If
          M = 0 or N, SEP = norm(T).
          If JOB = 'N' or 'E', SEP is not referenced.
[out]WORK
          WORK is REAL array, dimension (MAX(1,LWORK))
          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
[in]LWORK
          LWORK is INTEGER
          The dimension of the array WORK.
          If JOB = 'N', LWORK >= max(1,N);
          if JOB = 'E', LWORK >= max(1,M*(N-M));
          if JOB = 'V' or 'B', LWORK >= max(1,2*M*(N-M)).

          If LWORK = -1, then a workspace query is assumed; the routine
          only calculates the optimal size of the WORK array, returns
          this value as the first entry of the WORK array, and no error
          message related to LWORK is issued by XERBLA.
[out]IWORK
          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
[in]LIWORK
          LIWORK is INTEGER
          The dimension of the array IWORK.
          If JOB = 'N' or 'E', LIWORK >= 1;
          if JOB = 'V' or 'B', LIWORK >= max(1,M*(N-M)).

          If LIWORK = -1, then a workspace query is assumed; the
          routine only calculates the optimal size of the IWORK array,
          returns this value as the first entry of the IWORK array, and
          no error message related to LIWORK is issued by XERBLA.
[out]INFO
          INFO is INTEGER
          = 0: successful exit
          < 0: if INFO = -i, the i-th argument had an illegal value
          = 1: reordering of T failed because some eigenvalues are too
               close to separate (the problem is very ill-conditioned);
               T may have been partially reordered, and WR and WI
               contain the eigenvalues in the same order as in T; S and
               SEP (if requested) are set to zero.
Author
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
  STRSEN first collects the selected eigenvalues by computing an
  orthogonal transformation Z to move them to the top left corner of T.
  In other words, the selected eigenvalues are the eigenvalues of T11
  in:

          Z**T * T * Z = ( T11 T12 ) n1
                         (  0  T22 ) n2
                            n1  n2

  where N = n1+n2 and Z**T means the transpose of Z. The first n1 columns
  of Z span the specified invariant subspace of T.

  If T has been obtained from the real Schur factorization of a matrix
  A = Q*T*Q**T, then the reordered real Schur factorization of A is given
  by A = (Q*Z)*(Z**T*T*Z)*(Q*Z)**T, and the first n1 columns of Q*Z span
  the corresponding invariant subspace of A.

  The reciprocal condition number of the average of the eigenvalues of
  T11 may be returned in S. S lies between 0 (very badly conditioned)
  and 1 (very well conditioned). It is computed as follows. First we
  compute R so that

                         P = ( I  R ) n1
                             ( 0  0 ) n2
                               n1 n2

  is the projector on the invariant subspace associated with T11.
  R is the solution of the Sylvester equation:

                        T11*R - R*T22 = T12.

  Let F-norm(M) denote the Frobenius-norm of M and 2-norm(M) denote
  the two-norm of M. Then S is computed as the lower bound

                      (1 + F-norm(R)**2)**(-1/2)

  on the reciprocal of 2-norm(P), the true reciprocal condition number.
  S cannot underestimate 1 / 2-norm(P) by more than a factor of
  sqrt(N).

  An approximate error bound for the computed average of the
  eigenvalues of T11 is

                         EPS * norm(T) / S

  where EPS is the machine precision.

  The reciprocal condition number of the right invariant subspace
  spanned by the first n1 columns of Z (or of Q*Z) is returned in SEP.
  SEP is defined as the separation of T11 and T22:

                     sep( T11, T22 ) = sigma-min( C )

  where sigma-min(C) is the smallest singular value of the
  n1*n2-by-n1*n2 matrix

     C  = kprod( I(n2), T11 ) - kprod( transpose(T22), I(n1) )

  I(m) is an m by m identity matrix, and kprod denotes the Kronecker
  product. We estimate sigma-min(C) by the reciprocal of an estimate of
  the 1-norm of inverse(C). The true reciprocal 1-norm of inverse(C)
  cannot differ from sigma-min(C) by more than a factor of sqrt(n1*n2).

  When SEP is small, small changes in T can cause large changes in
  the invariant subspace. An approximate bound on the maximum angular
  error in the computed right invariant subspace is

                      EPS * norm(T) / SEP

Definition at line 312 of file strsen.f.

314*
315* -- LAPACK computational routine --
316* -- LAPACK is a software package provided by Univ. of Tennessee, --
317* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
318*
319* .. Scalar Arguments ..
320 CHARACTER COMPQ, JOB
321 INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
322 REAL S, SEP
323* ..
324* .. Array Arguments ..
325 LOGICAL SELECT( * )
326 INTEGER IWORK( * )
327 REAL Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
328 $ WR( * )
329* ..
330*
331* =====================================================================
332*
333* .. Parameters ..
334 REAL ZERO, ONE
335 parameter( zero = 0.0e+0, one = 1.0e+0 )
336* ..
337* .. Local Scalars ..
338 LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
339 $ WANTSP
340 INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
341 $ NN
342 REAL EST, RNORM, SCALE
343* ..
344* .. Local Arrays ..
345 INTEGER ISAVE( 3 )
346* ..
347* .. External Functions ..
348 LOGICAL LSAME
349 REAL SLANGE
350 EXTERNAL lsame, slange
351* ..
352* .. External Subroutines ..
353 EXTERNAL slacn2, slacpy, strexc, strsyl, xerbla
354* ..
355* .. Intrinsic Functions ..
356 INTRINSIC abs, max, sqrt
357* ..
358* .. Executable Statements ..
359*
360* Decode and test the input parameters
361*
362 wantbh = lsame( job, 'B' )
363 wants = lsame( job, 'E' ) .OR. wantbh
364 wantsp = lsame( job, 'V' ) .OR. wantbh
365 wantq = lsame( compq, 'V' )
366*
367 info = 0
368 lquery = ( lwork.EQ.-1 )
369 IF( .NOT.lsame( job, 'N' ) .AND. .NOT.wants .AND. .NOT.wantsp )
370 $ THEN
371 info = -1
372 ELSE IF( .NOT.lsame( compq, 'N' ) .AND. .NOT.wantq ) THEN
373 info = -2
374 ELSE IF( n.LT.0 ) THEN
375 info = -4
376 ELSE IF( ldt.LT.max( 1, n ) ) THEN
377 info = -6
378 ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
379 info = -8
380 ELSE
381*
382* Set M to the dimension of the specified invariant subspace,
383* and test LWORK and LIWORK.
384*
385 m = 0
386 pair = .false.
387 DO 10 k = 1, n
388 IF( pair ) THEN
389 pair = .false.
390 ELSE
391 IF( k.LT.n ) THEN
392 IF( t( k+1, k ).EQ.zero ) THEN
393 IF( SELECT( k ) )
394 $ m = m + 1
395 ELSE
396 pair = .true.
397 IF( SELECT( k ) .OR. SELECT( k+1 ) )
398 $ m = m + 2
399 END IF
400 ELSE
401 IF( SELECT( n ) )
402 $ m = m + 1
403 END IF
404 END IF
405 10 CONTINUE
406*
407 n1 = m
408 n2 = n - m
409 nn = n1*n2
410*
411 IF( wantsp ) THEN
412 lwmin = max( 1, 2*nn )
413 liwmin = max( 1, nn )
414 ELSE IF( lsame( job, 'N' ) ) THEN
415 lwmin = max( 1, n )
416 liwmin = 1
417 ELSE IF( lsame( job, 'E' ) ) THEN
418 lwmin = max( 1, nn )
419 liwmin = 1
420 END IF
421*
422 IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
423 info = -15
424 ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
425 info = -17
426 END IF
427 END IF
428*
429 IF( info.EQ.0 ) THEN
430 work( 1 ) = lwmin
431 iwork( 1 ) = liwmin
432 END IF
433*
434 IF( info.NE.0 ) THEN
435 CALL xerbla( 'STRSEN', -info )
436 RETURN
437 ELSE IF( lquery ) THEN
438 RETURN
439 END IF
440*
441* Quick return if possible.
442*
443 IF( m.EQ.n .OR. m.EQ.0 ) THEN
444 IF( wants )
445 $ s = one
446 IF( wantsp )
447 $ sep = slange( '1', n, n, t, ldt, work )
448 GO TO 40
449 END IF
450*
451* Collect the selected blocks at the top-left corner of T.
452*
453 ks = 0
454 pair = .false.
455 DO 20 k = 1, n
456 IF( pair ) THEN
457 pair = .false.
458 ELSE
459 swap = SELECT( k )
460 IF( k.LT.n ) THEN
461 IF( t( k+1, k ).NE.zero ) THEN
462 pair = .true.
463 swap = swap .OR. SELECT( k+1 )
464 END IF
465 END IF
466 IF( swap ) THEN
467 ks = ks + 1
468*
469* Swap the K-th block to position KS.
470*
471 ierr = 0
472 kk = k
473 IF( k.NE.ks )
474 $ CALL strexc( compq, n, t, ldt, q, ldq, kk, ks, work,
475 $ ierr )
476 IF( ierr.EQ.1 .OR. ierr.EQ.2 ) THEN
477*
478* Blocks too close to swap: exit.
479*
480 info = 1
481 IF( wants )
482 $ s = zero
483 IF( wantsp )
484 $ sep = zero
485 GO TO 40
486 END IF
487 IF( pair )
488 $ ks = ks + 1
489 END IF
490 END IF
491 20 CONTINUE
492*
493 IF( wants ) THEN
494*
495* Solve Sylvester equation for R:
496*
497* T11*R - R*T22 = scale*T12
498*
499 CALL slacpy( 'F', n1, n2, t( 1, n1+1 ), ldt, work, n1 )
500 CALL strsyl( 'N', 'N', -1, n1, n2, t, ldt, t( n1+1, n1+1 ),
501 $ ldt, work, n1, scale, ierr )
502*
503* Estimate the reciprocal of the condition number of the cluster
504* of eigenvalues.
505*
506 rnorm = slange( 'F', n1, n2, work, n1, work )
507 IF( rnorm.EQ.zero ) THEN
508 s = one
509 ELSE
510 s = scale / ( sqrt( scale*scale / rnorm+rnorm )*
511 $ sqrt( rnorm ) )
512 END IF
513 END IF
514*
515 IF( wantsp ) THEN
516*
517* Estimate sep(T11,T22).
518*
519 est = zero
520 kase = 0
521 30 CONTINUE
522 CALL slacn2( nn, work( nn+1 ), work, iwork, est, kase, isave )
523 IF( kase.NE.0 ) THEN
524 IF( kase.EQ.1 ) THEN
525*
526* Solve T11*R - R*T22 = scale*X.
527*
528 CALL strsyl( 'N', 'N', -1, n1, n2, t, ldt,
529 $ t( n1+1, n1+1 ), ldt, work, n1, scale,
530 $ ierr )
531 ELSE
532*
533* Solve T11**T*R - R*T22**T = scale*X.
534*
535 CALL strsyl( 'T', 'T', -1, n1, n2, t, ldt,
536 $ t( n1+1, n1+1 ), ldt, work, n1, scale,
537 $ ierr )
538 END IF
539 GO TO 30
540 END IF
541*
542 sep = scale / est
543 END IF
544*
545 40 CONTINUE
546*
547* Store the output eigenvalues in WR and WI.
548*
549 DO 50 k = 1, n
550 wr( k ) = t( k, k )
551 wi( k ) = zero
552 50 CONTINUE
553 DO 60 k = 1, n - 1
554 IF( t( k+1, k ).NE.zero ) THEN
555 wi( k ) = sqrt( abs( t( k, k+1 ) ) )*
556 $ sqrt( abs( t( k+1, k ) ) )
557 wi( k+1 ) = -wi( k )
558 END IF
559 60 CONTINUE
560*
561 work( 1 ) = lwmin
562 iwork( 1 ) = liwmin
563*
564 RETURN
565*
566* End of STRSEN
567*
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
real function slange(NORM, M, N, A, LDA, WORK)
SLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: slange.f:114
subroutine slacn2(N, V, X, ISGN, EST, KASE, ISAVE)
SLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: slacn2.f:136
subroutine strexc(COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK, INFO)
STREXC
Definition: strexc.f:148
subroutine strsyl(TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO)
STRSYL
Definition: strsyl.f:164
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