 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ dtrsen()

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

DTRSEN

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Purpose:
``` DTRSEN 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 DHSEQR), 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (N)` [out] WI ``` WI is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION 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.```
Further Details:
```  DTRSEN 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 311 of file dtrsen.f.

313 *
314 * -- LAPACK computational routine --
315 * -- LAPACK is a software package provided by Univ. of Tennessee, --
316 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
317 *
318 * .. Scalar Arguments ..
319  CHARACTER COMPQ, JOB
320  INTEGER INFO, LDQ, LDT, LIWORK, LWORK, M, N
321  DOUBLE PRECISION S, SEP
322 * ..
323 * .. Array Arguments ..
324  LOGICAL SELECT( * )
325  INTEGER IWORK( * )
326  DOUBLE PRECISION Q( LDQ, * ), T( LDT, * ), WI( * ), WORK( * ),
327  \$ WR( * )
328 * ..
329 *
330 * =====================================================================
331 *
332 * .. Parameters ..
333  DOUBLE PRECISION ZERO, ONE
334  parameter( zero = 0.0d+0, one = 1.0d+0 )
335 * ..
336 * .. Local Scalars ..
337  LOGICAL LQUERY, PAIR, SWAP, WANTBH, WANTQ, WANTS,
338  \$ WANTSP
339  INTEGER IERR, K, KASE, KK, KS, LIWMIN, LWMIN, N1, N2,
340  \$ NN
341  DOUBLE PRECISION EST, RNORM, SCALE
342 * ..
343 * .. Local Arrays ..
344  INTEGER ISAVE( 3 )
345 * ..
346 * .. External Functions ..
347  LOGICAL LSAME
348  DOUBLE PRECISION DLANGE
349  EXTERNAL lsame, dlange
350 * ..
351 * .. External Subroutines ..
352  EXTERNAL dlacn2, dlacpy, dtrexc, dtrsyl, xerbla
353 * ..
354 * .. Intrinsic Functions ..
355  INTRINSIC abs, max, sqrt
356 * ..
357 * .. Executable Statements ..
358 *
359 * Decode and test the input parameters
360 *
361  wantbh = lsame( job, 'B' )
362  wants = lsame( job, 'E' ) .OR. wantbh
363  wantsp = lsame( job, 'V' ) .OR. wantbh
364  wantq = lsame( compq, 'V' )
365 *
366  info = 0
367  lquery = ( lwork.EQ.-1 )
368  IF( .NOT.lsame( job, 'N' ) .AND. .NOT.wants .AND. .NOT.wantsp )
369  \$ THEN
370  info = -1
371  ELSE IF( .NOT.lsame( compq, 'N' ) .AND. .NOT.wantq ) THEN
372  info = -2
373  ELSE IF( n.LT.0 ) THEN
374  info = -4
375  ELSE IF( ldt.LT.max( 1, n ) ) THEN
376  info = -6
377  ELSE IF( ldq.LT.1 .OR. ( wantq .AND. ldq.LT.n ) ) THEN
378  info = -8
379  ELSE
380 *
381 * Set M to the dimension of the specified invariant subspace,
382 * and test LWORK and LIWORK.
383 *
384  m = 0
385  pair = .false.
386  DO 10 k = 1, n
387  IF( pair ) THEN
388  pair = .false.
389  ELSE
390  IF( k.LT.n ) THEN
391  IF( t( k+1, k ).EQ.zero ) THEN
392  IF( SELECT( k ) )
393  \$ m = m + 1
394  ELSE
395  pair = .true.
396  IF( SELECT( k ) .OR. SELECT( k+1 ) )
397  \$ m = m + 2
398  END IF
399  ELSE
400  IF( SELECT( n ) )
401  \$ m = m + 1
402  END IF
403  END IF
404  10 CONTINUE
405 *
406  n1 = m
407  n2 = n - m
408  nn = n1*n2
409 *
410  IF( wantsp ) THEN
411  lwmin = max( 1, 2*nn )
412  liwmin = max( 1, nn )
413  ELSE IF( lsame( job, 'N' ) ) THEN
414  lwmin = max( 1, n )
415  liwmin = 1
416  ELSE IF( lsame( job, 'E' ) ) THEN
417  lwmin = max( 1, nn )
418  liwmin = 1
419  END IF
420 *
421  IF( lwork.LT.lwmin .AND. .NOT.lquery ) THEN
422  info = -15
423  ELSE IF( liwork.LT.liwmin .AND. .NOT.lquery ) THEN
424  info = -17
425  END IF
426  END IF
427 *
428  IF( info.EQ.0 ) THEN
429  work( 1 ) = lwmin
430  iwork( 1 ) = liwmin
431  END IF
432 *
433  IF( info.NE.0 ) THEN
434  CALL xerbla( 'DTRSEN', -info )
435  RETURN
436  ELSE IF( lquery ) THEN
437  RETURN
438  END IF
439 *
440 * Quick return if possible.
441 *
442  IF( m.EQ.n .OR. m.EQ.0 ) THEN
443  IF( wants )
444  \$ s = one
445  IF( wantsp )
446  \$ sep = dlange( '1', n, n, t, ldt, work )
447  GO TO 40
448  END IF
449 *
450 * Collect the selected blocks at the top-left corner of T.
451 *
452  ks = 0
453  pair = .false.
454  DO 20 k = 1, n
455  IF( pair ) THEN
456  pair = .false.
457  ELSE
458  swap = SELECT( k )
459  IF( k.LT.n ) THEN
460  IF( t( k+1, k ).NE.zero ) THEN
461  pair = .true.
462  swap = swap .OR. SELECT( k+1 )
463  END IF
464  END IF
465  IF( swap ) THEN
466  ks = ks + 1
467 *
468 * Swap the K-th block to position KS.
469 *
470  ierr = 0
471  kk = k
472  IF( k.NE.ks )
473  \$ CALL dtrexc( compq, n, t, ldt, q, ldq, kk, ks, work,
474  \$ ierr )
475  IF( ierr.EQ.1 .OR. ierr.EQ.2 ) THEN
476 *
477 * Blocks too close to swap: exit.
478 *
479  info = 1
480  IF( wants )
481  \$ s = zero
482  IF( wantsp )
483  \$ sep = zero
484  GO TO 40
485  END IF
486  IF( pair )
487  \$ ks = ks + 1
488  END IF
489  END IF
490  20 CONTINUE
491 *
492  IF( wants ) THEN
493 *
494 * Solve Sylvester equation for R:
495 *
496 * T11*R - R*T22 = scale*T12
497 *
498  CALL dlacpy( 'F', n1, n2, t( 1, n1+1 ), ldt, work, n1 )
499  CALL dtrsyl( 'N', 'N', -1, n1, n2, t, ldt, t( n1+1, n1+1 ),
500  \$ ldt, work, n1, scale, ierr )
501 *
502 * Estimate the reciprocal of the condition number of the cluster
503 * of eigenvalues.
504 *
505  rnorm = dlange( 'F', n1, n2, work, n1, work )
506  IF( rnorm.EQ.zero ) THEN
507  s = one
508  ELSE
509  s = scale / ( sqrt( scale*scale / rnorm+rnorm )*
510  \$ sqrt( rnorm ) )
511  END IF
512  END IF
513 *
514  IF( wantsp ) THEN
515 *
516 * Estimate sep(T11,T22).
517 *
518  est = zero
519  kase = 0
520  30 CONTINUE
521  CALL dlacn2( nn, work( nn+1 ), work, iwork, est, kase, isave )
522  IF( kase.NE.0 ) THEN
523  IF( kase.EQ.1 ) THEN
524 *
525 * Solve T11*R - R*T22 = scale*X.
526 *
527  CALL dtrsyl( 'N', 'N', -1, n1, n2, t, ldt,
528  \$ t( n1+1, n1+1 ), ldt, work, n1, scale,
529  \$ ierr )
530  ELSE
531 *
532 * Solve T11**T*R - R*T22**T = scale*X.
533 *
534  CALL dtrsyl( 'T', 'T', -1, n1, n2, t, ldt,
535  \$ t( n1+1, n1+1 ), ldt, work, n1, scale,
536  \$ ierr )
537  END IF
538  GO TO 30
539  END IF
540 *
541  sep = scale / est
542  END IF
543 *
544  40 CONTINUE
545 *
546 * Store the output eigenvalues in WR and WI.
547 *
548  DO 50 k = 1, n
549  wr( k ) = t( k, k )
550  wi( k ) = zero
551  50 CONTINUE
552  DO 60 k = 1, n - 1
553  IF( t( k+1, k ).NE.zero ) THEN
554  wi( k ) = sqrt( abs( t( k, k+1 ) ) )*
555  \$ sqrt( abs( t( k+1, k ) ) )
556  wi( k+1 ) = -wi( k )
557  END IF
558  60 CONTINUE
559 *
560  work( 1 ) = lwmin
561  iwork( 1 ) = liwmin
562 *
563  RETURN
564 *
565 * End of DTRSEN
566 *
subroutine dlacpy(UPLO, M, N, A, LDA, B, LDB)
DLACPY copies all or part of one two-dimensional array to another.
Definition: dlacpy.f:103
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
double precision function dlange(NORM, M, N, A, LDA, WORK)
DLANGE returns the value of the 1-norm, Frobenius norm, infinity-norm, or the largest absolute value ...
Definition: dlange.f:114
subroutine dlacn2(N, V, X, ISGN, EST, KASE, ISAVE)
DLACN2 estimates the 1-norm of a square matrix, using reverse communication for evaluating matrix-vec...
Definition: dlacn2.f:136
subroutine dtrexc(COMPQ, N, T, LDT, Q, LDQ, IFST, ILST, WORK, INFO)
DTREXC
Definition: dtrexc.f:148
subroutine dtrsyl(TRANA, TRANB, ISGN, M, N, A, LDA, B, LDB, C, LDC, SCALE, INFO)
DTRSYL
Definition: dtrsyl.f:164
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