LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ 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

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.```
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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