 LAPACK  3.10.1 LAPACK: Linear Algebra PACKage

## ◆ chseqr()

 subroutine chseqr ( character JOB, character COMPZ, integer N, integer ILO, integer IHI, complex, dimension( ldh, * ) H, integer LDH, complex, dimension( * ) W, complex, dimension( ldz, * ) Z, integer LDZ, complex, dimension( * ) WORK, integer LWORK, integer INFO )

CHSEQR

Purpose:
```    CHSEQR computes the eigenvalues of a Hessenberg matrix H
and, optionally, the matrices T and Z from the Schur decomposition
H = Z T Z**H, where T is an upper triangular matrix (the
Schur form), and Z is the unitary matrix of Schur vectors.

Optionally Z may be postmultiplied into an input unitary
matrix Q so that this routine can give the Schur factorization
of a matrix A which has been reduced to the Hessenberg form H
by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*T*(QZ)**H.```
Parameters
 [in] JOB ``` JOB is CHARACTER*1 = 'E': compute eigenvalues only; = 'S': compute eigenvalues and the Schur form T.``` [in] COMPZ ``` COMPZ is CHARACTER*1 = 'N': no Schur vectors are computed; = 'I': Z is initialized to the unit matrix and the matrix Z of Schur vectors of H is returned; = 'V': Z must contain an unitary matrix Q on entry, and the product Q*Z is returned.``` [in] N ``` N is INTEGER The order of the matrix H. N >= 0.``` [in] ILO ` ILO is INTEGER` [in] IHI ``` IHI is INTEGER It is assumed that H is already upper triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally set by a previous call to CGEBAL, and then passed to ZGEHRD when the matrix output by CGEBAL is reduced to Hessenberg form. Otherwise ILO and IHI should be set to 1 and N respectively. If N > 0, then 1 <= ILO <= IHI <= N. If N = 0, then ILO = 1 and IHI = 0.``` [in,out] H ``` H is COMPLEX array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and JOB = 'S', H contains the upper triangular matrix T from the Schur decomposition (the Schur form). If INFO = 0 and JOB = 'E', the contents of H are unspecified on exit. (The output value of H when INFO > 0 is given under the description of INFO below.) Unlike earlier versions of CHSEQR, this subroutine may explicitly H(i,j) = 0 for i > j and j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.``` [in] LDH ``` LDH is INTEGER The leading dimension of the array H. LDH >= max(1,N).``` [out] W ``` W is COMPLEX array, dimension (N) The computed eigenvalues. If JOB = 'S', the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with W(i) = H(i,i).``` [in,out] Z ``` Z is COMPLEX array, dimension (LDZ,N) If COMPZ = 'N', Z is not referenced. If COMPZ = 'I', on entry Z need not be set and on exit, if INFO = 0, Z contains the unitary matrix Z of the Schur vectors of H. If COMPZ = 'V', on entry Z must contain an N-by-N matrix Q, which is assumed to be equal to the unit matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit, if INFO = 0, Z contains Q*Z. Normally Q is the unitary matrix generated by CUNGHR after the call to CGEHRD which formed the Hessenberg matrix H. (The output value of Z when INFO > 0 is given under the description of INFO below.)``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. if COMPZ = 'I' or COMPZ = 'V', then LDZ >= MAX(1,N). Otherwise, LDZ >= 1.``` [out] WORK ``` WORK is COMPLEX array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns an estimate of the optimal value for LWORK.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,N) is sufficient and delivers very good and sometimes optimal performance. However, LWORK as large as 11*N may be required for optimal performance. A workspace query is recommended to determine the optimal workspace size. If LWORK = -1, then CHSEQR does a workspace query. In this case, CHSEQR checks the input parameters and estimates the optimal workspace size for the given values of N, ILO and IHI. The estimate is returned in WORK(1). No error message related to LWORK is issued by XERBLA. Neither H nor Z are accessed.``` [out] INFO ``` INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value > 0: if INFO = i, CHSEQR failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of W contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO > 0 and JOB = 'E', then on exit, the remaining unconverged eigenvalues are the eigen- values of the upper Hessenberg matrix rows and columns ILO through INFO of the final, output value of H. If INFO > 0 and JOB = 'S', then on exit (*) (initial value of H)*U = U*(final value of H) where U is a unitary matrix. The final value of H is upper Hessenberg and triangular in rows and columns INFO+1 through IHI. If INFO > 0 and COMPZ = 'V', then on exit (final value of Z) = (initial value of Z)*U where U is the unitary matrix in (*) (regard- less of the value of JOB.) If INFO > 0 and COMPZ = 'I', then on exit (final value of Z) = U where U is the unitary matrix in (*) (regard- less of the value of JOB.) If INFO > 0 and COMPZ = 'N', then Z is not accessed.```
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
Further Details:
```             Default values supplied by
ILAENV(ISPEC,'CHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
It is suggested that these defaults be adjusted in order
to attain best performance in each particular
computational environment.

ISPEC=12: The CLAHQR vs CLAQR0 crossover point.
Default: 75. (Must be at least 11.)

ISPEC=13: Recommended deflation window size.
This depends on ILO, IHI and NS.  NS is the
number of simultaneous shifts returned
by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
The default for (IHI-ILO+1) <= 500 is NS.
The default for (IHI-ILO+1) >  500 is 3*NS/2.

ISPEC=14: Nibble crossover point. (See IPARMQ for
details.)  Default: 14% of deflation window
size.

ISPEC=15: Number of simultaneous shifts in a multishift
QR iteration.

If IHI-ILO+1 is ...

greater than      ...but less    ... the
or equal to ...      than        default is

1               30          NS =   2(+)
30               60          NS =   4(+)
60              150          NS =  10(+)
150              590          NS =  **
590             3000          NS =  64
3000             6000          NS = 128
6000             infinity      NS = 256

(+)  By default some or all matrices of this order
are passed to the implicit double shift routine
CLAHQR and this parameter is ignored.  See
ISPEC=12 above and comments in IPARMQ for
details.

(**)  The asterisks (**) indicate an ad-hoc
function of N increasing from 10 to 64.

ISPEC=16: Select structured matrix multiply.
If the number of simultaneous shifts (specified
by ISPEC=15) is less than 14, then the default
for ISPEC=16 is 0.  Otherwise the default for
ISPEC=16 is 2.```
References:
```  K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
Performance, SIAM Journal of Matrix Analysis, volume 23, pages
929--947, 2002.
```

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948–973, 2002.

Definition at line 297 of file chseqr.f.

299 *
300 * -- LAPACK computational routine --
301 * -- LAPACK is a software package provided by Univ. of Tennessee, --
302 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
303 *
304 * .. Scalar Arguments ..
305  INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
306  CHARACTER COMPZ, JOB
307 * ..
308 * .. Array Arguments ..
309  COMPLEX H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
310 * ..
311 *
312 * =====================================================================
313 *
314 * .. Parameters ..
315 *
316 * ==== Matrices of order NTINY or smaller must be processed by
317 * . CLAHQR because of insufficient subdiagonal scratch space.
318 * . (This is a hard limit.) ====
319  INTEGER NTINY
320  parameter( ntiny = 15 )
321 *
322 * ==== NL allocates some local workspace to help small matrices
323 * . through a rare CLAHQR failure. NL > NTINY = 15 is
324 * . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
325 * . mended. (The default value of NMIN is 75.) Using NL = 49
326 * . allows up to six simultaneous shifts and a 16-by-16
327 * . deflation window. ====
328  INTEGER NL
329  parameter( nl = 49 )
330  COMPLEX ZERO, ONE
331  parameter( zero = ( 0.0e0, 0.0e0 ),
332  \$ one = ( 1.0e0, 0.0e0 ) )
333  REAL RZERO
334  parameter( rzero = 0.0e0 )
335 * ..
336 * .. Local Arrays ..
337  COMPLEX HL( NL, NL ), WORKL( NL )
338 * ..
339 * .. Local Scalars ..
340  INTEGER KBOT, NMIN
341  LOGICAL INITZ, LQUERY, WANTT, WANTZ
342 * ..
343 * .. External Functions ..
344  INTEGER ILAENV
345  LOGICAL LSAME
346  EXTERNAL ilaenv, lsame
347 * ..
348 * .. External Subroutines ..
349  EXTERNAL ccopy, clacpy, clahqr, claqr0, claset, xerbla
350 * ..
351 * .. Intrinsic Functions ..
352  INTRINSIC cmplx, max, min, real
353 * ..
354 * .. Executable Statements ..
355 *
356 * ==== Decode and check the input parameters. ====
357 *
358  wantt = lsame( job, 'S' )
359  initz = lsame( compz, 'I' )
360  wantz = initz .OR. lsame( compz, 'V' )
361  work( 1 ) = cmplx( real( max( 1, n ) ), rzero )
362  lquery = lwork.EQ.-1
363 *
364  info = 0
365  IF( .NOT.lsame( job, 'E' ) .AND. .NOT.wantt ) THEN
366  info = -1
367  ELSE IF( .NOT.lsame( compz, 'N' ) .AND. .NOT.wantz ) THEN
368  info = -2
369  ELSE IF( n.LT.0 ) THEN
370  info = -3
371  ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
372  info = -4
373  ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
374  info = -5
375  ELSE IF( ldh.LT.max( 1, n ) ) THEN
376  info = -7
377  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.max( 1, n ) ) ) THEN
378  info = -10
379  ELSE IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
380  info = -12
381  END IF
382 *
383  IF( info.NE.0 ) THEN
384 *
385 * ==== Quick return in case of invalid argument. ====
386 *
387  CALL xerbla( 'CHSEQR', -info )
388  RETURN
389 *
390  ELSE IF( n.EQ.0 ) THEN
391 *
392 * ==== Quick return in case N = 0; nothing to do. ====
393 *
394  RETURN
395 *
396  ELSE IF( lquery ) THEN
397 *
398 * ==== Quick return in case of a workspace query ====
399 *
400  CALL claqr0( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi, z,
401  \$ ldz, work, lwork, info )
402 * ==== Ensure reported workspace size is backward-compatible with
403 * . previous LAPACK versions. ====
404  work( 1 ) = cmplx( max( real( work( 1 ) ), real( max( 1,
405  \$ n ) ) ), rzero )
406  RETURN
407 *
408  ELSE
409 *
410 * ==== copy eigenvalues isolated by CGEBAL ====
411 *
412  IF( ilo.GT.1 )
413  \$ CALL ccopy( ilo-1, h, ldh+1, w, 1 )
414  IF( ihi.LT.n )
415  \$ CALL ccopy( n-ihi, h( ihi+1, ihi+1 ), ldh+1, w( ihi+1 ), 1 )
416 *
417 * ==== Initialize Z, if requested ====
418 *
419  IF( initz )
420  \$ CALL claset( 'A', n, n, zero, one, z, ldz )
421 *
422 * ==== Quick return if possible ====
423 *
424  IF( ilo.EQ.ihi ) THEN
425  w( ilo ) = h( ilo, ilo )
426  RETURN
427  END IF
428 *
429 * ==== CLAHQR/CLAQR0 crossover point ====
430 *
431  nmin = ilaenv( 12, 'CHSEQR', job( : 1 ) // compz( : 1 ), n,
432  \$ ilo, ihi, lwork )
433  nmin = max( ntiny, nmin )
434 *
435 * ==== CLAQR0 for big matrices; CLAHQR for small ones ====
436 *
437  IF( n.GT.nmin ) THEN
438  CALL claqr0( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi,
439  \$ z, ldz, work, lwork, info )
440  ELSE
441 *
442 * ==== Small matrix ====
443 *
444  CALL clahqr( wantt, wantz, n, ilo, ihi, h, ldh, w, ilo, ihi,
445  \$ z, ldz, info )
446 *
447  IF( info.GT.0 ) THEN
448 *
449 * ==== A rare CLAHQR failure! CLAQR0 sometimes succeeds
450 * . when CLAHQR fails. ====
451 *
452  kbot = info
453 *
454  IF( n.GE.nl ) THEN
455 *
456 * ==== Larger matrices have enough subdiagonal scratch
457 * . space to call CLAQR0 directly. ====
458 *
459  CALL claqr0( wantt, wantz, n, ilo, kbot, h, ldh, w,
460  \$ ilo, ihi, z, ldz, work, lwork, info )
461 *
462  ELSE
463 *
464 * ==== Tiny matrices don't have enough subdiagonal
465 * . scratch space to benefit from CLAQR0. Hence,
466 * . tiny matrices must be copied into a larger
467 * . array before calling CLAQR0. ====
468 *
469  CALL clacpy( 'A', n, n, h, ldh, hl, nl )
470  hl( n+1, n ) = zero
471  CALL claset( 'A', nl, nl-n, zero, zero, hl( 1, n+1 ),
472  \$ nl )
473  CALL claqr0( wantt, wantz, nl, ilo, kbot, hl, nl, w,
474  \$ ilo, ihi, z, ldz, workl, nl, info )
475  IF( wantt .OR. info.NE.0 )
476  \$ CALL clacpy( 'A', n, n, hl, nl, h, ldh )
477  END IF
478  END IF
479  END IF
480 *
481 * ==== Clear out the trash, if necessary. ====
482 *
483  IF( ( wantt .OR. info.NE.0 ) .AND. n.GT.2 )
484  \$ CALL claset( 'L', n-2, n-2, zero, zero, h( 3, 1 ), ldh )
485 *
486 * ==== Ensure reported workspace size is backward-compatible with
487 * . previous LAPACK versions. ====
488 *
489  work( 1 ) = cmplx( max( real( max( 1, n ) ),
490  \$ real( work( 1 ) ) ), rzero )
491  END IF
492 *
493 * ==== End of CHSEQR ====
494 *
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
ILAENV
Definition: ilaenv.f:162
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:60
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:53
subroutine ccopy(N, CX, INCX, CY, INCY)
CCOPY
Definition: ccopy.f:81
subroutine claset(UPLO, M, N, ALPHA, BETA, A, LDA)
CLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values.
Definition: claset.f:106
subroutine claqr0(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
CLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur de...
Definition: claqr0.f:240
subroutine clahqr(WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILOZ, IHIZ, Z, LDZ, INFO)
CLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix,...
Definition: clahqr.f:195
subroutine clacpy(UPLO, M, N, A, LDA, B, LDB)
CLACPY copies all or part of one two-dimensional array to another.
Definition: clacpy.f:103
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