LAPACK  3.6.1 LAPACK: Linear Algebra PACKage
 subroutine shseqr ( character JOB, character COMPZ, integer N, integer ILO, integer IHI, real, dimension( ldh, * ) H, integer LDH, real, dimension( * ) WR, real, dimension( * ) WI, real, dimension( ldz, * ) Z, integer LDZ, real, dimension( * ) WORK, integer LWORK, integer INFO )

SHSEQR

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

Optionally Z may be postmultiplied into an input orthogonal
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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.```
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 orthogonal matrix Q on entry, and the product Q*Z is returned.``` [in] N ``` N is INTEGER The order of the matrix H. N .GE. 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 SGEBAL, and then passed to ZGEHRD when the matrix output by SGEBAL is reduced to Hessenberg form. Otherwise ILO and IHI should be set to 1 and N respectively. If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.``` [in,out] H ``` H is REAL array, dimension (LDH,N) On entry, the upper Hessenberg matrix H. On exit, if INFO = 0 and JOB = 'S', then H contains the upper quasi-triangular matrix T from the Schur decomposition (the Schur form); 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of eigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the contents of H are unspecified on exit. (The output value of H when INFO.GT.0 is given under the description of INFO below.) Unlike earlier versions of SHSEQR, this subroutine may explicitly H(i,j) = 0 for i.GT.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 .GE. max(1,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 computed eigenvalues. If two eigenvalues are computed as a complex conjugate pair, they are stored in consecutive elements of WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in the same order as on the diagonal of the Schur form returned in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and WI(i+1) = -WI(i).``` [in,out] Z ``` Z is REAL 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 orthogonal 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 orthogonal matrix generated by SORGHR after the call to SGEHRD which formed the Hessenberg matrix H. (The output value of Z when INFO.GT.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.GE.MAX(1,N). Otherwize, LDZ.GE.1.``` [out] WORK ``` WORK is REAL 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 .GE. 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 SHSEQR does a workspace query. In this case, SHSEQR 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 .LT. 0: if INFO = -i, the i-th argument had an illegal value .GT. 0: if INFO = i, SHSEQR failed to compute all of the eigenvalues. Elements 1:ilo-1 and i+1:n of WR and WI contain those eigenvalues which have been successfully computed. (Failures are rare.) If INFO .GT. 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 .GT. 0 and JOB = 'S', then on exit (*) (initial value of H)*U = U*(final value of H) where U is an orthogonal matrix. The final value of H is upper Hessenberg and quasi-triangular in rows and columns INFO+1 through IHI. If INFO .GT. 0 and COMPZ = 'V', then on exit (final value of Z) = (initial value of Z)*U where U is the orthogonal matrix in (*) (regard- less of the value of JOB.) If INFO .GT. 0 and COMPZ = 'I', then on exit (final value of Z) = U where U is the orthogonal matrix in (*) (regard- less of the value of JOB.) If INFO .GT. 0 and COMPZ = 'N', then Z is not accessed.```
Date
November 2011
Contributors:
Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA
Further Details:
```             Default values supplied by
ILAENV(ISPEC,'SHSEQR',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 SLAHQR vs SLAQR0 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).LE.500 is NS.
The default for (IHI-ILO+1).GT.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
SLAHQR 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 318 of file shseqr.f.

318 *
319 * -- LAPACK computational routine (version 3.4.0) --
320 * -- LAPACK is a software package provided by Univ. of Tennessee, --
321 * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
322 * November 2011
323 *
324 * .. Scalar Arguments ..
325  INTEGER ihi, ilo, info, ldh, ldz, lwork, n
326  CHARACTER compz, job
327 * ..
328 * .. Array Arguments ..
329  REAL h( ldh, * ), wi( * ), work( * ), wr( * ),
330  \$ z( ldz, * )
331 * ..
332 *
333 * =====================================================================
334 *
335 * .. Parameters ..
336 *
337 * ==== Matrices of order NTINY or smaller must be processed by
338 * . SLAHQR because of insufficient subdiagonal scratch space.
339 * . (This is a hard limit.) ====
340  INTEGER ntiny
341  parameter ( ntiny = 11 )
342 *
343 * ==== NL allocates some local workspace to help small matrices
344 * . through a rare SLAHQR failure. NL .GT. NTINY = 11 is
345 * . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
346 * . mended. (The default value of NMIN is 75.) Using NL = 49
347 * . allows up to six simultaneous shifts and a 16-by-16
348 * . deflation window. ====
349  INTEGER nl
350  parameter ( nl = 49 )
351  REAL zero, one
352  parameter ( zero = 0.0e0, one = 1.0e0 )
353 * ..
354 * .. Local Arrays ..
355  REAL hl( nl, nl ), workl( nl )
356 * ..
357 * .. Local Scalars ..
358  INTEGER i, kbot, nmin
359  LOGICAL initz, lquery, wantt, wantz
360 * ..
361 * .. External Functions ..
362  INTEGER ilaenv
363  LOGICAL lsame
364  EXTERNAL ilaenv, lsame
365 * ..
366 * .. External Subroutines ..
367  EXTERNAL slacpy, slahqr, slaqr0, slaset, xerbla
368 * ..
369 * .. Intrinsic Functions ..
370  INTRINSIC max, min, real
371 * ..
372 * .. Executable Statements ..
373 *
374 * ==== Decode and check the input parameters. ====
375 *
376  wantt = lsame( job, 'S' )
377  initz = lsame( compz, 'I' )
378  wantz = initz .OR. lsame( compz, 'V' )
379  work( 1 ) = REAL( MAX( 1, N ) )
380  lquery = lwork.EQ.-1
381 *
382  info = 0
383  IF( .NOT.lsame( job, 'E' ) .AND. .NOT.wantt ) THEN
384  info = -1
385  ELSE IF( .NOT.lsame( compz, 'N' ) .AND. .NOT.wantz ) THEN
386  info = -2
387  ELSE IF( n.LT.0 ) THEN
388  info = -3
389  ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
390  info = -4
391  ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
392  info = -5
393  ELSE IF( ldh.LT.max( 1, n ) ) THEN
394  info = -7
395  ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.max( 1, n ) ) ) THEN
396  info = -11
397  ELSE IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
398  info = -13
399  END IF
400 *
401  IF( info.NE.0 ) THEN
402 *
403 * ==== Quick return in case of invalid argument. ====
404 *
405  CALL xerbla( 'SHSEQR', -info )
406  RETURN
407 *
408  ELSE IF( n.EQ.0 ) THEN
409 *
410 * ==== Quick return in case N = 0; nothing to do. ====
411 *
412  RETURN
413 *
414  ELSE IF( lquery ) THEN
415 *
416 * ==== Quick return in case of a workspace query ====
417 *
418  CALL slaqr0( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, ilo,
419  \$ ihi, z, ldz, work, lwork, info )
420 * ==== Ensure reported workspace size is backward-compatible with
421 * . previous LAPACK versions. ====
422  work( 1 ) = max( REAL( MAX( 1, N ) ), work( 1 ) )
423  RETURN
424 *
425  ELSE
426 *
427 * ==== copy eigenvalues isolated by SGEBAL ====
428 *
429  DO 10 i = 1, ilo - 1
430  wr( i ) = h( i, i )
431  wi( i ) = zero
432  10 CONTINUE
433  DO 20 i = ihi + 1, n
434  wr( i ) = h( i, i )
435  wi( i ) = zero
436  20 CONTINUE
437 *
438 * ==== Initialize Z, if requested ====
439 *
440  IF( initz )
441  \$ CALL slaset( 'A', n, n, zero, one, z, ldz )
442 *
443 * ==== Quick return if possible ====
444 *
445  IF( ilo.EQ.ihi ) THEN
446  wr( ilo ) = h( ilo, ilo )
447  wi( ilo ) = zero
448  RETURN
449  END IF
450 *
451 * ==== SLAHQR/SLAQR0 crossover point ====
452 *
453  nmin = ilaenv( 12, 'SHSEQR', job( : 1 ) // compz( : 1 ), n,
454  \$ ilo, ihi, lwork )
455  nmin = max( ntiny, nmin )
456 *
457 * ==== SLAQR0 for big matrices; SLAHQR for small ones ====
458 *
459  IF( n.GT.nmin ) THEN
460  CALL slaqr0( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, ilo,
461  \$ ihi, z, ldz, work, lwork, info )
462  ELSE
463 *
464 * ==== Small matrix ====
465 *
466  CALL slahqr( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, ilo,
467  \$ ihi, z, ldz, info )
468 *
469  IF( info.GT.0 ) THEN
470 *
471 * ==== A rare SLAHQR failure! SLAQR0 sometimes succeeds
472 * . when SLAHQR fails. ====
473 *
474  kbot = info
475 *
476  IF( n.GE.nl ) THEN
477 *
478 * ==== Larger matrices have enough subdiagonal scratch
479 * . space to call SLAQR0 directly. ====
480 *
481  CALL slaqr0( wantt, wantz, n, ilo, kbot, h, ldh, wr,
482  \$ wi, ilo, ihi, z, ldz, work, lwork, info )
483 *
484  ELSE
485 *
486 * ==== Tiny matrices don't have enough subdiagonal
487 * . scratch space to benefit from SLAQR0. Hence,
488 * . tiny matrices must be copied into a larger
489 * . array before calling SLAQR0. ====
490 *
491  CALL slacpy( 'A', n, n, h, ldh, hl, nl )
492  hl( n+1, n ) = zero
493  CALL slaset( 'A', nl, nl-n, zero, zero, hl( 1, n+1 ),
494  \$ nl )
495  CALL slaqr0( wantt, wantz, nl, ilo, kbot, hl, nl, wr,
496  \$ wi, ilo, ihi, z, ldz, workl, nl, info )
497  IF( wantt .OR. info.NE.0 )
498  \$ CALL slacpy( 'A', n, n, hl, nl, h, ldh )
499  END IF
500  END IF
501  END IF
502 *
503 * ==== Clear out the trash, if necessary. ====
504 *
505  IF( ( wantt .OR. info.NE.0 ) .AND. n.GT.2 )
506  \$ CALL slaset( 'L', n-2, n-2, zero, zero, h( 3, 1 ), ldh )
507 *
508 * ==== Ensure reported workspace size is backward-compatible with
509 * . previous LAPACK versions. ====
510 *
511  work( 1 ) = max( REAL( MAX( 1, N ) ), work( 1 ) )
512  END IF
513 *
514 * ==== End of SHSEQR ====
515 *
subroutine slahqr(WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, INFO)
SLAHQR computes the eigenvalues and Schur factorization of an upper Hessenberg matrix, using the double-shift/single-shift QR algorithm.
Definition: slahqr.f:209
subroutine slaqr0(WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK, LWORK, INFO)
SLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur de...
Definition: slaqr0.f:258
subroutine xerbla(SRNAME, INFO)
XERBLA
Definition: xerbla.f:62
subroutine slacpy(UPLO, M, N, A, LDA, B, LDB)
SLACPY copies all or part of one two-dimensional array to another.
Definition: slacpy.f:105
subroutine slaset(UPLO, M, N, ALPHA, BETA, A, LDA)
SLASET initializes the off-diagonal elements and the diagonal elements of a matrix to given values...
Definition: slaset.f:112
integer function ilaenv(ISPEC, NAME, OPTS, N1, N2, N3, N4)
Definition: tstiee.f:83
logical function lsame(CA, CB)
LSAME
Definition: lsame.f:55

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