LAPACK 3.12.0 LAPACK: Linear Algebra PACKage
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## ◆ shseqr()

 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 >= 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 > 0, then 1 <= ILO <= IHI <= 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) < 0. 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 SHSEQR, 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] 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) > 0 and WI(i+1) < 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 > 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 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 >= 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 < 0: if INFO = -i, the i-th argument had an illegal value > 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 > 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 an orthogonal matrix. The final value of H is upper Hessenberg and quasi-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 orthogonal 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 orthogonal 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,'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) <= 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
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 314 of file shseqr.f.

316*
317* -- LAPACK computational routine --
318* -- LAPACK is a software package provided by Univ. of Tennessee, --
319* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
320*
321* .. Scalar Arguments ..
322 INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
323 CHARACTER COMPZ, JOB
324* ..
325* .. Array Arguments ..
326 REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ),
327 \$ Z( LDZ, * )
328* ..
329*
330* =====================================================================
331*
332* .. Parameters ..
333*
334* ==== Matrices of order NTINY or smaller must be processed by
335* . SLAHQR because of insufficient subdiagonal scratch space.
336* . (This is a hard limit.) ====
337 INTEGER NTINY
338 parameter( ntiny = 15 )
339*
340* ==== NL allocates some local workspace to help small matrices
341* . through a rare SLAHQR failure. NL > NTINY = 15 is
342* . required and NL <= NMIN = ILAENV(ISPEC=12,...) is recom-
343* . mended. (The default value of NMIN is 75.) Using NL = 49
344* . allows up to six simultaneous shifts and a 16-by-16
345* . deflation window. ====
346 INTEGER NL
347 parameter( nl = 49 )
348 REAL ZERO, ONE
349 parameter( zero = 0.0e0, one = 1.0e0 )
350* ..
351* .. Local Arrays ..
352 REAL HL( NL, NL ), WORKL( NL )
353* ..
354* .. Local Scalars ..
355 INTEGER I, KBOT, NMIN
356 LOGICAL INITZ, LQUERY, WANTT, WANTZ
357* ..
358* .. External Functions ..
359 INTEGER ILAENV
360 LOGICAL LSAME
361 REAL SROUNDUP_LWORK
362 EXTERNAL ilaenv, lsame, sroundup_lwork
363* ..
364* .. External Subroutines ..
365 EXTERNAL slacpy, slahqr, slaqr0, slaset, xerbla
366* ..
367* .. Intrinsic Functions ..
368 INTRINSIC max, min, real
369* ..
370* .. Executable Statements ..
371*
372* ==== Decode and check the input parameters. ====
373*
374 wantt = lsame( job, 'S' )
375 initz = lsame( compz, 'I' )
376 wantz = initz .OR. lsame( compz, 'V' )
377 work( 1 ) = sroundup_lwork( max( 1, n ) )
378 lquery = lwork.EQ.-1
379*
380 info = 0
381 IF( .NOT.lsame( job, 'E' ) .AND. .NOT.wantt ) THEN
382 info = -1
383 ELSE IF( .NOT.lsame( compz, 'N' ) .AND. .NOT.wantz ) THEN
384 info = -2
385 ELSE IF( n.LT.0 ) THEN
386 info = -3
387 ELSE IF( ilo.LT.1 .OR. ilo.GT.max( 1, n ) ) THEN
388 info = -4
389 ELSE IF( ihi.LT.min( ilo, n ) .OR. ihi.GT.n ) THEN
390 info = -5
391 ELSE IF( ldh.LT.max( 1, n ) ) THEN
392 info = -7
393 ELSE IF( ldz.LT.1 .OR. ( wantz .AND. ldz.LT.max( 1, n ) ) ) THEN
394 info = -11
395 ELSE IF( lwork.LT.max( 1, n ) .AND. .NOT.lquery ) THEN
396 info = -13
397 END IF
398*
399 IF( info.NE.0 ) THEN
400*
401* ==== Quick return in case of invalid argument. ====
402*
403 CALL xerbla( 'SHSEQR', -info )
404 RETURN
405*
406 ELSE IF( n.EQ.0 ) THEN
407*
408* ==== Quick return in case N = 0; nothing to do. ====
409*
410 RETURN
411*
412 ELSE IF( lquery ) THEN
413*
414* ==== Quick return in case of a workspace query ====
415*
416 CALL slaqr0( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, ilo,
417 \$ ihi, z, ldz, work, lwork, info )
418* ==== Ensure reported workspace size is backward-compatible with
419* . previous LAPACK versions. ====
420 work( 1 ) = max( real( max( 1, n ) ), work( 1 ) )
421 RETURN
422*
423 ELSE
424*
425* ==== copy eigenvalues isolated by SGEBAL ====
426*
427 DO 10 i = 1, ilo - 1
428 wr( i ) = h( i, i )
429 wi( i ) = zero
430 10 CONTINUE
431 DO 20 i = ihi + 1, n
432 wr( i ) = h( i, i )
433 wi( i ) = zero
434 20 CONTINUE
435*
436* ==== Initialize Z, if requested ====
437*
438 IF( initz )
439 \$ CALL slaset( 'A', n, n, zero, one, z, ldz )
440*
441* ==== Quick return if possible ====
442*
443 IF( ilo.EQ.ihi ) THEN
444 wr( ilo ) = h( ilo, ilo )
445 wi( ilo ) = zero
446 RETURN
447 END IF
448*
449* ==== SLAHQR/SLAQR0 crossover point ====
450*
451 nmin = ilaenv( 12, 'SHSEQR', job( : 1 ) // compz( : 1 ), n,
452 \$ ilo, ihi, lwork )
453 nmin = max( ntiny, nmin )
454*
455* ==== SLAQR0 for big matrices; SLAHQR for small ones ====
456*
457 IF( n.GT.nmin ) THEN
458 CALL slaqr0( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, ilo,
459 \$ ihi, z, ldz, work, lwork, info )
460 ELSE
461*
462* ==== Small matrix ====
463*
464 CALL slahqr( wantt, wantz, n, ilo, ihi, h, ldh, wr, wi, ilo,
465 \$ ihi, z, ldz, info )
466*
467 IF( info.GT.0 ) THEN
468*
469* ==== A rare SLAHQR failure! SLAQR0 sometimes succeeds
470* . when SLAHQR fails. ====
471*
472 kbot = info
473*
474 IF( n.GE.nl ) THEN
475*
476* ==== Larger matrices have enough subdiagonal scratch
477* . space to call SLAQR0 directly. ====
478*
479 CALL slaqr0( wantt, wantz, n, ilo, kbot, h, ldh, wr,
480 \$ wi, ilo, ihi, z, ldz, work, lwork, info )
481*
482 ELSE
483*
484* ==== Tiny matrices don't have enough subdiagonal
485* . scratch space to benefit from SLAQR0. Hence,
486* . tiny matrices must be copied into a larger
487* . array before calling SLAQR0. ====
488*
489 CALL slacpy( 'A', n, n, h, ldh, hl, nl )
490 hl( n+1, n ) = zero
491 CALL slaset( 'A', nl, nl-n, zero, zero, hl( 1, n+1 ),
492 \$ nl )
493 CALL slaqr0( wantt, wantz, nl, ilo, kbot, hl, nl, wr,
494 \$ wi, ilo, ihi, z, ldz, workl, nl, info )
495 IF( wantt .OR. info.NE.0 )
496 \$ CALL slacpy( 'A', n, n, hl, nl, h, ldh )
497 END IF
498 END IF
499 END IF
500*
501* ==== Clear out the trash, if necessary. ====
502*
503 IF( ( wantt .OR. info.NE.0 ) .AND. n.GT.2 )
504 \$ CALL slaset( 'L', n-2, n-2, zero, zero, h( 3, 1 ), ldh )
505*
506* ==== Ensure reported workspace size is backward-compatible with
507* . previous LAPACK versions. ====
508*
509 work( 1 ) = max( real( max( 1, n ) ), work( 1 ) )
510 END IF
511*
512* ==== End of SHSEQR ====
513*
subroutine xerbla(srname, info)
Definition cblat2.f:3285
integer function ilaenv(ispec, name, opts, n1, n2, n3, n4)
ILAENV
Definition ilaenv.f:162
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 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,...
Definition slahqr.f:207
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:256
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:110
logical function lsame(ca, cb)
LSAME
Definition lsame.f:48
real function sroundup_lwork(lwork)
SROUNDUP_LWORK
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