*> \brief \b SHSEQR
*
* =========== DOCUMENTATION ===========
*
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* http://www.netlib.org/lapack/explore-html/
*
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*
* Definition:
* ===========
*
* SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
* LDZ, WORK, LWORK, INFO )
*
* .. Scalar Arguments ..
* INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
* CHARACTER COMPZ, JOB
* ..
* .. Array Arguments ..
* REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ),
* $ Z( LDZ, * )
* ..
*
*
*> \par Purpose:
* =============
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
* Arguments:
* ==========
*
*> \param[in] JOB
*> \verbatim
*> JOB is CHARACTER*1
*> = 'E': compute eigenvalues only;
*> = 'S': compute eigenvalues and the Schur form T.
*> \endverbatim
*>
*> \param[in] COMPZ
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] N
*> \verbatim
*> N is INTEGER
*> The order of the matrix H. N .GE. 0.
*> \endverbatim
*>
*> \param[in] ILO
*> \verbatim
*> ILO is INTEGER
*> \endverbatim
*>
*> \param[in] IHI
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in,out] H
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[in] LDH
*> \verbatim
*> LDH is INTEGER
*> The leading dimension of the array H. LDH .GE. max(1,N).
*> \endverbatim
*>
*> \param[out] WR
*> \verbatim
*> WR is REAL array, dimension (N)
*> \endverbatim
*>
*> \param[out] WI
*> \verbatim
*> 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).
*> \endverbatim
*>
*> \param[in,out] Z
*> \verbatim
*> 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.)
*> \endverbatim
*>
*> \param[in] LDZ
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[out] WORK
*> \verbatim
*> WORK is REAL array, dimension (LWORK)
*> On exit, if INFO = 0, WORK(1) returns an estimate of
*> the optimal value for LWORK.
*> \endverbatim
*>
*> \param[in] LWORK
*> \verbatim
*> 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.
*> \endverbatim
*>
*> \param[out] INFO
*> \verbatim
*> 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.
*> \endverbatim
*
* Authors:
* ========
*
*> \author Univ. of Tennessee
*> \author Univ. of California Berkeley
*> \author Univ. of Colorado Denver
*> \author NAG Ltd.
*
*> \date November 2011
*
*> \ingroup realOTHERcomputational
*
*> \par Contributors:
* ==================
*>
*> Karen Braman and Ralph Byers, Department of Mathematics,
*> University of Kansas, USA
*
*> \par Further Details:
* =====================
*>
*> \verbatim
*>
*> 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.
*> \endverbatim
*
*> \par 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.
*> \n
*> 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.
*
* =====================================================================
SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
$ LDZ, WORK, LWORK, INFO )
*
* -- LAPACK computational routine (version 3.4.0) --
* -- LAPACK is a software package provided by Univ. of Tennessee, --
* -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
* November 2011
*
* .. Scalar Arguments ..
INTEGER IHI, ILO, INFO, LDH, LDZ, LWORK, N
CHARACTER COMPZ, JOB
* ..
* .. Array Arguments ..
REAL H( LDH, * ), WI( * ), WORK( * ), WR( * ),
$ Z( LDZ, * )
* ..
*
* =====================================================================
*
* .. Parameters ..
*
* ==== Matrices of order NTINY or smaller must be processed by
* . SLAHQR because of insufficient subdiagonal scratch space.
* . (This is a hard limit.) ====
INTEGER NTINY
PARAMETER ( NTINY = 11 )
*
* ==== NL allocates some local workspace to help small matrices
* . through a rare SLAHQR failure. NL .GT. NTINY = 11 is
* . required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
* . mended. (The default value of NMIN is 75.) Using NL = 49
* . allows up to six simultaneous shifts and a 16-by-16
* . deflation window. ====
INTEGER NL
PARAMETER ( NL = 49 )
REAL ZERO, ONE
PARAMETER ( ZERO = 0.0e0, ONE = 1.0e0 )
* ..
* .. Local Arrays ..
REAL HL( NL, NL ), WORKL( NL )
* ..
* .. Local Scalars ..
INTEGER I, KBOT, NMIN
LOGICAL INITZ, LQUERY, WANTT, WANTZ
* ..
* .. External Functions ..
INTEGER ILAENV
LOGICAL LSAME
EXTERNAL ILAENV, LSAME
* ..
* .. External Subroutines ..
EXTERNAL SLACPY, SLAHQR, SLAQR0, SLASET, XERBLA
* ..
* .. Intrinsic Functions ..
INTRINSIC MAX, MIN, REAL
* ..
* .. Executable Statements ..
*
* ==== Decode and check the input parameters. ====
*
WANTT = LSAME( JOB, 'S' )
INITZ = LSAME( COMPZ, 'I' )
WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
WORK( 1 ) = REAL( MAX( 1, N ) )
LQUERY = LWORK.EQ.-1
*
INFO = 0
IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
INFO = -1
ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
INFO = -2
ELSE IF( N.LT.0 ) THEN
INFO = -3
ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
INFO = -4
ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
INFO = -5
ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
INFO = -7
ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
INFO = -11
ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
INFO = -13
END IF
*
IF( INFO.NE.0 ) THEN
*
* ==== Quick return in case of invalid argument. ====
*
CALL XERBLA( 'SHSEQR', -INFO )
RETURN
*
ELSE IF( N.EQ.0 ) THEN
*
* ==== Quick return in case N = 0; nothing to do. ====
*
RETURN
*
ELSE IF( LQUERY ) THEN
*
* ==== Quick return in case of a workspace query ====
*
CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
$ IHI, Z, LDZ, WORK, LWORK, INFO )
* ==== Ensure reported workspace size is backward-compatible with
* . previous LAPACK versions. ====
WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) )
RETURN
*
ELSE
*
* ==== copy eigenvalues isolated by SGEBAL ====
*
DO 10 I = 1, ILO - 1
WR( I ) = H( I, I )
WI( I ) = ZERO
10 CONTINUE
DO 20 I = IHI + 1, N
WR( I ) = H( I, I )
WI( I ) = ZERO
20 CONTINUE
*
* ==== Initialize Z, if requested ====
*
IF( INITZ )
$ CALL SLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
*
* ==== Quick return if possible ====
*
IF( ILO.EQ.IHI ) THEN
WR( ILO ) = H( ILO, ILO )
WI( ILO ) = ZERO
RETURN
END IF
*
* ==== SLAHQR/SLAQR0 crossover point ====
*
NMIN = ILAENV( 12, 'SHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
$ ILO, IHI, LWORK )
NMIN = MAX( NTINY, NMIN )
*
* ==== SLAQR0 for big matrices; SLAHQR for small ones ====
*
IF( N.GT.NMIN ) THEN
CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
$ IHI, Z, LDZ, WORK, LWORK, INFO )
ELSE
*
* ==== Small matrix ====
*
CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
$ IHI, Z, LDZ, INFO )
*
IF( INFO.GT.0 ) THEN
*
* ==== A rare SLAHQR failure! SLAQR0 sometimes succeeds
* . when SLAHQR fails. ====
*
KBOT = INFO
*
IF( N.GE.NL ) THEN
*
* ==== Larger matrices have enough subdiagonal scratch
* . space to call SLAQR0 directly. ====
*
CALL SLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, WR,
$ WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
*
ELSE
*
* ==== Tiny matrices don't have enough subdiagonal
* . scratch space to benefit from SLAQR0. Hence,
* . tiny matrices must be copied into a larger
* . array before calling SLAQR0. ====
*
CALL SLACPY( 'A', N, N, H, LDH, HL, NL )
HL( N+1, N ) = ZERO
CALL SLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
$ NL )
CALL SLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR,
$ WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
IF( WANTT .OR. INFO.NE.0 )
$ CALL SLACPY( 'A', N, N, HL, NL, H, LDH )
END IF
END IF
END IF
*
* ==== Clear out the trash, if necessary. ====
*
IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
$ CALL SLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
*
* ==== Ensure reported workspace size is backward-compatible with
* . previous LAPACK versions. ====
*
WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) )
END IF
*
* ==== End of SHSEQR ====
*
END