LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zhseqr.f
Go to the documentation of this file.
00001 *> \brief \b ZHSEQR
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download ZHSEQR + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/zhseqr.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/zhseqr.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/zhseqr.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ,
00022 *                          WORK, LWORK, INFO )
00023 * 
00024 *       .. Scalar Arguments ..
00025 *       INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
00026 *       CHARACTER          COMPZ, JOB
00027 *       ..
00028 *       .. Array Arguments ..
00029 *       COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
00030 *       ..
00031 *  
00032 *
00033 *> \par Purpose:
00034 *  =============
00035 *>
00036 *> \verbatim
00037 *>
00038 *>    ZHSEQR computes the eigenvalues of a Hessenberg matrix H
00039 *>    and, optionally, the matrices T and Z from the Schur decomposition
00040 *>    H = Z T Z**H, where T is an upper triangular matrix (the
00041 *>    Schur form), and Z is the unitary matrix of Schur vectors.
00042 *>
00043 *>    Optionally Z may be postmultiplied into an input unitary
00044 *>    matrix Q so that this routine can give the Schur factorization
00045 *>    of a matrix A which has been reduced to the Hessenberg form H
00046 *>    by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
00047 *> \endverbatim
00048 *
00049 *  Arguments:
00050 *  ==========
00051 *
00052 *> \param[in] JOB
00053 *> \verbatim
00054 *>          JOB is CHARACTER*1
00055 *>           = 'E':  compute eigenvalues only;
00056 *>           = 'S':  compute eigenvalues and the Schur form T.
00057 *> \endverbatim
00058 *>
00059 *> \param[in] COMPZ
00060 *> \verbatim
00061 *>          COMPZ is CHARACTER*1
00062 *>           = 'N':  no Schur vectors are computed;
00063 *>           = 'I':  Z is initialized to the unit matrix and the matrix Z
00064 *>                   of Schur vectors of H is returned;
00065 *>           = 'V':  Z must contain an unitary matrix Q on entry, and
00066 *>                   the product Q*Z is returned.
00067 *> \endverbatim
00068 *>
00069 *> \param[in] N
00070 *> \verbatim
00071 *>          N is INTEGER
00072 *>           The order of the matrix H.  N .GE. 0.
00073 *> \endverbatim
00074 *>
00075 *> \param[in] ILO
00076 *> \verbatim
00077 *>          ILO is INTEGER
00078 *> \endverbatim
00079 *>
00080 *> \param[in] IHI
00081 *> \verbatim
00082 *>          IHI is INTEGER
00083 *>
00084 *>           It is assumed that H is already upper triangular in rows
00085 *>           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
00086 *>           set by a previous call to ZGEBAL, and then passed to ZGEHRD
00087 *>           when the matrix output by ZGEBAL is reduced to Hessenberg
00088 *>           form. Otherwise ILO and IHI should be set to 1 and N
00089 *>           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
00090 *>           If N = 0, then ILO = 1 and IHI = 0.
00091 *> \endverbatim
00092 *>
00093 *> \param[in,out] H
00094 *> \verbatim
00095 *>          H is COMPLEX*16 array, dimension (LDH,N)
00096 *>           On entry, the upper Hessenberg matrix H.
00097 *>           On exit, if INFO = 0 and JOB = 'S', H contains the upper
00098 *>           triangular matrix T from the Schur decomposition (the
00099 *>           Schur form). If INFO = 0 and JOB = 'E', the contents of
00100 *>           H are unspecified on exit.  (The output value of H when
00101 *>           INFO.GT.0 is given under the description of INFO below.)
00102 *>
00103 *>           Unlike earlier versions of ZHSEQR, this subroutine may
00104 *>           explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
00105 *>           or j = IHI+1, IHI+2, ... N.
00106 *> \endverbatim
00107 *>
00108 *> \param[in] LDH
00109 *> \verbatim
00110 *>          LDH is INTEGER
00111 *>           The leading dimension of the array H. LDH .GE. max(1,N).
00112 *> \endverbatim
00113 *>
00114 *> \param[out] W
00115 *> \verbatim
00116 *>          W is COMPLEX*16 array, dimension (N)
00117 *>           The computed eigenvalues. If JOB = 'S', the eigenvalues are
00118 *>           stored in the same order as on the diagonal of the Schur
00119 *>           form returned in H, with W(i) = H(i,i).
00120 *> \endverbatim
00121 *>
00122 *> \param[in,out] Z
00123 *> \verbatim
00124 *>          Z is COMPLEX*16 array, dimension (LDZ,N)
00125 *>           If COMPZ = 'N', Z is not referenced.
00126 *>           If COMPZ = 'I', on entry Z need not be set and on exit,
00127 *>           if INFO = 0, Z contains the unitary matrix Z of the Schur
00128 *>           vectors of H.  If COMPZ = 'V', on entry Z must contain an
00129 *>           N-by-N matrix Q, which is assumed to be equal to the unit
00130 *>           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
00131 *>           if INFO = 0, Z contains Q*Z.
00132 *>           Normally Q is the unitary matrix generated by ZUNGHR
00133 *>           after the call to ZGEHRD which formed the Hessenberg matrix
00134 *>           H. (The output value of Z when INFO.GT.0 is given under
00135 *>           the description of INFO below.)
00136 *> \endverbatim
00137 *>
00138 *> \param[in] LDZ
00139 *> \verbatim
00140 *>          LDZ is INTEGER
00141 *>           The leading dimension of the array Z.  if COMPZ = 'I' or
00142 *>           COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.
00143 *> \endverbatim
00144 *>
00145 *> \param[out] WORK
00146 *> \verbatim
00147 *>          WORK is COMPLEX*16 array, dimension (LWORK)
00148 *>           On exit, if INFO = 0, WORK(1) returns an estimate of
00149 *>           the optimal value for LWORK.
00150 *> \endverbatim
00151 *>
00152 *> \param[in] LWORK
00153 *> \verbatim
00154 *>          LWORK is INTEGER
00155 *>           The dimension of the array WORK.  LWORK .GE. max(1,N)
00156 *>           is sufficient and delivers very good and sometimes
00157 *>           optimal performance.  However, LWORK as large as 11*N
00158 *>           may be required for optimal performance.  A workspace
00159 *>           query is recommended to determine the optimal workspace
00160 *>           size.
00161 *>
00162 *>           If LWORK = -1, then ZHSEQR does a workspace query.
00163 *>           In this case, ZHSEQR checks the input parameters and
00164 *>           estimates the optimal workspace size for the given
00165 *>           values of N, ILO and IHI.  The estimate is returned
00166 *>           in WORK(1).  No error message related to LWORK is
00167 *>           issued by XERBLA.  Neither H nor Z are accessed.
00168 *> \endverbatim
00169 *>
00170 *> \param[out] INFO
00171 *> \verbatim
00172 *>          INFO is INTEGER
00173 *>             =  0:  successful exit
00174 *>           .LT. 0:  if INFO = -i, the i-th argument had an illegal
00175 *>                    value
00176 *>           .GT. 0:  if INFO = i, ZHSEQR failed to compute all of
00177 *>                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
00178 *>                and WI contain those eigenvalues which have been
00179 *>                successfully computed.  (Failures are rare.)
00180 *>
00181 *>                If INFO .GT. 0 and JOB = 'E', then on exit, the
00182 *>                remaining unconverged eigenvalues are the eigen-
00183 *>                values of the upper Hessenberg matrix rows and
00184 *>                columns ILO through INFO of the final, output
00185 *>                value of H.
00186 *>
00187 *>                If INFO .GT. 0 and JOB   = 'S', then on exit
00188 *>
00189 *>           (*)  (initial value of H)*U  = U*(final value of H)
00190 *>
00191 *>                where U is a unitary matrix.  The final
00192 *>                value of  H is upper Hessenberg and triangular in
00193 *>                rows and columns INFO+1 through IHI.
00194 *>
00195 *>                If INFO .GT. 0 and COMPZ = 'V', then on exit
00196 *>
00197 *>                  (final value of Z)  =  (initial value of Z)*U
00198 *>
00199 *>                where U is the unitary matrix in (*) (regard-
00200 *>                less of the value of JOB.)
00201 *>
00202 *>                If INFO .GT. 0 and COMPZ = 'I', then on exit
00203 *>                      (final value of Z)  = U
00204 *>                where U is the unitary matrix in (*) (regard-
00205 *>                less of the value of JOB.)
00206 *>
00207 *>                If INFO .GT. 0 and COMPZ = 'N', then Z is not
00208 *>                accessed.
00209 *> \endverbatim
00210 *
00211 *  Authors:
00212 *  ========
00213 *
00214 *> \author Univ. of Tennessee 
00215 *> \author Univ. of California Berkeley 
00216 *> \author Univ. of Colorado Denver 
00217 *> \author NAG Ltd. 
00218 *
00219 *> \date November 2011
00220 *
00221 *> \ingroup complex16OTHERcomputational
00222 *
00223 *> \par Contributors:
00224 *  ==================
00225 *>
00226 *>       Karen Braman and Ralph Byers, Department of Mathematics,
00227 *>       University of Kansas, USA
00228 *
00229 *> \par Further Details:
00230 *  =====================
00231 *>
00232 *> \verbatim
00233 *>
00234 *>             Default values supplied by
00235 *>             ILAENV(ISPEC,'ZHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
00236 *>             It is suggested that these defaults be adjusted in order
00237 *>             to attain best performance in each particular
00238 *>             computational environment.
00239 *>
00240 *>            ISPEC=12: The ZLAHQR vs ZLAQR0 crossover point.
00241 *>                      Default: 75. (Must be at least 11.)
00242 *>
00243 *>            ISPEC=13: Recommended deflation window size.
00244 *>                      This depends on ILO, IHI and NS.  NS is the
00245 *>                      number of simultaneous shifts returned
00246 *>                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
00247 *>                      The default for (IHI-ILO+1).LE.500 is NS.
00248 *>                      The default for (IHI-ILO+1).GT.500 is 3*NS/2.
00249 *>
00250 *>            ISPEC=14: Nibble crossover point. (See IPARMQ for
00251 *>                      details.)  Default: 14% of deflation window
00252 *>                      size.
00253 *>
00254 *>            ISPEC=15: Number of simultaneous shifts in a multishift
00255 *>                      QR iteration.
00256 *>
00257 *>                      If IHI-ILO+1 is ...
00258 *>
00259 *>                      greater than      ...but less    ... the
00260 *>                      or equal to ...      than        default is
00261 *>
00262 *>                           1               30          NS =   2(+)
00263 *>                          30               60          NS =   4(+)
00264 *>                          60              150          NS =  10(+)
00265 *>                         150              590          NS =  **
00266 *>                         590             3000          NS =  64
00267 *>                        3000             6000          NS = 128
00268 *>                        6000             infinity      NS = 256
00269 *>
00270 *>                  (+)  By default some or all matrices of this order
00271 *>                       are passed to the implicit double shift routine
00272 *>                       ZLAHQR and this parameter is ignored.  See
00273 *>                       ISPEC=12 above and comments in IPARMQ for
00274 *>                       details.
00275 *>
00276 *>                 (**)  The asterisks (**) indicate an ad-hoc
00277 *>                       function of N increasing from 10 to 64.
00278 *>
00279 *>            ISPEC=16: Select structured matrix multiply.
00280 *>                      If the number of simultaneous shifts (specified
00281 *>                      by ISPEC=15) is less than 14, then the default
00282 *>                      for ISPEC=16 is 0.  Otherwise the default for
00283 *>                      ISPEC=16 is 2.
00284 *> \endverbatim
00285 *
00286 *> \par References:
00287 *  ================
00288 *>
00289 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00290 *>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
00291 *>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
00292 *>       929--947, 2002.
00293 *> \n
00294 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00295 *>       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
00296 *>       of Matrix Analysis, volume 23, pages 948--973, 2002.
00297 *
00298 *  =====================================================================
00299       SUBROUTINE ZHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ,
00300      $                   WORK, LWORK, INFO )
00301 *
00302 *  -- LAPACK computational routine (version 3.4.0) --
00303 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00304 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00305 *     November 2011
00306 *
00307 *     .. Scalar Arguments ..
00308       INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
00309       CHARACTER          COMPZ, JOB
00310 *     ..
00311 *     .. Array Arguments ..
00312       COMPLEX*16         H( LDH, * ), W( * ), WORK( * ), Z( LDZ, * )
00313 *     ..
00314 *
00315 *  =====================================================================
00316 *
00317 *     .. Parameters ..
00318 *
00319 *     ==== Matrices of order NTINY or smaller must be processed by
00320 *     .    ZLAHQR because of insufficient subdiagonal scratch space.
00321 *     .    (This is a hard limit.) ====
00322       INTEGER            NTINY
00323       PARAMETER          ( NTINY = 11 )
00324 *
00325 *     ==== NL allocates some local workspace to help small matrices
00326 *     .    through a rare ZLAHQR failure.  NL .GT. NTINY = 11 is
00327 *     .    required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
00328 *     .    mended.  (The default value of NMIN is 75.)  Using NL = 49
00329 *     .    allows up to six simultaneous shifts and a 16-by-16
00330 *     .    deflation window.  ====
00331       INTEGER            NL
00332       PARAMETER          ( NL = 49 )
00333       COMPLEX*16         ZERO, ONE
00334       PARAMETER          ( ZERO = ( 0.0d0, 0.0d0 ),
00335      $                   ONE = ( 1.0d0, 0.0d0 ) )
00336       DOUBLE PRECISION   RZERO
00337       PARAMETER          ( RZERO = 0.0d0 )
00338 *     ..
00339 *     .. Local Arrays ..
00340       COMPLEX*16         HL( NL, NL ), WORKL( NL )
00341 *     ..
00342 *     .. Local Scalars ..
00343       INTEGER            KBOT, NMIN
00344       LOGICAL            INITZ, LQUERY, WANTT, WANTZ
00345 *     ..
00346 *     .. External Functions ..
00347       INTEGER            ILAENV
00348       LOGICAL            LSAME
00349       EXTERNAL           ILAENV, LSAME
00350 *     ..
00351 *     .. External Subroutines ..
00352       EXTERNAL           XERBLA, ZCOPY, ZLACPY, ZLAHQR, ZLAQR0, ZLASET
00353 *     ..
00354 *     .. Intrinsic Functions ..
00355       INTRINSIC          DBLE, DCMPLX, MAX, MIN
00356 *     ..
00357 *     .. Executable Statements ..
00358 *
00359 *     ==== Decode and check the input parameters. ====
00360 *
00361       WANTT = LSAME( JOB, 'S' )
00362       INITZ = LSAME( COMPZ, 'I' )
00363       WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
00364       WORK( 1 ) = DCMPLX( DBLE( MAX( 1, N ) ), RZERO )
00365       LQUERY = LWORK.EQ.-1
00366 *
00367       INFO = 0
00368       IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
00369          INFO = -1
00370       ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
00371          INFO = -2
00372       ELSE IF( N.LT.0 ) THEN
00373          INFO = -3
00374       ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
00375          INFO = -4
00376       ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
00377          INFO = -5
00378       ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
00379          INFO = -7
00380       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
00381          INFO = -10
00382       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
00383          INFO = -12
00384       END IF
00385 *
00386       IF( INFO.NE.0 ) THEN
00387 *
00388 *        ==== Quick return in case of invalid argument. ====
00389 *
00390          CALL XERBLA( 'ZHSEQR', -INFO )
00391          RETURN
00392 *
00393       ELSE IF( N.EQ.0 ) THEN
00394 *
00395 *        ==== Quick return in case N = 0; nothing to do. ====
00396 *
00397          RETURN
00398 *
00399       ELSE IF( LQUERY ) THEN
00400 *
00401 *        ==== Quick return in case of a workspace query ====
00402 *
00403          CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI, Z,
00404      $                LDZ, WORK, LWORK, INFO )
00405 *        ==== Ensure reported workspace size is backward-compatible with
00406 *        .    previous LAPACK versions. ====
00407          WORK( 1 ) = DCMPLX( MAX( DBLE( WORK( 1 ) ), DBLE( MAX( 1,
00408      $               N ) ) ), RZERO )
00409          RETURN
00410 *
00411       ELSE
00412 *
00413 *        ==== copy eigenvalues isolated by ZGEBAL ====
00414 *
00415          IF( ILO.GT.1 )
00416      $      CALL ZCOPY( ILO-1, H, LDH+1, W, 1 )
00417          IF( IHI.LT.N )
00418      $      CALL ZCOPY( N-IHI, H( IHI+1, IHI+1 ), LDH+1, W( IHI+1 ), 1 )
00419 *
00420 *        ==== Initialize Z, if requested ====
00421 *
00422          IF( INITZ )
00423      $      CALL ZLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
00424 *
00425 *        ==== Quick return if possible ====
00426 *
00427          IF( ILO.EQ.IHI ) THEN
00428             W( ILO ) = H( ILO, ILO )
00429             RETURN
00430          END IF
00431 *
00432 *        ==== ZLAHQR/ZLAQR0 crossover point ====
00433 *
00434          NMIN = ILAENV( 12, 'ZHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
00435      $          ILO, IHI, LWORK )
00436          NMIN = MAX( NTINY, NMIN )
00437 *
00438 *        ==== ZLAQR0 for big matrices; ZLAHQR for small ones ====
00439 *
00440          IF( N.GT.NMIN ) THEN
00441             CALL ZLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI,
00442      $                   Z, LDZ, WORK, LWORK, INFO )
00443          ELSE
00444 *
00445 *           ==== Small matrix ====
00446 *
00447             CALL ZLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, W, ILO, IHI,
00448      $                   Z, LDZ, INFO )
00449 *
00450             IF( INFO.GT.0 ) THEN
00451 *
00452 *              ==== A rare ZLAHQR failure!  ZLAQR0 sometimes succeeds
00453 *              .    when ZLAHQR fails. ====
00454 *
00455                KBOT = INFO
00456 *
00457                IF( N.GE.NL ) THEN
00458 *
00459 *                 ==== Larger matrices have enough subdiagonal scratch
00460 *                 .    space to call ZLAQR0 directly. ====
00461 *
00462                   CALL ZLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, W,
00463      $                         ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
00464 *
00465                ELSE
00466 *
00467 *                 ==== Tiny matrices don't have enough subdiagonal
00468 *                 .    scratch space to benefit from ZLAQR0.  Hence,
00469 *                 .    tiny matrices must be copied into a larger
00470 *                 .    array before calling ZLAQR0. ====
00471 *
00472                   CALL ZLACPY( 'A', N, N, H, LDH, HL, NL )
00473                   HL( N+1, N ) = ZERO
00474                   CALL ZLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
00475      $                         NL )
00476                   CALL ZLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, W,
00477      $                         ILO, IHI, Z, LDZ, WORKL, NL, INFO )
00478                   IF( WANTT .OR. INFO.NE.0 )
00479      $               CALL ZLACPY( 'A', N, N, HL, NL, H, LDH )
00480                END IF
00481             END IF
00482          END IF
00483 *
00484 *        ==== Clear out the trash, if necessary. ====
00485 *
00486          IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
00487      $      CALL ZLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
00488 *
00489 *        ==== Ensure reported workspace size is backward-compatible with
00490 *        .    previous LAPACK versions. ====
00491 *
00492          WORK( 1 ) = DCMPLX( MAX( DBLE( MAX( 1, N ) ),
00493      $               DBLE( WORK( 1 ) ) ), RZERO )
00494       END IF
00495 *
00496 *     ==== End of ZHSEQR ====
00497 *
00498       END
 All Files Functions