LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
shseqr.f
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00001 *> \brief \b SHSEQR
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download SHSEQR + dependencies 
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00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/shseqr.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/shseqr.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
00022 *                          LDZ, 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 *       REAL               H( LDH, * ), WI( * ), WORK( * ), WR( * ),
00030 *      $                   Z( LDZ, * )
00031 *       ..
00032 *  
00033 *
00034 *> \par Purpose:
00035 *  =============
00036 *>
00037 *> \verbatim
00038 *>
00039 *>    SHSEQR computes the eigenvalues of a Hessenberg matrix H
00040 *>    and, optionally, the matrices T and Z from the Schur decomposition
00041 *>    H = Z T Z**T, where T is an upper quasi-triangular matrix (the
00042 *>    Schur form), and Z is the orthogonal matrix of Schur vectors.
00043 *>
00044 *>    Optionally Z may be postmultiplied into an input orthogonal
00045 *>    matrix Q so that this routine can give the Schur factorization
00046 *>    of a matrix A which has been reduced to the Hessenberg form H
00047 *>    by the orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.
00048 *> \endverbatim
00049 *
00050 *  Arguments:
00051 *  ==========
00052 *
00053 *> \param[in] JOB
00054 *> \verbatim
00055 *>          JOB is CHARACTER*1
00056 *>           = 'E':  compute eigenvalues only;
00057 *>           = 'S':  compute eigenvalues and the Schur form T.
00058 *> \endverbatim
00059 *>
00060 *> \param[in] COMPZ
00061 *> \verbatim
00062 *>          COMPZ is CHARACTER*1
00063 *>           = 'N':  no Schur vectors are computed;
00064 *>           = 'I':  Z is initialized to the unit matrix and the matrix Z
00065 *>                   of Schur vectors of H is returned;
00066 *>           = 'V':  Z must contain an orthogonal matrix Q on entry, and
00067 *>                   the product Q*Z is returned.
00068 *> \endverbatim
00069 *>
00070 *> \param[in] N
00071 *> \verbatim
00072 *>          N is INTEGER
00073 *>           The order of the matrix H.  N .GE. 0.
00074 *> \endverbatim
00075 *>
00076 *> \param[in] ILO
00077 *> \verbatim
00078 *>          ILO is INTEGER
00079 *> \endverbatim
00080 *>
00081 *> \param[in] IHI
00082 *> \verbatim
00083 *>          IHI is INTEGER
00084 *>
00085 *>           It is assumed that H is already upper triangular in rows
00086 *>           and columns 1:ILO-1 and IHI+1:N. ILO and IHI are normally
00087 *>           set by a previous call to SGEBAL, and then passed to ZGEHRD
00088 *>           when the matrix output by SGEBAL is reduced to Hessenberg
00089 *>           form. Otherwise ILO and IHI should be set to 1 and N
00090 *>           respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
00091 *>           If N = 0, then ILO = 1 and IHI = 0.
00092 *> \endverbatim
00093 *>
00094 *> \param[in,out] H
00095 *> \verbatim
00096 *>          H is REAL array, dimension (LDH,N)
00097 *>           On entry, the upper Hessenberg matrix H.
00098 *>           On exit, if INFO = 0 and JOB = 'S', then H contains the
00099 *>           upper quasi-triangular matrix T from the Schur decomposition
00100 *>           (the Schur form); 2-by-2 diagonal blocks (corresponding to
00101 *>           complex conjugate pairs of eigenvalues) are returned in
00102 *>           standard form, with H(i,i) = H(i+1,i+1) and
00103 *>           H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and JOB = 'E', the
00104 *>           contents of H are unspecified on exit.  (The output value of
00105 *>           H when INFO.GT.0 is given under the description of INFO
00106 *>           below.)
00107 *>
00108 *>           Unlike earlier versions of SHSEQR, this subroutine may
00109 *>           explicitly H(i,j) = 0 for i.GT.j and j = 1, 2, ... ILO-1
00110 *>           or j = IHI+1, IHI+2, ... N.
00111 *> \endverbatim
00112 *>
00113 *> \param[in] LDH
00114 *> \verbatim
00115 *>          LDH is INTEGER
00116 *>           The leading dimension of the array H. LDH .GE. max(1,N).
00117 *> \endverbatim
00118 *>
00119 *> \param[out] WR
00120 *> \verbatim
00121 *>          WR is REAL array, dimension (N)
00122 *> \endverbatim
00123 *>
00124 *> \param[out] WI
00125 *> \verbatim
00126 *>          WI is REAL array, dimension (N)
00127 *>
00128 *>           The real and imaginary parts, respectively, of the computed
00129 *>           eigenvalues. If two eigenvalues are computed as a complex
00130 *>           conjugate pair, they are stored in consecutive elements of
00131 *>           WR and WI, say the i-th and (i+1)th, with WI(i) .GT. 0 and
00132 *>           WI(i+1) .LT. 0. If JOB = 'S', the eigenvalues are stored in
00133 *>           the same order as on the diagonal of the Schur form returned
00134 *>           in H, with WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2
00135 *>           diagonal block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
00136 *>           WI(i+1) = -WI(i).
00137 *> \endverbatim
00138 *>
00139 *> \param[in,out] Z
00140 *> \verbatim
00141 *>          Z is REAL array, dimension (LDZ,N)
00142 *>           If COMPZ = 'N', Z is not referenced.
00143 *>           If COMPZ = 'I', on entry Z need not be set and on exit,
00144 *>           if INFO = 0, Z contains the orthogonal matrix Z of the Schur
00145 *>           vectors of H.  If COMPZ = 'V', on entry Z must contain an
00146 *>           N-by-N matrix Q, which is assumed to be equal to the unit
00147 *>           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,
00148 *>           if INFO = 0, Z contains Q*Z.
00149 *>           Normally Q is the orthogonal matrix generated by SORGHR
00150 *>           after the call to SGEHRD which formed the Hessenberg matrix
00151 *>           H. (The output value of Z when INFO.GT.0 is given under
00152 *>           the description of INFO below.)
00153 *> \endverbatim
00154 *>
00155 *> \param[in] LDZ
00156 *> \verbatim
00157 *>          LDZ is INTEGER
00158 *>           The leading dimension of the array Z.  if COMPZ = 'I' or
00159 *>           COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.
00160 *> \endverbatim
00161 *>
00162 *> \param[out] WORK
00163 *> \verbatim
00164 *>          WORK is REAL array, dimension (LWORK)
00165 *>           On exit, if INFO = 0, WORK(1) returns an estimate of
00166 *>           the optimal value for LWORK.
00167 *> \endverbatim
00168 *>
00169 *> \param[in] LWORK
00170 *> \verbatim
00171 *>          LWORK is INTEGER
00172 *>           The dimension of the array WORK.  LWORK .GE. max(1,N)
00173 *>           is sufficient and delivers very good and sometimes
00174 *>           optimal performance.  However, LWORK as large as 11*N
00175 *>           may be required for optimal performance.  A workspace
00176 *>           query is recommended to determine the optimal workspace
00177 *>           size.
00178 *>
00179 *>           If LWORK = -1, then SHSEQR does a workspace query.
00180 *>           In this case, SHSEQR checks the input parameters and
00181 *>           estimates the optimal workspace size for the given
00182 *>           values of N, ILO and IHI.  The estimate is returned
00183 *>           in WORK(1).  No error message related to LWORK is
00184 *>           issued by XERBLA.  Neither H nor Z are accessed.
00185 *> \endverbatim
00186 *>
00187 *> \param[out] INFO
00188 *> \verbatim
00189 *>          INFO is INTEGER
00190 *>             =  0:  successful exit
00191 *>           .LT. 0:  if INFO = -i, the i-th argument had an illegal
00192 *>                    value
00193 *>           .GT. 0:  if INFO = i, SHSEQR failed to compute all of
00194 *>                the eigenvalues.  Elements 1:ilo-1 and i+1:n of WR
00195 *>                and WI contain those eigenvalues which have been
00196 *>                successfully computed.  (Failures are rare.)
00197 *>
00198 *>                If INFO .GT. 0 and JOB = 'E', then on exit, the
00199 *>                remaining unconverged eigenvalues are the eigen-
00200 *>                values of the upper Hessenberg matrix rows and
00201 *>                columns ILO through INFO of the final, output
00202 *>                value of H.
00203 *>
00204 *>                If INFO .GT. 0 and JOB   = 'S', then on exit
00205 *>
00206 *>           (*)  (initial value of H)*U  = U*(final value of H)
00207 *>
00208 *>                where U is an orthogonal matrix.  The final
00209 *>                value of H is upper Hessenberg and quasi-triangular
00210 *>                in rows and columns INFO+1 through IHI.
00211 *>
00212 *>                If INFO .GT. 0 and COMPZ = 'V', then on exit
00213 *>
00214 *>                  (final value of Z)  =  (initial value of Z)*U
00215 *>
00216 *>                where U is the orthogonal matrix in (*) (regard-
00217 *>                less of the value of JOB.)
00218 *>
00219 *>                If INFO .GT. 0 and COMPZ = 'I', then on exit
00220 *>                      (final value of Z)  = U
00221 *>                where U is the orthogonal matrix in (*) (regard-
00222 *>                less of the value of JOB.)
00223 *>
00224 *>                If INFO .GT. 0 and COMPZ = 'N', then Z is not
00225 *>                accessed.
00226 *> \endverbatim
00227 *
00228 *  Authors:
00229 *  ========
00230 *
00231 *> \author Univ. of Tennessee 
00232 *> \author Univ. of California Berkeley 
00233 *> \author Univ. of Colorado Denver 
00234 *> \author NAG Ltd. 
00235 *
00236 *> \date November 2011
00237 *
00238 *> \ingroup realOTHERcomputational
00239 *
00240 *> \par Contributors:
00241 *  ==================
00242 *>
00243 *>       Karen Braman and Ralph Byers, Department of Mathematics,
00244 *>       University of Kansas, USA
00245 *
00246 *> \par Further Details:
00247 *  =====================
00248 *>
00249 *> \verbatim
00250 *>
00251 *>             Default values supplied by
00252 *>             ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).
00253 *>             It is suggested that these defaults be adjusted in order
00254 *>             to attain best performance in each particular
00255 *>             computational environment.
00256 *>
00257 *>            ISPEC=12: The SLAHQR vs SLAQR0 crossover point.
00258 *>                      Default: 75. (Must be at least 11.)
00259 *>
00260 *>            ISPEC=13: Recommended deflation window size.
00261 *>                      This depends on ILO, IHI and NS.  NS is the
00262 *>                      number of simultaneous shifts returned
00263 *>                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)
00264 *>                      The default for (IHI-ILO+1).LE.500 is NS.
00265 *>                      The default for (IHI-ILO+1).GT.500 is 3*NS/2.
00266 *>
00267 *>            ISPEC=14: Nibble crossover point. (See IPARMQ for
00268 *>                      details.)  Default: 14% of deflation window
00269 *>                      size.
00270 *>
00271 *>            ISPEC=15: Number of simultaneous shifts in a multishift
00272 *>                      QR iteration.
00273 *>
00274 *>                      If IHI-ILO+1 is ...
00275 *>
00276 *>                      greater than      ...but less    ... the
00277 *>                      or equal to ...      than        default is
00278 *>
00279 *>                           1               30          NS =   2(+)
00280 *>                          30               60          NS =   4(+)
00281 *>                          60              150          NS =  10(+)
00282 *>                         150              590          NS =  **
00283 *>                         590             3000          NS =  64
00284 *>                        3000             6000          NS = 128
00285 *>                        6000             infinity      NS = 256
00286 *>
00287 *>                  (+)  By default some or all matrices of this order
00288 *>                       are passed to the implicit double shift routine
00289 *>                       SLAHQR and this parameter is ignored.  See
00290 *>                       ISPEC=12 above and comments in IPARMQ for
00291 *>                       details.
00292 *>
00293 *>                 (**)  The asterisks (**) indicate an ad-hoc
00294 *>                       function of N increasing from 10 to 64.
00295 *>
00296 *>            ISPEC=16: Select structured matrix multiply.
00297 *>                      If the number of simultaneous shifts (specified
00298 *>                      by ISPEC=15) is less than 14, then the default
00299 *>                      for ISPEC=16 is 0.  Otherwise the default for
00300 *>                      ISPEC=16 is 2.
00301 *> \endverbatim
00302 *
00303 *> \par References:
00304 *  ================
00305 *>
00306 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00307 *>       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
00308 *>       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
00309 *>       929--947, 2002.
00310 *> \n
00311 *>       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
00312 *>       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
00313 *>       of Matrix Analysis, volume 23, pages 948--973, 2002.
00314 *
00315 *  =====================================================================
00316       SUBROUTINE SHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, WR, WI, Z,
00317      $                   LDZ, WORK, LWORK, INFO )
00318 *
00319 *  -- LAPACK computational routine (version 3.4.0) --
00320 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00321 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00322 *     November 2011
00323 *
00324 *     .. Scalar Arguments ..
00325       INTEGER            IHI, ILO, INFO, LDH, LDZ, LWORK, N
00326       CHARACTER          COMPZ, JOB
00327 *     ..
00328 *     .. Array Arguments ..
00329       REAL               H( LDH, * ), WI( * ), WORK( * ), WR( * ),
00330      $                   Z( LDZ, * )
00331 *     ..
00332 *
00333 *  =====================================================================
00334 *
00335 *     .. Parameters ..
00336 *
00337 *     ==== Matrices of order NTINY or smaller must be processed by
00338 *     .    SLAHQR because of insufficient subdiagonal scratch space.
00339 *     .    (This is a hard limit.) ====
00340       INTEGER            NTINY
00341       PARAMETER          ( NTINY = 11 )
00342 *
00343 *     ==== NL allocates some local workspace to help small matrices
00344 *     .    through a rare SLAHQR failure.  NL .GT. NTINY = 11 is
00345 *     .    required and NL .LE. NMIN = ILAENV(ISPEC=12,...) is recom-
00346 *     .    mended.  (The default value of NMIN is 75.)  Using NL = 49
00347 *     .    allows up to six simultaneous shifts and a 16-by-16
00348 *     .    deflation window.  ====
00349       INTEGER            NL
00350       PARAMETER          ( NL = 49 )
00351       REAL               ZERO, ONE
00352       PARAMETER          ( ZERO = 0.0e0, ONE = 1.0e0 )
00353 *     ..
00354 *     .. Local Arrays ..
00355       REAL               HL( NL, NL ), WORKL( NL )
00356 *     ..
00357 *     .. Local Scalars ..
00358       INTEGER            I, KBOT, NMIN
00359       LOGICAL            INITZ, LQUERY, WANTT, WANTZ
00360 *     ..
00361 *     .. External Functions ..
00362       INTEGER            ILAENV
00363       LOGICAL            LSAME
00364       EXTERNAL           ILAENV, LSAME
00365 *     ..
00366 *     .. External Subroutines ..
00367       EXTERNAL           SLACPY, SLAHQR, SLAQR0, SLASET, XERBLA
00368 *     ..
00369 *     .. Intrinsic Functions ..
00370       INTRINSIC          MAX, MIN, REAL
00371 *     ..
00372 *     .. Executable Statements ..
00373 *
00374 *     ==== Decode and check the input parameters. ====
00375 *
00376       WANTT = LSAME( JOB, 'S' )
00377       INITZ = LSAME( COMPZ, 'I' )
00378       WANTZ = INITZ .OR. LSAME( COMPZ, 'V' )
00379       WORK( 1 ) = REAL( MAX( 1, N ) )
00380       LQUERY = LWORK.EQ.-1
00381 *
00382       INFO = 0
00383       IF( .NOT.LSAME( JOB, 'E' ) .AND. .NOT.WANTT ) THEN
00384          INFO = -1
00385       ELSE IF( .NOT.LSAME( COMPZ, 'N' ) .AND. .NOT.WANTZ ) THEN
00386          INFO = -2
00387       ELSE IF( N.LT.0 ) THEN
00388          INFO = -3
00389       ELSE IF( ILO.LT.1 .OR. ILO.GT.MAX( 1, N ) ) THEN
00390          INFO = -4
00391       ELSE IF( IHI.LT.MIN( ILO, N ) .OR. IHI.GT.N ) THEN
00392          INFO = -5
00393       ELSE IF( LDH.LT.MAX( 1, N ) ) THEN
00394          INFO = -7
00395       ELSE IF( LDZ.LT.1 .OR. ( WANTZ .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
00396          INFO = -11
00397       ELSE IF( LWORK.LT.MAX( 1, N ) .AND. .NOT.LQUERY ) THEN
00398          INFO = -13
00399       END IF
00400 *
00401       IF( INFO.NE.0 ) THEN
00402 *
00403 *        ==== Quick return in case of invalid argument. ====
00404 *
00405          CALL XERBLA( 'SHSEQR', -INFO )
00406          RETURN
00407 *
00408       ELSE IF( N.EQ.0 ) THEN
00409 *
00410 *        ==== Quick return in case N = 0; nothing to do. ====
00411 *
00412          RETURN
00413 *
00414       ELSE IF( LQUERY ) THEN
00415 *
00416 *        ==== Quick return in case of a workspace query ====
00417 *
00418          CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
00419      $                IHI, Z, LDZ, WORK, LWORK, INFO )
00420 *        ==== Ensure reported workspace size is backward-compatible with
00421 *        .    previous LAPACK versions. ====
00422          WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) )
00423          RETURN
00424 *
00425       ELSE
00426 *
00427 *        ==== copy eigenvalues isolated by SGEBAL ====
00428 *
00429          DO 10 I = 1, ILO - 1
00430             WR( I ) = H( I, I )
00431             WI( I ) = ZERO
00432    10    CONTINUE
00433          DO 20 I = IHI + 1, N
00434             WR( I ) = H( I, I )
00435             WI( I ) = ZERO
00436    20    CONTINUE
00437 *
00438 *        ==== Initialize Z, if requested ====
00439 *
00440          IF( INITZ )
00441      $      CALL SLASET( 'A', N, N, ZERO, ONE, Z, LDZ )
00442 *
00443 *        ==== Quick return if possible ====
00444 *
00445          IF( ILO.EQ.IHI ) THEN
00446             WR( ILO ) = H( ILO, ILO )
00447             WI( ILO ) = ZERO
00448             RETURN
00449          END IF
00450 *
00451 *        ==== SLAHQR/SLAQR0 crossover point ====
00452 *
00453          NMIN = ILAENV( 12, 'SHSEQR', JOB( : 1 ) // COMPZ( : 1 ), N,
00454      $          ILO, IHI, LWORK )
00455          NMIN = MAX( NTINY, NMIN )
00456 *
00457 *        ==== SLAQR0 for big matrices; SLAHQR for small ones ====
00458 *
00459          IF( N.GT.NMIN ) THEN
00460             CALL SLAQR0( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
00461      $                   IHI, Z, LDZ, WORK, LWORK, INFO )
00462          ELSE
00463 *
00464 *           ==== Small matrix ====
00465 *
00466             CALL SLAHQR( WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILO,
00467      $                   IHI, Z, LDZ, INFO )
00468 *
00469             IF( INFO.GT.0 ) THEN
00470 *
00471 *              ==== A rare SLAHQR failure!  SLAQR0 sometimes succeeds
00472 *              .    when SLAHQR fails. ====
00473 *
00474                KBOT = INFO
00475 *
00476                IF( N.GE.NL ) THEN
00477 *
00478 *                 ==== Larger matrices have enough subdiagonal scratch
00479 *                 .    space to call SLAQR0 directly. ====
00480 *
00481                   CALL SLAQR0( WANTT, WANTZ, N, ILO, KBOT, H, LDH, WR,
00482      $                         WI, ILO, IHI, Z, LDZ, WORK, LWORK, INFO )
00483 *
00484                ELSE
00485 *
00486 *                 ==== Tiny matrices don't have enough subdiagonal
00487 *                 .    scratch space to benefit from SLAQR0.  Hence,
00488 *                 .    tiny matrices must be copied into a larger
00489 *                 .    array before calling SLAQR0. ====
00490 *
00491                   CALL SLACPY( 'A', N, N, H, LDH, HL, NL )
00492                   HL( N+1, N ) = ZERO
00493                   CALL SLASET( 'A', NL, NL-N, ZERO, ZERO, HL( 1, N+1 ),
00494      $                         NL )
00495                   CALL SLAQR0( WANTT, WANTZ, NL, ILO, KBOT, HL, NL, WR,
00496      $                         WI, ILO, IHI, Z, LDZ, WORKL, NL, INFO )
00497                   IF( WANTT .OR. INFO.NE.0 )
00498      $               CALL SLACPY( 'A', N, N, HL, NL, H, LDH )
00499                END IF
00500             END IF
00501          END IF
00502 *
00503 *        ==== Clear out the trash, if necessary. ====
00504 *
00505          IF( ( WANTT .OR. INFO.NE.0 ) .AND. N.GT.2 )
00506      $      CALL SLASET( 'L', N-2, N-2, ZERO, ZERO, H( 3, 1 ), LDH )
00507 *
00508 *        ==== Ensure reported workspace size is backward-compatible with
00509 *        .    previous LAPACK versions. ====
00510 *
00511          WORK( 1 ) = MAX( REAL( MAX( 1, N ) ), WORK( 1 ) )
00512       END IF
00513 *
00514 *     ==== End of SHSEQR ====
00515 *
00516       END
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