LAPACK 3.3.0

cstedc.f

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00001       SUBROUTINE CSTEDC( COMPZ, N, D, E, Z, LDZ, WORK, LWORK, RWORK,
00002      $                   LRWORK, IWORK, LIWORK, INFO )
00003 *
00004 *  -- LAPACK routine (version 3.2) --
00005 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00006 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00007 *     November 2006
00008 *
00009 *     .. Scalar Arguments ..
00010       CHARACTER          COMPZ
00011       INTEGER            INFO, LDZ, LIWORK, LRWORK, LWORK, N
00012 *     ..
00013 *     .. Array Arguments ..
00014       INTEGER            IWORK( * )
00015       REAL               D( * ), E( * ), RWORK( * )
00016       COMPLEX            WORK( * ), Z( LDZ, * )
00017 *     ..
00018 *
00019 *  Purpose
00020 *  =======
00021 *
00022 *  CSTEDC computes all eigenvalues and, optionally, eigenvectors of a
00023 *  symmetric tridiagonal matrix using the divide and conquer method.
00024 *  The eigenvectors of a full or band complex Hermitian matrix can also
00025 *  be found if CHETRD or CHPTRD or CHBTRD has been used to reduce this
00026 *  matrix to tridiagonal form.
00027 *
00028 *  This code makes very mild assumptions about floating point
00029 *  arithmetic. It will work on machines with a guard digit in
00030 *  add/subtract, or on those binary machines without guard digits
00031 *  which subtract like the Cray X-MP, Cray Y-MP, Cray C-90, or Cray-2.
00032 *  It could conceivably fail on hexadecimal or decimal machines
00033 *  without guard digits, but we know of none.  See SLAED3 for details.
00034 *
00035 *  Arguments
00036 *  =========
00037 *
00038 *  COMPZ   (input) CHARACTER*1
00039 *          = 'N':  Compute eigenvalues only.
00040 *          = 'I':  Compute eigenvectors of tridiagonal matrix also.
00041 *          = 'V':  Compute eigenvectors of original Hermitian matrix
00042 *                  also.  On entry, Z contains the unitary matrix used
00043 *                  to reduce the original matrix to tridiagonal form.
00044 *
00045 *  N       (input) INTEGER
00046 *          The dimension of the symmetric tridiagonal matrix.  N >= 0.
00047 *
00048 *  D       (input/output) REAL array, dimension (N)
00049 *          On entry, the diagonal elements of the tridiagonal matrix.
00050 *          On exit, if INFO = 0, the eigenvalues in ascending order.
00051 *
00052 *  E       (input/output) REAL array, dimension (N-1)
00053 *          On entry, the subdiagonal elements of the tridiagonal matrix.
00054 *          On exit, E has been destroyed.
00055 *
00056 *  Z       (input/output) COMPLEX array, dimension (LDZ,N)
00057 *          On entry, if COMPZ = 'V', then Z contains the unitary
00058 *          matrix used in the reduction to tridiagonal form.
00059 *          On exit, if INFO = 0, then if COMPZ = 'V', Z contains the
00060 *          orthonormal eigenvectors of the original Hermitian matrix,
00061 *          and if COMPZ = 'I', Z contains the orthonormal eigenvectors
00062 *          of the symmetric tridiagonal matrix.
00063 *          If  COMPZ = 'N', then Z is not referenced.
00064 *
00065 *  LDZ     (input) INTEGER
00066 *          The leading dimension of the array Z.  LDZ >= 1.
00067 *          If eigenvectors are desired, then LDZ >= max(1,N).
00068 *
00069 *  WORK    (workspace/output) COMPLEX    array, dimension (MAX(1,LWORK))
00070 *          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00071 *
00072 *  LWORK   (input) INTEGER
00073 *          The dimension of the array WORK.
00074 *          If COMPZ = 'N' or 'I', or N <= 1, LWORK must be at least 1.
00075 *          If COMPZ = 'V' and N > 1, LWORK must be at least N*N.
00076 *          Note that for COMPZ = 'V', then if N is less than or
00077 *          equal to the minimum divide size, usually 25, then LWORK need
00078 *          only be 1.
00079 *
00080 *          If LWORK = -1, then a workspace query is assumed; the routine
00081 *          only calculates the optimal sizes of the WORK, RWORK and
00082 *          IWORK arrays, returns these values as the first entries of
00083 *          the WORK, RWORK and IWORK arrays, and no error message
00084 *          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
00085 *
00086 *  RWORK   (workspace/output) REAL array, dimension (MAX(1,LRWORK))
00087 *          On exit, if INFO = 0, RWORK(1) returns the optimal LRWORK.
00088 *
00089 *  LRWORK  (input) INTEGER
00090 *          The dimension of the array RWORK.
00091 *          If COMPZ = 'N' or N <= 1, LRWORK must be at least 1.
00092 *          If COMPZ = 'V' and N > 1, LRWORK must be at least
00093 *                         1 + 3*N + 2*N*lg N + 3*N**2 ,
00094 *                         where lg( N ) = smallest integer k such
00095 *                         that 2**k >= N.
00096 *          If COMPZ = 'I' and N > 1, LRWORK must be at least
00097 *                         1 + 4*N + 2*N**2 .
00098 *          Note that for COMPZ = 'I' or 'V', then if N is less than or
00099 *          equal to the minimum divide size, usually 25, then LRWORK
00100 *          need only be max(1,2*(N-1)).
00101 *
00102 *          If LRWORK = -1, then a workspace query is assumed; the
00103 *          routine only calculates the optimal sizes of the WORK, RWORK
00104 *          and IWORK arrays, returns these values as the first entries
00105 *          of the WORK, RWORK and IWORK arrays, and no error message
00106 *          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
00107 *
00108 *  IWORK   (workspace/output) INTEGER array, dimension (MAX(1,LIWORK))
00109 *          On exit, if INFO = 0, IWORK(1) returns the optimal LIWORK.
00110 *
00111 *  LIWORK  (input) INTEGER
00112 *          The dimension of the array IWORK.
00113 *          If COMPZ = 'N' or N <= 1, LIWORK must be at least 1.
00114 *          If COMPZ = 'V' or N > 1,  LIWORK must be at least
00115 *                                    6 + 6*N + 5*N*lg N.
00116 *          If COMPZ = 'I' or N > 1,  LIWORK must be at least
00117 *                                    3 + 5*N .
00118 *          Note that for COMPZ = 'I' or 'V', then if N is less than or
00119 *          equal to the minimum divide size, usually 25, then LIWORK
00120 *          need only be 1.
00121 *
00122 *          If LIWORK = -1, then a workspace query is assumed; the
00123 *          routine only calculates the optimal sizes of the WORK, RWORK
00124 *          and IWORK arrays, returns these values as the first entries
00125 *          of the WORK, RWORK and IWORK arrays, and no error message
00126 *          related to LWORK or LRWORK or LIWORK is issued by XERBLA.
00127 *
00128 *  INFO    (output) INTEGER
00129 *          = 0:  successful exit.
00130 *          < 0:  if INFO = -i, the i-th argument had an illegal value.
00131 *          > 0:  The algorithm failed to compute an eigenvalue while
00132 *                working on the submatrix lying in rows and columns
00133 *                INFO/(N+1) through mod(INFO,N+1).
00134 *
00135 *  Further Details
00136 *  ===============
00137 *
00138 *  Based on contributions by
00139 *     Jeff Rutter, Computer Science Division, University of California
00140 *     at Berkeley, USA
00141 *
00142 *  =====================================================================
00143 *
00144 *     .. Parameters ..
00145       REAL               ZERO, ONE, TWO
00146       PARAMETER          ( ZERO = 0.0E0, ONE = 1.0E0, TWO = 2.0E0 )
00147 *     ..
00148 *     .. Local Scalars ..
00149       LOGICAL            LQUERY
00150       INTEGER            FINISH, I, ICOMPZ, II, J, K, LGN, LIWMIN, LL,
00151      $                   LRWMIN, LWMIN, M, SMLSIZ, START
00152       REAL               EPS, ORGNRM, P, TINY
00153 *     ..
00154 *     .. External Functions ..
00155       LOGICAL            LSAME
00156       INTEGER            ILAENV
00157       REAL               SLAMCH, SLANST
00158       EXTERNAL           ILAENV, LSAME, SLAMCH, SLANST
00159 *     ..
00160 *     .. External Subroutines ..
00161       EXTERNAL           XERBLA, CLACPY, CLACRM, CLAED0, CSTEQR, CSWAP,
00162      $                   SLASCL, SLASET, SSTEDC, SSTEQR, SSTERF
00163 *     ..
00164 *     .. Intrinsic Functions ..
00165       INTRINSIC          ABS, INT, LOG, MAX, MOD, REAL, SQRT
00166 *     ..
00167 *     .. Executable Statements ..
00168 *
00169 *     Test the input parameters.
00170 *
00171       INFO = 0
00172       LQUERY = ( LWORK.EQ.-1 .OR. LRWORK.EQ.-1 .OR. LIWORK.EQ.-1 )
00173 *
00174       IF( LSAME( COMPZ, 'N' ) ) THEN
00175          ICOMPZ = 0
00176       ELSE IF( LSAME( COMPZ, 'V' ) ) THEN
00177          ICOMPZ = 1
00178       ELSE IF( LSAME( COMPZ, 'I' ) ) THEN
00179          ICOMPZ = 2
00180       ELSE
00181          ICOMPZ = -1
00182       END IF
00183       IF( ICOMPZ.LT.0 ) THEN
00184          INFO = -1
00185       ELSE IF( N.LT.0 ) THEN
00186          INFO = -2
00187       ELSE IF( ( LDZ.LT.1 ) .OR.
00188      $         ( ICOMPZ.GT.0 .AND. LDZ.LT.MAX( 1, N ) ) ) THEN
00189          INFO = -6
00190       END IF
00191 *
00192       IF( INFO.EQ.0 ) THEN
00193 *
00194 *        Compute the workspace requirements
00195 *
00196          SMLSIZ = ILAENV( 9, 'CSTEDC', ' ', 0, 0, 0, 0 )
00197          IF( N.LE.1 .OR. ICOMPZ.EQ.0 ) THEN
00198             LWMIN = 1
00199             LIWMIN = 1
00200             LRWMIN = 1
00201          ELSE IF( N.LE.SMLSIZ ) THEN
00202             LWMIN = 1
00203             LIWMIN = 1
00204             LRWMIN = 2*( N - 1 )
00205          ELSE IF( ICOMPZ.EQ.1 ) THEN
00206             LGN = INT( LOG( REAL( N ) ) / LOG( TWO ) )
00207             IF( 2**LGN.LT.N )
00208      $         LGN = LGN + 1
00209             IF( 2**LGN.LT.N )
00210      $         LGN = LGN + 1
00211             LWMIN = N*N
00212             LRWMIN = 1 + 3*N + 2*N*LGN + 3*N**2
00213             LIWMIN = 6 + 6*N + 5*N*LGN
00214          ELSE IF( ICOMPZ.EQ.2 ) THEN
00215             LWMIN = 1
00216             LRWMIN = 1 + 4*N + 2*N**2
00217             LIWMIN = 3 + 5*N
00218          END IF
00219          WORK( 1 ) = LWMIN
00220          RWORK( 1 ) = LRWMIN
00221          IWORK( 1 ) = LIWMIN
00222 *
00223          IF( LWORK.LT.LWMIN .AND. .NOT.LQUERY ) THEN
00224             INFO = -8
00225          ELSE IF( LRWORK.LT.LRWMIN .AND. .NOT.LQUERY ) THEN
00226             INFO = -10
00227          ELSE IF( LIWORK.LT.LIWMIN .AND. .NOT.LQUERY ) THEN
00228             INFO = -12
00229          END IF
00230       END IF
00231 *
00232       IF( INFO.NE.0 ) THEN
00233          CALL XERBLA( 'CSTEDC', -INFO )
00234          RETURN
00235       ELSE IF( LQUERY ) THEN
00236          RETURN
00237       END IF
00238 *
00239 *     Quick return if possible
00240 *
00241       IF( N.EQ.0 )
00242      $   RETURN
00243       IF( N.EQ.1 ) THEN
00244          IF( ICOMPZ.NE.0 )
00245      $      Z( 1, 1 ) = ONE
00246          RETURN
00247       END IF
00248 *
00249 *     If the following conditional clause is removed, then the routine
00250 *     will use the Divide and Conquer routine to compute only the
00251 *     eigenvalues, which requires (3N + 3N**2) real workspace and
00252 *     (2 + 5N + 2N lg(N)) integer workspace.
00253 *     Since on many architectures SSTERF is much faster than any other
00254 *     algorithm for finding eigenvalues only, it is used here
00255 *     as the default. If the conditional clause is removed, then
00256 *     information on the size of workspace needs to be changed.
00257 *
00258 *     If COMPZ = 'N', use SSTERF to compute the eigenvalues.
00259 *
00260       IF( ICOMPZ.EQ.0 ) THEN
00261          CALL SSTERF( N, D, E, INFO )
00262          GO TO 70
00263       END IF
00264 *
00265 *     If N is smaller than the minimum divide size (SMLSIZ+1), then
00266 *     solve the problem with another solver.
00267 *
00268       IF( N.LE.SMLSIZ ) THEN
00269 *
00270          CALL CSTEQR( COMPZ, N, D, E, Z, LDZ, RWORK, INFO )
00271 *
00272       ELSE
00273 *
00274 *        If COMPZ = 'I', we simply call SSTEDC instead.
00275 *
00276          IF( ICOMPZ.EQ.2 ) THEN
00277             CALL SLASET( 'Full', N, N, ZERO, ONE, RWORK, N )
00278             LL = N*N + 1
00279             CALL SSTEDC( 'I', N, D, E, RWORK, N,
00280      $                   RWORK( LL ), LRWORK-LL+1, IWORK, LIWORK, INFO )
00281             DO 20 J = 1, N
00282                DO 10 I = 1, N
00283                   Z( I, J ) = RWORK( ( J-1 )*N+I )
00284    10          CONTINUE
00285    20       CONTINUE
00286             GO TO 70
00287          END IF
00288 *
00289 *        From now on, only option left to be handled is COMPZ = 'V',
00290 *        i.e. ICOMPZ = 1.
00291 *
00292 *        Scale.
00293 *
00294          ORGNRM = SLANST( 'M', N, D, E )
00295          IF( ORGNRM.EQ.ZERO )
00296      $      GO TO 70
00297 *
00298          EPS = SLAMCH( 'Epsilon' )
00299 *
00300          START = 1
00301 *
00302 *        while ( START <= N )
00303 *
00304    30    CONTINUE
00305          IF( START.LE.N ) THEN
00306 *
00307 *           Let FINISH be the position of the next subdiagonal entry
00308 *           such that E( FINISH ) <= TINY or FINISH = N if no such
00309 *           subdiagonal exists.  The matrix identified by the elements
00310 *           between START and FINISH constitutes an independent
00311 *           sub-problem.
00312 *
00313             FINISH = START
00314    40       CONTINUE
00315             IF( FINISH.LT.N ) THEN
00316                TINY = EPS*SQRT( ABS( D( FINISH ) ) )*
00317      $                    SQRT( ABS( D( FINISH+1 ) ) )
00318                IF( ABS( E( FINISH ) ).GT.TINY ) THEN
00319                   FINISH = FINISH + 1
00320                   GO TO 40
00321                END IF
00322             END IF
00323 *
00324 *           (Sub) Problem determined.  Compute its size and solve it.
00325 *
00326             M = FINISH - START + 1
00327             IF( M.GT.SMLSIZ ) THEN
00328 *
00329 *              Scale.
00330 *
00331                ORGNRM = SLANST( 'M', M, D( START ), E( START ) )
00332                CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, M, 1, D( START ), M,
00333      $                      INFO )
00334                CALL SLASCL( 'G', 0, 0, ORGNRM, ONE, M-1, 1, E( START ),
00335      $                      M-1, INFO )
00336 *
00337                CALL CLAED0( N, M, D( START ), E( START ), Z( 1, START ),
00338      $                      LDZ, WORK, N, RWORK, IWORK, INFO )
00339                IF( INFO.GT.0 ) THEN
00340                   INFO = ( INFO / ( M+1 )+START-1 )*( N+1 ) +
00341      $                   MOD( INFO, ( M+1 ) ) + START - 1
00342                   GO TO 70
00343                END IF
00344 *
00345 *              Scale back.
00346 *
00347                CALL SLASCL( 'G', 0, 0, ONE, ORGNRM, M, 1, D( START ), M,
00348      $                      INFO )
00349 *
00350             ELSE
00351                CALL SSTEQR( 'I', M, D( START ), E( START ), RWORK, M,
00352      $                      RWORK( M*M+1 ), INFO )
00353                CALL CLACRM( N, M, Z( 1, START ), LDZ, RWORK, M, WORK, N,
00354      $                      RWORK( M*M+1 ) )
00355                CALL CLACPY( 'A', N, M, WORK, N, Z( 1, START ), LDZ )
00356                IF( INFO.GT.0 ) THEN
00357                   INFO = START*( N+1 ) + FINISH
00358                   GO TO 70
00359                END IF
00360             END IF
00361 *
00362             START = FINISH + 1
00363             GO TO 30
00364          END IF
00365 *
00366 *        endwhile
00367 *
00368 *        If the problem split any number of times, then the eigenvalues
00369 *        will not be properly ordered.  Here we permute the eigenvalues
00370 *        (and the associated eigenvectors) into ascending order.
00371 *
00372          IF( M.NE.N ) THEN
00373 *
00374 *           Use Selection Sort to minimize swaps of eigenvectors
00375 *
00376             DO 60 II = 2, N
00377                I = II - 1
00378                K = I
00379                P = D( I )
00380                DO 50 J = II, N
00381                   IF( D( J ).LT.P ) THEN
00382                      K = J
00383                      P = D( J )
00384                   END IF
00385    50          CONTINUE
00386                IF( K.NE.I ) THEN
00387                   D( K ) = D( I )
00388                   D( I ) = P
00389                   CALL CSWAP( N, Z( 1, I ), 1, Z( 1, K ), 1 )
00390                END IF
00391    60       CONTINUE
00392          END IF
00393       END IF
00394 *
00395    70 CONTINUE
00396       WORK( 1 ) = LWMIN
00397       RWORK( 1 ) = LRWMIN
00398       IWORK( 1 ) = LIWMIN
00399 *
00400       RETURN
00401 *
00402 *     End of CSTEDC
00403 *
00404       END
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