LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
zhetrd.f
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00001 *> \brief \b ZHETRD
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER          UPLO
00025 *       INTEGER            INFO, LDA, LWORK, N
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       DOUBLE PRECISION   D( * ), E( * )
00029 *       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
00030 *       ..
00031 *  
00032 *
00033 *> \par Purpose:
00034 *  =============
00035 *>
00036 *> \verbatim
00037 *>
00038 *> ZHETRD reduces a complex Hermitian matrix A to real symmetric
00039 *> tridiagonal form T by a unitary similarity transformation:
00040 *> Q**H * A * Q = T.
00041 *> \endverbatim
00042 *
00043 *  Arguments:
00044 *  ==========
00045 *
00046 *> \param[in] UPLO
00047 *> \verbatim
00048 *>          UPLO is CHARACTER*1
00049 *>          = 'U':  Upper triangle of A is stored;
00050 *>          = 'L':  Lower triangle of A is stored.
00051 *> \endverbatim
00052 *>
00053 *> \param[in] N
00054 *> \verbatim
00055 *>          N is INTEGER
00056 *>          The order of the matrix A.  N >= 0.
00057 *> \endverbatim
00058 *>
00059 *> \param[in,out] A
00060 *> \verbatim
00061 *>          A is COMPLEX*16 array, dimension (LDA,N)
00062 *>          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
00063 *>          N-by-N upper triangular part of A contains the upper
00064 *>          triangular part of the matrix A, and the strictly lower
00065 *>          triangular part of A is not referenced.  If UPLO = 'L', the
00066 *>          leading N-by-N lower triangular part of A contains the lower
00067 *>          triangular part of the matrix A, and the strictly upper
00068 *>          triangular part of A is not referenced.
00069 *>          On exit, if UPLO = 'U', the diagonal and first superdiagonal
00070 *>          of A are overwritten by the corresponding elements of the
00071 *>          tridiagonal matrix T, and the elements above the first
00072 *>          superdiagonal, with the array TAU, represent the unitary
00073 *>          matrix Q as a product of elementary reflectors; if UPLO
00074 *>          = 'L', the diagonal and first subdiagonal of A are over-
00075 *>          written by the corresponding elements of the tridiagonal
00076 *>          matrix T, and the elements below the first subdiagonal, with
00077 *>          the array TAU, represent the unitary matrix Q as a product
00078 *>          of elementary reflectors. See Further Details.
00079 *> \endverbatim
00080 *>
00081 *> \param[in] LDA
00082 *> \verbatim
00083 *>          LDA is INTEGER
00084 *>          The leading dimension of the array A.  LDA >= max(1,N).
00085 *> \endverbatim
00086 *>
00087 *> \param[out] D
00088 *> \verbatim
00089 *>          D is DOUBLE PRECISION array, dimension (N)
00090 *>          The diagonal elements of the tridiagonal matrix T:
00091 *>          D(i) = A(i,i).
00092 *> \endverbatim
00093 *>
00094 *> \param[out] E
00095 *> \verbatim
00096 *>          E is DOUBLE PRECISION array, dimension (N-1)
00097 *>          The off-diagonal elements of the tridiagonal matrix T:
00098 *>          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
00099 *> \endverbatim
00100 *>
00101 *> \param[out] TAU
00102 *> \verbatim
00103 *>          TAU is COMPLEX*16 array, dimension (N-1)
00104 *>          The scalar factors of the elementary reflectors (see Further
00105 *>          Details).
00106 *> \endverbatim
00107 *>
00108 *> \param[out] WORK
00109 *> \verbatim
00110 *>          WORK is COMPLEX*16 array, dimension (MAX(1,LWORK))
00111 *>          On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
00112 *> \endverbatim
00113 *>
00114 *> \param[in] LWORK
00115 *> \verbatim
00116 *>          LWORK is INTEGER
00117 *>          The dimension of the array WORK.  LWORK >= 1.
00118 *>          For optimum performance LWORK >= N*NB, where NB is the
00119 *>          optimal blocksize.
00120 *>
00121 *>          If LWORK = -1, then a workspace query is assumed; the routine
00122 *>          only calculates the optimal size of the WORK array, returns
00123 *>          this value as the first entry of the WORK array, and no error
00124 *>          message related to LWORK is issued by XERBLA.
00125 *> \endverbatim
00126 *>
00127 *> \param[out] INFO
00128 *> \verbatim
00129 *>          INFO is INTEGER
00130 *>          = 0:  successful exit
00131 *>          < 0:  if INFO = -i, the i-th argument had an illegal value
00132 *> \endverbatim
00133 *
00134 *  Authors:
00135 *  ========
00136 *
00137 *> \author Univ. of Tennessee 
00138 *> \author Univ. of California Berkeley 
00139 *> \author Univ. of Colorado Denver 
00140 *> \author NAG Ltd. 
00141 *
00142 *> \date November 2011
00143 *
00144 *> \ingroup complex16HEcomputational
00145 *
00146 *> \par Further Details:
00147 *  =====================
00148 *>
00149 *> \verbatim
00150 *>
00151 *>  If UPLO = 'U', the matrix Q is represented as a product of elementary
00152 *>  reflectors
00153 *>
00154 *>     Q = H(n-1) . . . H(2) H(1).
00155 *>
00156 *>  Each H(i) has the form
00157 *>
00158 *>     H(i) = I - tau * v * v**H
00159 *>
00160 *>  where tau is a complex scalar, and v is a complex vector with
00161 *>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
00162 *>  A(1:i-1,i+1), and tau in TAU(i).
00163 *>
00164 *>  If UPLO = 'L', the matrix Q is represented as a product of elementary
00165 *>  reflectors
00166 *>
00167 *>     Q = H(1) H(2) . . . H(n-1).
00168 *>
00169 *>  Each H(i) has the form
00170 *>
00171 *>     H(i) = I - tau * v * v**H
00172 *>
00173 *>  where tau is a complex scalar, and v is a complex vector with
00174 *>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
00175 *>  and tau in TAU(i).
00176 *>
00177 *>  The contents of A on exit are illustrated by the following examples
00178 *>  with n = 5:
00179 *>
00180 *>  if UPLO = 'U':                       if UPLO = 'L':
00181 *>
00182 *>    (  d   e   v2  v3  v4 )              (  d                  )
00183 *>    (      d   e   v3  v4 )              (  e   d              )
00184 *>    (          d   e   v4 )              (  v1  e   d          )
00185 *>    (              d   e  )              (  v1  v2  e   d      )
00186 *>    (                  d  )              (  v1  v2  v3  e   d  )
00187 *>
00188 *>  where d and e denote diagonal and off-diagonal elements of T, and vi
00189 *>  denotes an element of the vector defining H(i).
00190 *> \endverbatim
00191 *>
00192 *  =====================================================================
00193       SUBROUTINE ZHETRD( UPLO, N, A, LDA, D, E, TAU, WORK, LWORK, INFO )
00194 *
00195 *  -- LAPACK computational routine (version 3.4.0) --
00196 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00197 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00198 *     November 2011
00199 *
00200 *     .. Scalar Arguments ..
00201       CHARACTER          UPLO
00202       INTEGER            INFO, LDA, LWORK, N
00203 *     ..
00204 *     .. Array Arguments ..
00205       DOUBLE PRECISION   D( * ), E( * )
00206       COMPLEX*16         A( LDA, * ), TAU( * ), WORK( * )
00207 *     ..
00208 *
00209 *  =====================================================================
00210 *
00211 *     .. Parameters ..
00212       DOUBLE PRECISION   ONE
00213       PARAMETER          ( ONE = 1.0D+0 )
00214       COMPLEX*16         CONE
00215       PARAMETER          ( CONE = ( 1.0D+0, 0.0D+0 ) )
00216 *     ..
00217 *     .. Local Scalars ..
00218       LOGICAL            LQUERY, UPPER
00219       INTEGER            I, IINFO, IWS, J, KK, LDWORK, LWKOPT, NB,
00220      $                   NBMIN, NX
00221 *     ..
00222 *     .. External Subroutines ..
00223       EXTERNAL           XERBLA, ZHER2K, ZHETD2, ZLATRD
00224 *     ..
00225 *     .. Intrinsic Functions ..
00226       INTRINSIC          MAX
00227 *     ..
00228 *     .. External Functions ..
00229       LOGICAL            LSAME
00230       INTEGER            ILAENV
00231       EXTERNAL           LSAME, ILAENV
00232 *     ..
00233 *     .. Executable Statements ..
00234 *
00235 *     Test the input parameters
00236 *
00237       INFO = 0
00238       UPPER = LSAME( UPLO, 'U' )
00239       LQUERY = ( LWORK.EQ.-1 )
00240       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00241          INFO = -1
00242       ELSE IF( N.LT.0 ) THEN
00243          INFO = -2
00244       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00245          INFO = -4
00246       ELSE IF( LWORK.LT.1 .AND. .NOT.LQUERY ) THEN
00247          INFO = -9
00248       END IF
00249 *
00250       IF( INFO.EQ.0 ) THEN
00251 *
00252 *        Determine the block size.
00253 *
00254          NB = ILAENV( 1, 'ZHETRD', UPLO, N, -1, -1, -1 )
00255          LWKOPT = N*NB
00256          WORK( 1 ) = LWKOPT
00257       END IF
00258 *
00259       IF( INFO.NE.0 ) THEN
00260          CALL XERBLA( 'ZHETRD', -INFO )
00261          RETURN
00262       ELSE IF( LQUERY ) THEN
00263          RETURN
00264       END IF
00265 *
00266 *     Quick return if possible
00267 *
00268       IF( N.EQ.0 ) THEN
00269          WORK( 1 ) = 1
00270          RETURN
00271       END IF
00272 *
00273       NX = N
00274       IWS = 1
00275       IF( NB.GT.1 .AND. NB.LT.N ) THEN
00276 *
00277 *        Determine when to cross over from blocked to unblocked code
00278 *        (last block is always handled by unblocked code).
00279 *
00280          NX = MAX( NB, ILAENV( 3, 'ZHETRD', UPLO, N, -1, -1, -1 ) )
00281          IF( NX.LT.N ) THEN
00282 *
00283 *           Determine if workspace is large enough for blocked code.
00284 *
00285             LDWORK = N
00286             IWS = LDWORK*NB
00287             IF( LWORK.LT.IWS ) THEN
00288 *
00289 *              Not enough workspace to use optimal NB:  determine the
00290 *              minimum value of NB, and reduce NB or force use of
00291 *              unblocked code by setting NX = N.
00292 *
00293                NB = MAX( LWORK / LDWORK, 1 )
00294                NBMIN = ILAENV( 2, 'ZHETRD', UPLO, N, -1, -1, -1 )
00295                IF( NB.LT.NBMIN )
00296      $            NX = N
00297             END IF
00298          ELSE
00299             NX = N
00300          END IF
00301       ELSE
00302          NB = 1
00303       END IF
00304 *
00305       IF( UPPER ) THEN
00306 *
00307 *        Reduce the upper triangle of A.
00308 *        Columns 1:kk are handled by the unblocked method.
00309 *
00310          KK = N - ( ( N-NX+NB-1 ) / NB )*NB
00311          DO 20 I = N - NB + 1, KK + 1, -NB
00312 *
00313 *           Reduce columns i:i+nb-1 to tridiagonal form and form the
00314 *           matrix W which is needed to update the unreduced part of
00315 *           the matrix
00316 *
00317             CALL ZLATRD( UPLO, I+NB-1, NB, A, LDA, E, TAU, WORK,
00318      $                   LDWORK )
00319 *
00320 *           Update the unreduced submatrix A(1:i-1,1:i-1), using an
00321 *           update of the form:  A := A - V*W**H - W*V**H
00322 *
00323             CALL ZHER2K( UPLO, 'No transpose', I-1, NB, -CONE,
00324      $                   A( 1, I ), LDA, WORK, LDWORK, ONE, A, LDA )
00325 *
00326 *           Copy superdiagonal elements back into A, and diagonal
00327 *           elements into D
00328 *
00329             DO 10 J = I, I + NB - 1
00330                A( J-1, J ) = E( J-1 )
00331                D( J ) = A( J, J )
00332    10       CONTINUE
00333    20    CONTINUE
00334 *
00335 *        Use unblocked code to reduce the last or only block
00336 *
00337          CALL ZHETD2( UPLO, KK, A, LDA, D, E, TAU, IINFO )
00338       ELSE
00339 *
00340 *        Reduce the lower triangle of A
00341 *
00342          DO 40 I = 1, N - NX, NB
00343 *
00344 *           Reduce columns i:i+nb-1 to tridiagonal form and form the
00345 *           matrix W which is needed to update the unreduced part of
00346 *           the matrix
00347 *
00348             CALL ZLATRD( UPLO, N-I+1, NB, A( I, I ), LDA, E( I ),
00349      $                   TAU( I ), WORK, LDWORK )
00350 *
00351 *           Update the unreduced submatrix A(i+nb:n,i+nb:n), using
00352 *           an update of the form:  A := A - V*W**H - W*V**H
00353 *
00354             CALL ZHER2K( UPLO, 'No transpose', N-I-NB+1, NB, -CONE,
00355      $                   A( I+NB, I ), LDA, WORK( NB+1 ), LDWORK, ONE,
00356      $                   A( I+NB, I+NB ), LDA )
00357 *
00358 *           Copy subdiagonal elements back into A, and diagonal
00359 *           elements into D
00360 *
00361             DO 30 J = I, I + NB - 1
00362                A( J+1, J ) = E( J )
00363                D( J ) = A( J, J )
00364    30       CONTINUE
00365    40    CONTINUE
00366 *
00367 *        Use unblocked code to reduce the last or only block
00368 *
00369          CALL ZHETD2( UPLO, N-I+1, A( I, I ), LDA, D( I ), E( I ),
00370      $                TAU( I ), IINFO )
00371       END IF
00372 *
00373       WORK( 1 ) = LWKOPT
00374       RETURN
00375 *
00376 *     End of ZHETRD
00377 *
00378       END
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