LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
chetd2.f
Go to the documentation of this file.
00001 *> \brief \b CHETD2
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
00008 *> \htmlonly
00009 *> Download CHETD2 + dependencies 
00010 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.tgz?format=tgz&filename=/lapack/lapack_routine/chetd2.f"> 
00011 *> [TGZ]</a> 
00012 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.zip?format=zip&filename=/lapack/lapack_routine/chetd2.f"> 
00013 *> [ZIP]</a> 
00014 *> <a href="http://www.netlib.org/cgi-bin/netlibfiles.txt?format=txt&filename=/lapack/lapack_routine/chetd2.f"> 
00015 *> [TXT]</a>
00016 *> \endhtmlonly 
00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE CHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
00022 * 
00023 *       .. Scalar Arguments ..
00024 *       CHARACTER          UPLO
00025 *       INTEGER            INFO, LDA, N
00026 *       ..
00027 *       .. Array Arguments ..
00028 *       REAL               D( * ), E( * )
00029 *       COMPLEX            A( LDA, * ), TAU( * )
00030 *       ..
00031 *  
00032 *
00033 *> \par Purpose:
00034 *  =============
00035 *>
00036 *> \verbatim
00037 *>
00038 *> CHETD2 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 *>          Specifies whether the upper or lower triangular part of the
00050 *>          Hermitian matrix A is stored:
00051 *>          = 'U':  Upper triangular
00052 *>          = 'L':  Lower triangular
00053 *> \endverbatim
00054 *>
00055 *> \param[in] N
00056 *> \verbatim
00057 *>          N is INTEGER
00058 *>          The order of the matrix A.  N >= 0.
00059 *> \endverbatim
00060 *>
00061 *> \param[in,out] A
00062 *> \verbatim
00063 *>          A is COMPLEX array, dimension (LDA,N)
00064 *>          On entry, the Hermitian matrix A.  If UPLO = 'U', the leading
00065 *>          n-by-n upper triangular part of A contains the upper
00066 *>          triangular part of the matrix A, and the strictly lower
00067 *>          triangular part of A is not referenced.  If UPLO = 'L', the
00068 *>          leading n-by-n lower triangular part of A contains the lower
00069 *>          triangular part of the matrix A, and the strictly upper
00070 *>          triangular part of A is not referenced.
00071 *>          On exit, if UPLO = 'U', the diagonal and first superdiagonal
00072 *>          of A are overwritten by the corresponding elements of the
00073 *>          tridiagonal matrix T, and the elements above the first
00074 *>          superdiagonal, with the array TAU, represent the unitary
00075 *>          matrix Q as a product of elementary reflectors; if UPLO
00076 *>          = 'L', the diagonal and first subdiagonal of A are over-
00077 *>          written by the corresponding elements of the tridiagonal
00078 *>          matrix T, and the elements below the first subdiagonal, with
00079 *>          the array TAU, represent the unitary matrix Q as a product
00080 *>          of elementary reflectors. See Further Details.
00081 *> \endverbatim
00082 *>
00083 *> \param[in] LDA
00084 *> \verbatim
00085 *>          LDA is INTEGER
00086 *>          The leading dimension of the array A.  LDA >= max(1,N).
00087 *> \endverbatim
00088 *>
00089 *> \param[out] D
00090 *> \verbatim
00091 *>          D is REAL array, dimension (N)
00092 *>          The diagonal elements of the tridiagonal matrix T:
00093 *>          D(i) = A(i,i).
00094 *> \endverbatim
00095 *>
00096 *> \param[out] E
00097 *> \verbatim
00098 *>          E is REAL array, dimension (N-1)
00099 *>          The off-diagonal elements of the tridiagonal matrix T:
00100 *>          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
00101 *> \endverbatim
00102 *>
00103 *> \param[out] TAU
00104 *> \verbatim
00105 *>          TAU is COMPLEX array, dimension (N-1)
00106 *>          The scalar factors of the elementary reflectors (see Further
00107 *>          Details).
00108 *> \endverbatim
00109 *>
00110 *> \param[out] INFO
00111 *> \verbatim
00112 *>          INFO is INTEGER
00113 *>          = 0:  successful exit
00114 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
00115 *> \endverbatim
00116 *
00117 *  Authors:
00118 *  ========
00119 *
00120 *> \author Univ. of Tennessee 
00121 *> \author Univ. of California Berkeley 
00122 *> \author Univ. of Colorado Denver 
00123 *> \author NAG Ltd. 
00124 *
00125 *> \date November 2011
00126 *
00127 *> \ingroup complexHEcomputational
00128 *
00129 *> \par Further Details:
00130 *  =====================
00131 *>
00132 *> \verbatim
00133 *>
00134 *>  If UPLO = 'U', the matrix Q is represented as a product of elementary
00135 *>  reflectors
00136 *>
00137 *>     Q = H(n-1) . . . H(2) H(1).
00138 *>
00139 *>  Each H(i) has the form
00140 *>
00141 *>     H(i) = I - tau * v * v**H
00142 *>
00143 *>  where tau is a complex scalar, and v is a complex vector with
00144 *>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
00145 *>  A(1:i-1,i+1), and tau in TAU(i).
00146 *>
00147 *>  If UPLO = 'L', the matrix Q is represented as a product of elementary
00148 *>  reflectors
00149 *>
00150 *>     Q = H(1) H(2) . . . H(n-1).
00151 *>
00152 *>  Each H(i) has the form
00153 *>
00154 *>     H(i) = I - tau * v * v**H
00155 *>
00156 *>  where tau is a complex scalar, and v is a complex vector with
00157 *>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
00158 *>  and tau in TAU(i).
00159 *>
00160 *>  The contents of A on exit are illustrated by the following examples
00161 *>  with n = 5:
00162 *>
00163 *>  if UPLO = 'U':                       if UPLO = 'L':
00164 *>
00165 *>    (  d   e   v2  v3  v4 )              (  d                  )
00166 *>    (      d   e   v3  v4 )              (  e   d              )
00167 *>    (          d   e   v4 )              (  v1  e   d          )
00168 *>    (              d   e  )              (  v1  v2  e   d      )
00169 *>    (                  d  )              (  v1  v2  v3  e   d  )
00170 *>
00171 *>  where d and e denote diagonal and off-diagonal elements of T, and vi
00172 *>  denotes an element of the vector defining H(i).
00173 *> \endverbatim
00174 *>
00175 *  =====================================================================
00176       SUBROUTINE CHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )
00177 *
00178 *  -- LAPACK computational routine (version 3.4.0) --
00179 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00180 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00181 *     November 2011
00182 *
00183 *     .. Scalar Arguments ..
00184       CHARACTER          UPLO
00185       INTEGER            INFO, LDA, N
00186 *     ..
00187 *     .. Array Arguments ..
00188       REAL               D( * ), E( * )
00189       COMPLEX            A( LDA, * ), TAU( * )
00190 *     ..
00191 *
00192 *  =====================================================================
00193 *
00194 *     .. Parameters ..
00195       COMPLEX            ONE, ZERO, HALF
00196       PARAMETER          ( ONE = ( 1.0E+0, 0.0E+0 ),
00197      $                   ZERO = ( 0.0E+0, 0.0E+0 ),
00198      $                   HALF = ( 0.5E+0, 0.0E+0 ) )
00199 *     ..
00200 *     .. Local Scalars ..
00201       LOGICAL            UPPER
00202       INTEGER            I
00203       COMPLEX            ALPHA, TAUI
00204 *     ..
00205 *     .. External Subroutines ..
00206       EXTERNAL           CAXPY, CHEMV, CHER2, CLARFG, XERBLA
00207 *     ..
00208 *     .. External Functions ..
00209       LOGICAL            LSAME
00210       COMPLEX            CDOTC
00211       EXTERNAL           LSAME, CDOTC
00212 *     ..
00213 *     .. Intrinsic Functions ..
00214       INTRINSIC          MAX, MIN, REAL
00215 *     ..
00216 *     .. Executable Statements ..
00217 *
00218 *     Test the input parameters
00219 *
00220       INFO = 0
00221       UPPER = LSAME( UPLO, 'U' )
00222       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00223          INFO = -1
00224       ELSE IF( N.LT.0 ) THEN
00225          INFO = -2
00226       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00227          INFO = -4
00228       END IF
00229       IF( INFO.NE.0 ) THEN
00230          CALL XERBLA( 'CHETD2', -INFO )
00231          RETURN
00232       END IF
00233 *
00234 *     Quick return if possible
00235 *
00236       IF( N.LE.0 )
00237      $   RETURN
00238 *
00239       IF( UPPER ) THEN
00240 *
00241 *        Reduce the upper triangle of A
00242 *
00243          A( N, N ) = REAL( A( N, N ) )
00244          DO 10 I = N - 1, 1, -1
00245 *
00246 *           Generate elementary reflector H(i) = I - tau * v * v**H
00247 *           to annihilate A(1:i-1,i+1)
00248 *
00249             ALPHA = A( I, I+1 )
00250             CALL CLARFG( I, ALPHA, A( 1, I+1 ), 1, TAUI )
00251             E( I ) = ALPHA
00252 *
00253             IF( TAUI.NE.ZERO ) THEN
00254 *
00255 *              Apply H(i) from both sides to A(1:i,1:i)
00256 *
00257                A( I, I+1 ) = ONE
00258 *
00259 *              Compute  x := tau * A * v  storing x in TAU(1:i)
00260 *
00261                CALL CHEMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
00262      $                     TAU, 1 )
00263 *
00264 *              Compute  w := x - 1/2 * tau * (x**H * v) * v
00265 *
00266                ALPHA = -HALF*TAUI*CDOTC( I, TAU, 1, A( 1, I+1 ), 1 )
00267                CALL CAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
00268 *
00269 *              Apply the transformation as a rank-2 update:
00270 *                 A := A - v * w**H - w * v**H
00271 *
00272                CALL CHER2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
00273      $                     LDA )
00274 *
00275             ELSE
00276                A( I, I ) = REAL( A( I, I ) )
00277             END IF
00278             A( I, I+1 ) = E( I )
00279             D( I+1 ) = A( I+1, I+1 )
00280             TAU( I ) = TAUI
00281    10    CONTINUE
00282          D( 1 ) = A( 1, 1 )
00283       ELSE
00284 *
00285 *        Reduce the lower triangle of A
00286 *
00287          A( 1, 1 ) = REAL( A( 1, 1 ) )
00288          DO 20 I = 1, N - 1
00289 *
00290 *           Generate elementary reflector H(i) = I - tau * v * v**H
00291 *           to annihilate A(i+2:n,i)
00292 *
00293             ALPHA = A( I+1, I )
00294             CALL CLARFG( N-I, ALPHA, A( MIN( I+2, N ), I ), 1, TAUI )
00295             E( I ) = ALPHA
00296 *
00297             IF( TAUI.NE.ZERO ) THEN
00298 *
00299 *              Apply H(i) from both sides to A(i+1:n,i+1:n)
00300 *
00301                A( I+1, I ) = ONE
00302 *
00303 *              Compute  x := tau * A * v  storing y in TAU(i:n-1)
00304 *
00305                CALL CHEMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
00306      $                     A( I+1, I ), 1, ZERO, TAU( I ), 1 )
00307 *
00308 *              Compute  w := x - 1/2 * tau * (x**H * v) * v
00309 *
00310                ALPHA = -HALF*TAUI*CDOTC( N-I, TAU( I ), 1, A( I+1, I ),
00311      $                 1 )
00312                CALL CAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
00313 *
00314 *              Apply the transformation as a rank-2 update:
00315 *                 A := A - v * w**H - w * v**H
00316 *
00317                CALL CHER2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
00318      $                     A( I+1, I+1 ), LDA )
00319 *
00320             ELSE
00321                A( I+1, I+1 ) = REAL( A( I+1, I+1 ) )
00322             END IF
00323             A( I+1, I ) = E( I )
00324             D( I ) = A( I, I )
00325             TAU( I ) = TAUI
00326    20    CONTINUE
00327          D( N ) = A( N, N )
00328       END IF
00329 *
00330       RETURN
00331 *
00332 *     End of CHETD2
00333 *
00334       END
 All Files Functions