LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
dsytd2.f
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00001 *> \brief \b DSYTD2
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 DSYTD2( 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 *       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * )
00029 *       ..
00030 *  
00031 *
00032 *> \par Purpose:
00033 *  =============
00034 *>
00035 *> \verbatim
00036 *>
00037 *> DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
00038 *> form T by an orthogonal similarity transformation: Q**T * A * Q = T.
00039 *> \endverbatim
00040 *
00041 *  Arguments:
00042 *  ==========
00043 *
00044 *> \param[in] UPLO
00045 *> \verbatim
00046 *>          UPLO is CHARACTER*1
00047 *>          Specifies whether the upper or lower triangular part of the
00048 *>          symmetric matrix A is stored:
00049 *>          = 'U':  Upper triangular
00050 *>          = 'L':  Lower triangular
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 DOUBLE PRECISION array, dimension (LDA,N)
00062 *>          On entry, the symmetric 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 orthogonal
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 orthogonal 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 DOUBLE PRECISION array, dimension (N-1)
00104 *>          The scalar factors of the elementary reflectors (see Further
00105 *>          Details).
00106 *> \endverbatim
00107 *>
00108 *> \param[out] INFO
00109 *> \verbatim
00110 *>          INFO is INTEGER
00111 *>          = 0:  successful exit
00112 *>          < 0:  if INFO = -i, the i-th argument had an illegal value.
00113 *> \endverbatim
00114 *
00115 *  Authors:
00116 *  ========
00117 *
00118 *> \author Univ. of Tennessee 
00119 *> \author Univ. of California Berkeley 
00120 *> \author Univ. of Colorado Denver 
00121 *> \author NAG Ltd. 
00122 *
00123 *> \date November 2011
00124 *
00125 *> \ingroup doubleSYcomputational
00126 *
00127 *> \par Further Details:
00128 *  =====================
00129 *>
00130 *> \verbatim
00131 *>
00132 *>  If UPLO = 'U', the matrix Q is represented as a product of elementary
00133 *>  reflectors
00134 *>
00135 *>     Q = H(n-1) . . . H(2) H(1).
00136 *>
00137 *>  Each H(i) has the form
00138 *>
00139 *>     H(i) = I - tau * v * v**T
00140 *>
00141 *>  where tau is a real scalar, and v is a real vector with
00142 *>  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
00143 *>  A(1:i-1,i+1), and tau in TAU(i).
00144 *>
00145 *>  If UPLO = 'L', the matrix Q is represented as a product of elementary
00146 *>  reflectors
00147 *>
00148 *>     Q = H(1) H(2) . . . H(n-1).
00149 *>
00150 *>  Each H(i) has the form
00151 *>
00152 *>     H(i) = I - tau * v * v**T
00153 *>
00154 *>  where tau is a real scalar, and v is a real vector with
00155 *>  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
00156 *>  and tau in TAU(i).
00157 *>
00158 *>  The contents of A on exit are illustrated by the following examples
00159 *>  with n = 5:
00160 *>
00161 *>  if UPLO = 'U':                       if UPLO = 'L':
00162 *>
00163 *>    (  d   e   v2  v3  v4 )              (  d                  )
00164 *>    (      d   e   v3  v4 )              (  e   d              )
00165 *>    (          d   e   v4 )              (  v1  e   d          )
00166 *>    (              d   e  )              (  v1  v2  e   d      )
00167 *>    (                  d  )              (  v1  v2  v3  e   d  )
00168 *>
00169 *>  where d and e denote diagonal and off-diagonal elements of T, and vi
00170 *>  denotes an element of the vector defining H(i).
00171 *> \endverbatim
00172 *>
00173 *  =====================================================================
00174       SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
00175 *
00176 *  -- LAPACK computational routine (version 3.4.0) --
00177 *  -- LAPACK is a software package provided by Univ. of Tennessee,    --
00178 *  -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..--
00179 *     November 2011
00180 *
00181 *     .. Scalar Arguments ..
00182       CHARACTER          UPLO
00183       INTEGER            INFO, LDA, N
00184 *     ..
00185 *     .. Array Arguments ..
00186       DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * )
00187 *     ..
00188 *
00189 *  =====================================================================
00190 *
00191 *     .. Parameters ..
00192       DOUBLE PRECISION   ONE, ZERO, HALF
00193       PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0,
00194      $                   HALF = 1.0D0 / 2.0D0 )
00195 *     ..
00196 *     .. Local Scalars ..
00197       LOGICAL            UPPER
00198       INTEGER            I
00199       DOUBLE PRECISION   ALPHA, TAUI
00200 *     ..
00201 *     .. External Subroutines ..
00202       EXTERNAL           DAXPY, DLARFG, DSYMV, DSYR2, XERBLA
00203 *     ..
00204 *     .. External Functions ..
00205       LOGICAL            LSAME
00206       DOUBLE PRECISION   DDOT
00207       EXTERNAL           LSAME, DDOT
00208 *     ..
00209 *     .. Intrinsic Functions ..
00210       INTRINSIC          MAX, MIN
00211 *     ..
00212 *     .. Executable Statements ..
00213 *
00214 *     Test the input parameters
00215 *
00216       INFO = 0
00217       UPPER = LSAME( UPLO, 'U' )
00218       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00219          INFO = -1
00220       ELSE IF( N.LT.0 ) THEN
00221          INFO = -2
00222       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00223          INFO = -4
00224       END IF
00225       IF( INFO.NE.0 ) THEN
00226          CALL XERBLA( 'DSYTD2', -INFO )
00227          RETURN
00228       END IF
00229 *
00230 *     Quick return if possible
00231 *
00232       IF( N.LE.0 )
00233      $   RETURN
00234 *
00235       IF( UPPER ) THEN
00236 *
00237 *        Reduce the upper triangle of A
00238 *
00239          DO 10 I = N - 1, 1, -1
00240 *
00241 *           Generate elementary reflector H(i) = I - tau * v * v**T
00242 *           to annihilate A(1:i-1,i+1)
00243 *
00244             CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
00245             E( I ) = A( I, I+1 )
00246 *
00247             IF( TAUI.NE.ZERO ) THEN
00248 *
00249 *              Apply H(i) from both sides to A(1:i,1:i)
00250 *
00251                A( I, I+1 ) = ONE
00252 *
00253 *              Compute  x := tau * A * v  storing x in TAU(1:i)
00254 *
00255                CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
00256      $                     TAU, 1 )
00257 *
00258 *              Compute  w := x - 1/2 * tau * (x**T * v) * v
00259 *
00260                ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 )
00261                CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
00262 *
00263 *              Apply the transformation as a rank-2 update:
00264 *                 A := A - v * w**T - w * v**T
00265 *
00266                CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
00267      $                     LDA )
00268 *
00269                A( I, I+1 ) = E( I )
00270             END IF
00271             D( I+1 ) = A( I+1, I+1 )
00272             TAU( I ) = TAUI
00273    10    CONTINUE
00274          D( 1 ) = A( 1, 1 )
00275       ELSE
00276 *
00277 *        Reduce the lower triangle of A
00278 *
00279          DO 20 I = 1, N - 1
00280 *
00281 *           Generate elementary reflector H(i) = I - tau * v * v**T
00282 *           to annihilate A(i+2:n,i)
00283 *
00284             CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
00285      $                   TAUI )
00286             E( I ) = A( I+1, I )
00287 *
00288             IF( TAUI.NE.ZERO ) THEN
00289 *
00290 *              Apply H(i) from both sides to A(i+1:n,i+1:n)
00291 *
00292                A( I+1, I ) = ONE
00293 *
00294 *              Compute  x := tau * A * v  storing y in TAU(i:n-1)
00295 *
00296                CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
00297      $                     A( I+1, I ), 1, ZERO, TAU( I ), 1 )
00298 *
00299 *              Compute  w := x - 1/2 * tau * (x**T * v) * v
00300 *
00301                ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ),
00302      $                 1 )
00303                CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
00304 *
00305 *              Apply the transformation as a rank-2 update:
00306 *                 A := A - v * w**T - w * v**T
00307 *
00308                CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
00309      $                     A( I+1, I+1 ), LDA )
00310 *
00311                A( I+1, I ) = E( I )
00312             END IF
00313             D( I ) = A( I, I )
00314             TAU( I ) = TAUI
00315    20    CONTINUE
00316          D( N ) = A( N, N )
00317       END IF
00318 *
00319       RETURN
00320 *
00321 *     End of DSYTD2
00322 *
00323       END
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