LAPACK  3.4.1
LAPACK: Linear Algebra PACKage
ssytd2.f
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00001 *> \brief \b SSYTD2
00002 *
00003 *  =========== DOCUMENTATION ===========
00004 *
00005 * Online html documentation available at 
00006 *            http://www.netlib.org/lapack/explore-html/ 
00007 *
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00017 *
00018 *  Definition:
00019 *  ===========
00020 *
00021 *       SUBROUTINE SSYTD2( 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               A( LDA, * ), D( * ), E( * ), TAU( * )
00029 *       ..
00030 *  
00031 *
00032 *> \par Purpose:
00033 *  =============
00034 *>
00035 *> \verbatim
00036 *>
00037 *> SSYTD2 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 REAL 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 REAL 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 REAL 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 REAL 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 realSYcomputational
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 SSYTD2( 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       REAL               A( LDA, * ), D( * ), E( * ), TAU( * )
00187 *     ..
00188 *
00189 *  =====================================================================
00190 *
00191 *     .. Parameters ..
00192       REAL               ONE, ZERO, HALF
00193       PARAMETER          ( ONE = 1.0, ZERO = 0.0, HALF = 1.0 / 2.0 )
00194 *     ..
00195 *     .. Local Scalars ..
00196       LOGICAL            UPPER
00197       INTEGER            I
00198       REAL               ALPHA, TAUI
00199 *     ..
00200 *     .. External Subroutines ..
00201       EXTERNAL           SAXPY, SLARFG, SSYMV, SSYR2, XERBLA
00202 *     ..
00203 *     .. External Functions ..
00204       LOGICAL            LSAME
00205       REAL               SDOT
00206       EXTERNAL           LSAME, SDOT
00207 *     ..
00208 *     .. Intrinsic Functions ..
00209       INTRINSIC          MAX, MIN
00210 *     ..
00211 *     .. Executable Statements ..
00212 *
00213 *     Test the input parameters
00214 *
00215       INFO = 0
00216       UPPER = LSAME( UPLO, 'U' )
00217       IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
00218          INFO = -1
00219       ELSE IF( N.LT.0 ) THEN
00220          INFO = -2
00221       ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
00222          INFO = -4
00223       END IF
00224       IF( INFO.NE.0 ) THEN
00225          CALL XERBLA( 'SSYTD2', -INFO )
00226          RETURN
00227       END IF
00228 *
00229 *     Quick return if possible
00230 *
00231       IF( N.LE.0 )
00232      $   RETURN
00233 *
00234       IF( UPPER ) THEN
00235 *
00236 *        Reduce the upper triangle of A
00237 *
00238          DO 10 I = N - 1, 1, -1
00239 *
00240 *           Generate elementary reflector H(i) = I - tau * v * v**T
00241 *           to annihilate A(1:i-1,i+1)
00242 *
00243             CALL SLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
00244             E( I ) = A( I, I+1 )
00245 *
00246             IF( TAUI.NE.ZERO ) THEN
00247 *
00248 *              Apply H(i) from both sides to A(1:i,1:i)
00249 *
00250                A( I, I+1 ) = ONE
00251 *
00252 *              Compute  x := tau * A * v  storing x in TAU(1:i)
00253 *
00254                CALL SSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
00255      $                     TAU, 1 )
00256 *
00257 *              Compute  w := x - 1/2 * tau * (x**T * v) * v
00258 *
00259                ALPHA = -HALF*TAUI*SDOT( I, TAU, 1, A( 1, I+1 ), 1 )
00260                CALL SAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
00261 *
00262 *              Apply the transformation as a rank-2 update:
00263 *                 A := A - v * w**T - w * v**T
00264 *
00265                CALL SSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
00266      $                     LDA )
00267 *
00268                A( I, I+1 ) = E( I )
00269             END IF
00270             D( I+1 ) = A( I+1, I+1 )
00271             TAU( I ) = TAUI
00272    10    CONTINUE
00273          D( 1 ) = A( 1, 1 )
00274       ELSE
00275 *
00276 *        Reduce the lower triangle of A
00277 *
00278          DO 20 I = 1, N - 1
00279 *
00280 *           Generate elementary reflector H(i) = I - tau * v * v**T
00281 *           to annihilate A(i+2:n,i)
00282 *
00283             CALL SLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
00284      $                   TAUI )
00285             E( I ) = A( I+1, I )
00286 *
00287             IF( TAUI.NE.ZERO ) THEN
00288 *
00289 *              Apply H(i) from both sides to A(i+1:n,i+1:n)
00290 *
00291                A( I+1, I ) = ONE
00292 *
00293 *              Compute  x := tau * A * v  storing y in TAU(i:n-1)
00294 *
00295                CALL SSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
00296      $                     A( I+1, I ), 1, ZERO, TAU( I ), 1 )
00297 *
00298 *              Compute  w := x - 1/2 * tau * (x**T * v) * v
00299 *
00300                ALPHA = -HALF*TAUI*SDOT( N-I, TAU( I ), 1, A( I+1, I ),
00301      $                 1 )
00302                CALL SAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
00303 *
00304 *              Apply the transformation as a rank-2 update:
00305 *                 A := A - v * w**T - w * v**T
00306 *
00307                CALL SSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
00308      $                     A( I+1, I+1 ), LDA )
00309 *
00310                A( I+1, I ) = E( I )
00311             END IF
00312             D( I ) = A( I, I )
00313             TAU( I ) = TAUI
00314    20    CONTINUE
00315          D( N ) = A( N, N )
00316       END IF
00317 *
00318       RETURN
00319 *
00320 *     End of SSYTD2
00321 *
00322       END
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