dsytd2.f

          SUBROUTINE DSYTD2( UPLO, N, A, LDA, D, E, TAU, INFO )
    *
    *  -- LAPACK routine (version 3.1) --
    *     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
    *     November 2006
    *
    *     .. Scalar Arguments ..
          CHARACTER          UPLO
          INTEGER            INFO, LDA, N
    *     ..
    *     .. Array Arguments ..
          DOUBLE PRECISION   A( LDA, * ), D( * ), E( * ), TAU( * )
    *     ..
    *
    *  Purpose
    *  =======
    *
    *  DSYTD2 reduces a real symmetric matrix A to symmetric tridiagonal
    *  form T by an orthogonal similarity transformation: Q' * A * Q = T.
    *
    *  Arguments
    *  =========
    *
    *  UPLO    (input) CHARACTER*1
    *          Specifies whether the upper or lower triangular part of the
    *          symmetric matrix A is stored:
    *          = 'U':  Upper triangular
    *          = 'L':  Lower triangular
    *
    *  N       (input) INTEGER
    *          The order of the matrix A.  N >= 0.
    *
    *  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
    *          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
    *          n-by-n upper triangular part of A contains the upper
    *          triangular part of the matrix A, and the strictly lower
    *          triangular part of A is not referenced.  If UPLO = 'L', the
    *          leading n-by-n lower triangular part of A contains the lower
    *          triangular part of the matrix A, and the strictly upper
    *          triangular part of A is not referenced.
    *          On exit, if UPLO = 'U', the diagonal and first superdiagonal
    *          of A are overwritten by the corresponding elements of the
    *          tridiagonal matrix T, and the elements above the first
    *          superdiagonal, with the array TAU, represent the orthogonal
    *          matrix Q as a product of elementary reflectors; if UPLO
    *          = 'L', the diagonal and first subdiagonal of A are over-
    *          written by the corresponding elements of the tridiagonal
    *          matrix T, and the elements below the first subdiagonal, with
    *          the array TAU, represent the orthogonal matrix Q as a product
    *          of elementary reflectors. See Further Details.
    *
    *  LDA     (input) INTEGER
    *          The leading dimension of the array A.  LDA >= max(1,N).
    *
    *  D       (output) DOUBLE PRECISION array, dimension (N)
    *          The diagonal elements of the tridiagonal matrix T:
    *          D(i) = A(i,i).
    *
    *  E       (output) DOUBLE PRECISION array, dimension (N-1)
    *          The off-diagonal elements of the tridiagonal matrix T:
    *          E(i) = A(i,i+1) if UPLO = 'U', E(i) = A(i+1,i) if UPLO = 'L'.
    *
    *  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
    *          The scalar factors of the elementary reflectors (see Further
    *          Details).
    *
    *  INFO    (output) INTEGER
    *          = 0:  successful exit
    *          < 0:  if INFO = -i, the i-th argument had an illegal value.
    *
    *  Further Details
    *  ===============
    *
    *  If UPLO = 'U', the matrix Q is represented as a product of elementary
    *  reflectors
    *
    *     Q = H(n-1) . . . H(2) H(1).
    *
    *  Each H(i) has the form
    *
    *     H(i) = I - tau * v * v'
    *
    *  where tau is a real scalar, and v is a real vector with
    *  v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in
    *  A(1:i-1,i+1), and tau in TAU(i).
    *
    *  If UPLO = 'L', the matrix Q is represented as a product of elementary
    *  reflectors
    *
    *     Q = H(1) H(2) . . . H(n-1).
    *
    *  Each H(i) has the form
    *
    *     H(i) = I - tau * v * v'
    *
    *  where tau is a real scalar, and v is a real vector with
    *  v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in A(i+2:n,i),
    *  and tau in TAU(i).
    *
    *  The contents of A on exit are illustrated by the following examples
    *  with n = 5:
    *
    *  if UPLO = 'U':                       if UPLO = 'L':
    *
    *    (  d   e   v2  v3  v4 )              (  d                  )
    *    (      d   e   v3  v4 )              (  e   d              )
    *    (          d   e   v4 )              (  v1  e   d          )
    *    (              d   e  )              (  v1  v2  e   d      )
    *    (                  d  )              (  v1  v2  v3  e   d  )
    *
    *  where d and e denote diagonal and off-diagonal elements of T, and vi
    *  denotes an element of the vector defining H(i).
    *
    *  =====================================================================
    *
    *     .. Parameters ..
          DOUBLE PRECISION   ONE, ZERO, HALF
          PARAMETER          ( ONE = 1.0D0, ZERO = 0.0D0,
         $                   HALF = 1.0D0 / 2.0D0 )
    *     ..
    *     .. Local Scalars ..
          LOGICAL            UPPER
          INTEGER            I
          DOUBLE PRECISION   ALPHA, TAUI
    *     ..
    *     .. External Subroutines ..
          EXTERNAL           DAXPY, DLARFG, DSYMV, DSYR2, XERBLA
    *     ..
    *     .. External Functions ..
          LOGICAL            LSAME
          DOUBLE PRECISION   DDOT
          EXTERNAL           LSAME, DDOT
    *     ..
    *     .. Intrinsic Functions ..
          INTRINSIC          MAX, MIN
    *     ..
    *     .. Executable Statements ..
    *
    *     Test the input parameters
    *
          INFO = 0
          UPPER = LSAME( UPLO, 'U' )
          IF( .NOT.UPPER .AND. .NOT.LSAME( UPLO, 'L' ) ) THEN
             INFO = -1
          ELSE IF( N.LT.0 ) THEN
             INFO = -2
          ELSE IF( LDA.LT.MAX( 1, N ) ) THEN
             INFO = -4
          END IF
          IF( INFO.NE.0 ) THEN
             CALL XERBLA( 'DSYTD2', -INFO )
             RETURN
          END IF
    *
    *     Quick return if possible
    *
          IF( N.LE.0 )
         $   RETURN
    *
          IF( UPPER ) THEN
    *
    *        Reduce the upper triangle of A
    *
             DO 10 I = N - 1, 1, -1
    *
    *           Generate elementary reflector H(i) = I - tau * v * v'
    *           to annihilate A(1:i-1,i+1)
    *
                CALL DLARFG( I, A( I, I+1 ), A( 1, I+1 ), 1, TAUI )
                E( I ) = A( I, I+1 )
    *
                IF( TAUI.NE.ZERO ) THEN
    *
    *              Apply H(i) from both sides to A(1:i,1:i)
    *
                   A( I, I+1 ) = ONE
    *
    *              Compute  x := tau * A * v  storing x in TAU(1:i)
    *
                   CALL DSYMV( UPLO, I, TAUI, A, LDA, A( 1, I+1 ), 1, ZERO,
         $                     TAU, 1 )
    *
    *              Compute  w := x - 1/2 * tau * (x'*v) * v
    *
                   ALPHA = -HALF*TAUI*DDOT( I, TAU, 1, A( 1, I+1 ), 1 )
                   CALL DAXPY( I, ALPHA, A( 1, I+1 ), 1, TAU, 1 )
    *
    *              Apply the transformation as a rank-2 update:
    *                 A := A - v * w' - w * v'
    *
                   CALL DSYR2( UPLO, I, -ONE, A( 1, I+1 ), 1, TAU, 1, A,
         $                     LDA )
    *
                   A( I, I+1 ) = E( I )
                END IF
                D( I+1 ) = A( I+1, I+1 )
                TAU( I ) = TAUI
       10    CONTINUE
             D( 1 ) = A( 1, 1 )
          ELSE
    *
    *        Reduce the lower triangle of A
    *
             DO 20 I = 1, N - 1
    *
    *           Generate elementary reflector H(i) = I - tau * v * v'
    *           to annihilate A(i+2:n,i)
    *
                CALL DLARFG( N-I, A( I+1, I ), A( MIN( I+2, N ), I ), 1,
         $                   TAUI )
                E( I ) = A( I+1, I )
    *
                IF( TAUI.NE.ZERO ) THEN
    *
    *              Apply H(i) from both sides to A(i+1:n,i+1:n)
    *
                   A( I+1, I ) = ONE
    *
    *              Compute  x := tau * A * v  storing y in TAU(i:n-1)
    *
                   CALL DSYMV( UPLO, N-I, TAUI, A( I+1, I+1 ), LDA,
         $                     A( I+1, I ), 1, ZERO, TAU( I ), 1 )
    *
    *              Compute  w := x - 1/2 * tau * (x'*v) * v
    *
                   ALPHA = -HALF*TAUI*DDOT( N-I, TAU( I ), 1, A( I+1, I ),
         $                 1 )
                   CALL DAXPY( N-I, ALPHA, A( I+1, I ), 1, TAU( I ), 1 )
    *
    *              Apply the transformation as a rank-2 update:
    *                 A := A - v * w' - w * v'
    *
                   CALL DSYR2( UPLO, N-I, -ONE, A( I+1, I ), 1, TAU( I ), 1,
         $                     A( I+1, I+1 ), LDA )
    *
                   A( I+1, I ) = E( I )
                END IF
                D( I ) = A( I, I )
                TAU( I ) = TAUI
       20    CONTINUE
             D( N ) = A( N, N )
          END IF
    *
          RETURN
    *
    *     End of DSYTD2
    *
          END