dlatrd.f.html

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      SUBROUTINE <a name="DLATRD.1"></a><a href="dlatrd.f.html#DLATRD.1">DLATRD</a>( UPLO, N, NB, A, LDA, E, TAU, W, LDW )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK auxiliary routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      CHARACTER          UPLO
      INTEGER            LDA, LDW, N, NB
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      DOUBLE PRECISION   A( LDA, * ), E( * ), TAU( * ), W( LDW, * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="DLATRD.18"></a><a href="dlatrd.f.html#DLATRD.1">DLATRD</a> reduces NB rows and columns of a real symmetric matrix A to
</span><span class="comment">*</span><span class="comment">  symmetric tridiagonal form by an orthogonal similarity
</span><span class="comment">*</span><span class="comment">  transformation Q' * A * Q, and returns the matrices V and W which are
</span><span class="comment">*</span><span class="comment">  needed to apply the transformation to the unreduced part of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  If UPLO = 'U', <a name="DLATRD.23"></a><a href="dlatrd.f.html#DLATRD.1">DLATRD</a> reduces the last NB rows and columns of a
</span><span class="comment">*</span><span class="comment">  matrix, of which the upper triangle is supplied;
</span><span class="comment">*</span><span class="comment">  if UPLO = 'L', <a name="DLATRD.25"></a><a href="dlatrd.f.html#DLATRD.1">DLATRD</a> reduces the first NB rows and columns of a
</span><span class="comment">*</span><span class="comment">  matrix, of which the lower triangle is supplied.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  This is an auxiliary routine called by <a name="DSYTRD.28"></a><a href="dsytrd.f.html#DSYTRD.1">DSYTRD</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  UPLO    (input) CHARACTER*1
</span><span class="comment">*</span><span class="comment">          Specifies whether the upper or lower triangular part of the
</span><span class="comment">*</span><span class="comment">          symmetric matrix A is stored:
</span><span class="comment">*</span><span class="comment">          = 'U': Upper triangular
</span><span class="comment">*</span><span class="comment">          = 'L': Lower triangular
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The order of the matrix A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  NB      (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of rows and columns to be reduced.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment">          On entry, the symmetric matrix A.  If UPLO = 'U', the leading
</span><span class="comment">*</span><span class="comment">          n-by-n upper triangular part of A contains the upper
</span><span class="comment">*</span><span class="comment">          triangular part of the matrix A, and the strictly lower
</span><span class="comment">*</span><span class="comment">          triangular part of A is not referenced.  If UPLO = 'L', the
</span><span class="comment">*</span><span class="comment">          leading n-by-n lower triangular part of A contains the lower
</span><span class="comment">*</span><span class="comment">          triangular part of the matrix A, and the strictly upper
</span><span class="comment">*</span><span class="comment">          triangular part of A is not referenced.
</span><span class="comment">*</span><span class="comment">          On exit:
</span><span class="comment">*</span><span class="comment">          if UPLO = 'U', the last NB columns have been reduced to
</span><span class="comment">*</span><span class="comment">            tridiagonal form, with the diagonal elements overwriting
</span><span class="comment">*</span><span class="comment">            the diagonal elements of A; the elements above the diagonal
</span><span class="comment">*</span><span class="comment">            with the array TAU, represent the orthogonal matrix Q as a
</span><span class="comment">*</span><span class="comment">            product of elementary reflectors;
</span><span class="comment">*</span><span class="comment">          if UPLO = 'L', the first NB columns have been reduced to
</span><span class="comment">*</span><span class="comment">            tridiagonal form, with the diagonal elements overwriting
</span><span class="comment">*</span><span class="comment">            the diagonal elements of A; the elements below the diagonal
</span><span class="comment">*</span><span class="comment">            with the array TAU, represent the  orthogonal matrix Q as a
</span><span class="comment">*</span><span class="comment">            product of elementary reflectors.
</span><span class="comment">*</span><span class="comment">          See Further Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDA     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array A.  LDA &gt;= (1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  E       (output) DOUBLE PRECISION array, dimension (N-1)
</span><span class="comment">*</span><span class="comment">          If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
</span><span class="comment">*</span><span class="comment">          elements of the last NB columns of the reduced matrix;
</span><span class="comment">*</span><span class="comment">          if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
</span><span class="comment">*</span><span class="comment">          the first NB columns of the reduced matrix.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  TAU     (output) DOUBLE PRECISION array, dimension (N-1)
</span><span class="comment">*</span><span class="comment">          The scalar factors of the elementary reflectors, stored in
</span><span class="comment">*</span><span class="comment">          TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
</span><span class="comment">*</span><span class="comment">          See Further Details.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  W       (output) DOUBLE PRECISION array, dimension (LDW,NB)
</span><span class="comment">*</span><span class="comment">          The n-by-nb matrix W required to update the unreduced part
</span><span class="comment">*</span><span class="comment">          of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDW     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array W. LDW &gt;= max(1,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Further Details
</span><span class="comment">*</span><span class="comment">  ===============
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  If UPLO = 'U', the matrix Q is represented as a product of elementary
</span><span class="comment">*</span><span class="comment">  reflectors
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Q = H(n) H(n-1) . . . H(n-nb+1).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Each H(i) has the form
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     H(i) = I - tau * v * v'
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where tau is a real scalar, and v is a real vector with
</span><span class="comment">*</span><span class="comment">  v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
</span><span class="comment">*</span><span class="comment">  and tau in TAU(i-1).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  If UPLO = 'L', the matrix Q is represented as a product of elementary
</span><span class="comment">*</span><span class="comment">  reflectors
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Q = H(1) H(2) . . . H(nb).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Each H(i) has the form
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     H(i) = I - tau * v * v'
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  where tau is a real scalar, and v is a real vector with
</span><span class="comment">*</span><span class="comment">  v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
</span><span class="comment">*</span><span class="comment">  and tau in TAU(i).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The elements of the vectors v together form the n-by-nb matrix V
</span><span class="comment">*</span><span class="comment">  which is needed, with W, to apply the transformation to the unreduced
</span><span class="comment">*</span><span class="comment">  part of the matrix, using a symmetric rank-2k update of the form:
</span><span class="comment">*</span><span class="comment">  A := A - V*W' - W*V'.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The contents of A on exit are illustrated by the following examples
</span><span class="comment">*</span><span class="comment">  with n = 5 and nb = 2:
</span><span class="comment">*</span><span class="comment">
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