zlatrd.f(3)

NAME
zlatrd.f

SYNOPSIS

  Functions/Subroutines
subroutine zlatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
ZLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

Function/Subroutine Documentation

  subroutine zlatrd (character UPLO, integer N, integer NB, complex*16, dimension( lda, * ) A, integer LDA, double precision, dimension( * ) E, complex*16, dimension( * ) TAU, complex*16, dimension( ldw, * ) W, integer LDW)

Purpose:

     ZLATRD reduces NB rows and columns of a complex Hermitian matrix A to
     Hermitian tridiagonal form by a unitary similarity
     transformation Q**H * A * Q, and returns the matrices V and W which are
     needed to apply the transformation to the unreduced part of A.
     If UPLO = 'U', ZLATRD reduces the last NB rows and columns of a
     matrix, of which the upper triangle is supplied;
     if UPLO = 'L', ZLATRD reduces the first NB rows and columns of a
     matrix, of which the lower triangle is supplied.
     This is an auxiliary routine called by ZHETRD.

Parameters
    UPLO

              UPLO is CHARACTER*1
              Specifies whether the upper or lower triangular part of the
              Hermitian matrix A is stored:
              = 'U': Upper triangular
              = 'L': Lower triangular

    N

              N is INTEGER
              The order of the matrix A.

    NB

              NB is INTEGER
              The number of rows and columns to be reduced.

    A

              A is COMPLEX*16 array, dimension (LDA,N)
              On entry, the Hermitian 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 last NB columns have been reduced to
                tridiagonal form, with the diagonal elements overwriting
                the diagonal elements of A; the elements above the diagonal
                with the array TAU, represent the unitary matrix Q as a
                product of elementary reflectors;
              if UPLO = 'L', the first NB columns have been reduced to
                tridiagonal form, with the diagonal elements overwriting
                the diagonal elements of A; the elements below the diagonal
                with the array TAU, represent the  unitary matrix Q as a
                product of elementary reflectors.
              See Further Details.

    LDA

              LDA is INTEGER
              The leading dimension of the array A.  LDA >= max(1,N).

    E

              E is DOUBLE PRECISION array, dimension (N-1)
              If UPLO = 'U', E(n-nb:n-1) contains the superdiagonal
              elements of the last NB columns of the reduced matrix;
              if UPLO = 'L', E(1:nb) contains the subdiagonal elements of
              the first NB columns of the reduced matrix.

    TAU

              TAU is COMPLEX*16 array, dimension (N-1)
              The scalar factors of the elementary reflectors, stored in
              TAU(n-nb:n-1) if UPLO = 'U', and in TAU(1:nb) if UPLO = 'L'.
              See Further Details.

    W

              W is COMPLEX*16 array, dimension (LDW,NB)
              The n-by-nb matrix W required to update the unreduced part
              of A.

    LDW

              LDW is INTEGER
              The leading dimension of the array W. LDW >= max(1,N).

Author
    Univ. of Tennessee

    Univ. of California Berkeley

    Univ. of Colorado Denver

    NAG Ltd.

Further Details:

      If UPLO = 'U', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(n) H(n-1) . . . H(n-nb+1).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(i:n) = 0 and v(i-1) = 1; v(1:i-1) is stored on exit in A(1:i-1,i),
      and tau in TAU(i-1).
      If UPLO = 'L', the matrix Q is represented as a product of elementary
      reflectors
         Q = H(1) H(2) . . . H(nb).
      Each H(i) has the form
         H(i) = I - tau * v * v**H
      where tau is a complex scalar, and v is a complex vector with
      v(1:i) = 0 and v(i+1) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
      and tau in TAU(i).
      The elements of the vectors v together form the n-by-nb matrix V
      which is needed, with W, to apply the transformation to the unreduced
      part of the matrix, using a Hermitian rank-2k update of the form:
      A := A - V*W**H - W*V**H.
      The contents of A on exit are illustrated by the following examples
      with n = 5 and nb = 2:
      if UPLO = 'U':                       if UPLO = 'L':
        (  a   a   a   v4  v5 )              (  d                  )
        (      a   a   v4  v5 )              (  1   d              )
        (          a   1   v5 )              (  v1  1   a          )
        (              d   1  )              (  v1  v2  a   a      )
        (                  d  )              (  v1  v2  a   a   a  )
      where d denotes a diagonal element of the reduced matrix, a denotes
      an element of the original matrix that is unchanged, and vi denotes
      an element of the vector defining H(i).

Definition at line 198 of file zlatrd.f.

Author

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