Functions/Subroutines
subroutine	clatrd (UPLO, N, NB, A, LDA, E, TAU, W, LDW)
	CLATRD 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 clatrd	(	character	UPLO,
		integer	N,
		integer	NB,
		complex, dimension( lda, * )	A,
		integer	LDA,
		real, dimension( * )	E,
		complex, dimension( * )	TAU,
		complex, dimension( ldw, * )	W,
		integer	LDW
	)

CLATRD reduces the first nb rows and columns of a symmetric/Hermitian matrix A to real tridiagonal form by an unitary similarity transformation.

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Purpose:

 CLATRD 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', CLATRD reduces the last NB rows and columns of a
 matrix, of which the upper triangle is supplied;
 if UPLO = 'L', CLATRD reduces the first NB rows and columns of a
 matrix, of which the lower triangle is supplied.

 This is an auxiliary routine called by CHETRD.

Parameters:

[in]	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
[in]	N	N is INTEGER The order of the matrix A.
[in]	NB	NB is INTEGER The number of rows and columns to be reduced.
[in,out]	A	A is COMPLEX 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.
[in]	LDA	LDA is INTEGER The leading dimension of the array A. LDA >= max(1,N).
[out]	E	E is REAL 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.
[out]	TAU	TAU is COMPLEX 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.
[out]	W	W is COMPLEX array, dimension (LDW,NB) The n-by-nb matrix W required to update the unreduced part of A.
[in]	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.

Date:: September 2012

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 200 of file clatrd.f.

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