IRIX Base Documentation 2001 May/SGI IRIX Base Documentation 2001 May.iso/usr/share/catman/p_man/cat3/complib/zhetd2.z/zhetd2

Text File | 1998-10-30 | 5.3 KB | 199 lines

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     ZHETD2 - reduce a complex Hermitian matrix A to real symmetric
     tridiagonal form T by a unitary similarity transformation

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     SUBROUTINE ZHETD2( UPLO, N, A, LDA, D, E, TAU, INFO )

         CHARACTER      UPLO

         INTEGER        INFO, LDA, N

         DOUBLE         PRECISION D( * ), E( * )

         COMPLEX*16     A( LDA, * ), TAU( * )

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     ZHETD2 reduces a complex Hermitian matrix A to real symmetric tridiagonal
     form T by a unitary similarity transformation:  Q' * A * Q = T.


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

     N       (input) INTEGER
             The order of the matrix A.  N >= 0.

     A       (input/output) 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 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 unitary 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 unitary 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).



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     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) COMPLEX*16 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.

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     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 complex scalar, and v is a complex 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 complex scalar, and v is a complex 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).



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