SRC/zlahr2.f(3)            Library Functions Manual            SRC/zlahr2.f(3)

NAME
       SRC/zlahr2.f

SYNOPSIS
   Functions/Subroutines
       subroutine zlahr2 (n, k, nb, a, lda, tau, t, ldt, y, ldy)
           ZLAHR2 reduces the specified number of first columns of a general
           rectangular matrix A so that elements below the specified
           subdiagonal are zero, and returns auxiliary matrices which are
           needed to apply the transformation to the unreduced part of A.

Function/Subroutine Documentation
   subroutine zlahr2 (integer n, integer k, integer nb, complex*16, dimension(
       lda, * ) a, integer lda, complex*16, dimension( nb ) tau, complex*16,
       dimension( ldt, nb ) t, integer ldt, complex*16, dimension( ldy, nb )
       y, integer ldy)
       ZLAHR2 reduces the specified number of first columns of a general
       rectangular matrix A so that elements below the specified subdiagonal
       are zero, and returns auxiliary matrices which are needed to apply the
       transformation to the unreduced part of A.

       Purpose:


           !>
           !> ZLAHR2 reduces the first NB columns of A complex general n-BY-(n-k+1)
           !> matrix A so that elements below the k-th subdiagonal are zero. The
           !> reduction is performed by an unitary similarity transformation
           !> Q**H * A * Q. The routine returns the matrices V and T which determine
           !> Q as a block reflector I - V*T*V**H, and also the matrix Y = A * V * T.
           !>
           !> This is an auxiliary routine called by ZGEHRD.
           !>

       Parameters
           N

           !>          N is INTEGER
           !>          The order of the matrix A.
           !>

           K

           !>          K is INTEGER
           !>          The offset for the reduction. Elements below the k-th
           !>          subdiagonal in the first NB columns are reduced to zero.
           !>          K < N.
           !>

           NB

           !>          NB is INTEGER
           !>          The number of columns to be reduced.
           !>

           A

           !>          A is COMPLEX*16 array, dimension (LDA,N-K+1)
           !>          On entry, the n-by-(n-k+1) general matrix A.
           !>          On exit, the elements on and above the k-th subdiagonal in
           !>          the first NB columns are overwritten with the corresponding
           !>          elements of the reduced matrix; the elements below the k-th
           !>          subdiagonal, with the array TAU, represent the matrix Q as a
           !>          product of elementary reflectors. The other columns of A are
           !>          unchanged. See Further Details.
           !>

           LDA

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

           TAU

           !>          TAU is COMPLEX*16 array, dimension (NB)
           !>          The scalar factors of the elementary reflectors. See Further
           !>          Details.
           !>

           T

           !>          T is COMPLEX*16 array, dimension (LDT,NB)
           !>          The upper triangular matrix T.
           !>

           LDT

           !>          LDT is INTEGER
           !>          The leading dimension of the array T.  LDT >= NB.
           !>

           Y

           !>          Y is COMPLEX*16 array, dimension (LDY,NB)
           !>          The n-by-nb matrix Y.
           !>

           LDY

           !>          LDY is INTEGER
           !>          The leading dimension of the array Y. LDY >= N.
           !>

       Author
           Univ. of Tennessee


           Univ. of California Berkeley


           Univ. of Colorado Denver


           NAG Ltd.

       Further Details:


           !>
           !>  The matrix Q is represented as a product of nb 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+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in
           !>  A(i+k+1:n,i), and tau in TAU(i).
           !>
           !>  The elements of the vectors v together form the (n-k+1)-by-nb matrix
           !>  V which is needed, with T and Y, to apply the transformation to the
           !>  unreduced part of the matrix, using an update of the form:
           !>  A := (I - V*T*V**H) * (A - Y*V**H).
           !>
           !>  The contents of A on exit are illustrated by the following example
           !>  with n = 7, k = 3 and nb = 2:
           !>
           !>     ( a   a   a   a   a )
           !>     ( a   a   a   a   a )
           !>     ( a   a   a   a   a )
           !>     ( h   h   a   a   a )
           !>     ( v1  h   a   a   a )
           !>     ( v1  v2  a   a   a )
           !>     ( v1  v2  a   a   a )
           !>
           !>  where a denotes an element of the original matrix A, h denotes a
           !>  modified element of the upper Hessenberg matrix H, and vi denotes an
           !>  element of the vector defining H(i).
           !>
           !>  This subroutine is a slight modification of LAPACK-3.0's ZLAHRD
           !>  incorporating improvements proposed by Quintana-Orti and Van de
           !>  Gejin. Note that the entries of A(1:K,2:NB) differ from those
           !>  returned by the original LAPACK-3.0's ZLAHRD routine. (This
           !>  subroutine is not backward compatible with LAPACK-3.0's ZLAHRD.)
           !>

       References:
           Gregorio Quintana-Orti and Robert van de Geijn, "Improving the
           performance of reduction to Hessenberg form," ACM Transactions on
           Mathematical Software, 32(2):180-194, June 2006.

       Definition at line 180 of file zlahr2.f.

Author
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LAPACK                          Version 3.12.0                 SRC/zlahr2.f(3)