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 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 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:
Definition at line 180 of file zlahr2.f.
Author
Generated automatically by Doxygen for LAPACK from the source code.
Version 3.12.0 | LAPACK |