Dlahr2.f -
subroutine dlahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
DLAHR2 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.
DLAHR2 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:
DLAHR2 reduces the first NB columns of A real general n-BY-(n-k+1) matrix A so that elements below the k-th subdiagonal are zero. The reduction is performed by an orthogonal similarity transformation Q**T * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V**T, and also the matrix Y = A * V * T. This is an auxiliary routine called by DGEHRD.
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 DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.
T
T is DOUBLE PRECISION 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 DOUBLE PRECISION 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.
Date:
September 2012
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**T
where tau is a real scalar, and v is a real 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**T) * (A - Y*V**T).
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 DLAHRD
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 DLAHRD routine. (This
subroutine is not backward compatible with LAPACK-3.0's DLAHRD.)
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 182 of file dlahr2.f.
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