slahr2.f man page

slahr2.f —

Synopsis

Functions/Subroutines

subroutine slahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
SLAHR2 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 slahr2 (integerN, integerK, integerNB, real, dimension( lda, * )A, integerLDA, real, dimension( nb )TAU, real, dimension( ldt, nb )T, integerLDT, real, dimension( ldy, nb )Y, integerLDY)

SLAHR2 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:

SLAHR2 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 SGEHRD.

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 REAL 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 REAL array, dimension (NB)
The scalar factors of the elementary reflectors. See Further
Details.

T

T is REAL 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 REAL 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 slahr2.f.

Author

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Referenced By

slahr2(3) is an alias of slahr2.f(3).

Sat Nov 16 2013 Version 3.4.2 LAPACK