clahr2-f(3)

clahr2.f –

Synopsis

Functions/Subroutines

subroutine clahr2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)
CLAHR2 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 clahr2 (integerN, integerK, integerNB, complex, dimension( lda, * )A, integerLDA, complex, dimension( nb )TAU, complex, dimension( ldt, nb )T, integerLDT, complex, dimension( ldy, nb )Y, integerLDY)

CLAHR2 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:

 CLAHR2 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 CGEHRD.

Parameters:

 N is INTEGER
 The order of the matrix A.

 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 is INTEGER
 The number of columns to be reduced.

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

 T is COMPLEX array, dimension (LDT,NB)
 The upper triangular matrix T.

LDT

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

 Y is COMPLEX 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**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 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 clahr2.f.

Author

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