slahrd(l) [redhat man page]
SLAHRD(l) ) SLAHRD(l) NAME
SLAHRD - reduce 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 SYNOPSIS
SUBROUTINE SLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) INTEGER K, LDA, LDT, LDY, N, NB REAL A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB ) PURPOSE
SLAHRD 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' * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V', and also the matrix Y = A * V * T. This is an auxiliary routine called by SGEHRD. ARGUMENTS
N (input) INTEGER The order of the matrix A. K (input) INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. NB (input) INTEGER The number of columns to be reduced. A (input/output) 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 (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). TAU (output) REAL array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. T (output) REAL array, dimension (LDT,NB) The upper triangular matrix T. LDT (input) INTEGER The leading dimension of the array T. LDT >= NB. Y (output) REAL array, dimension (LDY,NB) The n-by-nb matrix Y. LDY (input) INTEGER The leading dimension of the array Y. LDY >= N. 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' 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') * (A - Y*V'). The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: ( a h a a a ) ( a h a a a ) ( a h 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 ele- ment of the vector defining H(i). LAPACK version 3.0 15 June 2000 SLAHRD(l)
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zlahrd.f(3) LAPACK zlahrd.f(3) NAME
zlahrd.f - SYNOPSIS
Functions/Subroutines subroutine zlahrd (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY) ZLAHRD Function/Subroutine Documentation subroutine zlahrd (integerN, integerK, integerNB, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( nb )TAU, complex*16, dimension( ldt, nb )T, integerLDT, complex*16, dimension( ldy, nb )Y, integerLDY) ZLAHRD Purpose: ZLAHRD 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 a 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 OBSOLETE auxiliary routine. This routine will be 'deprecated' in a future release. Please use the new routine ZLAHR2 instead. 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. 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 >= max(1,N). Author: Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Date: November 2011 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 h a a a ) ( a h a a a ) ( a h 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). Definition at line 170 of file zlahrd.f. Author Generated automatically by Doxygen for LAPACK from the source code. Version 3.4.1 Sun May 26 2013 zlahrd.f(3)