.TH ZLAHRD 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) " .SH NAME ZLAHRD - 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 .SH SYNOPSIS .TP 19 SUBROUTINE ZLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) .TP 19 .ti +4 INTEGER K, LDA, LDT, LDY, N, NB .TP 19 .ti +4 COMPLEX*16 A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB ) .SH 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\(aq * A * Q. The routine returns the matrices V and T which determine Q as a block reflector I - V*T*V\(aq, and also the matrix Y = A * V * T. This is an OBSOLETE auxiliary routine. .br This routine will be \(aqdeprecated\(aq in a future release. .br Please use the new routine ZLAHR2 instead. .br .SH ARGUMENTS .TP 8 N (input) INTEGER The order of the matrix A. .TP 8 K (input) INTEGER The offset for the reduction. Elements below the k-th subdiagonal in the first NB columns are reduced to zero. .TP 8 NB (input) INTEGER The number of columns to be reduced. .TP 8 A (input/output) 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 (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). .TP 8 TAU (output) COMPLEX*16 array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. .TP 8 T (output) COMPLEX*16 array, dimension (LDT,NB) The upper triangular matrix T. .TP 8 LDT (input) INTEGER The leading dimension of the array T. LDT >= NB. .TP 8 Y (output) COMPLEX*16 array, dimension (LDY,NB) The n-by-nb matrix Y. .TP 8 LDY (input) INTEGER The leading dimension of the array Y. LDY >= max(1,N). .SH FURTHER DETAILS The matrix Q is represented as a product of nb elementary reflectors Q = H(1) H(2) . . . H(nb). .br Each H(i) has the form .br H(i) = I - tau * v * v\(aq .br 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). .br 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\(aq) * (A - Y*V\(aq). .br The contents of A on exit are illustrated by the following example with n = 7, k = 3 and nb = 2: .br ( a h a a a ) .br ( a h a a a ) .br ( a h a a a ) .br ( h h a a a ) .br ( v1 h a a a ) .br ( v1 v2 a a a ) .br ( v1 v2 a a a ) .br 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). .br