Provided by: liblapack-doc_3.3.1-1_all

**NAME**

LAPACK-3 - 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

**SYNOPSIS**

SUBROUTINE CLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY ) INTEGER K, LDA, LDT, LDY, N, NB COMPLEX A( LDA, * ), T( LDT, NB ), TAU( NB ), Y( LDY, NB )

**PURPOSE**

CLAHRD 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 CLAHR2 instead.

**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) 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 (input) INTEGER The leading dimension of the array A. LDA >= max(1,N). TAU (output) COMPLEX array, dimension (NB) The scalar factors of the elementary reflectors. See Further Details. T (output) COMPLEX array, dimension (LDT,NB) The upper triangular matrix T. LDT (input) INTEGER The leading dimension of the array T. LDT >= NB. Y (output) COMPLEX array, dimension (LDY,NB) The n-by-nb matrix Y. LDY (input) INTEGER The leading dimension of the array Y. LDY >= max(1,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**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). LAPACK auxiliary routine (version 3.3.1) April 2011 CLAHRD(3lapack)