CLAHR2(1) LAPACK auxiliary routine (version 3.2) CLAHR2(1)[top]NAMECLAHR2 - 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 zeroSYNOPSISSUBROUTINE CLAHR2( 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 )PURPOSECLAHR2 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' * 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 CGEHRD.ARGUMENTSN (input) INTEGER The order of the matrix A. K (input) INTEGER The offset for the reduction. Elements below the k-th subdiago‐ nal in the first NB columns are reduced to zero. K < N. 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 ele‐ ments 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 reflec‐ tors. 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 >= N.FURTHER DETAILSThe 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 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 unre‐ duced part of the matrix, using an update of the form: A := (IV*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 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 modi‐ fied element of the upper Hessenberg matrix H, and vi denotes an ele‐ ment of the vector defining H(i). This file is a slight modification of LAPACK-3.0's CLAHRD 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 routine. This function is not backward compatible with LAPACK3.0. LAPACK auxiliary routine (versioNovember 2008 CLAHR2(1)-

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