CLAHR2(3)      LAPACK routine of NEC Numeric Library Collection      CLAHR2(3)



NAME
       CLAHR2

SYNOPSIS
       SUBROUTINE CLAHR2 (N, K, NB, A, LDA, TAU, T, LDT, Y, LDY)



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.




ARGUMENTS
           N         (input)
                     N is INTEGER
                     The order of the matrix A.

           K         (input)
                     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        (input)
                     NB is INTEGER
                     The number of columns to be reduced.

           A         (input/output)
                     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       (input)
                     LDA is INTEGER
                     The leading dimension of the array A.  LDA >= max(1,N).

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

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

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

           Y         (output)
                     Y is COMPLEX array, dimension (LDY,NB)
                     The n-by-nb matrix Y.

           LDY       (input)
                     LDY is 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**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.)



LAPACK routine                  31 October 2017                      CLAHR2(3)