

SUBROUTINE PZLAHRD(  N, K, NB, A, IA, JA, DESCA, TAU, T, Y, IY, JY, DESCY, WORK ) 
INTEGER IA, IY, JA, JY, K, N, NB  
INTEGER DESCA( * ), DESCY( * )  
COMPLEX*16 A( * ), T( * ), TAU( * ), WORK( * ), Y( * )  
PZLAHRD reduces the first NB columns of a complex general Nby(NK+1) distributed matrix A(IA:IA+N1,JA:JA+NK) so that elements below the kth 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 PZGEHRD. In the following comments sub( A ) denotes A(IA:IA+N1,JA:JA+N1).
N (global input) INTEGER The number of rows and columns to be operated on, i.e. the order of the distributed submatrix sub( A ). N >= 0. K (global input) INTEGER The offset for the reduction. Elements below the kth subdiagonal in the first NB columns are reduced to zero. NB (global input) INTEGER The number of columns to be reduced. A (local input/local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_A, LOCc(JA+NK)). On entry, this array contains the the local pieces of the Nby(NK+1) general distributed matrix A(IA:IA+N1,JA:JA+NK). On exit, the elements on and above the kth subdiagonal in the first NB columns are overwritten with the corresponding elements of the reduced distributed matrix; the elements below the kth subdiagonal, with the array TAU, represent the matrix Q as a product of elementary reflectors. The other columns of A(IA:IA+N1,JA:JA+NK) are unchanged. See Further Details. IA (global input) INTEGER The row index in the global array A indicating the first row of sub( A ). JA (global input) INTEGER The column index in the global array A indicating the first column of sub( A ). DESCA (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix A. TAU (local output) COMPLEX*16 array, dimension LOCc(JA+N2) The scalar factors of the elementary reflectors (see Further Details). TAU is tied to the distributed matrix A. T (local output) COMPLEX*16 array, dimension (NB_A,NB_A) The upper triangular matrix T. Y (local output) COMPLEX*16 pointer into the local memory to an array of dimension (LLD_Y,NB_A). On exit, this array contains the local pieces of the NbyNB distributed matrix Y. LLD_Y >= LOCr(IA+N1). IY (global input) INTEGER The row index in the global array Y indicating the first row of sub( Y ). JY (global input) INTEGER The column index in the global array Y indicating the first column of sub( Y ). DESCY (global and local input) INTEGER array of dimension DLEN_. The array descriptor for the distributed matrix Y. WORK (local workspace) COMPLEX*16 array, dimension (NB)
The matrix Q is represented as a product of nb elementary reflectorsQ = 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+k1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(ia+i+k:ia+n1,ja+i1), and tau in TAU(ja+i1).
The elements of the vectors v together form the (nk+1)bynb 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(ia:ia+n1,ja:ja+nk) := (IV*T*V’)*(A(ia:ia+n1,ja:ja+nk)Y*V’).
The contents of A(ia:ia+n1,ja:ja+nk) 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(ia:ia+n1,ja:ja+nk), h denotes a modified element of the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).
ScaLAPACK version 1.7  PZLAHRD (l)  13 August 2001 
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