CGEHRD man page on IRIX

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CGEHRD(3F)							    CGEHRD(3F)

NAME
     CGEHRD - reduce a complex general matrix A to upper Hessenberg form H by
     a unitary similarity transformation

SYNOPSIS
     SUBROUTINE CGEHRD( N, ILO, IHI, A, LDA, TAU, WORK, LWORK, INFO )

	 INTEGER	IHI, ILO, INFO, LDA, LWORK, N

	 COMPLEX	A( LDA, * ), TAU( * ), WORK( LWORK )

PURPOSE
     CGEHRD reduces a complex general matrix A to upper Hessenberg form H by a
     unitary similarity transformation:	 Q' * A * Q = H .

ARGUMENTS
     N	     (input) INTEGER
	     The order of the matrix A.	 N >= 0.

     ILO     (input) INTEGER
	     IHI     (input) INTEGER It is assumed that A is already upper
	     triangular in rows and columns 1:ILO-1 and IHI+1:N. ILO and IHI
	     are normally set by a previous call to CGEBAL; otherwise they
	     should be set to 1 and N respectively. See Further Details.

     A	     (input/output) COMPLEX array, dimension (LDA,N)
	     On entry, the N-by-N general matrix to be reduced.	 On exit, the
	     upper triangle and the first subdiagonal of A are overwritten
	     with the upper Hessenberg matrix H, and the elements below the
	     first subdiagonal, with the array TAU, represent the unitary
	     matrix Q as a product of elementary reflectors. See Further
	     Details.  LDA     (input) INTEGER The leading dimension of the
	     array A.  LDA >= max(1,N).

     TAU     (output) COMPLEX array, dimension (N-1)
	     The scalar factors of the elementary reflectors (see Further
	     Details). Elements 1:ILO-1 and IHI:N-1 of TAU are set to zero.

     WORK    (workspace/output) COMPLEX array, dimension (LWORK)
	     On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

     LWORK   (input) INTEGER
	     The length of the array WORK.  LWORK >= max(1,N).	For optimum
	     performance LWORK >= N*NB, where NB is the optimal blocksize.

     INFO    (output) INTEGER
	     = 0:  successful exit
	     < 0:  if INFO = -i, the i-th argument had an illegal value.

									Page 1

CGEHRD(3F)							    CGEHRD(3F)

FURTHER DETAILS
     The matrix Q is represented as a product of (ihi-ilo) elementary
     reflectors

	Q = H(ilo) H(ilo+1) . . . H(ihi-1).

     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) = 0,
     v(i+1) = 1 and v(ihi+1:n) = 0; v(i+2:ihi) is stored on exit in
     A(i+2:ihi,i), and tau in TAU(i).

     The contents of A are illustrated by the following example, with n = 7,
     ilo = 2 and ihi = 6:

     on entry,			      on exit,

     ( a   a   a   a   a   a   a )    (	 a   a	 h   h	 h   h	 a ) (	   a
     a	 a   a	 a   a )    (	   a   h   h   h   h   a ) (	 a   a	 a   a
     a	 a )	(      h   h   h   h   h   h ) (     a	 a   a	 a   a	 a )
     (	    v2	h   h	h   h	h ) (	  a   a	  a   a	  a   a )    (	    v2
     v3	 h   h	 h   h ) (     a   a   a   a   a   a )	  (	 v2  v3	 v4  h
     h	 h ) (			       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).

									Page 2

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