dorghr(l) [redhat man page]

DORGHR(l)								 )								 DORGHR(l)

NAME

       DORGHR  -  generate  a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order N, as returned by
       DGEHRD

SYNOPSIS

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

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

	   DOUBLE	  PRECISION A( LDA, * ), TAU( * ), WORK( * )

PURPOSE

       DORGHR generates a real orthogonal matrix Q which is defined as the product of IHI-ILO elementary reflectors of order  N,  as  returned	by
       DGEHRD: Q = H(ilo) H(ilo+1) . . . H(ihi-1).

ARGUMENTS

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

       ILO     (input) INTEGER
	       IHI	(input)  INTEGER  ILO  and  IHI must have the same values as in the previous call of DGEHRD. Q is equal to the unit matrix
	       except in the submatrix Q(ilo+1:ihi,ilo+1:ihi).	1 <= ILO <= IHI <= N, if N > 0; ILO=1 and IHI=0, if N=0.

       A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
	       On entry, the vectors which define the elementary reflectors, as returned by DGEHRD.  On exit, the N-by-N orthogonal matrix Q.

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

       TAU     (input) DOUBLE PRECISION array, dimension (N-1)
	       TAU(i) must contain the scalar factor of the elementary reflector H(i), as returned by DGEHRD.

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

       LWORK   (input) INTEGER
	       The dimension of the array WORK. LWORK >= IHI-ILO.  For optimum performance LWORK >= (IHI-ILO)*NB, where NB is the  optimal  block-
	       size.

	       If  LWORK  =  -1,  then	a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this
	       value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

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

LAPACK version 3.0						   15 June 2000 							 DORGHR(l)

DGEHRD(l)								 )								 DGEHRD(l)

NAME

       DGEHRD - reduce a real general matrix A to upper Hessenberg form H by an orthogonal similarity transformation

SYNOPSIS

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

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

	   DOUBLE	  PRECISION A( LDA, * ), TAU( * ), WORK( * )

PURPOSE

       DGEHRD reduces a real general matrix A to upper Hessenberg form H by an orthogonal 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 DGEBAL; otherwise they should be set to 1 and N respectively. See Further Details.

       A       (input/output) DOUBLE PRECISION 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 orthogonal 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) DOUBLE PRECISION 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) DOUBLE PRECISION 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.

	       If  LWORK  =  -1,  then	a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this
	       value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

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

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 real scalar, and v is a real 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 ele-
       ment of the vector defining H(i).

LAPACK version 3.0						   15 June 2000 							 DGEHRD(l)