dhgeqz()

subroutine dhgeqz	(	character	job,
		character	compq,
		character	compz,
		integer	n,
		integer	ilo,
		integer	ihi,
		double precision, dimension( ldh, * )	h,
		integer	ldh,
		double precision, dimension( ldt, * )	t,
		integer	ldt,
		double precision, dimension( * )	alphar,
		double precision, dimension( * )	alphai,
		double precision, dimension( * )	beta,
		double precision, dimension( ldq, * )	q,
		integer	ldq,
		double precision, dimension( ldz, * )	z,
		integer	ldz,
		double precision, dimension( * )	work,
		integer	lwork,
		integer	info )

DHGEQZ

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Purpose:

!>
!> DHGEQZ computes the eigenvalues of a real matrix pair (H,T),
!> where H is an upper Hessenberg matrix and T is upper triangular,
!> using the double-shift QZ method.
!> Matrix pairs of this type are produced by the reduction to
!> generalized upper Hessenberg form of a real matrix pair (A,B):
!>
!>    A = Q1*H*Z1**T,  B = Q1*T*Z1**T,
!>
!> as computed by DGGHRD.
!>
!> If JOB='S', then the Hessenberg-triangular pair (H,T) is
!> also reduced to generalized Schur form,
!>
!>    H = Q*S*Z**T,  T = Q*P*Z**T,
!>
!> where Q and Z are orthogonal matrices, P is an upper triangular
!> matrix, and S is a quasi-triangular matrix with 1-by-1 and 2-by-2
!> diagonal blocks.
!>
!> The 1-by-1 blocks correspond to real eigenvalues of the matrix pair
!> (H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of
!> eigenvalues.
!>
!> Additionally, the 2-by-2 upper triangular diagonal blocks of P
!> corresponding to 2-by-2 blocks of S are reduced to positive diagonal
!> form, i.e., if S(j+1,j) is non-zero, then P(j+1,j) = P(j,j+1) = 0,
!> P(j,j) > 0, and P(j+1,j+1) > 0.
!>
!> Optionally, the orthogonal matrix Q from the generalized Schur
!> factorization may be postmultiplied into an input matrix Q1, and the
!> orthogonal matrix Z may be postmultiplied into an input matrix Z1.
!> If Q1 and Z1 are the orthogonal matrices from DGGHRD that reduced
!> the matrix pair (A,B) to generalized upper Hessenberg form, then the
!> output matrices Q1*Q and Z1*Z are the orthogonal factors from the
!> generalized Schur factorization of (A,B):
!>
!>    A = (Q1*Q)*S*(Z1*Z)**T,  B = (Q1*Q)*P*(Z1*Z)**T.
!>
!> To avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently,
!> of (A,B)) are computed as a pair of values (alpha,beta), where alpha is
!> complex and beta real.
!> If beta is nonzero, lambda = alpha / beta is an eigenvalue of the
!> generalized nonsymmetric eigenvalue problem (GNEP)
!>    A*x = lambda*B*x
!> and if alpha is nonzero, mu = beta / alpha is an eigenvalue of the
!> alternate form of the GNEP
!>    mu*A*y = B*y.
!> Real eigenvalues can be read directly from the generalized Schur
!> form:
!>   alpha = S(i,i), beta = P(i,i).
!>
!> Ref: C.B. Moler & G.W. Stewart, , SIAM J. Numer. Anal., 10(1973),
!>      pp. 241--256.
!> 

Parameters

[in]	JOB	!> JOB is CHARACTER*1 !> = 'E': Compute eigenvalues only; !> = 'S': Compute eigenvalues and the Schur form. !>
[in]	COMPQ	!> COMPQ is CHARACTER*1 !> = 'N': Left Schur vectors (Q) are not computed; !> = 'I': Q is initialized to the unit matrix and the matrix Q !> of left Schur vectors of (H,T) is returned; !> = 'V': Q must contain an orthogonal matrix Q1 on entry and !> the product Q1*Q is returned. !>
[in]	COMPZ	!> COMPZ is CHARACTER*1 !> = 'N': Right Schur vectors (Z) are not computed; !> = 'I': Z is initialized to the unit matrix and the matrix Z !> of right Schur vectors of (H,T) is returned; !> = 'V': Z must contain an orthogonal matrix Z1 on entry and !> the product Z1*Z is returned. !>
[in]	N	!> N is INTEGER !> The order of the matrices H, T, Q, and Z. N >= 0. !>
[in]	ILO	!> ILO is INTEGER !>
[in]	IHI	!> IHI is INTEGER !> ILO and IHI mark the rows and columns of H which are in !> Hessenberg form. It is assumed that A is already upper !> triangular in rows and columns 1:ILO-1 and IHI+1:N. !> If N > 0, 1 <= ILO <= IHI <= N; if N = 0, ILO=1 and IHI=0. !>
[in,out]	H	!> H is DOUBLE PRECISION array, dimension (LDH, N) !> On entry, the N-by-N upper Hessenberg matrix H. !> On exit, if JOB = 'S', H contains the upper quasi-triangular !> matrix S from the generalized Schur factorization. !> If JOB = 'E', the diagonal blocks of H match those of S, but !> the rest of H is unspecified. !>
[in]	LDH	!> LDH is INTEGER !> The leading dimension of the array H. LDH >= max( 1, N ). !>
[in,out]	T	!> T is DOUBLE PRECISION array, dimension (LDT, N) !> On entry, the N-by-N upper triangular matrix T. !> On exit, if JOB = 'S', T contains the upper triangular !> matrix P from the generalized Schur factorization; !> 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S !> are reduced to positive diagonal form, i.e., if H(j+1,j) is !> non-zero, then T(j+1,j) = T(j,j+1) = 0, T(j,j) > 0, and !> T(j+1,j+1) > 0. !> If JOB = 'E', the diagonal blocks of T match those of P, but !> the rest of T is unspecified. !>
[in]	LDT	!> LDT is INTEGER !> The leading dimension of the array T. LDT >= max( 1, N ). !>
[out]	ALPHAR	!> ALPHAR is DOUBLE PRECISION array, dimension (N) !> The real parts of each scalar alpha defining an eigenvalue !> of GNEP. !>
[out]	ALPHAI	!> ALPHAI is DOUBLE PRECISION array, dimension (N) !> The imaginary parts of each scalar alpha defining an !> eigenvalue of GNEP. !> If ALPHAI(j) is zero, then the j-th eigenvalue is real; if !> positive, then the j-th and (j+1)-st eigenvalues are a !> complex conjugate pair, with ALPHAI(j+1) = -ALPHAI(j). !>
[out]	BETA	!> BETA is DOUBLE PRECISION array, dimension (N) !> The scalars beta that define the eigenvalues of GNEP. !> Together, the quantities alpha = (ALPHAR(j),ALPHAI(j)) and !> beta = BETA(j) represent the j-th eigenvalue of the matrix !> pair (A,B), in one of the forms lambda = alpha/beta or !> mu = beta/alpha. Since either lambda or mu may overflow, !> they should not, in general, be computed. !>
[in,out]	Q	!> Q is DOUBLE PRECISION array, dimension (LDQ, N) !> On entry, if COMPQ = 'V', the orthogonal matrix Q1 used in !> the reduction of (A,B) to generalized Hessenberg form. !> On exit, if COMPQ = 'I', the orthogonal matrix of left Schur !> vectors of (H,T), and if COMPQ = 'V', the orthogonal matrix !> of left Schur vectors of (A,B). !> Not referenced if COMPQ = 'N'. !>
[in]	LDQ	!> LDQ is INTEGER !> The leading dimension of the array Q. LDQ >= 1. !> If COMPQ='V' or 'I', then LDQ >= N. !>
[in,out]	Z	!> Z is DOUBLE PRECISION array, dimension (LDZ, N) !> On entry, if COMPZ = 'V', the orthogonal matrix Z1 used in !> the reduction of (A,B) to generalized Hessenberg form. !> On exit, if COMPZ = 'I', the orthogonal matrix of !> right Schur vectors of (H,T), and if COMPZ = 'V', the !> orthogonal matrix of right Schur vectors of (A,B). !> Not referenced if COMPZ = 'N'. !>
[in]	LDZ	!> LDZ is INTEGER !> The leading dimension of the array Z. LDZ >= 1. !> If COMPZ='V' or 'I', then LDZ >= N. !>
[out]	WORK	!> WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) !> On exit, if INFO >= 0, WORK(1) returns the optimal LWORK. !>
[in]	LWORK	!> LWORK is INTEGER !> The dimension of the array WORK. LWORK >= max(1,N). !> !> 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. !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -i, the i-th argument had an illegal value !> = 1,...,N: the QZ iteration did not converge. (H,T) is not !> in Schur form, but ALPHAR(i), ALPHAI(i), and !> BETA(i), i=INFO+1,...,N should be correct. !> = N+1,...,2*N: the shift calculation failed. (H,T) is not !> in Schur form, but ALPHAR(i), ALPHAI(i), and !> BETA(i), i=INFO-N+1,...,N should be correct. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

!>
!>  Iteration counters:
!>
!>  JITER  -- counts iterations.
!>  IITER  -- counts iterations run since ILAST was last
!>            changed.  This is therefore reset only when a 1-by-1 or
!>            2-by-2 block deflates off the bottom.
!>