zlaqr0()

subroutine zlaqr0	(	logical	wantt,
		logical	wantz,
		integer	n,
		integer	ilo,
		integer	ihi,
		complex*16, dimension( ldh, * )	h,
		integer	ldh,
		complex*16, dimension( * )	w,
		integer	iloz,
		integer	ihiz,
		complex*16, dimension( ldz, * )	z,
		integer	ldz,
		complex*16, dimension( * )	work,
		integer	lwork,
		integer	info )

ZLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from the Schur decomposition.

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Purpose:

!>
!>    ZLAQR0 computes the eigenvalues of a Hessenberg matrix H
!>    and, optionally, the matrices T and Z from the Schur decomposition
!>    H = Z T Z**H, where T is an upper triangular matrix (the
!>    Schur form), and Z is the unitary matrix of Schur vectors.
!>
!>    Optionally Z may be postmultiplied into an input unitary
!>    matrix Q so that this routine can give the Schur factorization
!>    of a matrix A which has been reduced to the Hessenberg form H
!>    by the unitary matrix Q:  A = Q*H*Q**H = (QZ)*H*(QZ)**H.
!> 

Parameters

[in]	WANTT	!> WANTT is LOGICAL !> = .TRUE. : the full Schur form T is required; !> = .FALSE.: only eigenvalues are required. !>
[in]	WANTZ	!> WANTZ is LOGICAL !> = .TRUE. : the matrix of Schur vectors Z is required; !> = .FALSE.: Schur vectors are not required. !>
[in]	N	!> N is INTEGER !> The order of the matrix H. N >= 0. !>
[in]	ILO	!> ILO is INTEGER !>
[in]	IHI	!> IHI is INTEGER !> !> It is assumed that H is already upper triangular in rows !> and columns 1:ILO-1 and IHI+1:N and, if ILO > 1, !> H(ILO,ILO-1) is zero. ILO and IHI are normally set by a !> previous call to ZGEBAL, and then passed to ZGEHRD when the !> matrix output by ZGEBAL is reduced to Hessenberg form. !> Otherwise, ILO and IHI should be set to 1 and N, !> respectively. If N > 0, then 1 <= ILO <= IHI <= N. !> If N = 0, then ILO = 1 and IHI = 0. !>
[in,out]	H	!> H is COMPLEX*16 array, dimension (LDH,N) !> On entry, the upper Hessenberg matrix H. !> On exit, if INFO = 0 and WANTT is .TRUE., then H !> contains the upper triangular matrix T from the Schur !> decomposition (the Schur form). If INFO = 0 and WANT is !> .FALSE., then the contents of H are unspecified on exit. !> (The output value of H when INFO > 0 is given under the !> description of INFO below.) !> !> This subroutine may explicitly set H(i,j) = 0 for i > j and !> j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N. !>
[in]	LDH	!> LDH is INTEGER !> The leading dimension of the array H. LDH >= max(1,N). !>
[out]	W	!> W is COMPLEX*16 array, dimension (N) !> The computed eigenvalues of H(ILO:IHI,ILO:IHI) are stored !> in W(ILO:IHI). If WANTT is .TRUE., then the eigenvalues are !> stored in the same order as on the diagonal of the Schur !> form returned in H, with W(i) = H(i,i). !>
[in]	ILOZ	!> ILOZ is INTEGER !>
[in]	IHIZ	!> IHIZ is INTEGER !> Specify the rows of Z to which transformations must be !> applied if WANTZ is .TRUE.. !> 1 <= ILOZ <= ILO; IHI <= IHIZ <= N. !>
[in,out]	Z	!> Z is COMPLEX*16 array, dimension (LDZ,IHI) !> If WANTZ is .FALSE., then Z is not referenced. !> If WANTZ is .TRUE., then Z(ILO:IHI,ILOZ:IHIZ) is !> replaced by Z(ILO:IHI,ILOZ:IHIZ)*U where U is the !> orthogonal Schur factor of H(ILO:IHI,ILO:IHI). !> (The output value of Z when INFO > 0 is given under !> the description of INFO below.) !>
[in]	LDZ	!> LDZ is INTEGER !> The leading dimension of the array Z. if WANTZ is .TRUE. !> then LDZ >= MAX(1,IHIZ). Otherwise, LDZ >= 1. !>
[out]	WORK	!> WORK is COMPLEX*16 array, dimension LWORK !> On exit, if LWORK = -1, WORK(1) returns an estimate of !> the optimal value for LWORK. !>
[in]	LWORK	!> LWORK is INTEGER !> The dimension of the array WORK. LWORK >= max(1,N) !> is sufficient, but LWORK typically as large as 6*N may !> be required for optimal performance. A workspace query !> to determine the optimal workspace size is recommended. !> !> If LWORK = -1, then ZLAQR0 does a workspace query. !> In this case, ZLAQR0 checks the input parameters and !> estimates the optimal workspace size for the given !> values of N, ILO and IHI. The estimate is returned !> in WORK(1). No error message related to LWORK is !> issued by XERBLA. Neither H nor Z are accessed. !>
[out]	INFO	!> INFO is INTEGER !> = 0: successful exit !> > 0: if INFO = i, ZLAQR0 failed to compute all of !> the eigenvalues. Elements 1:ilo-1 and i+1:n of WR !> and WI contain those eigenvalues which have been !> successfully computed. (Failures are rare.) !> !> If INFO > 0 and WANT is .FALSE., then on exit, !> the remaining unconverged eigenvalues are the eigen- !> values of the upper Hessenberg matrix rows and !> columns ILO through INFO of the final, output !> value of H. !> !> If INFO > 0 and WANTT is .TRUE., then on exit !> !> (*) (initial value of H)*U = U*(final value of H) !> !> where U is a unitary matrix. The final !> value of H is upper Hessenberg and triangular in !> rows and columns INFO+1 through IHI. !> !> If INFO > 0 and WANTZ is .TRUE., then on exit !> !> (final value of Z(ILO:IHI,ILOZ:IHIZ) !> = (initial value of Z(ILO:IHI,ILOZ:IHIZ)*U !> !> where U is the unitary matrix in (*) (regard- !> less of the value of WANTT.) !> !> If INFO > 0 and WANTZ is .FALSE., then Z is not !> accessed. !>

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Contributors:

Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas, USA

References:

 K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
 Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
 Performance, SIAM Journal of Matrix Analysis, volume 23, pages
 929--947, 2002.

K. Braman, R. Byers and R. Mathias, The Multi-Shift QR Algorithm Part II: Aggressive Early Deflation, SIAM Journal of Matrix Analysis, volume 23, pages 948–973, 2002.