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slaqr0.f(3)				      LAPACK				      slaqr0.f(3)

NAME
       slaqr0.f -

SYNOPSIS
   Functions/Subroutines
       subroutine slaqr0 (WANTT, WANTZ, N, ILO, IHI, H, LDH, WR, WI, ILOZ, IHIZ, Z, LDZ, WORK,
	   LWORK, INFO)
	   SLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices
	   from the Schur decomposition.

Function/Subroutine Documentation
   subroutine slaqr0 (logicalWANTT, logicalWANTZ, integerN, integerILO, integerIHI, real,
       dimension( ldh, * )H, integerLDH, real, dimension( * )WR, real, dimension( * )WI,
       integerILOZ, integerIHIZ, real, dimension( ldz, * )Z, integerLDZ, real, dimension( *
       )WORK, integerLWORK, integerINFO)
       SLAQR0 computes the eigenvalues of a Hessenberg matrix, and optionally the matrices from
       the Schur decomposition.

       Purpose:

	       SLAQR0 computes the eigenvalues of a Hessenberg matrix H
	       and, optionally, the matrices T and Z from the Schur decomposition
	       H = Z T Z**T, where T is an upper quasi-triangular matrix (the
	       Schur form), and Z is the orthogonal matrix of Schur vectors.

	       Optionally Z may be postmultiplied into an input orthogonal
	       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 orthogonal matrix Q:  A = Q*H*Q**T = (QZ)*T*(QZ)**T.

       Parameters:
	   WANTT

		     WANTT is LOGICAL
		     = .TRUE. : the full Schur form T is required;
		     = .FALSE.: only eigenvalues are required.

	   WANTZ

		     WANTZ is LOGICAL
		     = .TRUE. : the matrix of Schur vectors Z is required;
		     = .FALSE.: Schur vectors are not required.

	   N

		     N is INTEGER
		      The order of the matrix H.  N .GE. 0.

	   ILO

		     ILO is INTEGER

	   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.GT.1,
		      H(ILO,ILO-1) is zero. ILO and IHI are normally set by a
		      previous call to SGEBAL, and then passed to SGEHRD when the
		      matrix output by SGEBAL is reduced to Hessenberg form.
		      Otherwise, ILO and IHI should be set to 1 and N,
		      respectively.  If N.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
		      If N = 0, then ILO = 1 and IHI = 0.

	   H

		     H is REAL 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 quasi-triangular matrix T from the Schur
		      decomposition (the Schur form); 2-by-2 diagonal blocks
		      (corresponding to complex conjugate pairs of eigenvalues)
		      are returned in standard form, with H(i,i) = H(i+1,i+1)
		      and H(i+1,i)*H(i,i+1).LT.0. If INFO = 0 and WANTT is
		      .FALSE., then the contents of H are unspecified on exit.
		      (The output value of H when INFO.GT.0 is given under the
		      description of INFO below.)

		      This subroutine may explicitly set H(i,j) = 0 for i.GT.j and
		      j = 1, 2, ... ILO-1 or j = IHI+1, IHI+2, ... N.

	   LDH

		     LDH is INTEGER
		      The leading dimension of the array H. LDH .GE. max(1,N).

	   WR

		     WR is REAL array, dimension (IHI)

	   WI

		     WI is REAL array, dimension (IHI)
		      The real and imaginary parts, respectively, of the computed
		      eigenvalues of H(ILO:IHI,ILO:IHI) are stored in WR(ILO:IHI)
		      and WI(ILO:IHI). If two eigenvalues are computed as a
		      complex conjugate pair, they are stored in consecutive
		      elements of WR and WI, say the i-th and (i+1)th, with
		      WI(i) .GT. 0 and WI(i+1) .LT. 0. 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
		      WR(i) = H(i,i) and, if H(i:i+1,i:i+1) is a 2-by-2 diagonal
		      block, WI(i) = sqrt(-H(i+1,i)*H(i,i+1)) and
		      WI(i+1) = -WI(i).

	   ILOZ

		     ILOZ is INTEGER

	   IHIZ

		     IHIZ is INTEGER
		      Specify the rows of Z to which transformations must be
		      applied if WANTZ is .TRUE..
		      1 .LE. ILOZ .LE. ILO; IHI .LE. IHIZ .LE. N.

	   Z

		     Z is REAL 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.GT.0 is given under
		      the description of INFO below.)

	   LDZ

		     LDZ is INTEGER
		      The leading dimension of the array Z.  if WANTZ is .TRUE.
		      then LDZ.GE.MAX(1,IHIZ).	Otherwize, LDZ.GE.1.

	   WORK

		     WORK is REAL array, dimension LWORK
		      On exit, if LWORK = -1, WORK(1) returns an estimate of
		      the optimal value for LWORK.

	   LWORK

		     LWORK is INTEGER
		      The dimension of the array WORK.	LWORK .GE. 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 SLAQR0 does a workspace query.
		      In this case, SLAQR0 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.

	   INFO

		     INFO is INTEGER
			=  0:  successful exit
		      .GT. 0:  if INFO = i, SLAQR0 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 .GT. 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 .GT. 0 and WANTT is .TRUE., then on exit

		      (*)  (initial value of H)*U  = U*(final value of H)

			   where U is an orthogonal matrix.  The final
			   value of H is upper Hessenberg and quasi-triangular
			   in rows and columns INFO+1 through IHI.

			   If INFO .GT. 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 orthogonal matrix in (*) (regard-
			   less of the value of WANTT.)

			   If INFO .GT. 0 and WANTZ is .FALSE., then Z is not
			   accessed.

       Author:
	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

       Date:
	   September 2012

       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.

       Definition at line 256 of file slaqr0.f.

Author
       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      slaqr0.f(3)
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