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.

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.

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

December 2016

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

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.