SHSEQR 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.

JOB

          JOB is CHARACTER*1


           = 'E':  compute eigenvalues only;


           = 'S':  compute eigenvalues and the Schur form T.

COMPZ

          COMPZ is CHARACTER*1


           = 'N':  no Schur vectors are computed;


           = 'I':  Z is initialized to the unit matrix and the matrix Z


                   of Schur vectors of H is returned;


           = 'V':  Z must contain an orthogonal matrix Q on entry, and


                   the product Q*Z is returned.

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. ILO and IHI are normally


           set by a previous call to SGEBAL, and then passed to ZGEHRD


           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 JOB = 'S', 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 JOB = 'E', 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.)


           Unlike earlier versions of SHSEQR, this subroutine may


           explicitly 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 (N)

WI

          WI is REAL array, dimension (N)


           The real and imaginary parts, respectively, of the computed


           eigenvalues. 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 JOB = 'S', 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).

Z

          Z is REAL array, dimension (LDZ,N)


           If COMPZ = 'N', Z is not referenced.


           If COMPZ = 'I', on entry Z need not be set and on exit,


           if INFO = 0, Z contains the orthogonal matrix Z of the Schur


           vectors of H.  If COMPZ = 'V', on entry Z must contain an


           N-by-N matrix Q, which is assumed to be equal to the unit


           matrix except for the submatrix Z(ILO:IHI,ILO:IHI). On exit,


           if INFO = 0, Z contains Q*Z.


           Normally Q is the orthogonal matrix generated by SORGHR


           after the call to SGEHRD which formed the Hessenberg matrix


           H. (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 COMPZ = 'I' or


           COMPZ = 'V', then LDZ.GE.MAX(1,N).  Otherwize, LDZ.GE.1.

WORK

          WORK is REAL array, dimension (LWORK)


           On exit, if INFO = 0, 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 and delivers very good and sometimes


           optimal performance.  However, LWORK as large as 11*N


           may be required for optimal performance.  A workspace


           query is recommended to determine the optimal workspace


           size.


           If LWORK = -1, then SHSEQR does a workspace query.


           In this case, SHSEQR 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


           .LT. 0:  if INFO = -i, the i-th argument had an illegal


                    value


           .GT. 0:  if INFO = i, SHSEQR 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 JOB = 'E', 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 JOB   = 'S', 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 COMPZ = 'V', then on exit


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


                where U is the orthogonal matrix in (*) (regard-


                less of the value of JOB.)


                If INFO .GT. 0 and COMPZ = 'I', then on exit


                      (final value of Z)  = U


                where U is the orthogonal matrix in (*) (regard-


                less of the value of JOB.)


                If INFO .GT. 0 and COMPZ = 'N', 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

             Default values supplied by


             ILAENV(ISPEC,'SHSEQR',JOB(:1)//COMPZ(:1),N,ILO,IHI,LWORK).


             It is suggested that these defaults be adjusted in order


             to attain best performance in each particular


             computational environment.


            ISPEC=12: The SLAHQR vs SLAQR0 crossover point.


                      Default: 75. (Must be at least 11.)


            ISPEC=13: Recommended deflation window size.


                      This depends on ILO, IHI and NS.  NS is the


                      number of simultaneous shifts returned


                      by ILAENV(ISPEC=15).  (See ISPEC=15 below.)


                      The default for (IHI-ILO+1).LE.500 is NS.


                      The default for (IHI-ILO+1).GT.500 is 3*NS/2.


            ISPEC=14: Nibble crossover point. (See IPARMQ for


                      details.)  Default: 14% of deflation window


                      size.


            ISPEC=15: Number of simultaneous shifts in a multishift


                      QR iteration.


                      If IHI-ILO+1 is ...


                      greater than      ...but less    ... the


                      or equal to ...      than        default is


                           1               30          NS =   2(+)


                          30               60          NS =   4(+)


                          60              150          NS =  10(+)


                         150              590          NS =  **


                         590             3000          NS =  64


                        3000             6000          NS = 128


                        6000             infinity      NS = 256


                  (+)  By default some or all matrices of this order


                       are passed to the implicit double shift routine


                       SLAHQR and this parameter is ignored.  See


                       ISPEC=12 above and comments in IPARMQ for


                       details.


                 (**)  The asterisks (**) indicate an ad-hoc


                       function of N increasing from 10 to 64.


            ISPEC=16: Select structured matrix multiply.


                      If the number of simultaneous shifts (specified


                      by ISPEC=15) is less than 14, then the default


                      for ISPEC=16 is 0.  Otherwise the default for


                      ISPEC=16 is 2.

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.