hseqr(3)

NAME
hseqr - hseqr: Hessenberg eig, QR iteration

SYNOPSIS

  Functions
subroutine chseqr (job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info)
CHSEQR
subroutine dhseqr (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)
DHSEQR
subroutine shseqr (job, compz, n, ilo, ihi, h, ldh, wr, wi, z, ldz, work, lwork, info)
SHSEQR
subroutine zhseqr (job, compz, n, ilo, ihi, h, ldh, w, z, ldz, work, lwork, info)
ZHSEQR

Detailed Description

Function Documentation

  subroutine chseqr (character job, character compz, integer n, integer ilo, integer ihi, complex, dimension( ldh, * ) h, integer ldh, complex, dimension( * ) w, complex, dimension( ldz, * ) z, integer ldz, complex, dimension( * ) work, integer lwork, integer info)

CHSEQR

Purpose:

        CHSEQR 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)*T*(QZ)**H.

Parameters
    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 unitary matrix Q on entry, and
                       the product Q*Z is returned.

    N

              N is INTEGER
               The order of the matrix H.  N >= 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 CGEBAL, and then passed to ZGEHRD
               when the matrix output by CGEBAL 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.

    H

              H is COMPLEX array, dimension (LDH,N)
               On entry, the upper Hessenberg matrix H.
               On exit, if INFO = 0 and JOB = 'S', H contains the upper
               triangular matrix T from the Schur decomposition (the
               Schur form). If INFO = 0 and JOB = 'E', the contents of
               H are unspecified on exit.  (The output value of H when
               INFO > 0 is given under the description of INFO below.)
               Unlike earlier versions of CHSEQR, this subroutine may
               explicitly H(i,j) = 0 for i > 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 >= max(1,N).

    W

              W is COMPLEX array, dimension (N)
               The computed eigenvalues. If JOB = 'S', 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).

    Z

              Z is COMPLEX 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 unitary 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 unitary matrix generated by CUNGHR
               after the call to CGEHRD which formed the Hessenberg matrix
               H. (The output value of Z when INFO > 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 >= MAX(1,N).  Otherwise, LDZ >= 1.

    WORK

              WORK is COMPLEX 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 >= 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 CHSEQR does a workspace query.
               In this case, CHSEQR 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
                 < 0:  if INFO = -i, the i-th argument had an illegal
                        value
                 > 0:  if INFO = i, CHSEQR failed to compute all of
                    the eigenvalues.  Elements 1:ilo-1 and i+1:n of W
                    contain those eigenvalues which have been
                    successfully computed.  (Failures are rare.)
                    If INFO > 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 > 0 and JOB   = 'S', 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 COMPZ = 'V', then on exit
                      (final value of Z)  =  (initial value of Z)*U
                    where U is the unitary matrix in (*) (regard-
                    less of the value of JOB.)
                    If INFO > 0 and COMPZ = 'I', then on exit
                          (final value of Z)  = U
                    where U is the unitary matrix in (*) (regard-
                    less of the value of JOB.)
                    If INFO > 0 and COMPZ = 'N', 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

Further Details:

                 Default values supplied by
                 ILAENV(ISPEC,'CHSEQR',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 CLAHQR vs CLAQR0 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) <= 500 is NS.
                          The default for (IHI-ILO+1) >  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
                           CLAHQR 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.

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 297 of file chseqr.f.

  subroutine dhseqr (character job, character compz, integer n, integer ilo, integer ihi, double precision, dimension( ldh, * ) h, integer ldh, double precision, dimension( * ) wr, double precision, dimension( * ) wi, double precision, dimension( ldz, * ) z, integer ldz, double precision, dimension( * ) work, integer lwork, integer info)

DHSEQR

Purpose:

        DHSEQR 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
    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 >= 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 DGEBAL, and then passed to ZGEHRD
               when the matrix output by DGEBAL 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.

    H

              H is DOUBLE PRECISION 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) < 0. If INFO = 0 and JOB = 'E', the
               contents of H are unspecified on exit.  (The output value of
               H when INFO > 0 is given under the description of INFO
               below.)
               Unlike earlier versions of DHSEQR, this subroutine may
               explicitly H(i,j) = 0 for i > 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 >= max(1,N).

    WR

              WR is DOUBLE PRECISION array, dimension (N)

    WI

              WI is DOUBLE PRECISION 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) > 0 and
               WI(i+1) < 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 DOUBLE PRECISION 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 DORGHR
               after the call to DGEHRD which formed the Hessenberg matrix
               H. (The output value of Z when INFO > 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 >= MAX(1,N).  Otherwise, LDZ >= 1.

    WORK

              WORK is DOUBLE PRECISION 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 >= 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 DHSEQR does a workspace query.
               In this case, DHSEQR 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
                 < 0:  if INFO = -i, the i-th argument had an illegal
                        value
                 > 0:  if INFO = i, DHSEQR 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 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 > 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 > 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 > 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 > 0 and COMPZ = 'N', 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

Further Details:

                 Default values supplied by
                 ILAENV(ISPEC,'DHSEQR',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 DLAHQR vs DLAQR0 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) <= 500 is NS.
                          The default for (IHI-ILO+1) >  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
                           DLAHQR 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.

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 314 of file dhseqr.f.

  subroutine shseqr (character job, character compz, integer n, integer ilo, integer ihi, real, dimension( ldh, * ) h, integer ldh, real, dimension( * ) wr, real, dimension( * ) wi, real, dimension( ldz, * ) z, integer ldz, real, dimension( * ) work, integer lwork, integer info)

SHSEQR

Purpose:

        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.

Parameters
    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 >= 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 > 0, then 1 <= ILO <= IHI <= 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) < 0. If INFO = 0 and JOB = 'E', the
               contents of H are unspecified on exit.  (The output value of
               H when INFO > 0 is given under the description of INFO
               below.)
               Unlike earlier versions of SHSEQR, this subroutine may
               explicitly H(i,j) = 0 for i > 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 >= 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) > 0 and
               WI(i+1) < 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 > 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 >= MAX(1,N).  Otherwise, LDZ >= 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 >= 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
                 < 0:  if INFO = -i, the i-th argument had an illegal
                        value
                 > 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 > 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 > 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 > 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 > 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 > 0 and COMPZ = 'N', 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

Further Details:

                 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) <= 500 is NS.
                          The default for (IHI-ILO+1) >  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.

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 314 of file shseqr.f.

  subroutine zhseqr (character job, character compz, integer n, integer ilo, integer ihi, complex*16, dimension( ldh, * ) h, integer ldh, complex*16, dimension( * ) w, complex*16, dimension( ldz, * ) z, integer ldz, complex*16, dimension( * ) work, integer lwork, integer info)

ZHSEQR

Purpose:

        ZHSEQR 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)*T*(QZ)**H.

Parameters
    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 unitary matrix Q on entry, and
                       the product Q*Z is returned.

    N

              N is INTEGER
               The order of the matrix H.  N >= 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 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.

    H

              H is COMPLEX*16 array, dimension (LDH,N)
               On entry, the upper Hessenberg matrix H.
               On exit, if INFO = 0 and JOB = 'S', H contains the upper
               triangular matrix T from the Schur decomposition (the
               Schur form). If INFO = 0 and JOB = 'E', the contents of
               H are unspecified on exit.  (The output value of H when
               INFO > 0 is given under the description of INFO below.)
               Unlike earlier versions of ZHSEQR, this subroutine may
               explicitly H(i,j) = 0 for i > 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 >= max(1,N).

    W

              W is COMPLEX*16 array, dimension (N)
               The computed eigenvalues. If JOB = 'S', 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).

    Z

              Z is COMPLEX*16 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 unitary 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 unitary matrix generated by ZUNGHR
               after the call to ZGEHRD which formed the Hessenberg matrix
               H. (The output value of Z when INFO > 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 >= MAX(1,N).  Otherwise, LDZ >= 1.

    WORK

              WORK is COMPLEX*16 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 >= 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 ZHSEQR does a workspace query.
               In this case, ZHSEQR 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
                 < 0:  if INFO = -i, the i-th argument had an illegal
                        value
                 > 0:  if INFO = i, ZHSEQR failed to compute all of
                    the eigenvalues.  Elements 1:ilo-1 and i+1:n of W
                    contain those eigenvalues which have been
                    successfully computed.  (Failures are rare.)
                    If INFO > 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 > 0 and JOB   = 'S', 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 COMPZ = 'V', then on exit
                      (final value of Z)  =  (initial value of Z)*U
                    where U is the unitary matrix in (*) (regard-
                    less of the value of JOB.)
                    If INFO > 0 and COMPZ = 'I', then on exit
                          (final value of Z)  = U
                    where U is the unitary matrix in (*) (regard-
                    less of the value of JOB.)
                    If INFO > 0 and COMPZ = 'N', 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

Further Details:

                 Default values supplied by
                 ILAENV(ISPEC,'ZHSEQR',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 ZLAHQR vs ZLAQR0 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) <= 500 is NS.
                          The default for (IHI-ILO+1) >  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
                           ZLAHQR 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.

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 297 of file zhseqr.f.

Author

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