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.

  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.

  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.

  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.

Author

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