# COMPLEX routines for upper Hessenberg matrix

## chsein

```USAGE:
m, ifaill, ifailr, info, w, vl, vr = NumRu::Lapack.chsein( side, eigsrc, initv, select, h, w, vl, vr, [:usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CHSEIN( SIDE, EIGSRC, INITV, SELECT, N, H, LDH, W, VL, LDVL, VR, LDVR, MM, M, WORK, RWORK, IFAILL, IFAILR, INFO )

*  Purpose
*  =======
*
*  CHSEIN uses inverse iteration to find specified right and/or left
*  eigenvectors of a complex upper Hessenberg matrix H.
*
*  The right eigenvector x and the left eigenvector y of the matrix H
*  corresponding to an eigenvalue w are defined by:
*
*               H * x = w * x,     y**h * H = w * y**h
*
*  where y**h denotes the conjugate transpose of the vector y.
*

*  Arguments
*  =========
*
*  SIDE    (input) CHARACTER*1
*          = 'R': compute right eigenvectors only;
*          = 'L': compute left eigenvectors only;
*          = 'B': compute both right and left eigenvectors.
*
*  EIGSRC  (input) CHARACTER*1
*          Specifies the source of eigenvalues supplied in W:
*          = 'Q': the eigenvalues were found using CHSEQR; thus, if
*                 H has zero subdiagonal elements, and so is
*                 block-triangular, then the j-th eigenvalue can be
*                 assumed to be an eigenvalue of the block containing
*                 the j-th row/column.  This property allows CHSEIN to
*                 perform inverse iteration on just one diagonal block.
*          = 'N': no assumptions are made on the correspondence
*                 between eigenvalues and diagonal blocks.  In this
*                 case, CHSEIN must always perform inverse iteration
*                 using the whole matrix H.
*
*  INITV   (input) CHARACTER*1
*          = 'N': no initial vectors are supplied;
*          = 'U': user-supplied initial vectors are stored in the arrays
*                 VL and/or VR.
*
*  SELECT  (input) LOGICAL array, dimension (N)
*          Specifies the eigenvectors to be computed. To select the
*          eigenvector corresponding to the eigenvalue W(j),
*          SELECT(j) must be set to .TRUE..
*
*  N       (input) INTEGER
*          The order of the matrix H.  N >= 0.
*
*  H       (input) COMPLEX array, dimension (LDH,N)
*          The upper Hessenberg matrix H.
*
*  LDH     (input) INTEGER
*          The leading dimension of the array H.  LDH >= max(1,N).
*
*  W       (input/output) COMPLEX array, dimension (N)
*          On entry, the eigenvalues of H.
*          On exit, the real parts of W may have been altered since
*          close eigenvalues are perturbed slightly in searching for
*          independent eigenvectors.
*
*  VL      (input/output) COMPLEX array, dimension (LDVL,MM)
*          On entry, if INITV = 'U' and SIDE = 'L' or 'B', VL must
*          contain starting vectors for the inverse iteration for the
*          left eigenvectors; the starting vector for each eigenvector
*          must be in the same column in which the eigenvector will be
*          stored.
*          On exit, if SIDE = 'L' or 'B', the left eigenvectors
*          specified by SELECT will be stored consecutively in the
*          columns of VL, in the same order as their eigenvalues.
*          If SIDE = 'R', VL is not referenced.
*
*  LDVL    (input) INTEGER
*          The leading dimension of the array VL.
*          LDVL >= max(1,N) if SIDE = 'L' or 'B'; LDVL >= 1 otherwise.
*
*  VR      (input/output) COMPLEX array, dimension (LDVR,MM)
*          On entry, if INITV = 'U' and SIDE = 'R' or 'B', VR must
*          contain starting vectors for the inverse iteration for the
*          right eigenvectors; the starting vector for each eigenvector
*          must be in the same column in which the eigenvector will be
*          stored.
*          On exit, if SIDE = 'R' or 'B', the right eigenvectors
*          specified by SELECT will be stored consecutively in the
*          columns of VR, in the same order as their eigenvalues.
*          If SIDE = 'L', VR is not referenced.
*
*  LDVR    (input) INTEGER
*          The leading dimension of the array VR.
*          LDVR >= max(1,N) if SIDE = 'R' or 'B'; LDVR >= 1 otherwise.
*
*  MM      (input) INTEGER
*          The number of columns in the arrays VL and/or VR. MM >= M.
*
*  M       (output) INTEGER
*          The number of columns in the arrays VL and/or VR required to
*          store the eigenvectors (= the number of .TRUE. elements in
*          SELECT).
*
*  WORK    (workspace) COMPLEX array, dimension (N*N)
*
*  RWORK   (workspace) REAL array, dimension (N)
*
*  IFAILL  (output) INTEGER array, dimension (MM)
*          If SIDE = 'L' or 'B', IFAILL(i) = j > 0 if the left
*          eigenvector in the i-th column of VL (corresponding to the
*          eigenvalue w(j)) failed to converge; IFAILL(i) = 0 if the
*          eigenvector converged satisfactorily.
*          If SIDE = 'R', IFAILL is not referenced.
*
*  IFAILR  (output) INTEGER array, dimension (MM)
*          If SIDE = 'R' or 'B', IFAILR(i) = j > 0 if the right
*          eigenvector in the i-th column of VR (corresponding to the
*          eigenvalue w(j)) failed to converge; IFAILR(i) = 0 if the
*          eigenvector converged satisfactorily.
*          If SIDE = 'L', IFAILR is not referenced.
*
*  INFO    (output) INTEGER
*          = 0:  successful exit
*          < 0:  if INFO = -i, the i-th argument had an illegal value
*          > 0:  if INFO = i, i is the number of eigenvectors which
*                failed to converge; see IFAILL and IFAILR for further
*                details.
*

*  Further Details
*  ===============
*
*  Each eigenvector is normalized so that the element of largest
*  magnitude has magnitude 1; here the magnitude of a complex number
*  (x,y) is taken to be |x|+|y|.
*
*  =====================================================================
*

```
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## chseqr

```USAGE:
w, work, info, h, z = NumRu::Lapack.chseqr( job, compz, ilo, ihi, h, z, ldz, [:lwork => lwork, :usage => usage, :help => help])

FORTRAN MANUAL
SUBROUTINE CHSEQR( JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO )

*     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)*H*(QZ)**H.
*

*     Arguments
*     =========
*
*     JOB   (input) CHARACTER*1
*           = 'E':  compute eigenvalues only;
*           = 'S':  compute eigenvalues and the Schur form T.
*
*     COMPZ (input) 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     (input) INTEGER
*           The order of the matrix H.  N .GE. 0.
*
*     ILO   (input) INTEGER
*     IHI   (input) 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 CGEHRD
*           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.GT.0, then 1.LE.ILO.LE.IHI.LE.N.
*           If N = 0, then ILO = 1 and IHI = 0.
*
*     H     (input/output) 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.GT.0 is given under the description of INFO below.)
*
*           Unlike earlier versions of CHSEQR, 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   (input) INTEGER
*           The leading dimension of the array H. LDH .GE. max(1,N).
*
*     W        (output) 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     (input/output) 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.GT.0 is given under
*           the description of INFO below.)
*
*     LDZ   (input) 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  (workspace/output) COMPLEX array, dimension (LWORK)
*           On exit, if INFO = 0, WORK(1) returns an estimate of
*           the optimal value for LWORK.
*
*     LWORK (input) 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 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  (output) INTEGER
*             =  0:  successful exit
*           .LT. 0:  if INFO = -i, the i-th argument had an illegal
*                    value
*           .GT. 0:  if INFO = i, CHSEQR 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 a unitary matrix.  The final
*                value of  H is upper Hessenberg and 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 unitary 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 unitary matrix in (*) (regard-
*                less of the value of JOB.)
*
*                If INFO .GT. 0 and COMPZ = 'N', then Z is not
*                accessed.
*

*     ================================================================
*             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).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
*                       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.
*
*     ================================================================
*     Based on contributions by
*        Karen Braman and Ralph Byers, Department of Mathematics,
*        University of Kansas, USA
*
*     ================================================================
*     References:
*       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*       Algorithm Part I: Maintaining Well Focused Shifts, and Level 3
*       Performance, SIAM Journal of Matrix Analysis, volume 23, pages
*       929--947, 2002.
*
*       K. Braman, R. Byers and R. Mathias, The Multi-Shift QR
*       Algorithm Part II: Aggressive Early Deflation, SIAM Journal
*       of Matrix Analysis, volume 23, pages 948--973, 2002.
*
*     ================================================================

```
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