LAPACK  3.4.2 LAPACK: Linear Algebra PACKage
zhseqr.f File Reference

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## Functions/Subroutines

subroutine zhseqr (JOB, COMPZ, N, ILO, IHI, H, LDH, W, Z, LDZ, WORK, LWORK, INFO)
ZHSEQR

## Function/Subroutine Documentation

 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)*H*(QZ)**H.```
Parameters:
 [in] JOB ``` JOB is CHARACTER*1 = 'E': compute eigenvalues only; = 'S': compute eigenvalues and the Schur form T.``` [in] 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.``` [in] N ``` N is INTEGER The order of the matrix H. N .GE. 0.``` [in] ILO ` ILO is INTEGER` [in] 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.GT.0, then 1.LE.ILO.LE.IHI.LE.N. If N = 0, then ILO = 1 and IHI = 0.``` [in,out] 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.GT.0 is given under the description of INFO below.) Unlike earlier versions of ZHSEQR, 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.``` [in] LDH ``` LDH is INTEGER The leading dimension of the array H. LDH .GE. max(1,N).``` [out] 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).``` [in,out] 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.GT.0 is given under the description of INFO below.)``` [in] LDZ ``` LDZ is INTEGER The leading dimension of the array Z. if COMPZ = 'I' or COMPZ = 'V', then LDZ.GE.MAX(1,N). Otherwize, LDZ.GE.1.``` [out] WORK ``` WORK is COMPLEX*16 array, dimension (LWORK) On exit, if INFO = 0, WORK(1) returns an estimate of the optimal value for LWORK.``` [in] LWORK ``` LWORK is INTEGER The dimension of the array WORK. LWORK .GE. max(1,N) is sufficient and delivers very good and sometimes optimal performance. However, LWORK as large as 11*N may be required for optimal performance. A workspace query is recommended to determine the optimal workspace size. If LWORK = -1, then 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.``` [out] INFO ``` INFO is INTEGER = 0: successful exit .LT. 0: if INFO = -i, the i-th argument had an illegal value .GT. 0: if INFO = i, ZHSEQR 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.```
Date:
November 2011
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).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
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 299 of file zhseqr.f.

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