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NAG[f08psc] NAG[nag_zhseqr] - Eigenvalues and Schur factorization of complex upper Hessenberg matrix reduced from complex general matrix
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Calling Sequence
f08psc(job, compz, ilo, ihi, h, w, z, 'n'=n, 'fail'=fail)
nag_zhseqr(. . .)
Parameters
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job - String;
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On entry: indicates whether eigenvalues only or the Schur form  is required.
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 Eigenvalues only are required.
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 The Schur form  is required.
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Constraint:   "Nag_EigVals" or "Nag_Schur". .
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compz - String;
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On entry: indicates whether the Schur vectors are to be computed.
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 No Schur vectors are computed (and the array z is not referenced).
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 The Schur vectors of  are computed (and the array z is initialized by the function).
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Constraint:   "Nag_NotZ", "Nag_UpdateZ" or "Nag_InitZ". .
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ilo - integer;
ihi - integer;
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On entry: if the matrix  has been balanced by f08nvc (nag_zgebal), then ilo and ihi must contain the values returned by that function. Otherwise, ilo must be set to 1 and ihi to n.
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Constraint:  and  . .
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h - Matrix(1..dim1, 1..dim2, datatype=complex[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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On exit: if  , the array contains no useful information.
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w - Vector(1..dim, datatype=complex[8]);
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Note: the dimension, dim, of the array w must be at least  .
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On exit: the computed eigenvalues, unless  (in which case see Section [Error Indicators and Warnings]). The eigenvalues are stored in the same order as on the diagonal of the Schur form  (if computed).
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z - Matrix(1..dim1, 1..dim2, datatype=complex[8], order=order);
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Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.
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On entry: if  , z must contain the unitary matrix  from the reduction to Hessenberg form.
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If  , z need not be set.
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If  , z is not referenced.
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'n'=n - integer; (optional)
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Default value: the dimension of the array h.
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On entry:  , the order of the matrix  .
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Constraint:  . .
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'fail'=fail - table; (optional)
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The NAG error argument, see the documentation for NagError.
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Description
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Purpose
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nag_zhseqr (f08psc) computes all the eigenvalues and, optionally, the Schur factorization of a complex Hessenberg matrix or a complex general matrix which has been reduced to Hessenberg form.
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Description
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nag_zhseqr (f08psc) computes all the eigenvalues and, optionally, the Schur factorization of a complex upper Hessenberg matrix  :
where  is an upper triangular matrix (the Schur form of  ), and  is the unitary matrix whose columns are the Schur vectors ![z[i]](/support/helpjp/helpview.aspx?si=10459/file08286/math277.png) . The diagonal elements of  are the eigenvalues of  .
The function may also be used to compute the Schur factorization of a complex general matrix  which has been reduced to upper Hessenberg form  :
![[[A,=QHQ^H, Q ],[,=(Q,Z)T(Q,Z)^H.]]](/support/helpjp/helpview.aspx?si=10459/file08286/math290.png)
In this case, after f08nsc (nag_zgehrd) has been called to reduce  to Hessenberg form, f08ntc (nag_zunghr) must be called to form  explicitly;  is then passed to nag_zhseqr (f08psc), which must be called with  .
The function can also take advantage of a previous call to f08nvc (nag_zgebal) which may have balanced the original matrix before reducing it to Hessenberg form, so that the Hessenberg matrix  has the structure:
![([[H[11],H[12],H[13]],[,H[22],H[23]],[,,H[33]]])](/support/helpjp/helpview.aspx?si=10459/file08286/math315.png)
where ![H[11]](/support/helpjp/helpview.aspx?si=10459/file08286/math318.png) and ![H[33]](/support/helpjp/helpview.aspx?si=10459/file08286/math320.png) are upper triangular. If so, only the central diagonal block ![H[22]](/support/helpjp/helpview.aspx?si=10459/file08286/math322.png) (in rows and columns ![i[lo]](/support/helpjp/helpview.aspx?si=10459/file08286/math324.png) to ![i[hi]](/support/helpjp/helpview.aspx?si=10459/file08286/math326.png) ) needs to be further reduced to Schur form (the blocks ![H[12]](/support/helpjp/helpview.aspx?si=10459/file08286/math328.png) and ![H[23]](/support/helpjp/helpview.aspx?si=10459/file08286/math330.png) are also affected). Therefore the values of ![i[lo]](/support/helpjp/helpview.aspx?si=10459/file08286/math332.png) and ![i[hi]](/support/helpjp/helpview.aspx?si=10459/file08286/math334.png) can be supplied to nag_zhseqr (f08psc) directly. Also, f08nwc (nag_zgebak) must be called after this function to permute the Schur vectors of the balanced matrix to those of the original matrix. If f08nvc (nag_zgebal) has not been called however, then ![i[lo]](/support/helpjp/helpview.aspx?si=10459/file08286/math342.png) must be set to 1 and ![i[hi]](/support/helpjp/helpview.aspx?si=10459/file08286/math344.png) to  . Note that if the Schur factorization of  is required, f08nvc (nag_zgebal) must not be called with   "Nag_Schur" or "Nag_DoBoth", because the balancing transformation is not unitary.
nag_zhseqr (f08psc) uses a multishift form of the upper Hessenberg  algorithm, due to Bai and Demmel (1989). The Schur vectors are normalized so that  = 1](/support/helpjp/helpview.aspx?si=10459/file08286/math361.png) , but are determined only to within a complex factor of absolute value 1.
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Error Indicators and Warnings
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"NE_ALLOC_FAIL"
Dynamic memory allocation failed.
"NE_BAD_PARAM"
On entry, argument  had an illegal value.
"NE_CONVERGENCE"
The algorithm has failed to find all the eigenvalues after a total of  iterations.
"NE_INT"
On entry,  . Constraint:  .
"NE_INT_3"
On entry,  ,  ,  . Constraint:  and  .
"NE_INTERNAL_ERROR"
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please consult NAG for assistance.
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Further Comments
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The total number of real floating-point operations depends on how rapidly the algorithm converges, but is typically about:
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 if only eigenvalues are computed;
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 if the Schur form is computed;
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 if the full Schur factorization is computed.
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The real analogue of this function is f08pec (nag_dhseqr).
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Examples
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job := "Nag_Schur":
compz := "Nag_InitZ":
n := 4:
ilo := 1:
ihi := 4:
h := Matrix([[-3.97 -5.04*I , -1.1318 -2.5693*I , -4.6027 -0.1426*I , -1.4249 +1.733*I ], [-5.4797 +0*I , 1.8585 -1.5502*I , 4.4145 -0.7638*I , -0.4805 -1.1976*I ], [0 +0*I , 6.2673 +0*I , -0.4504 -0.029*I , -1.3467 +1.6579*I ], [0 +0*I , 0 +0*I , -3.5 +0*I , 2.5619 -3.3708*I ]], datatype=complex[8]):
h_orig := copy(h):
w := Vector(4, datatype=complex[8]):
z := Matrix([[0 +0*I , 0 +0*I , 0 +0*I , 0 +0*I ], [0 +0*I , 0 +0*I , 0 +0*I , 0 +0*I ], [0 +0*I , 0 +0*I , 0 +0*I , 0 +0*I ], [0 +0*I , 0 +0*I , 0 +0*I , 0 +0*I ]], datatype=complex[8]):
NAG:-f08psc(job, compz, ilo, ihi, h, w, z, 'n' = n):
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See Also
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Bai Z and Demmel J W (1989) On a block implementation of Hessenberg multishift  iteration Internat. J. High Speed Comput. 1 97–112
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
f08 Chapter Introduction.
NAG Toolbox Overview.
NAG Web Site.
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