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NAG[f08yxc] NAG[nag_ztgevc] - Left and right eigenvectors of a pair of complex upper triangular matrices
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Calling Sequence
f08yxc(side, how_many, select, a, b, vl, vr, m, 'n'=n, 'mm'=mm, 'fail'=fail)
nag_ztgevc(. . .)
Parameters
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side - String;
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On entry: specifies the required sets of generalized eigenvectors.
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 Only right eigenvectors are computed.
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 Only left eigenvectors are computed.
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 Both left and right eigenvectors are computed.
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Constraint:   "Nag_BothSides", "Nag_LeftSide" or "Nag_RightSide". .
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how_many - String;
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On entry: specifies further details of the required generalized eigenvectors.
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 All right and/or left eigenvectors are computed.
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 All right and/or left eigenvectors are computed; they are backtransformed using the input matrices supplied in arrays vr and/or vl.
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 Selected right and/or left eigenvectors, defined by the array select, are computed.
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Constraint:   "Nag_ComputeAll", "Nag_BackTransform" or "Nag_ComputeSelected". .
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select - Vector(1..dim, datatype=boolean);
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Note: the dimension, dim, of the array select must be at least
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 when  ;
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 otherwise.
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a - 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: the matrix  must be in upper triangular form. Usually, this is the matrix  returned by f08xsc (nag_zhgeqz).
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b - 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: the matrix  must be in upper triangular form with non-negative real diagonal elements. Usually, this is the matrix  returned by f08xsc (nag_zhgeqz)
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vl - 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   "Nag_LeftSide" or "Nag_BothSides", vl contains:
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if  , the matrix  ;
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if  , the left eigenvectors of  specified by select, stored consecutively in the rows or columns (depending on the value of storage order) of the array vl, in the same order as their corresponding eigenvalues.
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vr - 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   "Nag_RightSide" or "Nag_BothSides", vr contains:
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if  , the matrix  ;
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if  , the right eigenvectors of  specified by select, stored consecutively in the rows or columns (depending on the value of storage order) of the array vr, in the same order as their corresponding eigenvalues.
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m - assignable;
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Note: On exit the variable m will have a value of type integer.
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On exit: the number of columns in the arrays vl and/or vr actually used to store the eigenvectors. If   "Nag_ComputeAll" or "Nag_BackTransform", m is set to n. Each selected eigenvector occupies one column.
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'n'=n - integer; (optional)
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Default value: the first dimension of the arrays vl, vr.
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Constraint:  . .
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'mm'=mm - integer; (optional)
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Default value: the second dimension of the arrays vl, vr.
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On entry: the number of columns in the arrays vl and/or vr.
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if  , mm must not be less than the number of requested eigenvectors.
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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_ztgevc (f08yxc) computes some or all of the right and/or left generalized eigenvectors of a pair of complex upper triangular matrices  .
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Description
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nag_ztgevc (f08yxc) computes some or all of the right and/or left generalized eigenvectors of the matrix pair  which is assumed to be in upper triangular form. If the matrix pair  is not upper triangular then the function f08xsc (nag_zhgeqz) should be called before invoking nag_ztgevc (f08yxc).
The right generalized eigenvector  and the left generalized eigenvector  of  corresponding to a generalized eigenvalue  are defined by
and
If a generalized eigenvalue is determined as  , which is due to zero diagonal elements at the same locations in both  and  , a unit vector is returned as the corresponding eigenvector.
Note that the generalized eigenvalues are computed using f08xsc (nag_zhgeqz) but nag_ztgevc (f08yxc) does not explicitly require the generalized eigenvalues to compute eigenvectors. The ordering of the eigenvectors is based on the ordering of the eigenvalues as computed by nag_ztgevc (f08yxc).
If all eigenvectors are requested, the function may either return the matrices  and/or  of right or left eigenvectors of  , or the products  and/or  , where  and  are two matrices supplied by you. Usually,  and  are chosen as the unitary matrices returned by f08xsc (nag_zhgeqz). Equivalently,  and  are the left and right Schur vectors of the matrix pair supplied to f08xsc (nag_zhgeqz). In that case,  and  are the left and right generalized eigenvectors, respectively, of the matrix pair supplied to f08xsc (nag_zhgeqz).
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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_CONSTRAINT"
On entry, ![select[j]=< >](/support/helpjp/helpview.aspx?si=10490/file08315/math505.png) . Constraint: ![select[j]](/support/helpjp/helpview.aspx?si=10490/file08315/math507.png)  true or false, for  .
"NE_ENUM_INT_2"
On entry,  ,  ,  . Constraint: if  , mm must not be less than the number of requested eigenvectors.
On entry,  ,  ,  . Constraint: if   "Nag_ComputeAll" or "Nag_BackTransform",  .
"NE_INT"
On entry,  . Constraint:  .
"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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Accuracy
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It is beyond the scope of this manual to summarize the accuracy of the solution of the generalized eigenvalue problem. Interested readers should consult Section 4.11 of the LAPACK Users' Guide (see Anderson et al. (1999)) and Chapter 6 of Stewart and Sun (1990).
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Further Comments
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nag_ztgevc (f08yxc) is the sixth step in the solution of the complex generalized eigenvalue problem and is usually called after f08xsc (nag_zhgeqz).
The real analogue of this function is f08ykc (nag_dtgevc).
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Examples
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side := "Nag_BothSides":
how_many := "Nag_BackTransform":
n := 4:
mm := 4:
selector := Vector([false], datatype=boolean):
a := Matrix([[-0.1438233530385214 +0.3741062911974961*I , 1.61978125186467 +0.5366170936785039*I , -5.735642590516833 -3.142957175229157*I , 0.8798344594474323 +2.121055303267374*I ], [0 +0*I , 0.2823856561928748 +0.5208777880043785*I , 0.8621110088889227 -2.167166334598111*I , -3.106018675330568 +1.841506903478933*I ], [0 +0*I , 0 +0*I , 1.855074244893337 -0.1373369717845307*I , -1.098834831815082 -0.9880107323733517*I ], [0 +0*I , 0 +0*I , 0 +0*I , 0.07048820152838589 -0.1296769105942377*I ]], datatype=complex[8]):
b := Matrix([[0.2263375141379954 +0*I , -0.002717330036084967 -0.742347364611636*I , -3.468409696159829 +0.1539788635359349*I , 0.7560522313684802 +1.232848952545635*I ], [0 +0*I , 0.5722925376877438 +0*I , 0.5726907354403189 -2.024908875044406*I , -3.138311880819075 +1.830164861889455*I ], [0 +0*I , 0 +0*I , 2.75062607182066 +0*I , -1.220622047233081 -1.233359591775959*I ], [0 +0*I , 0 +0*I , 0 +0*I , 0.1539006001169539 +0*I ]], datatype=complex[8]):
vl := Matrix([[-0.7090128348336837 -0.461476605295773*I , -0.03302144109783005 +0.09031754537465141*I , -0.1628796424652427 -0.492448017279966*I , 0.07738808120167312 -0.008290082415805361*I ], [-0.4837138711442768 +0.01888381070268122*I , 0.1585276550385816 +0.2160986717612115*I , 0.6753888500844321 +0.4579080836762148*I , -0.1663170609273308 +0.01853004417349101*I ], [0.01301179598327535 +0.04484513137596107*I , 0.07439715940099322 +0.2069093549321268*I , 0.101387554212626 +0.09091777594277581*I , 0.9636186134385221 -0.04865057630715301*I ], [0.1975761880057742 +0.09374269385944103*I , -0.03434567875120512 +0.932414407837031*I , -0.05299665545468768 -0.2092394246122089*I , -0.1815547581349471 -0.04525252686037*I ]], datatype=complex[8]):
vr := Matrix([[0.1125541434588202 -0.2530015045809273*I , -0.8454652086827862 -0.04393763372337054*I , 0.3778872609370845 -0.1354773959002193*I , -0.1007041124568258 +0.1878447328930141*I ], [-0.1679732170998375 +0.187087665395137*I , 0.3287962269251319 +0.114727120860468*I , 0.8540175661380134 -0.1668164345930828*I , -0.2209931686448631 -0.09747936182932314*I ], [0.6207048630041594 -0.6731519289895066*I , 0.2900654196264742 +0.1373850965514009*I , 0.08053884907530008 -0.0674853925159091*I , -0.08875151592027866 -0.1991525172615829*I ], [-0.08915551057292437 +0.117252956207256*I , -0.2275026764742463 +0.08496301506044918*I , -0.2211484305422937 -0.1473972671191686*I , -0.5421617631561363 -0.7448187009937004*I ]], datatype=complex[8]):
NAG:-f08yxc(side, how_many, selector, a, b, vl, vr, m, 'n' = n, 'mm' = mm):
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See Also
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Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Moler C B and Stewart G W (1973) An algorithm for generalized matrix eigenproblems SIAM J. Numer. Anal. 10 241–256
Stewart G W and Sun J-G (1990) Matrix Perturbation Theory Academic Press, London
f08 Chapter Introduction.
NAG Toolbox Overview.
NAG Web Site.
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