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NAG[f08qvc] NAG[nag_ztrsyl] - Solve complex Sylvester matrix equation  ,  and  are upper triangular or conjugate-transposes
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Calling Sequence
f08qvc(trana, tranb, sign_type, a, b, c, scal, 'm'=m, 'n'=n, 'fail'=fail)
nag_ztrsyl(. . .)
Parameters
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trana - String;
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On entry: specifies the option  .
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  .
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  .
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Constraint:   "Nag_NoTrans" or "Nag_ConjTrans". .
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tranb - String;
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On entry: specifies the option  .
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  .
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  .
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Constraint:   "Nag_NoTrans" or "Nag_ConjTrans". .
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sign_type - String;
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On entry: indicates the form of the Sylvester equation.
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 The equation is of the form  .
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 The equation is of the form  .
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Constraint:   "Nag_Plus" or "Nag_Minus". .
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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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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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c - 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: is overwritten by the solution matrix  .
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scal - assignable;
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Note: On exit the variable scal will have a value of type float.
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On exit: the value of the scale factor  .
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'm'=m - integer; (optional)
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Default value: the first dimension of the array a and the second dimension of the array athe arrays a, c.
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Constraint:  . .
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'n'=n - integer; (optional)
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Default value: the dimension of the array b.
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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_ztrsyl (f08qvc) solves the complex triangular Sylvester matrix equation.
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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_INT"
On entry,  . Constraint:  .
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.
"NE_PERTURBED"
 and  have common or close eigenvalues, perturbed values of which were used to solve the equation.
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Accuracy
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Consider the equation  . (To apply the remarks to the equation  , simply replace  by  )
Let  be the computed solution and  the residual matrix:
Then the residual is always small:
![||R||[F]=O(epsilon)(||A||[F],+,||B||[F])||X˜||[F]](/support/helpjp/helpview.aspx?si=10472/file08295/math394.png)
However,  is not necessarily the exact solution of a slightly perturbed equation; in other words, the solution is not backwards stable.
For the forward error, the following bound holds:
![||X˜,-,X||[F]<=(||R||[F])/(sep(A,,,B))](/support/helpjp/helpview.aspx?si=10472/file08295/math402.png)
but this may be a considerable over estimate. See Golub and Van Loan (1996) for a definition of  , and Higham (1992) for further details.
These remarks also apply to the solution of a general Sylvester equation, as described in Section [Further Comments].
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Examples
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trana := "Nag_NoTrans":
tranb := "Nag_NoTrans":
sign_type := "Nag_Plus":
m := 4:
n := 4:
a := Matrix([[-6 -7*I , 0.36 -0.36*I , -0.19 +0.48*I , 0.88 -0.25*I ], [0 +0*I , -5 +2*I , -0.03 -0.72*I , -0.23 +0.13*I ], [0 +0*I , 0 +0*I , 8 -1*I , 0.9399999999999999 +0.53*I ], [0 +0*I , 0 +0*I , 0 +0*I , 3 -4*I ]], datatype=complex[8]):
b := Matrix([[0.5 -0.2*I , -0.29 -0.16*I , -0.37 +0.84*I , -0.55 +0.73*I ], [0 +0*I , -0.4 +0.9*I , 0.06 +0.22*I , -0.43 +0.17*I ], [0 +0*I , 0 +0*I , -0.9 -0.1*I , -0.89 -0.42*I ], [0 +0*I , 0 +0*I , 0 +0*I , 0.3 -0.7*I ]], datatype=complex[8]):
c := Matrix([[0.63 +0.35*I , 0.45 -0.5600000000000001*I , 0.08 -0.14*I , -0.17 -0.23*I ], [-0.17 +0.09*I , -0.07000000000000001 -0.31*I , 0.27 -0.54*I , 0.35 +1.21*I ], [-0.93 -0.44*I , -0.33 -0.35*I , 0.41 -0.03*I , 0.57 +0.84*I ], [0.54 +0.25*I , -0.62 -0.05*I , -0.52 -0.13*I , 0.11 -0.08*I ]], datatype=complex[8]):
NAG:-f08qvc(trana, tranb, sign_type, a, b, c, scal, 'm' = m, 'n' = n):
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See Also
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Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Higham N J (1992) Perturbation theory and backward error for  Numerical Analysis Report University of Manchester
f08 Chapter Introduction.
NAG Toolbox Overview.
NAG Web Site.
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