Download EQUIVALENT REAL FORMULATIONS FOR SOLVING COMPLEX

EQUIVALENT REAL FORMULATIONS FOR SOLVING
COMPLEX LINEAR SYSTEMS IN COMPUTATIONAL
ELECTROMAGNETICS
S. Tringali1 , G. Angiulli2 , G. Di Massa3
1
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DIMET Department, Università Mediterranea, 89100 Reggio Calabria, Italy
e-mail : [email protected]
DIMET Department, Università Mediterranea, 89100 Reggio Calabria, Italy
e-mail : [email protected]
2
DEIS Department, Università della Calabria, 87030 Rende (Cs), Italy
e-mail : [email protected]
Abstract: within this paper we make a comparison in
terms of performances between the GMRES method and
the K equivalent real formulation, when both applied to
the solution of complex linear systems like Cz = γ. As well
we point out how performances grow worse passing from
complex to real arithmetics, except that C fulfills suitable
conditions. Everything is verified over a set of matrices
coming out from discretization via MoM of the electrical
field integral equation in the electromagnetic scattering by
an ideal electric conductor.
Keywords: complex systems, real formulations, .
1
Introduction
By a practical point of view, in solving the integral
equation (EFIE in the following) describing the electromagnetic field backscattered by an ideal electrical
conductor we proceeds effectively by two steps: reducing the original problem to a linear equation system and subsequently inverting it by means of suitable numerical algorithms [1]. For general speaking, it
happens a projective method, as the one of moments
(MoM in the following) generates a system like
Cz = γ
(1)
where C is an n × n complex matrix, γ ∈ Cn is the
column of known terms and z ∈ Cn is the vector of
unknowns. Formally, the solution to the system is expressed by z = C −1 γ, where C −1 is the left inverse
of C. Almost the whole number of packets nowadays available employs real algebra to compute C −1
[2]. So this is enough to justify, whenever necessary,
any interest in developing effective tecniques capable
to convert a linear system like (1) into a purely real
form and comparing in terms of performances general
purpose solvers for the inversion problem either in real
arithmetic or in complex one. Specifically, by separation of real from imaginary, (1) can be restated in the
form (A + iB)(x + iy) = (α + iβ), where A, B ∈ Rn×n ;
α, β ∈ Rn and you let x = <(z) and y = =(z). Hence
equating coefficients on the left and right side through
R leads to the following real equivalent formulations,
expressed by 2 × 2 block matrices, which from now on
going we’ll refer to as K1-K4 formulations:
A −B
x
α
K1 :
=
B A
y
β
A B
x
α
K2 :
=
B −A
−y
β
B A
x
β
K3 :
=
A −B
y
α
B −A
x
β
K4 :
=
A B
−y
α
For future reference, we denote the matrix associated
with the K1 to K4 formulations by K1 to K4 , respectively. In fact, it’s to be said that, for any pair of
bases for R2n , there exists a different equivalent real
formulation of (1). This not standing, K1-K4 are
recognized to be among the most significant ones.
The convergence rate of an iterative method applied directly to an equivalent real formulation is often substantially worse than for the corresponding
complex iterative method. That’s due to the spectral
properties of the equivalent real formulation matrix,
which comes to be duplicated by a conjugate symmetry. From this point of view, a real approach seems
never appropiate as to solver performances. This not
standing, if C is hermitian and we use K1 or K4 formulations, the spectrum is preserved on passing from
complex to real arithmetic, with the only exception
the algebraic multiplicity of any eigenvalue belonging
to C is actually doubled [3]. Anyway this doesn’t
occur into trouble, when GMRES-based methods are
meant to be employed, because of their capability to
resolve simultaneously multiple eigenvalues. A further disadvantage of the real formulations K1-K4 is
1.1
Sparsity properties
If n is a positive integer and A ∈ Cn×n , we define a
sparsity index v(A) for A by letting v(A) equal to the
ratio between the total number of entries in A and
the number of its non-zero elements. That standing,
if r and i respectively denotes the number of purely
real and purely imaginary entries in the matrix of the
linear system (1) and K is the matrix of its real equivalent K formulation, we argue
v(K) = v(C) +
i+r
2n2
(3)
This produces an increasing sparsity of K with respect
to C which, despite a duplication of row and column
sizes induced by passing from complex to real algebra, turns into a benefit as to the rate of convergence
for GMRES-based iterative methods for a nice class
of problems (substantially, the ones in which C is hermitian and i + r is large) often encountered in electromagnetics. These facts are tested over some matrices
coming out from EFIE discretization via MoM and
are exemplified in the table (1) below.
Table 1: Numerical results.
Calculation time (s)
Total residue
Figure 1: Spectra of C (top) and K1 (bottom) before pre-
Complex form
K1 form
15,723
1,9204 e-013
9,369
2,751 e-013
condioning.
that they don’t preserve the sparsity pattern of C.
Anyway, this can be usefully bypassed adopting an
alternative formulation, which we call the K formulation, capable to preserve the nonzero pattern of the
block entries at the expense of doubling the size of
each dense submatrix. Here a block entry matrix is
a sparse matrix whose entries are all (small) dense
(sub)matrices.
So, in the K formulation, cij = aij + ibij corresponds, via the scalar K1 formulation, to the 2-by-2
block entry of the 2n-by-2n real matrix K given by
aij
bij
−bij
aij
References
[1] Peterson A., Ray S. R. Mittra R., “Computational
methods for electromagnetics”, IEEE Press, 2000
[2] Golub G., Van Loan C., “Matrix computations”, Hopkins University Press, 1996
[3] David Day, Michael A. Heroux, “Solving complexvalued linear systems via equivalent real formulations”,
Siam J. Sci. Comput., vol. 23, No. 2
(2)
and similarly for K2-K4 formulations. The properties of the K formulation defined enable us to implement efficient and robust preconditioned iterative
solvers for complex linear systems. We can efficiently
compute and apply the exact equivalent of a complexvalued preconditioner.
If the complex preconditioned linear system has
nice spectral properties, then the K formulation leads
to convergence that is competitive with the true complex solver and sometimes even better. In the following, we try to give a brief explanation of these facts
introducing some arguments based on sparsity.
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