Charles A. Klein and Raj Mittra

Abstract - A method of calculating the sensitivity of the solution of Af = g due to errors in either the vector g or the matrix A is presented. The usefulness of this method is demonstrated with problems from antenna theory, scattering theory, and remote sensing.

**Stability and Convergence of Movement Method Solutions
**

R. Mittra and C.A. Klein

chapter in book *, Numerical and Asymptotic
Techniques in Electromagnetics.* editted by Raj Mittra.,
Springer-Verlag, 1975, pp. 129-163.

Abstract - The material presented in this chapter is divided into two broad
areas. In the first part we discuss the problem of stability of integral equation
solutions derived via matrix methods. We introduce a quantity called the
"condition number" which is useful not only for identifying the unstable
regimes in the solution procedure but also for evaluating different techniques
that might be employed for extracting stable solutions from ill-conditioned
equations.
The second part of this chapter deals with the question of convergence
of solution of matrix equations. Several different tests for convergence are
examined in considerable detail for the thin-wire antenna problem and some
relevant guidelines based on these studies are presented.

**An Application of the "Condition Number" Concept to the Solution
of Scattering Problems in the Presence of the Interior Resonant Frequencies
**

Charles A. Klein and Raj Mittra

*IEEE Transactions on Antennas and Propagation*, May 1975. pp.
431-435

Abstract - For the problem of scattering by infinite conducting cylinders,
erroneous results are obtained at certain frequencies. This paper discusses the
nature of these resonant solutions, indicates how the matrix condition of
numbers can be used to detect these situations, and evaluates several
methods used to obtain correct results at resonances.

**The Use of Pivot Ratios as a Guide to Stability of Matrix Equations
Arising in the Method of Moments **

Raj Mittra and Charles A. Klein

*IEEE Transactions on Antennas and Propagation*, May 1975. pp.
448-450.

Abstract - In a previous communication the authors have shown that in
applying the method of moments to problems in electromagnetics the matrix
condition numbers can be used to indicate regions in which instability occurs.
However, the calculation of the condition numbers requires an explicit
matrix inverse which is costly. The pivot ratio, which is easily calculated
from information known within the Gaussian elimination scheme, indicates
instabilities as well as the condition number but without requiring an explicit
inverse. A number of examples illustrating the use of the pivot ratio are
presented.

**The Effect of Different Testing Functions in the Moment Method
Solution of Thin-Wire Antenna Problems **

C.A. Klein and R. Mittra

*IEEE Transactions on Antennas and Propagation*, March 1975.
pp. 258-261

Abstract - The use of piecewise sinusoids for expansion functions and
rectangular pulses for testing functions is described in the application of the
method of moments to thin-wire antennas and scatterers. This choice of
expansion and testing functions allows efficient calculation of matrix
elements and yields accurate results for certain widths of the testing function.
However, for other widths, although the standard criteria for the selection of
these functions are satisfied, erroneous results are obtained. The validity of
the moment method solution can be checked by examining the near-field.

**Forward Scattering for Square Cylinders in the Resonance Region
with Application to Aperture Blockage **

W.V.T. Rusch, Jorgen Appel-Hansen, Charles A. Klein, and Raj
Mittra

*IEEE Transactions on Antennas and Propagation*, Vol. AP-24,
No. 2, March 1976. pp. 182-189

Abstract - The relationship between the induced field ratio (IFR) of a cylinder
and aperture blocking of a constant-phase aperture by cylindrical struts is
discussed. An analytical technique is presented whereby the IFR of
rectangular cylinders can be calculated using the method-of-moments with
internal constraint points. An experimental technique using a forward-
scattering range is used to measure the IFR's of square and circular cylinders
in an anechoic chamber. These experimental results are compared with the
theory, and their implications on aperture blocking losses and boresight cross
polarization are discussed.

**A Graphical Aid for Extracting Circular-Polarized Components from
Spinning-Linear Patterns**

Charles A. Klein

*IEEE Transactions on Antennas and Propagation*, May 1977. pp.
451-452

Abstract - Gain measurements of circularly polarized antennas are often made
with spinning linearly polarized gain standards rather than circularly
polarized (CP) gain standards. This communication describes a graphical aid
for quickly extracting the CP components.

**Design of Shaped-Beam Antennas Through Minimax Gain
Optimization **

Charles A. Klein

*IEEE Transactions of Antennas and Propagation*, Vol. AP-32, No.
9, September 1984. pp. 963-968

Abstract - An essential part of realizing a shaped beam by using a multiple-
beam antenna is determining the excitation coefficient for each individual
port. An algorithm which optimizes the power gain in a minimax sense is
described. This criterion, although not as mathematically tractable as least
squares, is necessary when the specifications prescribe that the worst case be as
good as possible. It will be shown how the basic algorithm is very efficient
and how it permits simple modifications to allow the specification of one-
sided errors for the pattern, and direct optimization of gain slopes. A design
of a C-band shaped-beam satellite antenna provides an example.