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.