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Let A be a real matrix, then there exist orthogonal matrices of U and V and diagonal matrix ~ such that
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This is known as singular value decomposition. Using the L 2 norm for vectors, let the matrix norm be
The relations between matrix norm and singular values are
--1 ---l-t
the largest singular value of A =-1
is 1/an .
The condition number is
If the condition number is large, the matrix solution can be unstable. To change the condition number, pre-conditioning can be done. Let
The goal is to find a pre-conditioning matrix C
x'=Cx --1 x=C X'
Then ==-1 _ AC X' = b =-1 Multiply by C =-1==-1 I =-1_ C AC X = C b Let
(2.5.99) (2.5.100)
A =C
(2.5.103) (2.5.104)
=-1_ b = C b
Ax' =b
-, -
A judicious choice of the pre-conditioning matrix can drastically change the condition number. To make the computation of A efficient, C and C be simple matrices. must
Anderson, M. G. (1965), Scattering from bodies of revolution, IEEE Trans. Antennas Propagat., 13, 303-310. Barrett, R., M. Berry, T. F. Chan, J. Dammel, J. Donato, J. Dongarra, V. Eijkhout, R. Pozo, C. Romine, H. van der Vorst (1993), Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods, SIAM Publications, Philadelphia, PA. Borup, D. T. and O. P. Gandhi (1985), Calculation of high resolution SAR distributions in biological bodies using the FFT algorithm and conjugate gradient method, IEEE Trans. Microwave Theory Tech., 33, 417-419. Brigham, E. O. (1988), The Fast Fourier Transform and its Applications, Prientice-Hall, Englewood Cliffs, NJ. Catedra, M. F., E. Gago, and L. Nuno (1989), A numerical scheme to obtain the RCS of three-dimensional bodies of resonant size using the conjugate gradient method and the fast Fourier transform, IEEE Trans. Antennas Propagat., 37, 528-537. Chan, C. H. and R. Mittra (1987), Some recent developments in iterative techniques for solving electromagnetic boundary value problems, Radio Sci., 22(6), 929-934. Chew, W. C. (1990), Waves and Fields in Inhomogeneous Media, Van Nostrand Reinhold, New York. Gan, H. and W. C. Chew (1995), A discrete BCG-FFT algorithm for solving 3D inhomogeneous scatterer problems, J. Electromag. Waves and Appl., 9, 1339-1357. Glisson, A. W. and D. R. Wilton (1979), Simple and efficient numerical techniques for treating bodies of revolution, Technical Report 105, Engineering Experiment Station, The University of Mississippi, University, Mississippi. Glisson, A. W. and D. R. Wilton (1980), Simple and efficient numerical methods for problems of electromagnetic radiation and scattering from surfaces, IEEE Trans. Antennas Propagat., 28, 593-603. Goedecke, G. H. and S. G. O'Brien (1988), Scattering by irregular inhomogeneous particles via the digitized Green's function algorithm, Appl. Optics, 27, 2431-2438. Golub, G. H. and C. F. Van Loan (1996), Matrix Computations, 3rd edition, Johns Hopkins University Press, Baltimore, MD. Goodman, J. J., B. T. Drain, and P. J. Flatau (1991), Application of fast-Fourier-transform techniques to the discrete-dipole approximation, Optics Lett., 16(15), 1198-1200. Gradshteyn,1. S. and 1. M. Ryzhik (1965), Table of Integrals, Series and Products, Academic Press, New York. Harrington, R. F. (1968), Field Computation by Moment Method, Macmillan, New York. Hestenes, M. R. and E. Stiefel (1952), Methods of conjugate gradients for solving linear systems, J. Res. Nat. Bur. Standards, 49, 409-436. Jackson, J. D. (1975), Classical Electrodynamics, John Wiley & Sons, New York. Jin, J. M. and J. L. Volakis (1992), A biconjugate gradient FFT solution for scattering by planar plates, Electromagnetics, 12, 105-109. Joseph, J. (1990), Application of integral equation and finite difference method to electromagnetic scattering by two dimensional and boy of revolution geometries, Ph.D. thesis, Department of Electrical Engineering and Computer Science, University of UrbanaChampaign, Urbana, IL. Kas, A. and E. L. Yip (1987), Preconditioned conjugate gradient methods for solving electromagnetic problems, IEEE Trans. Antennas Propagat., 35,147-152.