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Solution of Linear Algebraic Equations part 5

A quick-and-dirty way to solve complex systems is to take the real and imaginary parts of (2.3.16), giving A·x−C·y=b (2.3.17) C·x+A·y=d which can be written as a 2N × 2N set of real equations | 50 Chapter2. Solution ofLinearAlgebraic Equations A quick-and-dirty way to solve complex systems is to take the real and imaginary parts of 2.3.16 giving A x C y b C x A y d 2.3.17 which can be written as a 2N x 2N set of real equations A -C x b C Al yl - d I 2.3.18 and then solved with ludcmp and lubksb in their present forms. This scheme is a factor of 2 inefficient in storage since A and C are stored twice. It is also a factor of 2 inefficient in time since the complex multiplies in a complexified version of the routines would each use 4 real multiplies while the solution of a 2N x 2N problem involves 8 times the work of an N x N one. If you can tolerate these factor-of-two inefficiencies then equation 2.3.18 is an easy way to proceed. CITED REFERENCES AND FURTHER READING Golub G.H. and Van Loan C.F. 1989 Matrix Computations 2nd ed. Baltimore Johns Hopkins University Press Chapter 4. Dongarra J.J. et al. 1979 LINPACK User s Guide Philadelphia S.I.A.M. . Forsythe G.E. Malcolm M.A. and Moler C.B. 1977 Computer Methods for Mathematical Computations Englewood Cliffs NJ Prentice-Hall 3.3 and p. 50. Forsythe G.E. and Moler C.B. 1967 Computer Solution of Linear Algebraic Systems Englewood Cliffs NJ Prentice-Hall Chapters 9 16 and 18. Westlake J.R. 1968 A Handbook ofNumerical Matrix Inversion and Solution ofLinear Equations New York Wiley . Stoer J. and Bulirsch R. 1980 Introduction to NumericalAnalysis New York Springer-Verlag 4.2. Ralston A. and Rabinowitz P. 1978 A First Course in Numerical Analysis 2nd ed. New York McGraw-Hill 9.11. Horn R.A. and Johnson C.R. 1985 MatrixAnalysis Cambridge Cambridge University Press . 2.4 Tridiagonal and Band Diagonal Systems of Equations The special case of a system of linear equations that is tridiagonal that is has nonzero elements only on the diagonal plus or minus one column is one that occurs frequently. Also common are systems that are band diagonal with nonzero elements only along a few diagonal lines adjacent to the main .