with higher order sometimes, but not always, giving higher accuracy. “Romberg integration,” which is discussed in §, is a general formalism for making use of integration methods of a variety of different orders, and we recommend it highly. | 130 Chapter4. Integration of Functions of various orders with higher order sometimes but not always giving higher accuracy. Romberg integration which is discussed in is a general formalism for making use of integration methods of a variety of different orders and we recommend it highly. Apart from the methods of this chapter and of Chapter 16 there are yet other methods for obtaining integrals. One important class is based on function approximation. We discuss explicitly the integration of functions by Chebyshev approximation Clenshaw-Curtis quadrature in . Although not explicitly discussed here you ought to be able to figure out how to do cubic spline quadrature using the output of the routine spline in . Hint Integrate equation over x analytically. See 1 . Some integrals related to Fourier transforms can be calculated using the fast Fourier transform FFT algorithm. This is discussed in . Multidimensional integrals are another whole multidimensional bag of worms. Section is an introductory discussion in this chapter the important technique of Monte-Carlo integration is treated in Chapter 7. CITED REFERENCES AND FURTHER READING Carnahan B. Luther . and Wilkes . 1969 Applied Numerical Methods New York Wiley Chapter 2. Isaacson E. and Keller . 1966 AnalysisofNumericalMethods New York Wiley Chapter 7. Acton . 1970 Numerical Methods That Work 1990 corrected edition Washington Mathematical Association of America Chapter 4. Stoer J. and Bulirsch R. 1980 Introduction to NumericalAnalysis New York Springer-Verlag Chapter 3. Ralston A. and Rabinowitz P. 1978 A First Course in Numerical Analysis 2nd ed. New York McGraw-Hill Chapter 4. Dahlquist G. and Bjorck A. 1974 Numerical Methods Englewood Cliffs NJ Prentice-Hall . Kahaner D. Moler C. and Nash S. 1989 Numerical Methods and Software Englewood Cliffs NJ Prentice Hall Chapter 5. Forsythe . Malcolm . and Moler . 1977 Computer Methods for Mathematical Computations Englewood .
