f(x, y, z). Multidimensional interpolation is often accomplished by a sequence of one-dimensional interpolations. We discuss this in §. | 108 Chapter 3. Interpolation and Extrapolation f x y z . Multidimensional interpolation is often accomplished by a sequence of one-dimensional interpolations. We discuss this in . CITED REFERENCES AND FURTHER READING Abramowitz M. and Stegun . 1964 Handbook of Mathematical Functions Applied Mathematics Series Volume 55 Washington National Bureau of Standards reprinted 1968 by Dover Publications New York . Stoer J. and Bulirsch R. 1980 Introduction to NumericalAnalysis New York Springer-Verlag Chapter 2. Acton . 1970 Numerical Methods That Work 1990 corrected edition Washington Mathematical Association of America Chapter 3. Kahaner D. Moler C. and Nash S. 1989 Numerical Methods and Software Englewood Cliffs NJ Prentice Hall Chapter 4. Johnson and Riess . 1982 Numerical Analysis 2nd ed. Reading MA Addison-Wesley Chapter 5. Ralston A. and Rabinowitz P. 1978 AFirstCourse in Numerical Analysis 2nd ed. New York McGraw-Hill Chapter 3. Isaacson E. and Keller . 1966 AnalysisofNumericalMethods New York Wiley Chapter 6. Polynomial Interpolation and Extrapolation Through any two points there is a unique line. Through any three points a unique quadratic. Et cetera. The interpolating polynomial of degree N - 1 through the N points y1 f x1 y2 f x2 . yN f xN is given explicitly by Lagrange s classical formula p x - x2 x - x3 . x - xn x - x1 x - x3 . x - xn xi - x2 xi - x3 . xi - xn y x2 - xi x2 - x3 . x-2 - xn y x - xi x - x2 . x - xn-1 xn - xi xn - x2 . xn - xn-i y There are N terms each a polynomial of degree N - 1 and each constructed to be zero at all of the xi except one at which it is constructed to be yi. It is not terribly wrong to implement the Lagrange formula straightforwardly but it is not terribly right either. The resulting algorithm gives no error estimate and it is also somewhat awkward to program. A much better algorithm for constructing the same unique interpolating polynomial is Neville s algorithm closely related to and .
