Abstract
We introduce general methods to evaluate functions and to find roots (or zeros) of functions of all kinds. We examine various approximation methods and study their respective numerical accuracy, by examining their backward error and the condition numbers for evaluation and rootfinding. ⊲
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Notes
- 1.
See Higham (2002) for an extended discussion of this issue.
- 2.
Bisection is explained in many numerical analysis texts, and it isn’t quite trivial: If you know a function is positive somewhere (say at x = 1) and negative somewhere else (say x = 0), have a look at it halfway between: If it’s zero there, you’ve found the root; if it’s positive, then the root is between 0 and \({}^{1}/_{2}\), and otherwise it’s between \({}^{1}/_{2}\) and 1. Repeat as necessary.
- 3.
Phrased this way, the proof works even for complex x with an appropriate definition of “average.” See the complex form of the Taylor series remainder given in Chap. 2.
- 4.
Note that the role of the variables x and y are reversed if we compare it to other relation of this type derived so far. This is because, as we mentioned, rootfinding is in some sense an inverse problem.
- 5.
A function is elementary if it can be constructed in a finite tower of Liouvillian extensions of logarithmic, exponential, or algebraic type: This means that \(\ln x =\int _{ 1}^{x}\frac{dt} {t}\), the exponential function, all rational polynomials, and all roots of polynomial functions are elementary. The trigonometric functions are just exponentials, for example, \(\sin x {= }^{({e}^{ix}-{e}^{-ix}) }/_{2i}\). A transcendental function is a function that is not algebraic; some elementary functions are transcendental. Transcendental functions that are not elementary (such as the Gamma function) are called special functions. See, for example, Geddes et al. (1992) for a fuller discussion.
- 6.
See Gil et al. (2007) for an excellent compendium of methods for numerical evaluation of many special functions of practical interest.
- 7.
See Muller et al. (2009) for a discussion of recent progress toward that laudable goal.
- 8.
- 9.
There is a difficulty in this example for very large x; see Problem 3.28.
- 10.
Henry Briggs (1561–1630) made the first truly useful tables of the real logarithm function. The calculations were carried out, by hand, to astonishing 14-digit accuracy; the tables contained tens of thousands of entries. Nowadays this computing feat seems superhuman; we doubt Robin could duplicate this, though as we will see, better methods will occur in this hypothetical than in the actuality of Briggs’ history.
- 11.
See the beautiful book Lanczos (1988). Lanczos’ method is indeed alive and well in modern numerical analysis, nowadays, but mostly for the solution of partial differential equations; our use of it here is twofold, as we will see.
- 12.
There is a very clever trick for recovering some of this information, which can be used to do nearly this; see Higham (2002). This trick relies on ensuring that the correlated rounding errors cancel out.
- 13.
Despite some of its shortcomings, the monomial basis is very good—near-optimal, in fact—if the coefficients don’t have too wide a dynamic range—on the unit disk | z | ≤ 1.
- 14.
Lawrence et al. (2012) concentrate heavily on finding a good initial guess function to start the iteration, and regularize the problem near the lines of discontinuity. Indeed, the bulk of the paper is on those two aspects. However, some time is spent on the iteration methods that might be used, as well. In addition to considering Newton iteration and Halley iteration, the paper also considers a family of higher-order methods and settles on one of them as being (marginally) the most efficient for that function.
- 15.
- 16.
- 17.
See, for example, the discussion in Corless and Assefa (2007).
- 18.
If we were to try to solve these equations in Matlab, we would have to first implement the Jacobian elliptic functions and their derivatives (see the Google project at http://code.google.com/p/elliptic/, and note that a partial implementation exists in vanilla Matlab, ellipj).
- 19.
We return to it when we study boundary value problems in Chap. 14.
- 20.
We will see more examples of approximation of functions using Chebfun in Chap. 8, and many more can be found at http://www2.maths.ox.ac.uk/chebfun/examples/approx/. A place that Chebfun also shines, however, is in rootfinding. You can find several very impressive examples at http://www2.maths.ox.ac.uk/chebfun/examples/roots/.
- 21.
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Corless, R.M., Fillion, N. (2013). Rootfinding and Function Evaluation. In: A Graduate Introduction to Numerical Methods. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8453-0_3
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