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In this book we will systematically use elementary mathematical concepts which the reader should know already, yet he or she might not recall them immediately.

Computing the of a real function (equivalently, the of the equation () = 0) is a problem that we encounter quite often in scienti fic computing. In general, this task cannot be accomplished in a finite number of operations. For instance, we have already seen in Section 1.4.1 that when is a generic polynomial of degree greater than four, there do not exist explicit formulae for the zeros. The situation is even more difficult when is not a polynomial.

Approximating a function consists of replacing it by another function of simpler form that may be used as its surrogate. This strategy is used frequently in numerical integration where, instead of computing ∫ (), one carries out the exact computation of ∫ (), being a function simple to integrate (e.g. a polynomial), as we will see in the next chapter. In other instances the function may be available only partially through its values at some selected points. In these cases we aim at constructing a continuous function that could represent the empirical law which is behind the finite set of data. We provide some examples which illustrate this kind of approach.

In this chapter we propose methods for the numerical approximation of derivatives and integrals of functions. Concerning integration, quite often for a generic function it is not possible to find a primitive in an explicit form. Even when a primitive is known, its use might not be easy.

In applied sciences, one is quite often led to face a linear system of the form

Given a square matrix A ∈ℂ, the eigenvalue problem consists in finding a scalar λ (real or complex) and a nonnull vector such that

A differential equation is an equation involving one or more derivatives of an unknown function. If all derivatives are taken with respect to a single independent variable we call it an , whereas we have a when partial derivatives are present.

Boundary-value problems are differential problems set in an interval (, ) of the real line or in an open multidimensional region Ω ⊂ ℝ ( = 2, 3) for which the value of the unknown solution (or its derivatives) is prescribed at the end-points and of the interval, or on the boundary ∂Ω of the multidimensional region.

Solution 1.1 Only the numbers of the form ±0.1 · 2 with = 0,1 and e = ±2,±1,0 belong to the set F(2, 2,−2, 2). For a given exponent, we can represent in this set only the two numbers 0.10 and 0.11, and their opposites. Consequently, the number of elements belonging to F(2, 2,−2, 2) is 20. Finally, ∈ = ½.