Numerical Methods Contents - SAM

Remark 7.0.2 (Interpolation and approximation: enabling technologies).
Approximation and interpolation are useful for several numerical tasks, like integration, differentiation
and computation of the solutions of differential equations, as well as for computer graphics: smooth
curves and surfaces.
8 Polynomial Interpolation
Remark 7.0.3 (Function representation).
this is a “foundations” part of the course
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8.1 Polynomials
Notation: Vector space of the polynomials of degree ≤ k, k ∈ N:
P k := {t ↦→ α k t k + α k−1 t k−1 + · · · + α 1 t + α 0 , α j ∈ K} . (8.1.1)
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General function f : D ⊂ R ↦→ K, D interval, contains an “infinite amount of information”.
Terminology:
the functions t ↦→ t k , k ∈ N 0 , are called monomials
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How to represent f on a computer?
t ↦→ α k t k + α k−1 t k−1 + · · · + α 0 = monomial representation of a polynomial.
➙ Idea: parametrization, a finite number of parameters α 1,...,α n , n ∈ N, characterizes f.
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Obvious: P k is a vector space. What is its dimension?
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Special case:
Representation with finite linear combination of basis functions
b j : D ⊂ R ↦→ K, j = 1, ...,n:
f = ∑ n
j=1 α jb j , α j ∈ K .
➙ f ∈ finite dimensional function space V n := Span {b 1 , ...,b n }.
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✬
Theorem 8.1.1 (Dimension of space of polynomials).
dim P k = k + 1 and P k ⊂ C ∞ (R).
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Proof. Dimension formula by linear independence of monomials.
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Why are polynomials important in computational mathematics ?
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Easy to compute, integrate and differentiate
Vector space & algebra
Analysis: Taylor polynomials & power series
Remark 8.1.1 (Polynomials in Matlab).
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MATLAB: α k t k + α k−1 t k−1 + · · · + α 0 ➙ Vector (α k , α k−1 , ...,α 0 ) (ordered!).
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Remark 8.1.2 (Horner scheme).
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p(t) = (t · · · t(t(α n t + α n−1 ) + α n−2 ) + · · · + α 1 ) + α 0 . (8.1.2)
Evaluation of a polynomial in monomial representation:
Horner scheme