Polynomials Polynomials Evaluation of polynomials involve only arithmetic operations, which can be done on today s digital computers. We consider polynomials with real coefficients and real variable. p n (x) = a n x n + a n 1 x n 1 + a 1 x + a 0, where Review n 0 Z, a j, x R, a n 0. A polynomial remains a polynomial with variable translation (shift of origin) or dilation(scaling). Any p n is bounded on finite intervals, unbounded on infinite intervals. Any p n (x) is continuous and differentiable for arbitrarily many times, i.e., P n (a, b) C (a, b), where P n denotes the set of polynomials with degree up to n. The set P n is a linear space of dimension n + 1. A natural basis of P n consists of the monic monomials 1, x, x 2,, x n, any polynomial in P n is a linear combination of the basis functions. 1
Polynomials Differentiation is a linear operator from P n to P n 1, indefinite integration is a linear operator from P n to P n+1, and a definite integration is a linear operator from P n to R. Two important theorems related to polynomials: - Taylor s theorem, - Weierstrass approximation theorem. Any p n (x) has n roots x i in the complex plane, and can be represented in the factored form p n (x) = i=1:n (x x i ). Complexity for evaluating a polynomial p n (x) in terms of arithmetic operations, - at any single point : O(n 2 ), O(n log(n)) and O(n), - at n points: O(n 3 ), O(n 2 log(n)), O(n 2 ), and O(n log 2 (n)), Approximation and interpolation with polynomials: 1. with monic monomial polynomials - Vandermonde matrices - Fast computation with Vandermonde matrices - Fast solution of Vandermonde systems 2. with special polynomials: - Chebyshev polynomials, - Hermite polynomials, - Legendre polynomials. 3. with piecewise polynomials - Splines, especially cubic splines, 2
- Fast computation with tridiagonal matrices, - Fast solution of tridiagonal systems, The applications include data smoothing, data modeling, data simplification. Other types of polynomials: complex variable, multiple variables. (to be continued) 1 2 1 Taylor, Brook, 1685-1731, England. First mathematics paper in 1708 on the center of oscillation of a body. 2 Weierstrass, Karl, 1815-1897, Germany. A high school teacher until welknown for his original work. 3
f n (x) = n where j=0 α j cos(jx) + β j sin(jx), x R, n 0 Z, α j, β j, R, α n + β n > 0. Review A trigonometric polynomial remains a trigonometric polynomial with variable translation (shift of origin) or dilation(scaling). Any f n is bounded on any interval (a, b) (find the bounds in terms of the coefficients). Any f n is continuous and differentiable for arbitrarily many times, i.e., T n (a, b) C (a, b), where T n denotes the set of trigonometric polynomials with degree up to n. Any f n is periodic on (, ) (find the period of f n ). The set T n is a linear space of dimension 2n+1. The following functions form an orthogonal basis of T n (0, 2π), {cos(jx), sin(kx) j = 0 : n, k = 1 : n.} with the inner product defined as f, g = 2π 0 4 f(x)g(x)dx.
Differentiation is a linear operator from T n to T n, indefinite integration is a linear operator from T n to T n, and a definite integration is a linear operator from T n to R. Any f n (x) of variable x can be expressed as a polynomial p n (cos(x)) of variable cos(x). Two important theorems related to trigonometric polynomials: Taylor s theorem, Fourier s theorem. Approximation and interpolation with trigonometric polynomials. Approximation and interpolation with piecewise trigonometric polynomials. The applications include data modeling, data analysis, data filtering, and data compression (e.g. COS transform, SIN transform, Fourier transform). Evaluation of cos(x) and sin(x) within required accuracy : - pre-select a set of reference points (seeds) in [0, π/4], - conversion between degrees and radians, and map x into the canonical interval [0, π/4], - find the closest seed, - determine a sufficient truncation degree in the Taylor series, - evaluate the approximating Taylor polynomial, - benchmark your code. 5
Complexity for evaluating a trigonometric polynomial f n (x) : - at any single point : - at n points: - at n equal-spaced points in [0, 2π): Euler s formula Representation in exponentials: e ix = cos(x) + i sin(x). f n (x) = γ 0 + n where j=1 γ je jx + γ j e jx, x R, n 0 Z, γ j, γ j C, γ n + γ n > 0. - Find relations between (α j, β j ) and (γ j, γ j ) - Denote by F n the set of functions as linear combinations of functions e jx, j = n : n, over the complex field. Then, T n F n. - Write evaluation of f n at n points in matrix-vector product form. - Consider the case that the evaluation points are the n-th roots of unity. (to be continued) 6