Interpolation by Spline Functions

Transcription

1 Interpolation by Spline Functions Com S 477/577 Sep High-degree polynomials tend to have large oscillations which are not the characteristics of the original data. To yield smooth interpolating curves cubic spline functions are often used. Roughly speaking a cubic spline is a set of polynomials of degree three that are smoothly connected at given supporting points. More precisely let a = x 0 < x 1 < < x n = b be a partition of the interval [ab]. They are also refer to as knots. A cubic spline S on the interval is a real function with the following properties: (a) S is twice continuously differentiable on [ab] or in short S C [ab]; (b) S coincides with a cubic polynomial on every subinterval [x i x i+1 ] i = 01...n 1. Thus a cubic spline consists of cubic polynomials pieced together in such a fashion that their values and those of their first two derivatives coincide at the knots x 0 x 1...x n. 1 Theoretical Foundations Consider a set F = {f 0 f 1...f n } of n+1 real numbers. We denote by S(F;) an interpolating spline function with S(F;x i ) = f i for i = 01...n. Such an interpolating spline function is not uniquely determined by the supporting points (x i f i ) i = 01...n. Roughly speaking there are still two degrees of freedom left. So we add an additional requirement: 1 S (F;a) = S (F;b) = 0. (1) This condition will ensure the uniqueness of the interpolating function. For this purpose we consider the sets K m [ab] integer m > 0 of real functions f on [ab] for which f (m 1) is absolutely continuous on [ab] and the square of f (m) is integrable on the interval. Clearly S K 3 [ab]. If f K [ab] then we can define f = b a f (x) dx. The following theorem states that spline functions have an important minimum-norm property. 1 Another alternative is to require that S (k) (F;a) = S (k) (F;b) for k = 01; that is S(F;) is periodic. A third alternative is to specify the values of S (F;a) and S (F;b). Areal functionf isabsolutely continuous on[ab]iffor everyǫ > 0thereexistδ > 0suchthat i f(bi) f(ai) < ǫ for every finite set of intervals [a ib i] with a a 1 < b 1 < < a n < b n b and i bi ai < δ. 1

2 Theorem 1 Given a partition a = x 0 < x 1 < < x n = b of the interval [ab] values F = {f 0...f n } and a function f K [ab] with f(x i ) = f i for i = 01...n then f S(F;) = f S(F;) 0 holds for every spline function S(F;) under the condition that S (F;a) = S (F;b) = 0. The minimum-norm property of the spline function implies that among all functions f in K [ab] with f(x i ) = f i i = 01...n the spline function S(F;) with S (F;a) = S (F;b) = 0 minimizes the integral f = b a f (x) dx. The spline function under condition (1) is often referred to as the natural spline function. Computing Interpolating Cubic Spline Functions Nowwedescribehowtodeterminecubicsplinefunctionsthatassumeprescribedvaluesf 0 f 1...f n at the knots a = x 0 < x 1 < < x n = b respectively and satisfy condition (1). Let = x j+1 x j j = 01...n 1. For convenience we adopt a short-hand notation M j = S (F;x j ) j = 01...n 1 where F = {f 0 f 1...f n } for the values of the second derivatives at the knots. We refer to M 0...M n as the moments of S(F;). As we will see below spline functions are readily characterized by their moments which can be calculated as the solution of a system of linear equations. The second derivative S (F;) of the spline function coincides with a linear function in each interval [x j x j+1 ] j = 0...n 1. Thus these linear functions can be described in terms of the moments M i : S (F;x) = M j x j+1 x Integrating the above equation yields x x j for x [x j x j+1 ]. S (F;x) = M j (x j+1 x) (x x j ) S(F;x) = M j (x j+1 x) 3 (x x j ) 3 +A j () +A j (x x j )+B j for x [x j x j+1 ] j = 01...n 1. Here A j B j are constants of integration for which we can obtain the following equations from S(F;x j ) = f j and S(F;x j+1 ) = f j+1 : M j+1 M j +B j = f j +A j +B j = f j+1.

3 Solving the above equations we have B j = f j M j (3) A j = f j+1 f j (M j+1 M j ). (4) This yields the following representation of the spline function in terms of its moments: for x [x j x j+1 ] where S(F;x) = α j +β j (x x j )+γ j (x x j ) +δ j (x x j ) 3 (5) α j = f j γ j = M j β j = S (F;x j ) = M j +A j = f j+1 f j M j δ j = S (Fx + j ) = M j+1 M j. Thus the interpolating spline function has been characterized by its moments M j. Next we will address how to calculate these moments. The continuity of S (F;) ensures that S (F;x j ) = S (F;x + j ). () This yields n 1 equations for the moments. Substituting equations (4) for A j in () gives S (F;x) = M j (x j+1 x) (x x j ) For j = 1...n 1 we have therefore S (F;x j ) = f j f j 1 Plugging these two equations into () yields + f j+1 f j (M j+1 M j ). + 3 M j + M j 1 S (F;x + j ) = f j+1 f j 3 M j M j+1 M j M j + 3 M j+1 = f j+1 f j f j f j 1 3

4 for i = 1...n 1. The above equations now can be written in a common format: where the coefficients are as follows µ j M j 1 +M j +λ j M j+1 = d j j = 1...n 1 λ j = + µ j = 1 λ j = d j = + ( fj+1 f j + f ) j f j 1 j = 1...n 1. There are n 1 equations for the n + 1 unknown moments. Two further equations can be derived from the moments at a and b: So we let M 0 = S (F;a) = 0; M n = S (F;b) = 0. λ 0 = 0 d 0 = 0 µ n = 0 d n = 0. This leads to the following system of linear equations for the moments: λ 0 0 µ 1 λ 1 µ λ n 1 0 µ n M 0 M 1 M n The coefficients λ i µ i d i are well defined. Note in particular that = λ i 0 µ i 0 λ i +µ i = 1 d 0 d 1 d n. (7) for all i = 01...n and that these coefficients depend only on the knots x j and not on the prescribed values f i. Theorem The system (7) of linear equations is nonsingular for any partition a = x 0 < x 1 < < x n = b of [ab]. The above theorem implies that the system (7) of equations has unique solutions. Consequently the problem of interpolation by cubic splines has a unique solution under the conditions S (F;a) = S (F;b) = 0. Example. Figure 1 shows the interpolation of two functions. In (a) the function f(x) = 4sinx is inter- 4

5 (a) (b) Figure 1: Cubic spline interpolations of (a) f(x) = 4sinx over [ 5π 5π The interpolating cubic splines are draw in green lines. ] and (b) p(x) = x4 10 over [ 44]. polated with the partition x i = (i 5)π i = by a cubic spline consisting of the following ten polynomials: x x x x 1.379x x x x x x x x x x x x x x x x x x x x x x x x x 3. In (b) the quartic polynomial p(x) = x4 10 is interpolated with the partition 4 < < 0 < < 4 by a cubic spline consisting of the following four polynomials: References x x x x x x x x x x x 3. [1] J. Stoer and R. Bulirsch. Introduction to Numerical Analysis. Springer-Verlag New York Inc. nd edition