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thing that is not usually available. Figure 5.10 Intervals and functions used for piecewise cubic interpolation. • Not-a-Knot Spline: s. 1. (x2) = s. 2. (x2) and s n?1. (xn) = s n. (xn). This is the default choice in MATLAB. Whichever choice is made, the resulting function s(x) provides a smooth interpolation of the given data points.
(xj), j = 2,,n ? 1 (S ? C. 2. [a, b]). f) two end conditions: examples i) S (x1) = S (xn)=0(natural or free spline); ii) S (x1) = f (x1),S (xn) = f (xn). (complete or clamped spline); iii) S. 1. = S n?1. =0(parabolically terminated); iv) S. 1. (x2) = S. 2. (x2), S n?2. (xn?1) = S n?1. (xn?1). (not-a-knot). Note: if. Sj(x) = aj +bj(x?xj)+cj(x?xj). 2.
Understanding the conditions that underlie a cubic fit. • Understanding the differences between natural, clamped, and not-a-knot end conditions. • Knowing how to fit a spline to data with MATLAB's built-in functions. U d t di h ltidi i. l i t l ti i. • Understanding how multidimensional interpolation is implemented with MATLAB.
5.1: Cubic Splines. Interpolating cubic splines need two additional conditions to be uniquely defined. Definition. [11.3] An cubic interpolatory spilne s is called a Not-a-Knot Spline s. 0. (x1) = s. 1. (x1) and s m?2. (xm?1) = s m?1. (xm?1). We consider here natural splines. MATLAB uses splines with a not-a-knot condition.
4 Mar 2002 6. 0. Not-a-knot Spline Without specifying any extra conditions at the end points (other than that the spline interpolates the data points there), the not-a-knot spline requires that the third derivative of the spline is continuous at x1 and xN 1. One can determine the equations for m0 and mN by requiring that s?
SPLINES. A cubic spline is a function defined piecewise with each piece being a cubic polynomial. Let the break points (knots) be x1 < x2 < < xn, and let y1,y2,,yn be the For example, we can interpolate a parabola r(x) through the data Not a Knot Spline In this type of spline, we obtain two additional condi- tions by
Piecewise Linear Polynomial. 6. A piecewise linear function for the spline of degree 1 can be written as where is a linear polynomial. The knots t. 0. , t. 1. ,, t n and the It is important that the spline of degree 1 be continuous at the knots,. i.e., the left limit and the right . 15. The interpolation does not look smooth at knots
In this lecture, we will only consider spline interpolation using linear splines. (splines (splines of degree 3). Generalization to splines of general order is relatively straightforward. 2 Linear Interpolating Splines. A simple piecewise polynomial fit is the .. conditions are expressed mathematically as the not–a–knot conditions.
22 Aug 2008 Piecewise Linear Interpolation (C0). ? Cubic Hermite Interpolation (C1). ? Cubic Spline Interpolation (C2). ? The equations for C2. ? The spline matrices for different boundary conditions . derivative p (x0) = q0 and p (xm+1) = qm+1. ? Natural q0 = qm+1 = 0. ? Not-a-knot p0 = p1 and pm?1 = pm.
2. • Runge's example. Numerical Analysis 2, lecture 4, slide 3. A linear spline is a continuous piecewise degree-1 polynomial knots interpolating linear spline si (x) = bi . sk = spline(x,f); sn = nspline(x,f); xx = linspace(x(1),x(end)); plot(xx,ppval(sk,xx),'r',xx,ppval(sn,xx),'b',x,f,'o') legend('not-a-knot','natural'). 0. 6. 0. 1 notaknot.

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