1. INTERPOLATION
For both irregulary spaced and evenly spaced data.

2. Linear Interpolation
x = a
x = b y x x

4. Linear Interpolation

5. Spline & Linear Interpolation

6. The function should be continuous at the joints.
Spline interpolation – we approximate the interpolation function y(x) over the interval [a, b] by dividing the interval into subregions.
The function should be continuous at the joints.
Define spline function y(x) of degree N with joints: b a
data need to be in ascending (or descending order); if not, it should be rearranged
Spline function y(x) has two properties:
a) In each interval ui-1  x  ui (i = 1,…, m), the function y(x) is a polynomial of degree < N.
b) At each interior joint, y(x) and its first N-1 derivatives are continuous.

7. The most common spline function is the cubic spline N = 3
Example. Consider a data series with elements (xi, yi), i =1, …, N
The first two derivatives y’(x) and y’’(x) of the interpolation function can be defined for all xi
The third derivative y’’’(x) is a constant for all x
At the segment junctions:
Continuity of the spline function
Continuity of the slope
Continuity of the curvature

8. Since y’’’(x) is a constant, y’’(x) must be linear, so that
Integrating twice, getting integration constants from a) continuity of the function and b) of the slope:

9. Cubic Spline
Fifth degree polynomial
Sixth degree polynomial
Eighth degree polynomial = cubic spline for this example

10. clear all
close all
x=[ ];
y=[ ];
plot(x,y,'o','MarkerFaceColor','red','MarkerSize',10)
axis([ ]) ; set(gca,'FontSize',18)
x1=[3:.25:9];
y1=interp1(x,y,x1);
hold on
plot(x1,y1,'b','LineWidth',3)
y2=interp1(x,y,x1,'spline');
plot(x1,y2,'r','LineWidth',4)
%Shape-preserving piecewise cubic interpolation.
%The interpolated value at a query point is based on a shape-preserving
%piecewise cubic interpolation of the values at neighboring grid points.
y3=interp1(x,y,x1,'pchip');
plot(x1,y3,'k','LineWidth',2)