Numerical Analysis : Several Interpolation Methods

[This is actually my project report for my Numerical Analysis course this semester]

* Cubic Spline Algorithm application in Typeface Design

As known Cubic Spline is usually the best choice in computer graphics applications, which have 2nd order continuity. If Typeface Design ([Knuth 1979] – this guy again !), each dimension (X & Y) should be applied with a Cubic Spline algorithm. Then, the curve nodes will be [Xi, Yi].

My work is as:

* Polynomial & Chebyshev & Cubic Spline Interpolation

The target function: f(x) = (1 + 6 * x ^ 2) ^ –1    (A typical Runge Function)

Polynomial Interpolation (Newton Form)

Ck = (Yk - Pk-1(xk)) / (Xk - X0)(Xk - X1)..(Xk - Xk-1) is used to calculate coefficiences;

u = u (X - Xi) + Ci (Horner Method) is used to compute the final results.

The Runge Phenomena is really obvious … this is why Polynomial is a bad choice for a wide range of interpolation.

Chebyshev:

Xi = cos[(i - 1) * PI / 20] (1 <= i <= 21)

Cubic Spline Interpolation:

Much better isn’t it : )