introduction to numerical analysis i math/cmpsc 455 splines
TRANSCRIPT
Introduction to Numerical Analysis I
MATH/CMPSC 455
Splines
SPLINE
Suppose that n+1 points has been specified and satisfy . A spline of degree k is a function such that:
On each subinterval , is a polynomial of degree
has a continuous (k-1)-th derivate on
Spline is a piecewise polynomial of degree at most k, and has continuous derivatives of all order up to k-1.
Example: Spline of degree 0
Example: Spline of degree 1
CUBIC SPLINE
A cubic spline is a piecewise cubic polynomial
x …
y …
is cubic polynomial (piecewise polynomial)
(Interpolation)
, (Continuity)
Question: Can we uniquely determine the cubic spline?
Unknowns (coefficients):
Conditions:
• Interpolation:• Continuity of 1st order derivative:• Continuity of 2nd order derivative:
Total:
We have two degrees of freedom!
DERIVE THE CUBIC SPLINE
Step 1: 2nd order derivative is piecewise linear;
(use the continuity of
2nd order derivative)
Step 2: Take integration twice, get the cubic spline with undetermined coefficient;
Step 3: Determine the coefficient of the low order terms; (use the
interpolation property)
Step 4: Determine the remaining coefficient
by solving a symmetric, tri-diagonal system;(use the continuity of 1st order derivative)
Where:
Nature Cubic Spline:
Clamped Cubic Spline: