1. Computers in Civil Engineering 53:081 Spring 2003 Lecture #15 Spline Interpolation

2. Newton Interpolation Problem In some cases Newton Interpolation (while going through all the data points) gives problematic results because of round-off errors and overshoot. This happens in the vicinity of an abrupt change. Simple, low order, piecewise polynomials (splines) give better results. Linear splines

3. Spline Concept Use low order (most often 3rd order), piecewise polynomials

4. Origin of Term Spline

5. Main drawback: lack of smoothness or, discontinuous derivatives – but it’s the best you can do with piecewise linear! Linear Splines

6. Linear Spline Interpolation

7. If we require that the m th order derivatives to be smooth, m+1 order spline must be used. For n+1 data points there are n intervals involved, thus, there are 3n coefficients to evaluate. Thus, 3n conditions are required. Quadratic Splines Quadratic (second order) splines, have first derivative continuous at knots.

8. Conditions Required I.The function values must be equal at the interior points ( 2n-2 conditions) II.The first and last functions must pass through the end points ( 2 conditions) III.The first derivatives at the interior points must be equal ( n-1 conditions) IV.Assume that the second derivative is zero at the first point ( 1 condition)

9. Quadratic Splines: Condition I The function values must be equal at the interior points ( 2n-2 conditions)

10. Quadratic Splines: Condition II The first and last functions must pass through the end points ( 2 conditions)

11. The first derivatives at the interior points must be equal ( n-1 conditions) Quadratic Splines: Condition III

12. Quadratic Splines Condition IV Assume that the second derivative is zero at the first point ( 1 condition) The first 2 points will be connected with a straight line.

13. Quadratic Splines The 3n unknowns can be obtained by solving a system of linear equations ( 3n by 3n ) See example 18.9 in the textbook for a worked example. 1(a) 3 2(a) 4 1(b) 2(b)

14. Cubic Splines

15. For n+1 data points, there are 4n unknowns to evaluate. Thus, 4n conditions are required: General form: I.The function values must be equal at the interior points ( 2n-2 conditions) II.The first and last functions must pass through the end points ( 2 conditions) III.The first derivatives at the interior points must be equal ( n-1 conditions) IV.The second derivatives at the interior points must be equal ( n-1 conditions) V.The second derivatives at the end points are zero (2 conditions)

16. Cubic Splines Solve 4n by 4n system of linear equations Solve n-1 by n-1 system of linear equations (derived using Lagrange polynomials) Two methods of solving for the unknown coefficients:

17. MATLAB Splines yy=spline(x,y,xx) Two functions for one-dimensional interpolation SPLINE: y=interp1(x,y,xi,’nearest’) INTERP1: y=interp1(x,y,xi,’linear’) y=interp1(x,y,xi,’spline’) y=interp1(x,y,xi,’cubic’)

18. Next: Numerical Integration