Scientific Computing Linear and Quadratic Splines

Splines Overview We have looked at two methods of finding interpolating polynomials for a set of data points – Lagrange polynomials and Newton’s method. While these methods are fairly simple to program, they have a serious drawback – for a large number of data points, the interpolating polynomial can be quite unstable.

Splines Overview Here is a set of 11 data points and the computer interpolating polynomial. Note the high degree of oscillation near the ends. This is an unfortunate property of interpolating polynomials of high degree!

Splines Overview Here is a picture of the same data, but with spline curves interpolating the data. Note how this interpolation is so much more smooth and stable than the previous example.

What is a Spline Curve? A spline curve is a curve that is made of a set of simple curves (lines, quadratics, cubics) that are joined together. A spline is piece-wise defined. That is, it is defined over a set of sub-intervals of a given interval. This set is called a partition. Definition: A partition of an interval [a,b] is a sequence of points between a and b such that a = t 0 < t 1 < … < t n = b The numbers t i (i = 1 to n-1) are called knots

Linear Spline Definition: A function S is a linear spline on [a,b] if – The domain of S is [a,b] – S is continuous on [a,b] – There is a partition of points on [a,b] such that S is a linear function on each sub-interval [t i, t i+1 ]

Linear Spline A linear spline is defined by its values at the set of knots. Given the table of values there is a unique linear spline with those values. On each sub-interval, [t i, t i+1 ], the linear spline is defined by a linear function: Then, S(t i )= y i and S(t i+1 ) = y i+1 (verify this)

Linear Spline Example:

Linear Spline Piece-Wise Linear Formula for Spline: Note: The fractions here are just the first divided differences for the data.

Matlab Linear Spline Function function v = piecelin2(x,y,u) %Piecewise linear interpolation. (Slightly modified from Moler text) % v = piecelin2(x,y,u) finds the piecewise linear value of S(x) % with S(x(j)) = y(j) and returns v = S(u). % First divided difference – “diff” is a built-in Matlab command delta = diff(y)./diff(x); % Find subinterval index k so that x(k) <= u < x(k+1) n = length(x); k = 1; for j = 2:n-1 if x(j) <= u; k = j; end % Evaluate spline at u s = u - x(k); v = y(k) + s*delta(k);

Matlab Linear Spline Example % x,y data in vector form x = 1:6; y = [ ]; % A series of values along the x-axis u = 1.0:.05:6.0; [m n] = size(u); % polyvals stores the linear spline values for the u-values splinevals = zeros(n); for i = 1:n % Compute each polynomial value splinevals(i) = piecelin2(x,y,u(i)); end % Plot the x-y data as circles ('o') and the polynomial data as '-' plot(x,y,'o',u,splinevals,'-')

Matlab Linear Spline Example

Quadratic Spline Definition: A function Q is a quadratic spline on [a,b] if – The domain of Q is [a,b] – Q is continuous on [a,b] – Q’ is continuous on [a,b] – There is a partition of points on [a,b] such that Q is a polynomial of degree <= 2 on each sub-interval [t i, t i+1 ]

Quadratic Spline Example: Class Exercise: Verify that Q satisfies all parts of the definition of a quadratic spline.

Formula for Quadratic Spline Need to find constants a i, b i, c i, such that

Formula for Quadratic Spline

Need to find constants a i, b i, c i, such that Thus, we need to determine 3n different constants.

Formula for Quadratic Spline We know that: – Q i (t i ) = y i – Q i (t i+1 ) = y i+1 – Q i ’ (t i+1 ) = Q i+1 ’ (t i+1 ) (Q’ is continuous at knots) This gives n + n + (n-1) =3n -1 equations for 3n unknowns! Need one more condition on Q. Could use one of these – a 0 = 0 (Q 0 is a line) – Q’(t 0 ) = z 0 (slope at start) – Q’’(t 0 ) = w 0 (curvature at start) We will use the second condition.

Formula for Quadratic Spline Let z i = Q i ’ (t i ) Recall: Q i (x) = a i (x-t i ) 2 + b i ( x-t i ) + c i Then, Q i (t i ) = y i -> c i = y i Also, Q i ’ (t i ) = z i -> b i = z i So, Q i (x) = a i (x-t i ) 2 + z i (x-t i ) + y i Since Q i ’(t i+1 ) = z i+1 we get 2a i (t i+1 -t i ) + z i = z i+1 Thus, a i = (z i+1 -z i )/2(t i+1 -t i )

Formula for Quadratic Spline We then get a formula for Q i (x) in terms of data points t i and y i and the derivative values z i Now, z 0 is given as data. How do we get other z i ?

Formula for Quadratic Spline

Algorithm: Given data (t i, y i ) and z 0 = derivative at Q(a), compute – 1) for i = 1, 2, …,n – 2)

Quadratic Spline Example: Data x = [0 1 4], y = [1 -2 1] Create Matlab Program piecequad. This is one of your homework problems. We add another input variable z0, so we have piecequad(x,y,z0,u) Output is Q i (u) for i = interval in which u lies. Our example, get – Q 0 (x)= -3x – Q 1 (x) = 7/3 (x-1) 2 – 7 (x-1) -2 (Verify this is correct!)

Quadratic Spline Example: Plot this using Matlab: x = [0 1 4]; y = [1 -2 1]; % A series of values along the x-axis u = 0.0:.05:4.0; [m n] = size(u); % polyvals stores the quadratic spline values for the u-values splinevals = zeros(n); for i = 1:n splinevals(i) = piecequad(x,y,1,u(i)); end % Plot the x-y data as circles ('o') and the polynomial data as '-' plot(x,y,'o',u,splinevals,'-')

Quadratic Spline Example: