Chapter 14 Curve Fitting : Polynomial Interpolation Gab Byung Chae

You’ve got a problem Estimation of intermediate values between data points

14.1 Introduction to Interpolation

(2) MATLAB Function : polyfit and polyval Only one straight line that connects two points. Only one parabola connects a set of three points.

Determining polynomial coefficients Interpolating polynomial of solve Ill-conditioned system (Vandermonde matrices)

4.1.2 MATLAB Functions >> format long >> T = [ ]; >> density = [ ]; >> p = polyfit(T, density,2) >> d = polyval(p,350)

14.2 Newton Interpolating polynomial

Graphical depiction of linear interpolation. The shaded areas indicate the similar triangles used to derive the Newton linear -interpolation formula [Eq. (14.5)]. Figure 14.2

FDD = finite divided difference The second FDD

Two linear interpolations to estimate ln 2. Note how the smaller interval provides a better estimate. Figure 14.3

Example 14.3 Problem : f(x) = ln x –x 1 = 1 f(x 1 ) =0 –x 2 = 4 f(x 2 ) = –x 3 = 6 f(x 3 ) = Solution : –b 1 = 0 –b 2 = ( – 0) /(4-1) = –b 3 = f 2 (x) = (x-1) – (x-1)(x-4) f 2 (x) = (x-1) – (x-1)(x-4)

The use of quadratic interpolation to estimate ln 2. The linear interpolation from x = 1 to 4 is also included for comparison. Figure 14.4

Newton Interpolation  Divided-difference table : Adding (3,14) and (4,22) <- a 1 <- a 2 <- a 3

Graphical depiction of the recursive nature of finite divided differences. This representation is referred to as a divided difference table. Figure 14.5

Example 14.4 Problem : f(x) = ln x –x 1 = 1 f(x 1 ) =0 –x 2 = 4 f(x 2 ) = –x 3 = 6 f(x 3 ) = –x 4 = 5 f(x 4 ) = f 3 (x) = b1 + b2(x-x1)+ b3(x-x1)(x-x2)+b4(x-x1)(x- x2)(x-x3) b3(x-x1)(x-x2)+b4(x-x1)(x- x2)(x-x3)

Solution : –b 1 = f(x 1 ) = 0 –b 2 = f[x 2, x 1 ] = ( – 0) /(4-1) = –f[x 3,x 2 ] = ( – ) /(6-4) = –f[x 4,x 3 ] = ( – ) /(5-6) = –b 3 = f[x 3,x 2,x 1 ] = ( – ) /(6-1) = = –f[x 4,x 3,x 2 ] = ( – ) /(5-4) = = –b 4 =f[x 4,x 3,x 2,x 1 ] = ( ) /(5-1) = =

f 3 (x) = (x-1) (x-1)(x-4) (x-1)(x-4) (x-1)(x- 4)(x-6) (x-1)(x- 4)(x-6)

function yint = Newtint(x,y,xx) % yint = Newtint(x,y,xx): % Newton interpolation. Uses an (n - 1)-order Newton % interpolating polynomial based on n data points (x, y) % to determine a value of the dependent variable (yint) % at a given value of the independent variable, xx. % input: % x = independent variable % y = dependent variable % xx = value of independent variable at which % interpolation is calculated % output: % yint = interpolated value of dependent variable

% compute the finite divided differences in the form of a % difference table n = length(x); if length(y)~=n, error('x and y must be same length'); end b = zeros(n,n); % assign dependent variables to the first column of b. b(:,1) = y(:); % the (:) ensures that y is a column vector. for j = 2:n for i = 1:n-j+1 for i = 1:n-j+1 b(i,j) = (b(i+1,j-1)-b(i,j-1))/(x(i+j-1)-x(i)); b(i,j) = (b(i+1,j-1)-b(i,j-1))/(x(i+j-1)-x(i)); end endend % use the finite divided differences to interpolate xt = 1; yint = b(1,1); for j = 1:n-1 xt = xt*(xx-x(j)); xt = xt*(xx-x(j)); yint = yint+b(1,j+1)*xt; yint = yint+b(1,j+1)*xt;end

14.3 Lagrange Interpolating polynomial weight coefficients The first order Lagrange interpolating polynomial Lagrange interpolating polynomial The n-th order Lagrange interpolating polynomial Lagrange interpolating polynomial

The second order Lagrange interpolating polynomial Lagrange interpolating polynomial

Example 14.5 Problem : Use a Lagrange interpolating polynomial of the first and second order to evaluate the density of unused motor oil at T = 15 o C –x 1 = 0 f(x 1 ) = 3.85 –x 2 = 20 f(x 2 ) = –x 3 = 40 f(x 3 ) = Solution : First-order at x=15: –f 1 (x)= Second-order : – f 2 (x)=

function yint = Lagrange(x,y,xx) % yint = Lagrange(x,y,xx): % Lagrange interpolation. Uses an (n - 1)-order Lagrange % interpolating polynomial based on n data points (x, y) % to determine a value of the dependent variable (yint) % at a given value of the independent variable, xx. % input: % x = independent variable % y = dependent variable % xx = value of independent variable at which the % interpolation is calculated % output: % yint = interpolated value of dependent variable MATLAB M-file:Lagrange

n = length(x); if length(y)~=n, error('x and y must be same length'); end s = 0; for i = 1:n product = y(i); for j = 1:n if i ~= j product = product*(xx-x(j))/(x(i)-x(j)); end s = s+product; end yint = s;

14.4 Inverse Interpolation (1,1) (3,3) (5,5) -> (?,2) Polynomial interpolation –Determine (n-1)-th order polynomial p(x) –solve p(t) = 2 Ex : (2, 0.5), (3,0.3333), and (4,0.25) –Find x so that f(x)=0.3 –f 2 (x) = x 2 – 0.375x –Solve 0.3 = x 2 – 0.375x for x –x = or

Extrapolation is the process of estimating a value of f(x) that lies outside the range of the known base points Extrapolation

Illustration of the possible divergence of an extrapolated prediction. The extrapolation is based on fitting a parabola through the first three known points. Figure 14.10

Example 14.6 Problem : Fit a seventh-order polynomial to the first 8 points (1920 to 1990). Use it to compute the population in 2000 by extrapolationand compare your prediction with the actual result. Solution : >> t = [1920 :10:1990]; >> t = [1920 :10:1990]; >> pop = [ ]; >> pop = [ ]; >> p = polyfit(t, pop, 7) >> p = polyfit(t, pop, 7) Warning message …… Warning message ……

>> ts = (t-1955)/35; >> P = polyfit(ts, pop, 7); >> polyval(p, ( )/35) >>tt=linspace(1920,2000); >>pp=polyval(p, (tt-1955)/35); Plot(t,pop, ‘o’, tt, pp)

Use of a seventh-order polynomial to make a prediction of U.S. population in 2000 based on data from 1920 through Figure 14.11

Oscillations Dangers of higher-order polynomial interpolation Ex : Ringe’s function

Comparison of Runge’s function (dashed line) with a fourth-order polynomial fit to 5 points sampled from the function. Figure 14.12

Comparison of Runge’s function (dashed line) with a tenth-order polynomial fit to 11 points sampled from the function. Figure 14.13