1 / 23

Numerical Computation

Numerical Computation. Lecture 11: Polynomial Interpolation United International College. Review. Polynomial Interpolation : Given data {(y k ,x k )}, find polynomial P(x) such that P(x k )= y k Lagrange Interpolation Newton Interpolation 1. P 0 (x) = y 0

plato
Download Presentation

Numerical Computation

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Numerical Computation Lecture 11: Polynomial Interpolation United International College

  2. Review • Polynomial Interpolation: Given data {(yk ,xk )}, find polynomial P(x) such that P(xk)= yk • Lagrange Interpolation • Newton Interpolation 1. P0(x) = y0 2. Pk+1 (x) = Pk (x) + ck (x – x0 )(x – x1 ) … (x – xk ) Set Pk+1 (xk+1) = yk+1 solve for ck (k=1,2,…,n) 3. P(x) = Pn (x)

  3. Today • We will cover: • More on Polynomial Interpolation • Readings: • Pav, section 5.1.3, 5.1.4 • Moler, section 3.1 • Reading Quiz!!

  4. Interpolation Definition: Given a set of (x,y) data points, Interpolation is the process of finding a function (usually a polynomial) that passes through these data points.

  5. Newton’s Nested Formula Recall: Newton’s method is iterative with Pk+1 (x) = Pk (x) + ck+1 (x – x0)(x – x1) … (x – xk) Now, Pk (x) satisfies a similar formula: Pk (x) = Pk-1 (x) + ck (x – x0)(x – x1) … (x – xk-1) So, Pk+1 (x) = Pk-1 (x) + [ck (x – x0)…(x – xk-1)] + + [ck+1 (x – x0)…(x – xk)] Thus, Pn (x) = c0 + [c1 (x – x0)] + … + [cn (x – x0 )…(x – xn-1 )]

  6. Newton Nested Formula Pn (x) = c0 + [c1 (x – x0)] + … + [cn (x – x0 )…(x – xn-1 )] We can write this in a compact form as Here we define the product for k =0 to be 1, so the first term is c0 , as it should be. We can rewrite this formula as (verify this!!) Pn (x) = c0 + (x – x0)[c1 + (x – x1)[c2 + (x – x2)[…]]]

  7. Newton Nested Formula Pn (x) = c0 + (x – x0)[c1 + (x – x1)[c2 + (x – x2)[…]]] This allows a better way of calculating Pn (t) for a value of t (or x): Then, vn = Pn (t) This is called Newton’s Nested formula for Pn

  8. Efficiency Newton’s Nested formula Requires 2n additions, n multiplications O(n) Lagrange Interpolation Requires at least n2 additions O(n2)

  9. Newton Divided Differences vn = Pn (t) This is efficient, but is there a fast way to calculate theck ‘s? Let us suppose that yk = f(xk ) for some function f.

  10. Newton Divided Differences Definition (Divided Differences). For a given collection of data { (xi ,f(xi) ) } a kth order divided difference is a function of k + 1 (not necessarily distinct) data values written as f[xi , xi+1 , … , xi+k]. 0th Order: f[xi ] = f(xi) = yi kth Order:

  11. Newton Divided Differences Theorem For Newton’s Interpolation formula Pn (x) = c0 + [c1 (x – x0)] + … + [cn (x – x0 )…(x – xn )] the coefficients ck can be calculated as ck = f[x0 , x1 , … , xk]

  12. Newton Divided Differences Algorithm for Calculating Divided Differences: Calculate the differences in “pyramid” fashion, until you reach the last one.

  13. Newton Divided Differences Example: Data (1,3), (1/2, -10), (3,2)

  14. Newton Divided Differences Example: Data (1,3), (1/2, -10), (3,2)

  15. Newton Divided Differences Example: Data (1,3), (1/2, -10), (3,2) Top row of the pyramid contains coefficients, c0 = 3, c1 = 26 , c2 = -53/5 Recall: Earlier work gave p(x) = 3+ 26 (x-1) – (53/5) (x-1)(x-1/2)

  16. Matlab - Newton Divided Differences Notes: We want to store the values in the “pyramid”. These can be stored in a matrix. We let F(i,k) = kth divided difference Differences are indexed from 0 to n, so F will have indexes from 1 to n+1. Vectors: x[], y[] for initial data, y = f(x).

  17. Matlab - Newton Divided Differences Notes: F(i,k) = kth divided difference Get F(i,k+1) = (F(i+1,k)-F(i,k))/(x(i+k)-x(i))

  18. Matlab - Newton Divided Differences function F = div_diff(x, y) % div_diff function computes all divided differences for % the data stored in x and y = f(x) n = length(x); m = length(y); if m~=n; error('x and y must be same size'); else F = zeros(n, n); for i = 1:n % define 0th divide difference F(i,1) = y(i); end for k = 1:n-1 % Get kth divided difference for i = 1:n-k F(i,k+1) = (F(i+1,k)-F(i,k))/(x(i+k)-x(i)); end end

  19. Matlab - Newton Divided Differences Example: x=[1 0.5 3]‘; y=[3 -10 2]'; div_diff(x,y) ans = 3.0000 26.0000 -10.6000 -10.0000 4.8000 0 2.0000 0 0 Top row (F(1,*)) holds coefficients ck .

  20. Matlab - Newton Nested Formula Need loop running “Backwards” For evaluation, we replace t by a value u.

  21. Matlab - Newton Nested Formula function v = newtonInterp(x,y,u) % v = newtonInterp(x,y,u) computes the Newton method % interpolation value p(u) for the given data x and y n = length(x); m = length(y); if m~=n; error('x and y must be same size'); else F = div_diff(x,y); % Compute Newton polynomial using nested format v = F(1,n); % v0 = cn for i = n-1:-1:1 v = F(1,i) + (u-x(i))*v; % nested formula end end

  22. Matlab - Newton Nested Formula Test: >> newtonInterp(x,y,1) ans = 3 >> newtonInterp(x,y,0.5) ans = -10 >> newtonInterp(x,y,3) ans = 2

  23. Class Exercise Here is a table of values for the temperature at a given depth for a lake. Find a good estimate of the temperature at a depth of 7.5 meters. Use a cubic interpolating polynomial to help solve this problem.

More Related