Math 104A Homework 2

 

General Instructions: Please write your homework papers neatly. You need to turn in both

your codes and descriptions on the appropriate runs you made by compressing and emailing them

to TA with your name and perm number as the file name. Write your own code, individually. Do

not copy codes!

1. (a) Write the Lagrangian form of the interpolating polynomial P2(x) corresponding to the

data in the table below:

xj f(xj )0 11 13 -5

(b) Use P2(x) you obtained in (a) to approximate f(2).

2. (a) Implement the Barycentric Formula for evaluating the interpolating polynomial for arbi￾trarily distributed nodes x0, ..., xn; you need to write a function or script that computes

the barycentric weights λ

(n)

j = 1/Πk6=j (xj jxk) first and another code to use these values

in the Barycentric Formula. Make sure to test your implementation.

(b) Consider the following table of data

xj f(xj )

0.00 0.0000

0.25 0.7071

0.50 1.0000

0.75 0.7071

1.25 -0.7071

1.50 -1.0000

Use your code in (a) to find P5(2) as an approximation of f(2).

3. The Runge Example. Let

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Using your Barycentric Formula code (Prob. 3) and (2) and (3) below, evaluate and plot the

interpolating polynomial of f(x) corresponding to

(a) the equidistributed nodes xj = =5 + j(10/n), j = 0, ..., n for n = 4, 8, and 12.

(b) the nodes xj = 5 cos( jπn

), j = 0, ..., n for n = 4, 8, 12, and 100.

(c) Repeat (a) for f(x) = e

for x ∈ [[5, 5] and comment on the result.

Remark 1. It can be shown that for equidistributed nodes one can use the barycentric weights

is the binomial coefficient (nchoosek(n,j) in Matlab). It can be shown that for the

Make sure to employ (2) and (3) in your Barycentric Formula code for this problem. To plot

the corresponding Pn(x) evaluate Pn(x) at a large number of points x¯ to have a good plotting

resolution, e.g. x¯k = =5+k(10/ne), k = 0, ..., ne with ne = 5000. Note that your Barycentric

Formula cannot be used to evaluate Pn(x) when x coincides with an interpolating node! Plot

also f for comparison. Compare (a) and (b) and comment on the result in view of what you

observed in Prob. 2.