Homework 3 Solution

Please submit to CANVAS a .zip le that includes the following Matlab functions:




Lagrange interp.m







test Lagrange interpolation.m







compute Lebesgue function.m







test Lebesgue function.m










Exercise 1 Write a function Lagrange interp.m that computes the Lagrangian interpolant of a given set of data points (xi; yi), i = 1, 2, :::. The function should be of the form







function [y] = Lagrange interp(xi,yi,x)







Input:




xi: vector of interpolation nodes




yi: vector of data points at interpolation nodes




x: vector of points at which we evaluate the polynomial interpolant




Output:




y: polynomial interpolant evaluated at x







Hint: Compare the output of your function with the output of the Matlab/Octave built-in function,




y=polyval(polyfit(xi,yi,length(xi)-1),x)




(see the Matlab/Octave documentation).







Exercise 2 Consider the nonlinear function




f(x) =
1
;
x 2 [ 1; 1]:
(1)
1 + 20x2






By using the Matlab function you coded in Exercise 1, determine the Lagrangian interpolant of f, i.e. the polynomial N f(x) that interpolates the set of data fxi; f(xi)gi=0;::;N in the following cases:



















1






1. Evenly-spaced grid with N + 1 points




j




xj = 1 + 2 N
;j = 0; ::; N
(2)






2. Unevenly-spaced grid with N + 1 points (Chebyshev-Gauss-Lobatto points)




xj = cos


j ; j = 0; 1; :::; N;
(3)
N






In particular, write a Matlab function test Lagrange interpolation.m







function [x,f,P1,P2,P3,P4]=test Lagrange interpolation()







that returns the follwing items:




x: vector of 1000 evenly-spaced nodes in [ 1; 1] (use the command x=linspace(-1,1,1000)).




f: vector representing (1) evaluated at x.




P1: Lagrangian interpolant of (1) built on the grid (2) with N = 8 nodes and evaluated at x.




P2: Lagrangian interpolant of (1) built on the grid (2) with N = 20 nodes and evaluated at x.




P1: Lagrangian interpolant of (1) built on the grid (3) with N = 8 nodes and evaluated at x.




P1: Lagrangian interpolant of (1) built on the grid (3) with N = 20 nodes and evaluated at x.







The function should also plot (1) (in blue) and the Lagrangian interpolants (in red) obtained by using both the evenly-spaced and the unevenly-spaced grids for the cases N = 8 and N = 20 (4 di erent gures). Each gure should include the graph of f(x), the data points fxi; f(xi)g and the interpolant N f(x) through those points.




Hint: See the code uploaded in CANVAS for examples of similar plots.




Exercise 3 Let fli(x)gi=0;::;N be the set of Lagrange characteristic polynomials associated with the nodes fxj gj=0;::;N . We have seen in class that the polynomial interpolation error is related to the Lebesgue function

N






Xj
jlj(x)j




N (x) =
(Lebesgue function);
(4)
=0






and the Lebesgue constant






N = max
N (x)
(Lebesgue constant).
(5)
x2[ 1;1]









Given the vector of interpolation nodes xi=[xi(1) ... xi(N+1)], write a Matlab/Octave function compute Lebesgue function.m that returns the Lebesgue function (4) evaluated at









2






1000 evenly-spaced nodes between xi(1) and xi(N+1). Such function should also return the Lebesgue constant (5).




function [lambda,L]=compute Lebesgue function(xi)







Input:




xi: vector of interpolation nodes xi=[xi(1) ... xi(N+1)]




Output:




lambda: Lebesgue function N (x) evaluated at 1000 evenly-spaced nodes between xi(1) and xi(N+1).

L: Lebesgue constant N .




Apply the function compute Lebesgue function(xi) to the four cases of evenly- and unevenly-spaced grids you studied in Exercise 2. To this end, write a function






function [L1,L2,L3,L4]=test Lebesgue function()







that plots the Lebesgue function (4) corresponding to the aforementioned four cases (in 4 di erent Figures), and returns the value of the Lebesgue constant for each case.




Remark: Recall, that the smaller the Lebesgue constant the smaller the approximation error of the Lagrangian polynomial interpolation. If fact, we have seen in class that




kf(x) N (x)k1
(1 + N ) infN kf(x) (x)k1
(6)


2P

































































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