COMP 350 Numerical Computing Assignment

Reference no: EM132400657

COMP 350 Numerical Computing Assignment - McGill University, Canada

Assignment - Polynomial Interpolation, Spline Interpolation, and Least Squares Approximation

Q1. Use your hand to find the Vandermonde form, the Lagrange form, and the Newton form of the interpolating polynomial for these data

x	1	2	3	4
y	2	0	-8	-37

Q2. (Programming by MATLAB) Let f(x) = 1/(1 + 25x²).

(a) Using 7 equally spaced nodes (or knots) on the interval [-1, 1],

i. Find the interpolating polynomial p(x) of degree 6 for f(x) by the Newton approach.

ii. Find the natural cubic spline function S(x) to interpolate f(x).

iii. Find the function g(x) = a + bx² + cx⁴to approximate f(x) by the least squares method.

Print the coefficients of p(x), S(x) and g(x).

Print the four values f(x), f(x) - p(x), f(x) - S(x), and f(x) - g(x) at 13 equally spaced points x_i = -1 + (1/6)i for i = 0, 1, . . . , 12.

Plot y = f(x), y = p(x), y = S(x) and y = g(x) on the same plot.

In additional to submitting your MATLAB programs, please put them in a PDF file so that it will be easy for the TA to read (note that the TA may or may not run your MATLAB programs).

(b) Using 7 Chebyshev nodes

x_i = cos((2i+1/2n+2)π), i = 0, 1, . . . , n, n = 6

to do the same things as in (a).

Note: You are not allowed to use MATLAB built-in functions polyfit, polyval, and spline.

Reference no: EM132400657

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