CS3331: Numerical Methods
Quiz 2
April. 15, 1997

1.

(4%)

(a)

What is the linear interpolation formula for a function that passes two points (a, f(a)) and (b, f(b))?

(b)

What is the error for the above linear interpolation formula?

2.

(6%) There are two ways to find an interpolation polynomial: Lagrange interpolation and power series formulas. What are the tow major advantages of using the Lagrange interpolation formula over the other?

3.

(2%) What is the MATLAB command for polynomial fitting?

4.

(6%)

(a)

(4%) What are the four shape functions v_i(x), $i = \mbox{1 to 4}$, of the Lagrange interpolation polynomial that passes the following four points: (1, y₁), (2, y₂), (3, y₃), and (4, y₄)? (You don't need to expand the shape functions.)

(b)

(2%) What is the Lagrange interpolation polynomial in terms of the four shape functions?

5.

(6%)

(a)

What is the general formula for the Chebyshev polynomial?

(b)

What are the roots for the Chebyshev polynomial?

(c)

What is the advantage of using the roots as the points for polynomial interpolation?

6.

(6%) A 2-input function passes the following four points:

i (x_i, y_i) z_i

1 (0, 0) 1

2 (3, 0) 4

3 (0, 2) 3

4 (3, 2) 6

Find the value at (1, 1) using the bilinear interpolation.