代写Mathematics程序，代写Final Written Project

日期：2018-05-15 01:03

Mathematics

Final Written Project

Spring, 2018

1. Determine the value n and h required to approximate ∫ 2

dx to within 10−5 using the

Trapezoidal rule, and the Simpson’s rule, respectively.

2. Let x˜ be an approximation of the solution to Ax = b. Why is ∥b − A˜x∥ = ∥r∥ not a good

measurement for the accuracy of the solution x˜? Give an example to support your argument.

Calculate the relative error and absolute error in your example.

3. Suppose that we are to determine a, b, c, d, and e for a quadrature formula

∫ 1−1f(x)dx = af(−1) + bf(0) + cf(1) + df′(−1) + ef′

(1)that gives exact results for a class of polynomials.

(a) What is the degree of this class of polynomials in general?

(b) Establish necessary equations to find the constants a, b, c, d and e that will produce an

exact quadrature formula for the class of polynomials specified in part (a).

(c) Solve the system of equations for a, b, c, d, e.

4. Let the system of equations be Ax = b with an exact solution x. Rewrite the equation so

that iterative solutions have the form of x

k = T xk−1 + c, where x

k

stands for the solution at

the k

th iteration.

(a) Under what condition would the iteration algorithm converge?

(b) With the convergence condition given in (a), how do we know that the solution is the

true solution of the system Ax = b? [Hint: Follow the notes in class]

5. (a) State the problem of the discrete least square approximation of a set of data {xi

, yi}m1

using polynomials, and explain how are the normal equations obtained. Do not derive the

normal equations.

(b) Let ϕ0(x) = 12

, ϕk(x) = cos kx, k = 1, 2, · · · , n and ϕn+k(x) = sin kx, k = 1, 2, · · · , n−1 be

the set of 2n trigonometric polynomials. Show that functions {ϕm} are orthogonal functions

with respect to the weight function w(x) = 1. [Hint: Follow the book calculation, and the

class notes].

(c) Let f(x) = x. Find the continuous least square approximation of f(x) = x on [−π, π]

using trig polynomial {ϕ0, ϕ1, · · · , ϕ2n−1} given above.

(d) State the problem of the continuous least square approximation of a function f(x) on

[a, b] using polynomials.

6. (a) What is the best way to place the interpolation nodes for the Lagrange polynomial interpolation

to minimize the absolute approximation error of a function f(x) on [-1, 1]? Explain

why.

1

(b) What if f(x) is defined on [1,3]? Show your method, and explain how the method in part

(a) can still be applied.

(c) Use the zeros of the monic Chebyshev polynomial T˜

4(x) to construct an interpolating

polynomial of degre 3 for f(x) = x ln x on [1, 3]. Find the bound of the maximum error of

the interpolating polynomial on the entire interval [1,3].

(d) Use 4 evenly spaced nodes {xi} on [1,3] with x0 = 1 and x3 = 3 to find a polynomial

interpolation of degree 3 for f(x) = x ln x on [1,3]. Find also the bound of the maximum

error for the interpolating polynomial on the entire interval [1,3].

(e) Compare the two error bounds on (c) and (d). Is the conclusion consistent with the

conclusion on Section 8.4 about the minimizing property of monic Chebyshev polynomials.