辅导Mathematics程序，辅导Final Written Project 讲解数据库SQL|讲解SPSS
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 ?0(x) = 12
, ?k(x) = cos kx, k = 1, 2, · · · , n and ?n+k(x) = sin kx, k = 1, 2, · · · , n?1 be
the set of 2n trigonometric polynomials. Show that functions {?m} 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 {?0, ?1, · · · , ?2n?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.