Reference no: EM132599742

MATH243 Introduction to Numerical Mathematics

1. Give two weaknesses of Newton-Raphson method of finding a root of the equation f (x) = 0.

2. Consider the function f (x) = x⁴ + x - 1.
(a) Use the intermediate value theorem to show that f (x) has a real root α in the interval [0.5, 1.0].
(b) Starting with the interval [0.5, 1.0], use interval bisection method twice to find an interval of width 0.125 which contain α.
(c) Taking 0.75 as a first approximation, apply the Newton-Raphson process twice to f (x) to obtain an approximate value of α. Give your answer to 3 decimal places.

3. (a) Using Gaussian elimination with partial pivoting and working to 2 decimal places, solve the system
1.12x1 - 1.42x2 - 0.75x3 = 0.57,
2.80x1 - 3.55x2 - 0.94x3 = 0.00,
1.31x1 - 1.45x2 + 0.43x3 = -1.54
(b) In a sentence, describe how scaled partial pivoting differs from partial pivoting.

4. Consider the system of linear equations
x1 + 2x2 + 3x3 = 2 2x1 - 3x2 + 2x3 = 9
3x1 + x2 - x3 = -1.
Solve the system using LU factorization, and find the determinant of the coefficient matrix using your LU factorization.

5. (a) Prove that if A and B are positive definite matrices then so is A+B.
(b) Find the first 3 iterations of the SOR method with ω = 1.6 for the following system, using
X(0) = (1, 2, 2)T ,
4x1 - x2 + x3 = 7,
8x1 - 8x2 + x3 = -21,
-2x1 + x2 + 5x3 = 15.
Ensure that you show all the calculations.

6. (a) Define the condition number of a matrix A.
(b) Explain what is meant by saying that the system of linear equations Ax = b is ill-conditioned.
(c) Calculate ||A||1 and ||A||2 for the matrix
A =            1   -2
                -3   4
7. (a) State Lagrange's formula for the interpolating polynomial of degree n or less, Pn(x) which passes through the points (xi, f (xi)), i = 0, 1, . . . , n, where all the points xi are distinct.

(b) The following is a partial tabulation of the function f (x) = ln(1 + x)

x           0.3          0.4           0.6          0.7

f (x)    0.2624      0.3365       0.4700     0.5306

i. Using all four of these points in Lagrange, formula (a) above, compute P3(x), the third degree Lagrange interpolating polynomial. (Do not expand).
ii. Use b(i) above to estimate ln(1.5)

8. (a) Define the divided differences f [xi, xi+1, . . . , xi+k] for a function f (x).
(b) Consider the quadratic polynomial
P2(x) = f [x0] + f [x0, x1](x - x0) + f [x0, x1, x2](x - x0)(x - x1).
Show that this polynomial interpolates f (x) at the points xi, f (xi)), i = 0, 1, 2. [6]
(c) Use divided differences to construct the quadratic polynomial P2(x) that passes through the points.
(0.1, 0.1248), (0.2, 0.2562), and (0.4, 0.6108). [4]

(d) Given that all these points lie on the curve y = f (x), use the polynomial P2(x) of the previous part to estimate f (0.3).

9. Consider a cubic spline interpolation
S0(x) = 1 + 2x - x³ for x ∈ [0, 1],

S1(x) = 2 + b(x-1) + c(x - 1)² + d(x - 1)³ for x ∈ [1, 2],

determine constants b, c, and d so that all conditions for a natural splines hold.

10. Use Simpson's rule with n = 6 to estimate the integral

₁∫⁴√1 + x³dx

11. Let g(x) = ln(x). Determine the values of n and h necessary to approximate ₁∫⁴g(x)dx to within 10^-3 using composite Simpson's rule.

12. Determine constants a, b, c, and d so that the quadrature formula

_-1∫¹ f (x) dx = a f (-1) + b f (1) + c f (-1) + d f (1)

has degree of precision three (3).

Attachment:- Introduction to Numerical Mathematics.rar