APPLIED LINEAR ALGEBRA ASSIGNMENT

QUESTION 1 -

1. An amount of money A, invested over a period of n years with an annual interest rate r, grows by an amount A (1 + r)ⁿ. Calculate the final balances for investments of R750, R1000, R3000, R5000 and R11999 over 10 years with an interest rate of 9%.

2. Suppose vectors a and b are defined a S follows

a = [2 -1  5  0]

b = [3  2 -1  4]

Evaluate by hand the vector C in the following statements

(a) C = a - b,

(b) C = b + a, -3,

(c) C=2* a + a  ^{^}b,

(d) C = b /a,       

(e) C = b a

3. Given the following algorithm, write an Octave/Matlab code of the algorithm

1 Initialize a

2 Initialize x to a/2

3 Repeat 6 times (Say)

Replace x by (x + a/x)/2

Display x

4 Stop

4. Decide which of the following numbers are not acceptable in Matlab/Octave, and state why

(a) 9.87

(b) 3.57*e²

(c) 3.57e^2.1

(d) 33.57e^-2

5. Write a program to calculate x, where

x = -b ± √(b² - 4ac)/2a

and a = 2, b = -10, c = 12

QUESTION 2 -

1. A formula to find a numerical approximation to the second derivative of a function f (x) is

f''(x) = d²f/dx² = f (x + h) - 2f (x) + f (x - h)/h²

with the approximation being better and better as h → 0 Write a function file deriv2 m that implements the formula. The inputs should be the function to be differentiated (remember the function handle construct @, the value of x, and the value of h) Use your code to estimate the second derivative of sin (x) at x = π/4 with h = 10^-1.

2. Write the code to evaluate the following series _n=1∑^∞ u_n in which un is not known explicitly but is given in terms of a recurrence relation. You should stop the summation when |u_n| < 10^-8

u_n+1 = (u_n-1)² + (u_n)² with u₁ = 0.5, u₂ = 0.6

3. In the following, R can take any value between 5 and 6. Write a code to calculate a value of I for R = 5.00, 5.01, 5.02,   ,5.99 and 6 I is defined as

I = 30/R√(R²+15π²)

QUESTION 3 -

1. Define the 100 x 100 square matrix A and the column vector b by

Aij = Iij + 1/(i-j)²+1, b_i = 1 + 2/i, 1 ≤ i, j ≤ 100

where I_ij is the 100 x 100 identity matrix (i.e., 1 on the main diagonal and 0 everywhere else).

Write the code to solve Ax = b for x using both the Gauss-Seidel method and the A\b construct.

2. The Hilbert matrix is a square n x n matrix defined by

H_ijⁿ = 1/i + j - 1

Define bⁿ to be a column vector of dimension n, and with each element 1. Construct bⁿ and then solve write the code that solve for xⁿ, Hⁿxⁿ = bⁿ.

3. Modify the power method so that it finds the smallest eigenvalue and corresponding eigenvector. Do this by evaluating

x_n+1 = A^-1x_n

rather than x_n+1 = Ax_n, When convergence occurs, it is to the eigenvector corresponding to the inverse of the smallest eigenvalue.

QUESTION 4 -

1. Write the code find the maximum value, as well as the point at which the maximum occurs, of

L= x₁ + 2x₂ + 3x₃

subject to the constraints                           

x₁ + x₂ + 2x₃ ≤ 8

3x₁ + 3x₂ + x₃ ≤ 9

x₁, x₂, x₃ ≥ 0

2. Write the code to find the minimum value, as well as the point at which the minimum occurs, of

L = -4x₁ - 5x₂ + 2x₃

subject to the constraints                           

-x₁ + 2x₂ + 2x₃ ≤ 10

2x₁ + 4x₂ + 5x₃ ≤ 25

x₁, x₂, x₃ ≥ 0