APPLIED LINEAR ALGEBRA ASSIGNMENT

QUESTION 1 -

(a) What do the following statements do?

(i) > edit

(ii) > ex1

(iii) >help fplot 

(b) A bank wants to calculate the interest due to its customers and add to those accounts The interest rate applicable depends on the initial balance, as follows

Initial balance                    Interest rate

Less than R1000                0%

R1000 to R5000                 5%

R5000 to R10,000              8%

R10,000 to R20,000           9%

Above R20,000                 10%

Write an Octave/Matlab code that implements and solves the above problem (Hint Use the following pseudo-if lop structure

if (condition)

statement

else if (condition)

else

statement

end if

(c) Give the mathematical meaning of the following Octave/Matlab operations

(i) <

(ii) <=

(iii) >

(iv) >=

(v) ==

(d) Write an Octave/Matlab code that solves the following problem

Evaluate the following series _n=1∑^∞U_n in which U_n is not known explicity 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), with U₁ = 0.5

QUESTION 2 -

(a) A MatLab/Octave code for optimising a linear programming problem is

[xmax,Lmax] =glpk(C, A, b, lb, ub, ctype, vartype, s)

Explain the role of each of the parameters lb, ub, ctype, vartype and s

(b) Use the simplex method to solve the following linear programming problem

Maximize

L = x₁ + 1.6x₂

subject to the constraints

0.33x₁ + x₂ < 4

x₁ + 0.66x₂ < 5

x₁, x₂ > 0

QUESTION 3 -

(a) Suppose we have invested some money that draws 10% interest per year, compounded annually We want to know how long it will take for the investment to double More specifically, we want a statement of the account each year until the balance has doubled Write a Matlab/Octave code that implements and solves this problem

The pseudocode for this problem is as follows

(1) Initialise balance, year, interest rate

(ii) Display headings

(iii) Repeat

Update year and balance according to interest rate,

Display year, balance,

Until balance exceeds twice initial balance

(iv) Stop

(b) The binomial coefficient is widely used in Mathematics and Statistics It is defined as the number of ways of choosing r objects out of n without regard to order and is given by

Write a Matlab/Octave code that computes the binomial coefficient when n = 10 and r = 3.

QUESTION 4 -

(a) Write a Matlab/Octave code that solves the following system of equations

2x - y + z = 4

x + y + z = 3

3x - y - z = 1,

using the left division operator "\" first and then using the Gauss-seidal iterative method.

(b) The Hilbert matrix is a square n x n matrix defined by

H_ij⁽ⁿ⁾ = 1/i + j -1

Give the Matlab/Octave code to find the matrix H⁽ⁿ⁾ with entries H_ij⁽ⁿ⁾ in terms of n. Then consider the case n = 13 (i.e., set n = 13 in your code)

(c) Write a Matlab/Octave function file that takes as input a matrix A, and tests whether or not the matrix is

(i) square, and

(ii) diagonally dominant, reporting the answers on the screen