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# Write a JAVA program using a simple menu based system to assist you in your investigation of three numerical integration techniques rectangular rule, trapezoidal rule and Simpsons rule

INSTRUCTIONS TO CANDIDATES

#### The project comprises two questions. Question Q1 is worth 65 and question Q2 is worth 35.

Computer Algorithms and Modelling

Q1.

An important computation used in Statistics for the normal distribution is the following

P=0.5+1x=ax=be-x22dx

Write a JAVA program using a simple menu based system to assist you in your investigation of three numerical integration techniques: rectangular rule, trapezoidal rule and Simpson’s rule for computing ND given above.

Analysis of results: This should include a well documented and clearly presented table for the case when a=0, b=1 and show the results using 4, 64, 128, 512 and 1024 strips, obtained by using the three numerical rules given above.

(3 methods x 2 marks per strip case x 5 strip cases = 3 x 2 x 5 = 30 marks.)

Conclusions:  Discuss what you consider to be the most accurate method, the most efficient method and also what is the approximation to P.

Deliverables:

Deliverable 1 – the report in PDF format should contain:

User documentation guide of how to use the program

Analysis of results (see above for detailed breakdown of what is required)

Conclusions (see above for detailed breakdown of what is required)

Deliverable 2 – the well commented JAVA source code

The correctly functioning Java source code uploaded as a zip file

Q2.

1. The following code segment performs the multiplication of a scalar (s) with a matrix (a) followed by the addition of another matrix (b) and stores the result in matrix c. Note that each matrix has the same number of rows and columns defined by the variable n.

// use for-i loop to process rows

for (i = 0, i < n; i++) {

// use for-j loop to process columns

for (j = 0; j < n; j++) {

// solutions are stored into array c

c[i][j] = s * a[i][j] + b[i][j];

}

}

Determine the computational count in terms of n for the code above and include the loop header cost in the count.

1. What is the order of complexity of the algorithm in (a) above using the big-O notation. Explain carefully how you determine this.

1. The cost of performing operations is provided in the processor specification as follows:
 operation Time taken (microseconds) add 1 subtract 1 multiply 2 divide 3 assign 0.5 compare 0.5

Using these timings calculate the time taken to execute the code in 2(a) when n=5, 500, 50000, 500000 and 5000000. You should show all working out in each calculation as well as providing the completed table below. Your answers should be in the time unit of seconds:

 n Time taken to execute code in (a) above (seconds) 5 500 50000 500000 5000000

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