Numerical Methods

Numerical Methods

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Numerical Methods

APPROXIMATION AND ROUND-OFF ERRORS

Experiment No. 2

I. OBJECTIVES

1. Develop algorithms for iterative methods in the approximation of values of functions using infinite series expansions.

2. Determine the true and approximate relative errors due to rounding-off for each iterative method, and establish a stopping criterion for each algorithm.

3. Evaluate the effect of the use of single and double precision floating-point numbers in the accuracy of the numerical solution.

II. MACHINE PROBLEMS

1. If it is known that

For , evaluate this series correct to seven significant figures. How many terms are required to achieve such accuracy?

2. Evaluate using two approaches

and

and compare with the true value of . Use 20 terms to evaluate the series and compute the true and relative errors as terms are added.

3. The infinite series

converges on a value of as approaches infinity. Evaluate this infinite series in single and double precision to calculate for by computing the sum from to and in reverse order. Also evaluate the true and approximate relative errors for each case.

4. Determine the roots of a quadratic equation having , , and . Use single and double precision arithmetic. Evaluate the true relative error and determine the cause of the discrepancy between the solutions. The true roots for the equation are and .

5. For 100 equally spaced values of between to , evaluate the expressions

and

Note that both expressions are mathematically equivalent. Using plotting tools, graph the function in the given interval. Which function is better in terms of resisting loss of significance?

III. METHODOLOGY

Machine Problem 2.1

Code 2.1.1 Pseudocode for Problem 2.1

x = 0.1

I=0

prev=0

Ea=100

If xEs Exit

pa = av

End Do

Display Ea, Et, av

Code 2.1.2 VBA Code for Problem 2.1

Sub machineproblem1()

Dim count As Double

Dim pa As Double

Dim av As Double

Dim tv As Double

Dim Ea As Double

Dim Et As Double

Dim Es As Double

count = 0

pa = 0

Ea = 100

Es = 0.5 * 10 ^ -5

x = 0.1

If Abs(x) < 1 Then

tv = 1 / (1 - x)

Et = ((tv - 1) / tv) * 100

Do While Abs(Et) > Es

av = pa + x ^ count

count = count + 1

Ea = ((av - pa) / av) * 100

Et = ((tv - av) / tv) * 100

Range("A" & count + 2) = count

Range("B" & count + 2) = av

Range("C" & count + 2) = Abs(Ea)

Range("D" & count + 2) = Abs(Et)

pa = av

Loop

MsgBox "Approximate Value is " & av

End If

End Sub

Code 2.1.3 MathScript Code for Problem 2.1

format long

x = 0.1;

if abs(x)Es

av = pa + x ^ count;

count = count + 1;

Ea = ((av - pa) / av) * 100;

Et = ((tv - av) / tv) * 100;

pa = av;

end

fprintf('Approximated Value is %f \n',av)

fprintf('Relative Error is %f percent \n',Ea)

fprintf('True Error is %f percent \n',Et)

fprintf('Number of Iteration %i iterations \n',count)

end

Machine Problem 2.2

Code 2.2.1 Pseudocode for Problem 2.2

x = 5

n = 20

DO

av = pa + (-1) ^ count * x ^ count / factorial(count)

count = count + 1

Ea = ((av - pa) / av) * 100

Et = ((tv - av) / tv) * 100

pa = av

IF count>n EXIT

END DO

Code 2.2.2 VBA Code for Problem 2.2

Sub machineproblem2()

Dim count1 As Double

Dim count2 As Double

x = 5

n = 20

count1 = 0

count2 = 0

tv = Exp(-x)

Range("B2").Value = tv

'Approach 1

Do While count1 < n

av = pa + (-1) ^ count1 * x ^ count1 / factorial(count1)

count1 = count1 + 1

Ea = ((av - pa) / av) * 100

Et = ((tv - av) / tv) * 100

Range("A" & count1 + 6) = count1

Range("B" & count1 + 6) = av

Range("C" & count1 + 6) = Abs(Ea)

Range("D" & count1 + 6) = Abs(Et)

pa = av

Loop

'Approach 2

Do While count2 < n

av2 = pa2 + x ^ count2 / factorial(count2 * 1)

count2 = count2 + 1

iav = 1 / av2

Ea2 = ((iav - ipa) / iav) * 100

Et2 = ((tv - iav) / tv) * 100

Range("F" & count2 + 6) = count2

Range("G" & count2 + 6) = iav

Range("H" & count2 + 6) = Abs(Ea2)

Range("I" & count2 + 6) = Abs(Et2)

ipa = iav

pa2 = av2

Loop

Range("B3").Value = Range("B26").Value

Range("B4").Value = Range("G26").Value

End Sub

Code 2.2.3 MathScript Code for Problem 2.2

APPROACH 1

y=5;

x=(1-y);

i=2;

table=ones(20,4);

true=.006737947;

while i>figure

>>subplot(2,2,1);

>> plot(x,y,'g');

>>title('sqrt(x+4)-sqrt(x+3)')

>>z=1./(sqrt(x+4)+sqrt(x+3));

>>subplot(2,2,2);

>> plot(x,z);

>>title('1/(sqrt(x+4)+sqrt(x+3))')

IV. RESULTS AND INTERPRETATION

Machine Problem 2.1 Power Series

VBA:

Figure 2.1.1 Once ran, the program will display the approximated values, the approximation errors and true errors in every iteration.

MATLAB:

Figure 2.1.2 Once ran, the program will display the approximated value, relative error, true error, and the number of iterations of the power series corrected with seven significant figures

Machine Problem 2.2

VBA:

Figure 2.2.1 Once ran, the program will display the data of the two approaches used in the program, with 20 iterations as stated in the problem.

MATLAB:

Figure 2.2.2 Once the mathscript file (MachineProblem2a) is ran in MATLAB, the program will display the 20 iterations with the corresponding approximate values, true error and relative error

Figure 2.2.3 For the second approach, once the mathscript file (MachineProblem2b) is ran in MATLAB, the program will display the 20 iterations with the corresponding approximate values, true error and relative error.

Machine Problem 2.3

VBA:

Figure 2.3.1 Once ran, the program will display the data in single and double data types in 10000 iterations.

Figure 2.3.2 Once ran, the program will display the data in single and double data types in 10000 iterations where k = 10000.

MATLAB:

Figure 2.3.3 Once the mathscript file (MachineProblem3a) is ran in MATLAB, the program will display the 10000 iterations with the corresponding approximate values, true error and relative error

Figure 2.3.4 For the reversed order, once the mathscript file (MachineProblem3b) is ran in MATLAB, the program will display the 10000 iterations with the corresponding approximate values, true error and relative error.

Machine Problem 2.4

VBA:

Figure 2.4.1 Once ran, the program will display the roots of the given parameters in single and double data types.

MATLAB:

Figure 2.4.2 Once the mathscript file (MachineProblem4) is ran in MATLAB, the program will display the roots of the equation in single and double data types.

Machine Problem 2.5

VBA:

Figure 2.5.1 - After configuring the x-scale of the graphs, the last step is labelling the axis, graphs, and the figure to relay informative data.

V. CONCLUSIONS AND RECOMMENDATIONS

Conclusion:

In the series of machine problem done. It was proven that problems involving Taylor series, Infinite Series, Loss of significance is easily done once ran using computer aided softwares such as MATLAB and Excel. And also data type plays a key role in numerical analysis. It was further proven in the Loss of Significance problem that number of decimal places or how precise the data affects the relative error of it.

Recommendations:

The students encountered problems in plotting data in Excel, specifically on how to employ through visual basic codes. The group recommends to the instructor to give background information on plotting in Excel and also on data types existing for data.

VI. REFERENCES

https://en.wikipedia.org/wiki/Loss_of_significance

https://msdn.microsoft.com/en-us/library/xay7978z.aspx

http://www.functionx.com/vbaexcel/Lesson03.htm

Machine Problem No.2 Approximation and Round-Off ErrorsPage 19