Errors in computation using MATLAB

1. PRACTICAL
Name- Saloni Singhal
M.Sc. (Statistics) II-Sem.
Roll No: 2046398
Course- MATH-409 L
Numerical Analysis Lab
Submitted To: Dr. S.C. Pandey
1.3

2. OBJECTIVE
Error Analysis in Computation: Round Off
and Truncation Errors
Problem Statement
1.Write a program (script file) for computation of Exponential function ex up to
4 terms in its series expansion. Calculate the value (true value of ex at x=0.001)
2.Evaluate the error in the computation of ex at x=0.001 (absolute error and
fractional relative error in percentage.)
3. Approximate the first derivative of tan(x) at x =1, and evaluate its relative
percentage error.

3. Theory
Computational Errors:
• Rounding off error occur as machine has limited capacity to store
exact number.
• For example: rational number having finite number of digits.
• The accumulated effect become significant after repeated operations.
They are of two types: 1.chopping
2.symmetry round off
• Truncation error arises when exact mathematical procedure is
approximated and process is truncated after a finite number of
iterations for computational simplicity.
• Example: when infinite series is to be added to arrive at exact result

4. Program
>> format long
>> n=0;
x=0.001;
y=0;
%expanding taylor series for ex
>> while n<=4
a=x^n/factorial(n);
n=n+1; y=y+a
end
y =
1
y =
1.001000000000000
y =
1.001000500000000

5. Program
Contd.
y =
1.001000500166667
y =
1.001000500166708
>> truevav=exp(0.001)
truevav =
1.001000500166708
>> err=abs(truevav-y)
err =
0

6. Error Analysis
As h grows smaller and smaller, f[x + h, x − h]
becomes a better and better approximation to
f(x) .If we plot the truncation error against h on a
log- scale (for linearity), we expect to see a
straight line. For small h values, the error is
dominated by roundoff rather than by truncation
error. An advantage of the higher order of
accuracy is that we can get very small truncation
errors even when h is not very small, and so we
tend to be able to reach a better optimal error
before cancellation effects start to dominate.

7. 2. Program Contd.
h=zeros(5,1)
%initial zero matrix for approximated value
approxval=zeros(5,1)
err=zeros(5,1)
e=zeros(5,1)
format long
for i=1:5; x=1;
h(i)=10^(-i);
trueval=(sec(x))^2;
%numerical diffential
approxval(i)=(tan(x+h(i))-tan(x))/h(i);
%relative error
err(i)=abs(trueval-approxval(i))
e(i)=(err(i)/trueval)*100
end
Another way to create a
matrix is to use a function,
such as ones, zeros,
or rand.

8. Conclusion
Effect of change of the step size (h) in the
approximation of the function tan(x)

9. Error Analysis
The secant of a function based at a and a +h, as well as the tangent at a.
h ( f (a +h)− f (a))/h E(f ;a,h)
10−1 4.073519 -0.6480711
10−2 3.4798299 -0.053110792
10−3 3.4308632 -0.00534437
10−4 3.4260524 -0.0053357
10−5 3.4255721 -5.37919*10−5
Round Off Error: E(f ;a,h) = f’(a)− f (a+h)− f (a) /h. We observe
that the approximation improves with decreasing h, as expected.
More precisely, when h is reduced by a factor of 10, the error is
reduced by the same factor.

10. Truncation Error
Expansion of f (a +h) about x = a using Taylor expansion, where ξh lies
in the interval (a,a+h). The formula may be rearranged to give an
expression for the error often referred to as the truncation error of the
approximation. It is bounded as:
Optimum step size(h)
Total error is given by:
To find the value of h which minimizes this expression, we differentiate with
respect to h and set the derivative to zero. We find 0 (h) = 0, we obtain the
approximate optimal value of h

11. References
• Class Codes by Prof. S.C. Pandey Sir
• MATLAB documentation
• Numerical Differentiation e-notes
• Introduction to Scientific Computing (CS 3220)
Bindel, Spring. 2012