lecture08.pdf - ERG2012A: Advanced Engineering Mathematics ...

ERG2012A: Advanced Engineering Mathematics (Syllabus B) L8 - Page 1 of 6
Lecture 8: Numerical Methods
Outline
• Basic concepts of numerical methods (Kreyszig: Section 17.1)
§ Real-number representation
§ Round-off
§ Loss of significant digits
§ Errors of numerical results
§ Error propagation
§ Algorithm
• Solutions of equations by iteration (Kreyszig: Section 17.2)
§ Fixed-point iteration method
Basic Concepts of Numerical Methods
Numerical methods are used to solve problems on computers or calculators by
numerical calculations. The results are given as a table of numbers and/or graphical
representations (figures).
The steps from a given problem to the final answer are usually as follows:
1. Setting up a mathematical model of the problem , such as a system of
equations, or a differential equation.
2. Choosing mathematical method , perhaps together with preliminary error
estimation and a choice of step size.
3. Programming , either using a computer languages, such as C and FORTRAN, or
using a computing environment, such as MATLAB, MAPLE, MATHEMATICA.
4. Doing the computation .
5. Interpreting the results in physical terms, including making decisions to rerun if
further results are needed.
Real-number Representation
Many digital computers have two ways of representing numbers:
(a) Fixed-point system , which represents all numbers with a fixed number of decimal
places. For example, each of the numbers 62.358, 0.013, 1.000 has 3 decimal
places.
(b) Floating-point system , which keeps the number of significant digits fixed
(whereas the decimal point is “floating”).
Significant digits of a number c is any given digit of c , except the zeros to the left
of the first nonzero digit that serve only to fix the position of the decimal point.
For example, each of the numbers 1360, 1.360, 0.001360 has 4 significant digits.
Also,
0 2 6 3
10 623.8 , 10 6.238 , 10 0.0006238 , 10 0.6238 × × × × have 4 significant
digits.
Theoretically, we can represent any nonzero number a as:
is an integer. 1 1 . 0 10 e m m a
e
< ≤ ⋅ ± =
The fractional part m is called mantissa and e is called the exponent .
On the computer, m and e are limited. For example, the IEEE floating point standard
for single precision is about 38 38 < < - e and for double precision 308 308 < < - e .
If in a computation a number outside that range occurs, this is called underflow when
the number is smaller and overflow when it is larger.

ERG2012A: Advanced Engineering Mathematics (Syllabus B) L8 - Page 2 of 6
Round-Off
To obtain a number with k decimal places, we can use either chopping or rounding
off.