Download Numerical Methods for Engineers

Prof. Gevelyn B. Itao
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
Decimal system – base 10
Base 5 system
Binary system – base 2
Octal system – base 8
Hexadecimal system – base 16
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
 Decimal or base-10:
10 different digits for
counting, i.e.,
0,1,2,3,4,5,6,7,8,9.
(Human and natural,
how many fingers do
you have?).
Binary or base-2:
only 2 digits for
counting, i.e., 0 and 1.
(All the computers
use binary number
system.)
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
It is easy and natural to realize a “0” and “1” by a
hardware device that has only two states – “off” and
“on”. A typical such device is a transistor. You can think
of a transistor as a light bulb; it can either be bright
when it is turned “on” or dark when it is turned “off”.
So a 8-bit binary number is represented by 8 light
bulbs in a computer.
1
0
1
0
1
1
0
1
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
In an arbitrary number base b, an m-digit integer Im is
written as:
Where b is the number base, and the digits dj is the
set of integers from 0 to (b-1).
The decimal value of an integer above is obtained from
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
The largest number that can be stored in m digits is
one in which every digit is equal to (b-1). The decimal
equivalent of this number is
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
Memory cells usually consists of 8-bit (1-byte) multiples,
with each position containing one binary digit or bit.
1. Sign-magnitude notation
(-1)Sign×Number
2. Two’s complement notation
Two’s complement of a negative integer
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
Floating point numbers or reals contain an
integer part and a fractional part.
An n-digit fractional part Pn may be written in
a number base b as
Obtaining the decimal value of Pn is
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
An n-digit fractional part Pn may be written in
a number base b as
Obtaining the decimal value of Pn is
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
(-1)Sign×Mantissa×bExponent
(b is the base of the number system used)
Mantissa is usually required to be normalized:
≤ Mantissa < 1
Example (b = 10):
0.0294 = 0.294×10-1
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
Convert the following decimal numbers to the
designated number systems and determine the percent
relative round-off error
1. (39.99)10 to binary number system in 10 significant
bits in the fractional part
2. (87.12)10 to hexadecimal number system in 10
significant hexadecimal digits in the fractional part
3. (0.236)10 to octal number system in 10 significant
octal digits
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
Convert the following decimal numbers to the
designated number systems and determine the percent
relative round-off error
1. (0.99)10 to binary number system in 8 significant bits
2. (1.6666)10 to octal number system in 10 significant
octal digits in the fractional part
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
Algebraic, transcendental (i.e., involving trigonometric and
exponential functions), ordinary differential equations, or
partial differential equations...
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology
:
:
:
:
:
:
:
:
:
:
Where xj (j=1,2,…m) denotes the unknown variable
aij (i=1,2,…n; j=1,2,…m) denotes the coefficients of
the unknown variable
bi (i=1,2,…n) denotes the non-homogeneous terms
ES 84 Numerical Methods for Engineers, Mindanao State University- Iligan Institute of Technology

One that produces either exact or
approximate solutions in closed form.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
1. A unique solution – a consistent set of solutions
2. No solution – an inconsistent set of equations
3. An infinite number of solutions – a redundant
set of equations
4. The trival solution xj = 0 – a set of
homogenous equations
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology
1. Direct methods – are systematic procedures,
based on algebraic elimination, that obtain the solution
in a fixed number of operations.
2. Iterative methods – obtain the solution
asymptotically by an iterative procedure. A trial solution
is assumed, the trial solution is substituted into the
system of equations to determine the mismatch in the
trial solution, and an improved solution is obtained
from the mismatch data.
ES 84 Numerical Methods for Engineers, Mindanao
State University- Iligan Institute of Technology