PH36010
Numerical Methods
Some more useful functions in
MathCAD
Piecewise continuous
functions
• So far, functions have been
continuous & ‘smooth’
• Real world has ‘abrupt’ changes
• MathCAD has functions which
can model these changes
Abrupt changes in the
real world
• The impulse given by a hammer.
• The magnetic field due to a current
in a circular conductor has different
equation inside and outside of the
conductor.
• A rocket motor produces a constant
thrust until the fuel runs out, then it
produces zero thrust.
• A potential well has abrupt changes
in potential at its edges.
The if() function
•
•
•
•
•
•
3 arguments
1 boolean expression (condition)
‘True’ value
‘False’ value
Condition evaluated
if ‘true’ (non-zero) return true
value else return false value
Boolean Expressions
• Use relation operators
(evaluation palette)
=,>,<,,,
• True=1 (anything non-zero)
• False=0
• Can use multiply & add for
AND/OR
Examples of if()
if( 1  0  999 42)  999
if( 5  6    1 )  1
Magnetic field due to
wire #1
B
B
 0 I
2  R
 0 I R
2
2  a
Field outside wire (R>a)
Field inside wire (R<a)
Extract common terms & combine
with if()
BWireR
( )
 0 I test
if R a  1  R
2 
R a2
Magnetic field due to
wire #2
I test  2 A
2 1 0
4
a
mm
BW ire( R)1 1 0 4
0
0
2
4
6
R
mm
8
10
The Kronecker delta
function: d(x,y)
• Provides an impulse.
• Returns 1 if x=y, 0 otherwise
• Equivalent to if(x=y,1,0)
x
10  9.8  10
.
1
d ( x 2 )
0
1
0
2
4
6
x
8
10
Heavyside Step function
F(x)
• Provides a step at x=0
• Equivalent to if(x>0,1,0)
2
2
F ( x 1)
1
0
0
10
10
5
0
x
5
10
10
Combining F(x) to form
pulses
t1
1
pulse( x)
t2
3
F (x
t1 ) F ( t2
x)
2
t1
p ulse( x)
t2
1
0
0
2
4
6
x
8
10
Truncation
floor(), ceil(), trunc()
• All functions take real & convert to
integer
• trunc  0
• floor  -
• ceil  
tru nc3.1
( ) 3
tru nc3.9
( ) 3
tru nc( 3.1)  3
tru nc( 3.9)  3
flo or( 3.1)  3
flo or( 3.9)  3
flo or( 3.1)  4
flo or( 3.9)  4
ceil( 3.1)  4
ceil( 3.9)  4
ceil( 3.1)  3
ceil( 3.9)  3
Random Numbers
• Useful for:
– Modelling statistical processes
– Monte Carlo simulation
• 2 methods of generation:
– rnd(x) – returns 1 number in range
0<x (Uniform distribution)
– All others return vector of numbers
with specified distribution
Random Numbers
• Lottery Predictor (!!)
NBalls
i
6
MaxNum
42
0  NBalls 1
Drawi
trunc( rnd( 42 ) )
1
T
Draw  ( 13 29 18 6 20 3 )
• Needs work
• Duplicate numbers
• Not guaranteed