PH36010 PH36010 Numerical MethodsNumerical Methods

Some more useful functions in Some more useful functions in MathCADMathCAD

Piecewise continuous functionsPiecewise continuous functions

• So far, functions have been So far, functions have been continuous & ‘smooth’continuous & ‘smooth’

• Real world has ‘abrupt’ changesReal world has ‘abrupt’ changes• MathCAD has functions which can MathCAD has functions which can

model these changesmodel these changes

Abrupt changes in the real Abrupt changes in the real worldworld

• The impulse given by a hammer.The impulse given by a hammer.• The magnetic field due to a current in a The magnetic field due to a current in a

circular conductor has different equation circular conductor has different equation inside and outside of the conductor.inside and outside of the conductor.

• A rocket motor produces a constant A rocket motor produces a constant thrust until the fuel runs out, then it thrust until the fuel runs out, then it produces zero thrust.produces zero thrust.

• A potential well has abrupt changes in A potential well has abrupt changes in potential at its edges.potential at its edges.

The if() functionThe if() function

• 3 arguments3 arguments• 1 boolean expression (condition)1 boolean expression (condition)• ‘‘True’ valueTrue’ value• ‘‘False’ valueFalse’ value• Condition evaluatedCondition evaluated• if ‘true’ (non-zero) return true value if ‘true’ (non-zero) return true value

else return false valueelse return false value

Boolean ExpressionsBoolean Expressions

• Use relation operators (evaluation Use relation operators (evaluation palette)palette)=,>,<,=,>,<,,,,,

• True=1 (anything non-zero)True=1 (anything non-zero)• False=0False=0• Can use multiply & add for Can use multiply & add for

AND/ORAND/OR

Examples of if()Examples of if()

if 1 0 999 42( ) 999

if 5 6 1( ) 1

Magnetic field due to wire Magnetic field due to wire #1#1

B 0 I

2 R

B 0 I R

2 a2

BWire R( ) 0 I test

2 if R a

1

R

R

a2

Field outside wire (R>a)Field outside wire (R>a)

Field inside wire (R<a)Field inside wire (R<a)

Extract common terms & combine Extract common terms & combine with if()with if()

Magnetic field due to wire Magnetic field due to wire #2#2

I test 2 A

0 2 4 6 8 100

1 10 4

2 10 4

BWire R( )

a

mm

R

mm

The Kronecker delta The Kronecker delta function: function: (x,y)(x,y)

• Provides an impulse. Provides an impulse. • Returns 1 if x=y, 0 otherwiseReturns 1 if x=y, 0 otherwise• Equivalent to if(x=y,1,0)Equivalent to if(x=y,1,0)

x 10 9.8 10 .

0 2 4 6 8 101

0

1

x 2( )

x

Heavyside Step function Heavyside Step function (x)(x)

• Provides a step at x=0Provides a step at x=0• Equivalent to if(x>0,1,0)Equivalent to if(x>0,1,0)

10 5 0 5 100

1

22

0

x 1( )

1010 x

Combining Combining (x) to form (x) to form pulsespulses

t1 1 t2 3

pulse x( ) x t1( ) t2 x( )

0 2 4 6 8 100

1

2

pulse x( )

t1 t2

x

Truncation Truncation floor(), ceil(), trunc()floor(), ceil(), trunc()

• All functions take real & convert to All functions take real & convert to integerinteger

• trunc trunc 0 0• floor floor - -• ceil ceil

trunc 3.1( ) 3 trunc 3.9( ) 3 trunc 3.1( ) 3 trunc 3.9( ) 3

floor 3.1( ) 3 floor 3.9( ) 3 floor 3.1( ) 4 floor 3.9( ) 4

ceil 3.1( ) 4 ceil 3.9( ) 4 ceil 3.1( ) 3 ceil 3.9( ) 3

Random NumbersRandom Numbers

• Useful for:Useful for:– Modelling statistical processesModelling statistical processes– Monte Carlo simulationMonte Carlo simulation

• 2 methods of generation:2 methods of generation:– rnd(x) – returns 1 number in range rnd(x) – returns 1 number in range

0<x (Uniform distribution)0<x (Uniform distribution)– All others return vector of numbers All others return vector of numbers

with specified distributionwith specified distribution

Random NumbersRandom Numbers

• Lottery Predictor (!!)Lottery Predictor (!!)NBalls 6 MaxNum 42

i 0 NBalls 1

Drawi trunc rnd 42( )( ) 1

DrawT

13 29 18 6 20 3( )

• Needs workNeeds work• Duplicate numbersDuplicate numbers• Not guaranteedNot guaranteed