Computer-Aided Analysis II
85-211
Dr. Mohammed Khalid
Why Study Numerical Methods?
• Numerical methods are extremely powerful
problem-solving tools
• Capable of handling large systems of equations,
non-linearities, and complicated geometries that
are not uncommon in engineering practices
• Allows us to solve problems that would be
impossible to solve analytically
• Computational vs. analytical approaches
© 2003-2006 Roberto Muscedere
Why Study Numerical Methods?
• Typically we would use software to solve
problems (e.g. MATLAB, Excel)
• Understanding the theory behind methods which
the CAD tools utilize, enables us to use the tools
properly and efficiently
• Some problems can not be solved using
existing tools
• May have to design your own methods to solve
particular problems
© 2003-2006 Roberto Muscedere
Numerical Methods
• Solution of a system of simultaneous linear
algebraic equations with real and complex
coefficients
• Matrix oriented methods
• Equations of one variable
• Roots of polynomials
•
•
•
•
Curve-fitting techniques
Numerical integration
Solution of ordinary differential equations
Solutions of non-linear algebraic equations
© 2003-2006 Roberto Muscedere
Methods
• Roots of equations
• Value of a variable or a parameter that satisfies
a single equation
f ( x) = ax + bx + c = 0
2
−x
f ( x) = e − x = 0
© 2003-2006 Roberto Muscedere
Methods
• Systems of Linear Algebraic Equations
• Similar to roots in terms of satisfying equations
• Satisfy a set of linear algebraic equations
f ( x) = a1 x1 + a2 x2 + a3 x3 + ... + an xn − c = 0
© 2003-2006 Roberto Muscedere
Methods
• Curve Fitting
• Fit curves to data points
• Two categories: regression and interpolation
• Regression is used where the is a significant
degree of error associated with the data
• Example: Experimental results
• Derive a single curve the represents the data
without necessarily matching all the points
© 2003-2006 Roberto Muscedere
Methods
• Integration
• Physical interpretation is the determination of the
“area under a curve”
• Applications in engineering practice
• Important part of the solution of Differential
Equations methods
t
d = ∫ v(t )dt
0
© 2003-2006 Roberto Muscedere
Methods
• Ordinary Differential Equations
• Great significance
• Physical laws depend on the rate of change of a
quantity rather than the magnitude or quantity
itself
• Example: Faraday’s Law
• Voltage drop across and inductor
di
VoltageDro p = L
dt
© 2003-2006 Roberto Muscedere
Where do you use Numerical
Methods?
• SPICE
• Simulation Program with Integrated Circuit
Emphasis
R
v1
L
v2
+
vIN
i
Reference node
© 2003-2006 Roberto Muscedere
C
vOUT
_
Learning Outcomes
• Introduce and use “C++”
• Object Orientated “C”
• Cover Numerical Methods and understand
their implementation using “C++”
• Make you a better problem solver
© 2003-2006 Roberto Muscedere
Questions?
© 2003-2006 Roberto Muscedere