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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