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    School of Electrical Engineering and Computing Department of Electronics and Communication

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    Keneni Alemayehu
    This document summarizes key topics from Chapter 3 of an engineering course, including solving systems of linear equations, polynomials, numerical differentiation and integration, and solving ordinary differential equations. Methods for each topic using MATLAB are presented, along with examples and exercises. Key MATLAB functions introduced include matrix inverse, polynomial fitting and evaluation, ODE solvers, and numerical integration methods.

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    School of Electrical Engineering and Computing Department of Electronics and Communication

    Uploaded by

    Keneni Alemayehu
    31 views12 pages
    This document summarizes key topics from Chapter 3 of an engineering course, including solving systems of linear equations, polynomials, numerical differentiation and integration, and solving ordinary differential equations. Methods for each topic using MATLAB are presented, along with examples and exercises. Key MATLAB functions introduced include matrix inverse, polynomial fitting and evaluation, ODE solvers, and numerical integration methods.

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    This document summarizes key topics from Chapter 3 of an engineering course, including solving systems of linear equations, polynomials, numerical differentiation and integration, and solving ordinary differential equations. Methods for each topic using MATLAB are presented, along with examples and exercises. Key MATLAB functions introduced include matrix inverse, polynomial fitting and evaluation, ODE solvers, and numerical integration methods.

    Copyright:

    © All Rights Reserved
    Download as PPTX, PDF, TXT or read online from Scribd
    Download as pptx, pdf, or txt
    31 views12 pages

    School of Electrical Engineering and Computing Department of Electronics and Communication

    Uploaded by

    Keneni Alemayehu
    This document summarizes key topics from Chapter 3 of an engineering course, including solving systems of linear equations, polynomials, numerical differentiation and integration, and solving ordinary differential equations. Methods for each topic using MATLAB are presented, along with examples and exercises. Key MATLAB functions introduced include matrix inverse, polynomial fitting and evaluation, ODE solvers, and numerical integration methods.

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    Adama Science and Technology University

    School of Electrical Engineering and Computing


    Department of Electronics and Communication

    Course Name: Engineering Application Software


    Course code ECE 2208
    Tewodros Endale
    2021
    Outline

    Chapter -3
    Solving Equations, Curve Fitting,
    and Numerical Techniques
    • Linear Algebra
    • Polynomials
    • Optimization
    • Differentiation/Integration
    • Differential Equations

    04/03/2023 Chapter Three 2


    Systems of Linear Equations

    • Given a system of linear equations More Linear Algebra


    r=rank(mat) % Rank of matrix
    d=det(mat) % Matrix determinant
    E=inv(mat) % Matrix inverse
    • solve with a single line of code • MATLAB has many built-in matrix
    decomposition methods
    a=[1 2 -3;-3 -1 1;1 -1 1];
    [V,D]=eig(X) % Eigenvalue
    b=[5;-8;0]; decomposition
    c=a\b [U,S,V]=svd(X) % Singular value
    decomposition

    04/03/2023 Chapter Three 3


    Exercise

    Solve the following systems of equations:

    System 1 System 2

    ANS
    A=[1 4;-3 1]; ANS
    b=[34;2]; A=[2 -2;-1 1;3 4];
    x=A\b; b=[4;3;2];
    y=inv(A)*b; x=A\b;
    rank(A)

    04/03/2023 Chapter Three 4


    Polynomials
    • Many functions can be well described by a high-
    order polynomial
    • MATLAB represents a polynomials by a vector of
    coefficients r=roots(P) %the roots of a
    polynomial
    • if vector P describes a polynomial
    P = poly(r) %Characteristic
    polynomial or polynomial with
    specified roots
    P(1) P(2) P(3) P(4) y = polyval(p,x) %Polynomial
    evaluation
    Example
    the polynomial
    Could be represented by
    P=[5 3 0 9]

    04/03/2023 Chapter Three 5


    Exercise
    • Find the best second-order polynomial that fits the points:
    (-1,0), (0,-1), (2,3)
    i.e

    ANS: 7

    X=[-1 0 2]; %data vectors 6

    Y=[0 -1 3]; %data vectors 5

    p2=polyfit(X,Y,2); 4

    plot(X,Y,'o', 'MarkerSize', 10); 3

    hold on; 2

    1
    x = -3:.01:3;
    0
    plot(x,polyval(p2,x), 'r--');
    -1
    04/03/2023 Chapter Three -3 -2 -1 0 1 26 3
    Numerical Differentiation

    • MATLAB can 'differentiate' numerically 1

    • see help 0.8

    0.6
    Exercise 0.4

    x=0:0.01:2*pi; 0.2

    y=sin(x); 0

    -0.2
    dydx=diff(y)./diff(x); -0.4

    plot(y) -0.6

    hold on -0.8

    -1
    plot(dydx) 0 100 200 300 400 500 600 700

    04/03/2023 Chapter Three 7


    Numerical Integration

    • MATLAB contains common integration • Trapezoidal rule (input is a vector)


    methods
    x=0:0.01:pi;
    • adaptive Simpson quadrature
    • input is a function z=trapz(x,sin(x))
    q = quad(fun,a,b)
    See help

    04/03/2023 Chapter Three 8


    ODE Solvers

    • MATLAB contains implementations of common ODE solvers


    • Using the correct ODE solver can save you lots of time and give more accurate results
    ode23 %Low-order solver.
    • Use when integrating over small intervals or when accuracy is less important than speed

    ode45 %High order(Runge-Kutta) solver


    • High accuracy and reasonable speed.
    • Most commonly used.

    ode15s %Stiff ODE solver(Gear's algorithm)


    • Use when the diff eq's have time constants that vary by orders of magnitude

    04/03/2023 Chapter Three 9


    Cont’d
    • solve the position and velocity % pendulum
    of the pendulum: % L is pendulum length
    % g is acceleration due to
    gravity
    function dxdt = pendulum (t,x)
    L=1
    theta=x(1);
    gamma=x(2);
    dtheta=gamma;
    dgamma=-(9.8/L)*sin(theta);
    dxdt=zeros(2,1);
    dxdt(1)=dtheta;
    dxdt(2)=dgamma;
    04/03/2023 Chapter Three 10
    Cont’d

    [t,x]=ode45('pendulum',[0 10],
    [0.9*pi;0]); 8
    Position
    plot(t,x(:,1)); 6 Velocity

    hold on; 4

    plot(t,x(:,2),'r'); 2

    legend('Position','Velocity'); 0

    -2

    -4

    -6

    -8
    0 1 2 3 4 5 6 7 8 9 10

    04/03/2023 Chapter Three 11


    Thank you

    04/03/2023 Chapter One 12

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