Computational Methods EML3041

Slides:

Advertisements

Similar presentations

Lecture 5 Newton-Raphson Method

Advertisements

5/19/ Runge 4 th Order Method Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker

Nonlinear Equations Your nonlinearity confuses me “The problem of not knowing what we missed is that we believe we haven't missed anything” – Stephen Chew.

Chapter 4 Roots of Equations

Chapter 1 Introduction The solutions of engineering problems can be obtained using analytical methods or numerical methods. Analytical differentiation.

Complex Numbers Section 2.1. Objectives Rewrite the square root of a negative number as a complex number. Write the complex conjugate of a complex number.

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 181 Interpolation Chapter 18 Estimation of intermediate.

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 Interpolation Chapter 18.

Lecture Notes Dr. Rakhmad Arief Siregar Universiti Malaysia Perlis

Section 8.1 Completing the Square. Factoring Before today the only way we had for solving quadratics was to factor. x 2 - 2x - 15 = 0 (x + 3)(x - 5) =

Table of Contents First, add (or subtract) to place the constant on the right side. Quadratic Equation: Solving by completing the square Example: Solve.

Notes Over 5.4 Imaginary Numbers.

9/19/2015http://numericalmethods.eng.usf.edu1 Introduction to Scientific Computing Major: All Engineering Majors Authors: Autar Kaw, Luke Snyder

Welcome to EML3041: Computational Methods. SIT IN FIRST FOUR ROWS ONLY My name is Dr. Kaw. Introduce yourself to the person on your left and right! Say.

Copyright © 2006 The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1 ~ Roots of Equations ~ Open Methods Chapter 6 Credit:

Lecture 6 Numerical Analysis. Solution of Non-Linear Equations Chapter 2.

Calculus 3.4 Manipulate real and complex numbers and solve equations AS

Derivation of the Quadratic Formula The following shows how the method of Completing the Square can be used to derive the Quadratic Formula. Start with.

 5-9 Complex Numbers Objective: Students will be able to add, subtract, multiply, and divide complex numbers.

Gaussian Elimination Mechanical Engineering Majors Author(s): Autar Kaw Transforming Numerical Methods Education for.

Mechanical Engineering Majors Authors: Autar Kaw

1 NUMERICAL METHOD. 2 Introduction Definition : The study of approximation techniques for solving mathematical problems, taking into account the extent.

3.5 Perform Basic Matrix Operations Add Matrices Subtract Matrices Solve Matric equations for x and y.

I am able to solve a quadratic equation using complex numbers. I am able to add, subtract, multiply, and divide complex numbers. Solve the equation.

NUMERICAL ANALYSIS I. Introduction Numerical analysis is concerned with the process by which mathematical problems are solved by the operations.

Newton-Raphson Method

Quadratic Equations An Introduction

Simplify each expression.

Introduction to Numerical Methods Mathematical Procedures

High Accuracy Differentiation Formulas

3.5: Solving Nonlinear Systems

Introduction to Scientific Computing

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

New head garb has been ordered

Gauss Quadrature Rule of Integration

FST Chapter 7 Review Questions.

Interpolation Estimation of intermediate values between precise data points. The most common method is: Although there is one and only one nth-order.

Simplify each expression.

Advanced Numerical Methods (S. A. Sahu) Code: AMC 51151

Completing the Square (3.2.3)

Computational Methods EML3041

Complex Numbers and Roots

Introduction to Scientific Computing

Mechanical Engineering Majors Authors: Autar Kaw, Charlie Barker

Numerical Analysis Lecture 45.

Computational Methods Autar Kaw University of South Florida

Quadratic Equations by Dr. Terri

Computational Methods EML3041

Newton-Raphson Method

Computers in Civil Engineering 53:081 Spring 2003

Trapezoidal Rule of Integration

Copyright © 2014, 2010, 2007 Pearson Education, Inc.

Newton-Raphson Method

Simplify each expression.

Quadratic Equations and Functions

Taylor Polynomials – Day 2

Bring it together.

Newton-Raphson Method

Mechanical Engineering Majors Authors: Autar Kaw, Jai Paul

Notes Over Using Radicals

Dear Power point User, This power point will be best viewed as a slideshow. At the top of the page click on slideshow, then click from the beginning.

Computational Methods EML3041

1 Newton’s Method.

CISE-301: Numerical Methods Topic 1: Introduction to Numerical Methods and Taylor Series Lectures 1-4: KFUPM CISE301_Topic1.

Presentation transcript:

Computational Methods EML3041 Bring it together Computational Methods EML3041

Why do we use numerical methods?

Why use Numerical Methods? To solve problems that cannot be solved exactly

Why use Numerical Methods? To solve problems that have exact solutions but are otherwise intractable!

Assembling a fulcrum for a bascule bridge Girder Trunnion Hub

What model should I use to calculate contraction of trunnion?

Finding the fluid temperature to get enough contraction

Finding the expression for thermal expansion coefficient

What is the temperature of the trunnion after half an hour? http://numericalmethods.eng.usf.edu

How long does it take a trunnion to cool down? http://numericalmethods.eng.usf.edu

Example A solid steel shaft at room temperature of 27°C is needed to be contracted so that it can be shrunk-fit into a hollow hub. It is placed in a refrigerated chamber that is maintained at −33°C. The rate of change of temperature of the solid shaft q is given by Using Euler’s method, find the temperature of the steel shaft after 86400 seconds. Take a step size of h = 43200 seconds. http://numericalmethods.eng.usf.edu

Temperature vs. Time http://numericalmethods.eng.usf.edu

What is the rate of change of heat stored in the cylinder? http://numericalmethods.eng.usf.edu

Can you identify? We talked about 7 mathematical procedures in this course. Write them down. Now connect the problem just discussed to these 7 mathematical procedures on each of the previous slides.

A 4-letter word that starts with F I am a fool! What one fool can do, another can!

The Newton-Raphson method of finding real roots of the nonlinear equation f(x)=0 can be derived by using the first two terms of the Taylor series. One would then expect that adding one more term of the Taylor series to the Newton-Raphson method would result in higher accuracy of the results for the same number of iterations as the Newton-Raphson method. However, this new approach of adding one more term of Taylor series is not feasible because one has to solve a quadratic equation to get the new estimate of the root of the equation. the estimates of the root of the equation may turn out to be complex numbers one needs to find the first derivative of f(x). one needs to find the second derivative of f(x).

(P × Q) × (R × S) ((P × Q) × R) × S P × ((Q × R) × S) It is desired to compute the following matrix product, P×Q×R×S. The sizes of the four matrices are: P is 1×4, Q is 4×20, R is 20×5, and S is 5×1. Amongst the choices given below, the most efficient (the least number of floating point operations of add, subtract, multiplication and divide) way to compute the matrix product is (P × Q) × (R × S) ((P × Q) × R) × S P × ((Q × R) × S) (P × (Q × R)) × S

THE END