Numerical Methods

Numerical Methods

Numerical Methods 263-2014 Prof. M. K. Banda Botany Building: 2-10. Prof. M. K. Banda (Tuks) WTW263 Semester II 1 / 27 Theme 1: Solving Nonlinear Equations Prof. M. K. Banda (Tuks) WTW263 Semester II 2

Theme 1: Solving Nonlinear Equations

Theme 1: Solving Nonlinear Equations Prof. Mapundi K. Banda (Tuks) WTW263 Semester II 1 / 22 Sources of Errors Finite decimal representation (Rounding): Finite decimal representation will be used to represent

Accelerating Convergence

Accelerating Convergence MATH 375 Numerical Analysis J. Robert Buchanan Department of Mathematics Fall 2013 Motivation We have seen that most fixed-point methods for root finding converge only linearly

Simple Iteration, cont d

Jim Lambers MAT 772 Fall Semester 2010-11 Lecture 2 Notes These notes correspond to Section 1.2 in the text. Simple Iteration, cont d In general, nonlinear equations cannot be solved in a finite sequence

Fixed-Point Iteration

Fixed-Point Iteration MATH 375 Numerical Analysis J. Robert Buchanan Department of Mathematics Fall 2013 Motivation Several root-finding algorithms operate by repeatedly evaluating a function until its

FIXED POINT ITERATION

FIXED POINT ITERATION The idea of the fixed point iteration methods is to first reformulate a equation to an equivalent fixed point problem: f (x) = 0 x = g(x) and then to use the iteration: with an initial

BEKG 2452 NUMERICAL METHODS Solution of Nonlinear Equations

BEKG 2452 NUMERICAL METHODS Solution of Nonlinear Equations Ser Lee Loh a, Wei Sen Loi a a Fakulti Kejuruteraan Elektrik Universiti Teknikal Malaysia Melaka Lesson Outcome Upon completion of this lesson,

Variable. Peter W. White Fall 2018 / Numerical Analysis. Department of Mathematics Tarleton State University

Newton s Iterative s Peter W. White white@tarleton.edu Department of Mathematics Tarleton State University Fall 2018 / Numerical Analysis Overview Newton s Iterative s Newton s Iterative s Newton s Iterative

Numerical Analysis Fall. Roots: Open Methods

Numerical Analysis 2015 Fall Roots: Open Methods Open Methods Open methods differ from bracketing methods, in that they require only a single starting value or two starting values that do not necessarily

COURSE Iterative methods for solving linear systems

COURSE 0 4.3. Iterative methods for solving linear systems Because of round-off errors, direct methods become less efficient than iterative methods for large systems (>00 000 variables). An iterative scheme

Numerical solutions of nonlinear systems of equations

Numerical solutions of nonlinear systems of equations Tsung-Ming Huang Department of Mathematics National Taiwan Normal University, Taiwan E-mail: min@math.ntnu.edu.tw August 28, 2011 Outline 1 Fixed points

CE 191: Civil and Environmental Engineering Systems Analysis. LEC 05 : Optimality Conditions

CE 191: Civil and Environmental Engineering Systems Analysis LEC : Optimality Conditions Professor Scott Moura Civil & Environmental Engineering University of California, Berkeley Fall 214 Prof. Moura

04-Economic Dispatch 2. EE570 Energy Utilization & Conservation Professor Henry Louie

04-Economic Dispatch EE570 Energy Utilization & Conservation Professor Henry Louie 1 Topics Example 1 Example Dr. Henry Louie Consider two generators with the following cost curves and constraints: C 1

SYSTEMS OF NONLINEAR EQUATIONS

SYSTEMS OF NONLINEAR EQUATIONS Widely used in the mathematical modeling of real world phenomena. We introduce some numerical methods for their solution. For better intuition, we examine systems of two

Math 3B: Lecture 11. Noah White. October 25, 2017

Math 3B: Lecture 11 Noah White October 25, 2017 Introduction Midterm 1 Introduction Midterm 1 Average is 73%. This is higher than I expected which is good. Introduction Midterm 1 Average is 73%. This is

Chapter 12: Iterative Methods

ES 40: Scientific and Engineering Computation. Uchechukwu Ofoegbu Temple University Chapter : Iterative Methods ES 40: Scientific and Engineering Computation. Gauss-Seidel Method The Gauss-Seidel method

CS 323: Numerical Analysis and Computing

CS 323: Numerical Analysis and Computing MIDTERM #2 Instructions: This is an open notes exam, i.e., you are allowed to consult any textbook, your class notes, homeworks, or any of the handouts from us.

Numerical Methods. Root Finding

Numerical Methods Solving Non Linear 1-Dimensional Equations Root Finding Given a real valued function f of one variable (say ), the idea is to find an such that: f() 0 1 Root Finding Eamples Find real

ECE580 Partial Solution to Problem Set 3

ECE580 Fall 2015 Solution to Problem Set 3 October 23, 2015 1 ECE580 Partial Solution to Problem Set 3 These problems are from the textbook by Chong and Zak, 4th edition, which is the textbook for the

Quiescent Steady State (DC) Analysis The Newton-Raphson Method

Quiescent Steady State (DC) Analysis The Newton-Raphson Method J. Roychowdhury, University of California at Berkeley Slide 1 Solving the System's DAEs DAEs: many types of solutions useful DC steady state:

NONLINEAR EQUATIONS AND TAYLOR S THEOREM

APPENDIX C NONLINEAR EQUATIONS AND TAYLOR S THEOREM C.1 INTRODUCTION In adjustment computations it is frequently necessary to deal with nonlinear equations. For example, some observation equations relate

Lecture 5. September 4, 2018 Math/CS 471: Introduction to Scientific Computing University of New Mexico

Lecture 5 September 4, 2018 Math/CS 471: Introduction to Scientific Computing University of New Mexico 1 Review: Office hours at regularly scheduled times this week Tuesday: 9:30am-11am Wed: 2:30pm-4:00pm

Section 1.6 Inverse Functions

0 Chapter 1 Section 1.6 Inverse Functions A fashion designer is travelling to Milan for a fashion show. He asks his assistant, Betty, what 7 degrees Fahrenheit is in Celsius, and after a quick search on

Chapter 1. Root Finding Methods. 1.1 Bisection method

Chapter 1 Root Finding Methods We begin by considering numerical solutions to the problem f(x) = 0 (1.1) Although the problem above is simple to state it is not always easy to solve analytically. This

L Hopital s Rule. We will use our knowledge of derivatives in order to evaluate limits that produce indeterminate forms.

L Hopital s Rule We will use our knowledge of derivatives in order to evaluate its that produce indeterminate forms. Main Idea x c f x g x If, when taking the it as x c, you get an INDETERMINATE FORM..

L Hopital s Rule. We will use our knowledge of derivatives in order to evaluate limits that produce indeterminate forms.

L Hopital s Rule We will use our knowledge of derivatives in order to evaluate its that produce indeterminate forms. Indeterminate Limits Main Idea x c f x g x If, when taking the it as x c, you get an

Constrained Optimization

1 / 22 Constrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University March 30, 2015 2 / 22 1. Equality constraints only 1.1 Reduced gradient 1.2 Lagrange

CHAPTER-II ROOTS OF EQUATIONS

CHAPTER-II ROOTS OF EQUATIONS 2.1 Introduction The roots or zeros of equations can be simply defined as the values of x that makes f(x) =0. There are many ways to solve for roots of equations. For some

Announcements. Topics: Homework:

Announcements Topics: - sections 7.3 (the definite integral +area), 7.4 (FTC), 7.5 (additional techniques of integration) * Read these sections and study solved examples in your textbook! Homework: - review

Week 6: Limits and Continuity.

Week 6: Limits and Continuity. MA161/MA1161: Semester 1 Calculus. Prof. Götz Pfeiffer School of Mathematics, Statistics and Applied Mathematics NUI Galway October 10-11, 2016 Recall: Limits. [Section 2.2

Section 9.8. First let s get some practice with determining the interval of convergence of power series.

First let s get some practice with determining the interval of convergence of power series. First let s get some practice with determining the interval of convergence of power series. Example (1) Determine

Bisection Method. and compute f (p 1 ). repeat with p 2 = a 2+b 2

Bisection Method Given continuous function f (x) on the interval [a, b] with f (a) f (b) < 0, there must be a root in (a, b). To find a root: set [a 1, b 1 ] = [a, b]. set p 1 = a 1+b 1 2 and compute f

ROOT FINDING REVIEW MICHELLE FENG

ROOT FINDING REVIEW MICHELLE FENG 1.1. Bisection Method. 1. Root Finding Methods (1) Very naive approach based on the Intermediate Value Theorem (2) You need to be looking in an interval with only one

Lesson 19 Factoring Polynomials

Fast Five Lesson 19 Factoring Polynomials Factor the number 38,754 (NO CALCULATOR) Divide 72,765 by 38 (NO CALCULATOR) Math 2 Honors - Santowski How would you know if 145 was a factor of 14,436,705? What

Numerical Analysis: Solutions of System of. Linear Equation. Natasha S. Sharma, PhD

Mathematical Question we are interested in answering numerically How to solve the following linear system for x Ax = b? where A is an n n invertible matrix and b is vector of length n. Notation: x denote

Lecture 14: Ordinary Differential Equation I. First Order

Lecture 14: Ordinary Differential Equation I. First Order 1. Key points Maple commands dsolve 2. Introduction We consider a function of one variable. An ordinary differential equations (ODE) specifies

Figure 1: Graph of y = x cos(x)

Chapter The Solution of Nonlinear Equations f(x) = 0 In this chapter we will study methods for find the solutions of functions of single variables, ie values of x such that f(x) = 0 For example, f(x) =

The Missing-Index Problem in Process Control

The Missing-Index Problem in Process Control Joseph G. Voelkel CQAS, KGCOE, RIT October 2009 Voelkel (RIT) Missing-Index Problem 10/09 1 / 35 Topics 1 Examples Introduction to the Problem 2 Models, Inference,

CHAPTER 10 Zeros of Functions

CHAPTER 10 Zeros of Functions An important part of the maths syllabus in secondary school is equation solving. This is important for the simple reason that equations are important a wide range of problems

An introduction to particle filters

An introduction to particle filters Andreas Svensson Department of Information Technology Uppsala University June 10, 2014 June 10, 2014, 1 / 16 Andreas Svensson - An introduction to particle filters Outline

Zeros of Functions. Chapter 10

Chapter 10 Zeros of Functions An important part of the mathematics syllabus in secondary school is equation solving. This is important for the simple reason that equations are important a wide range of

CS 323: Numerical Analysis and Computing

CS 323: Numerical Analysis and Computing MIDTERM #2 Instructions: This is an open notes exam, i.e., you are allowed to consult any textbook, your class notes, homeworks, or any of the handouts from us.

Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem

Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem min{f (x) : x R n }. The iterative algorithms that we will consider are of the form x k+1 = x k + t k d k, k = 0, 1,...

Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem

Lecture 4 - The Gradient Method Objective: find an optimal solution of the problem min{f (x) : x R n }. The iterative algorithms that we will consider are of the form x k+1 = x k + t k d k, k = 0, 1,...

CHAPTER 4 ROOTS OF EQUATIONS

CHAPTER 4 ROOTS OF EQUATIONS Chapter 3 : TOPIC COVERS (ROOTS OF EQUATIONS) Definition of Root of Equations Bracketing Method Graphical Method Bisection Method False Position Method Open Method One-Point

Fitting. PHY 688: Numerical Methods for (Astro)Physics

Fitting Fitting Data We get experimental/observational data as a sequence of times (or positions) and associate values N points: (x i, y i ) Often we have errors in our measurements at each of these values:

1. Algebra and Functions

1. Algebra and Functions 1.1.1 Equations and Inequalities 1.1.2 The Quadratic Formula 1.1.3 Exponentials and Logarithms 1.2 Introduction to Functions 1.3 Domain and Range 1.4.1 Graphing Functions 1.4.2

Midterm Review. Igor Yanovsky (Math 151A TA)

Midterm Review Igor Yanovsky (Math 5A TA) Root-Finding Methods Rootfinding methods are designed to find a zero of a function f, that is, to find a value of x such that f(x) =0 Bisection Method To apply

STOP, a i+ 1 is the desired root. )f(a i) > 0. Else If f(a i+ 1. Set a i+1 = a i+ 1 and b i+1 = b Else Set a i+1 = a i and b i+1 = a i+ 1

53 17. Lecture 17 Nonlinear Equations Essentially, the only way that one can solve nonlinear equations is by iteration. The quadratic formula enables one to compute the roots of p(x) = 0 when p P. Formulas

1. Method 1: bisection. The bisection methods starts from two points a 0 and b 0 such that

Chapter 4 Nonlinear equations 4.1 Root finding Consider the problem of solving any nonlinear relation g(x) = h(x) in the real variable x. We rephrase this problem as one of finding the zero (root) of a

Least Squares Regression

Least Squares Regression Chemical Engineering 2450 - Numerical Methods Given N data points x i, y i, i 1 N, and a function that we wish to fit to these data points, fx, we define S as the sum of the squared

CS 450 Numerical Analysis. Chapter 5: Nonlinear Equations

Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80

Solving Ill-Posed Cauchy Problems in Three Space Dimensions using Krylov Methods

Solving Ill-Posed Cauchy Problems in Three Space Dimensions using Krylov Methods Lars Eldén Department of Mathematics Linköping University, Sweden Joint work with Valeria Simoncini February 21 Lars Eldén

Math 551 Homework Assignment 3 Page 1 of 6

Math 551 Homework Assignment 3 Page 1 of 6 Name and section: ID number: E-mail: 1. Consider Newton s method for finding + α with α > 0 by finding the positive root of f(x) = x 2 α = 0. Assuming that x

Announcements. Topics: Homework:

Announcements Topics: - sections 7.4 (FTC), 7.5 (additional techniques of integration), 7.6 (applications of integration) * Read these sections and study solved examples in your textbook! Homework: - review

An Introduction to Expectation-Maximization

An Introduction to Expectation-Maximization Dahua Lin Abstract This notes reviews the basics about the Expectation-Maximization EM) algorithm, a popular approach to perform model estimation of the generative

Outline. Scientific Computing: An Introductory Survey. Nonlinear Equations. Nonlinear Equations. Examples: Nonlinear Equations

Methods for Systems of Methods for Systems of Outline Scientific Computing: An Introductory Survey Chapter 5 1 Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign

A nonlinear equation is any equation of the form. f(x) = 0. A nonlinear equation can have any number of solutions (finite, countable, uncountable)

Nonlinear equations Definition A nonlinear equation is any equation of the form where f is a nonlinear function. Nonlinear equations x 2 + x + 1 = 0 (f : R R) f(x) = 0 (x cos y, 2y sin x) = (0, 0) (f :

An Improved Hybrid Algorithm to Bisection Method and Newton-Raphson Method

Applied Mathematical Sciences, Vol. 11, 2017, no. 56, 2789-2797 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.710302 An Improved Hybrid Algorithm to Bisection Method and Newton-Raphson

Order of convergence

Order of convergence Linear and Quadratic Order of convergence Computing square root with Newton s Method Given a > 0, p def = a is positive root of equation Newton s Method p k+1 = p k p2 k a 2p k = 1

MA 1B PRACTICAL - HOMEWORK SET 3 SOLUTIONS. Solution. (d) We have matrix form Ax = b and vector equation 4

MA B PRACTICAL - HOMEWORK SET SOLUTIONS (Reading) ( pts)[ch, Problem (d), (e)] Solution (d) We have matrix form Ax = b and vector equation 4 i= x iv i = b, where v i is the ith column of A, and 4 A = 8

x 2 x n r n J(x + t(x x ))(x x )dt. For warming-up we start with methods for solving a single equation of one variable.

Maria Cameron 1. Fixed point methods for solving nonlinear equations We address the problem of solving an equation of the form (1) r(x) = 0, where F (x) : R n R n is a vector-function. Eq. (1) can be written

Welcome to Math Video Lessons. Stanley Ocken. Department of Mathematics The City College of New York Fall 2013

Welcome to Math 19500 Video Lessons Prof. Department of Mathematics The City College of New York Fall 013 An important feature of the following Beamer slide presentations is that you, the reader, move

Math 320: Real Analysis MWF 1pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible.

Math 320: Real Analysis MWF pm, Campion Hall 302 Homework 8 Solutions Please write neatly, and in complete sentences when possible. Do the following problems from the book: 4.3.5, 4.3.7, 4.3.8, 4.3.9,

Homework 2 solutions

Homework 2 solutions Section 2.1: Ex 1,2,3,4,6,11; AP 1 (18 points; Ex 3, AP 1 graded, 4 pts each; 2 pts to try the others) 1. Determine if each function has a unique fixed point on the specified interval.

Announcements. Topics: Homework:

Announcements Topics: - sections 7.5 (additional techniques of integration), 7.6 (applications of integration), * Read these sections and study solved examples in your textbook! Homework: - review lecture

CE 191: Civil & Environmental Engineering Systems Analysis. LEC 17 : Final Review

CE 191: Civil & Environmental Engineering Systems Analysis LEC 17 : Final Review Professor Scott Moura Civil & Environmental Engineering University of California, Berkeley Fall 2014 Prof. Moura UC Berkeley

Numerical Optimization

Unconstrained Optimization (II) Computer Science and Automation Indian Institute of Science Bangalore 560 012, India. NPTEL Course on Unconstrained Optimization Let f : R R Unconstrained problem min x

2.098/6.255/ Optimization Methods Practice True/False Questions

2.098/6.255/15.093 Optimization Methods Practice True/False Questions December 11, 2009 Part I For each one of the statements below, state whether it is true or false. Include a 1-3 line supporting sentence

Optimization. Escuela de Ingeniería Informática de Oviedo. (Dpto. de Matemáticas-UniOvi) Numerical Computation Optimization 1 / 30

Optimization Escuela de Ingeniería Informática de Oviedo (Dpto. de Matemáticas-UniOvi) Numerical Computation Optimization 1 / 30 Unconstrained optimization Outline 1 Unconstrained optimization 2 Constrained

Numerical Methods in Informatics

Numerical Methods in Informatics Lecture 2, 30.09.2016: Nonlinear Equations in One Variable http://www.math.uzh.ch/binf4232 Tulin Kaman Institute of Mathematics, University of Zurich E-mail: tulin.kaman@math.uzh.ch

CS 323: Numerical Analysis and Computing

CS 323: Numerical Analysis and Computing MIDTERM #1 Instructions: This is an open notes exam, i.e., you are allowed to consult any textbook, your class notes, homeworks, or any of the handouts from us.

Root Finding: Close Methods. Bisection and False Position Dr. Marco A. Arocha Aug, 2014

Root Finding: Close Methods Bisection and False Position Dr. Marco A. Arocha Aug, 2014 1 Roots Given function f(x), we seek x values for which f(x)=0 Solution x is the root of the equation or zero of the

lim Bounded above by M Converges to L M

Calculus 2 MONOTONE SEQUENCES If for all we say is nondecreasing. If for all we say is increasing. Similarly for nonincreasing decreasing. A sequence is said to be monotonic if it is either nondecreasing

EE Homework 5 - Solutions

EE054 - Homework 5 - Solutions 1. We know the general result that the -transform of α n 1 u[n] is with 1 α 1 ROC α < < and the -transform of α n 1 u[ n 1] is 1 α 1 with ROC 0 < α. Using this result, the

2. If the values for f(x) can be made as close as we like to L by choosing arbitrarily large. lim

Limits at Infinity and Horizontal Asymptotes As we prepare to practice graphing functions, we should consider one last piece of information about a function that will be helpful in drawing its graph the

- - - - - - - - - - - - - - - - - - DISCLAIMER - - - - - - - - - - - - - - - - - - General Information: This midterm is a sample midterm. This means: The sample midterm contains problems that are of similar,

Scientific Computing: An Introductory Survey

Scientific Computing: An Introductory Survey Chapter 5 Nonlinear Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction

Homework #6 Solutions

Problems Section.1: 6, 4, 40, 46 Section.:, 8, 10, 14, 18, 4, 0 Homework #6 Solutions.1.6. Determine whether the functions f (x) = cos x + sin x and g(x) = cos x sin x are linearly dependent or linearly

Solutions of Equations in One Variable. Newton s Method

Solutions of Equations in One Variable Newton s Method Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University c 2011 Brooks/Cole,

Report due date. Please note: report has to be handed in by Monday, May 16 noon.

Lecture 23 18.86 Report due date Please note: report has to be handed in by Monday, May 16 noon. Course evaluation: Please submit your feedback (you should have gotten an email) From the weak form to finite

Numerical Methods in Physics and Astrophysics

Kostas Kokkotas 2 October 20, 2014 2 http://www.tat.physik.uni-tuebingen.de/ kokkotas Kostas Kokkotas 3 TOPICS 1. Solving nonlinear equations 2. Solving linear systems of equations 3. Interpolation, approximation

Chapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs

Chapter 2: First Order DE 2.4 Linear vs. Nonlinear DEs First Order DE 2.4 Linear vs. Nonlinear DE We recall the general form of the First Oreder DEs (FODE): dy = f(t, y) (1) dt where f(t, y) is a function

PARTIAL FRACTION DECOMPOSITION. Mr. Velazquez Honors Precalculus

PARTIAL FRACTION DECOMPOSITION Mr. Velazquez Honors Precalculus ADDING AND SUBTRACTING RATIONAL EXPRESSIONS Recall that we can use multiplication and common denominators to write a sum or difference of

NEW DERIVATIVE FREE ITERATIVE METHOD FOR SOLVING NON-LINEAR EQUATIONS

NEW DERIVATIVE FREE ITERATIVE METHOD FOR SOLVING NON-LINEAR EQUATIONS Dr. Farooq Ahmad Principal, Govt. Degree College Darya Khan, Bhakkar, Punjab Education Department, PAKISTAN farooqgujar@gmail.com Sifat

MIDTERM REVIEW FOR MATH The limit

MIDTERM REVIEW FOR MATH 500 SHUANGLIN SHAO. The limit Define lim n a n = A: For any ε > 0, there exists N N such that for any n N, a n A < ε. The key in this definition is to realize that the choice of

Fixed point iteration Numerical Analysis Math 465/565

Fixed point iteration Numerical Analysis Math 465/565 1 Fixed Point Iteration Suppose we wanted to solve : f(x) = cos(x) x =0 or cos(x) =x We might consider a iteration of this type : x k+1 = cos(x k )

Computational Methods CMSC/AMSC/MAPL 460. Solving nonlinear equations and zero finding. Finding zeroes of functions

Computational Methods CMSC/AMSC/MAPL 460 Solving nonlinear equations and zero finding Ramani Duraiswami, Dept. of Computer Science Where does it arise? Finding zeroes of functions Solving functional equations

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH

UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH FACULTY OF SCIENCE AND ENGINEERING DEPARTMENT OF MATHEMATICS & STATISTICS END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA440 SEMESTER: Autumn 04 MODULE TITLE:

Solution Methods for Stochastic Programs

Solution Methods for Stochastic Programs Huseyin Topaloglu School of Operations Research and Information Engineering Cornell University ht88@cornell.edu August 14, 2010 1 Outline Cutting plane methods

NUMERICAL METHODS. x n+1 = 2x n x 2 n. In particular: which of them gives faster convergence, and why? [Work to four decimal places.

NUMERICAL METHODS 1. Rearranging the equation x 3 =.5 gives the iterative formula x n+1 = g(x n ), where g(x) = (2x 2 ) 1. (a) Starting with x = 1, compute the x n up to n = 6, and describe what is happening.

03-Economic Dispatch 1. EE570 Energy Utilization & Conservation Professor Henry Louie

03-Economic Dispatch 1 EE570 Energy Utilization & Conservation Professor Henry Louie 1 Topics Generator Curves Economic Dispatch (ED) Formulation ED (No Generator Limits, No Losses) ED (No Losses) ED Example

Numerical Methods in Physics and Astrophysics

Kostas Kokkotas 2 October 17, 2017 2 http://www.tat.physik.uni-tuebingen.de/ kokkotas Kostas Kokkotas 3 TOPICS 1. Solving nonlinear equations 2. Solving linear systems of equations 3. Interpolation, approximation

Interior-Point Methods for Linear Optimization

Interior-Point Methods for Linear Optimization Robert M. Freund and Jorge Vera March, 204 c 204 Robert M. Freund and Jorge Vera. All rights reserved. Linear Optimization with a Logarithmic Barrier Function

Goals for This Lecture:

Goals for This Lecture: Learn the Newton-Raphson method for finding real roots of real functions Learn the Bisection method for finding real roots of a real function Look at efficient implementations of

Lecture 10: Powers of Matrices, Difference Equations

Lecture 10: Powers of Matrices, Difference Equations Difference Equations A difference equation, also sometimes called a recurrence equation is an equation that defines a sequence recursively, i.e. each

Bisection and False Position Dr. Marco A. Arocha Aug, 2014

Bisection and False Position Dr. Marco A. Arocha Aug, 2014 1 Given function f, we seek x values for which f(x)=0 Solution x is the root of the equation or zero of the function f Problem is known as root

SECTION 5: POWER FLOW. ESE 470 Energy Distribution Systems

SECTION 5: POWER FLOW ESE 470 Energy Distribution Systems 2 Introduction Nodal Analysis 3 Consider the following circuit Three voltage sources VV sss, VV sss, VV sss Generic branch impedances Could be

Math 128A: Homework 2 Solutions

Math 128A: Homework 2 Solutions Due: June 28 1. In problems where high precision is not needed, the IEEE standard provides a specification for single precision numbers, which occupy 32 bits of storage.

Next topics: Solving systems of linear equations

Next topics: Solving systems of linear equations 1 Gaussian elimination (today) 2 Gaussian elimination with partial pivoting (Week 9) 3 The method of LU-decomposition (Week 10) 4 Iterative techniques: