Jim Lambers MAT 419/519 Summer Session Lecture 11 Notes

Size: px
Start display at page:

Download "Jim Lambers MAT 419/519 Summer Session Lecture 11 Notes"

Transcription

1 Jim Lambers MAT 49/59 Summer Session 20-2 Lecture Notes These notes correspond to Section 34 in the text Broyden s Method One of the drawbacks of using Newton s Method to solve a system of nonlinear equations g(x = 0 is the computational expense that must be incurred during each iteration to evaluate the partial derivatives of g at x (k, and then solve a system of linear equations involving the resulting Jacobian matrix The algorithm does not facilitate the re-use of data from previous iterations, and in some cases evaluation of the partial derivatives can be unnecessarily costly An alternative is to modify Newton s Method so that approximate partial derivatives are used, since the slightly slower convergence resulting from such an approximation is offset by the improved efficiency of each iteration However, simply replacing the analytical Jacobian matrix J g (x of g with a matrix consisting of finite difference approximations of the partial derivatives does not do much to reduce the cost of each iteration, because the cost of solving the system of linear equations is unchanged However, because J g (x consists of the partial derivatives evaluated at an element of a convergent sequence, intuitively Jacobian matrices from consecutive iterations are near one another in some sense, which suggests that it should be possible to cheaply update an approximate Jacobian matrix from iteration to iteration, in such a way that the inverse of the Jacobian matrix, which is what is really needed during each Newton iteration, can be updated efficiently as well This is the case when an n n matrix B has the form B = A + u v, where u and v are given vectors in R n, and u v is the outer product of u and v, defined by u v u v 2 u v n u v = uv T u 2 v u 2 v 2 u 2 v n = u n v u n v 2 u n v n This modification of A to obtain B is called a rank-one update This is because u v has rank one, since every column of u v is a scalar multiple of u To obtain B from A, we note that if Ax = u,

2 then Bx = (A + u vx = Ax + uv T x = u + u(v x = ( + v xu, which yields B u = + v A u A u On the other hand, if x is such that v A x = 0, then which yields BA x = (A + u va x = AA x + uv T A x = x + u(v A x = x, B x = A x This takes us to the following more general problem: given a matrix C, we wish to construct a matrix D such that the following conditions are satisfied: Dw = z, for given vectors w and z Dy = Cy, if y is orthogonal to a given vector g In our application, C = A, D = B, w = u, z = /( + v A ua u, and g = A T v To solve this problem, we set D = C + (z Cw g g w Then, if g y = 0, the second term in the definition of D vanishes, and we obtain Dy = Cy, but in computing Dw, we obtain factors of g w in the numerator and denominator that cancel, which yields Dw = Cw + (z Cw = z Applying this definition of D, we obtain [( B = A +v A + u A u A u v A u ] v A = A A (u va + v A u This formula for the inverse of a rank-one update is known as the Sherman-Morrison Formula We now return to the problem of approximating the Jacobian of g, and efficiently obtaining its inverse, at each iterate x (k We begin with an exact Jacobian, D 0 = J g (x (0, and use D 0 to compute the first iterate, x (, using Newton s Method as follows: Then, we use the approximation d (0 = D 0 g(x(0, x ( = x (0 + d (0 g (x g(x g(x 0 x x 0 Generalizing this approach to a system of equations, we seek an approximation D to J g (x ( that has these properties: 2

3 where D (x ( x (0 = g(x ( g(x (0 (the Secant Condition If z (x ( x (0 = 0, then D z = J g (x (0 z = D 0 z It follows from previous discussion that D = D 0 + y 0 D 0 d (0 d (0 d (0 d (0, d (0 = x ( x (0, y (0 = g(x ( g(x (0 However, it can be shown (Chapter 3, Exercise 5 that y 0 D 0 d (0 = g(x (, which yields the simplified formula D = D 0 + d (0 d (0 g(x( d (0 Once we have computed D 0, we can apply the Sherman-Morrison formula to obtain ( D0 d (0 d g(x ( d (0 D (0 0 ( + d (0 D0 d (0 d g(x ( ( (0 g(x ( d (0 D0 D = D 0 = D 0 D 0 d (0 d (0 + d (0 D = D0 (u(0 d (0 D0 d (0 (d (0 + u (0, 0 g(x( where u (0 = D 0 g(x( Then, as D is an approximation to J g (x (, we can obtain our next iterate x (2 as follows: D d ( = g(x (, x (2 = x ( + d ( Repeating this process, we obtain the following algorithm, which is known as Broyden s Method: Choose x (0 D 0 = J g (x (0 d (0 = D 0 g(x(0 x ( = x (0 + d (0 k = 0 while not converged do u (k = D k g(x(k+ c k = d (k (d (k + u (k D k+ = D k c k [u (k d (k ]D k k = k + 3

4 d (k = D k g(x(k x (k+ = x (k + d (k end Note that it is not necessary to compute D k for k ; only D k is needed It follows that no systems of linear equations need to be solved during an iteration; only matrix-vector multiplications are required, thus saving an order of magnitude of computational effort during each iteration compared to Newton s Method Example We consider the system of equations g(x = 0, where x 2 + y 2 + z 2 3 g(x, y, z = x 2 + y 2 z x + y + z 3 We will begin with one step of Newton s Method to solve this system of equations, with initial guess x (0 = (x (0, y (0, z (0 = (, 0, As computed in a previous example, J g (x, y, z = 2x 2y 2z 2x 2y Therefore, the Newton iterate x ( is obtained by solving the system of equations or, equivalently, D 0 d (0 = g(x (0 where D 0 = J g (x (0 = J g (x (0 (x ( x (0 = g(x (0, 2x (0 2y (0 2z (0 2x (0 2y (0 g(x (0 =, d (0 = (x (0 2 + (y (0 2 + (z (0 2 3 (x (0 2 + (y (0 2 z (0 x (0 + y (0 + z (0 3 Substituting (x (0, y (0, z (0 = (, 0, yields the system x ( y ( z ( 4 = x ( x (0 y ( y (0 z ( z (0,

5 This system has the solution d (0 = ( 2, 2, 0 which yields x( = ( 3 2, 2, Now, instead of computing J g (x (, we compute D = D 0 + d (0 d (0 g(x( d ( = /2 /2 /2 /2 / /4 /4 0 = /4 / = 5/2 /2 2 5/2 /2 This yields the system 5/2 /2 2 5/2 /2 x (2 3 2 y (2 2 z (2 = 2 2 0, which has the solution x (2 = ( 5 4, 3 4, This happens to be the same second iterate computed by Newton s Method, though this is not the case for a general function g(x In the next lecture, we will learn how to apply Broyden s Method to solve minimization problems Exercises Chapter 3, Exercise 4 2 Chapter 3, Exercise 5 5

Jim Lambers MAT 610 Summer Session Lecture 1 Notes

Jim Lambers MAT 610 Summer Session Lecture 1 Notes Jim Lambers MAT 60 Summer Session 2009-0 Lecture Notes Introduction This course is about numerical linear algebra, which is the study of the approximate solution of fundamental problems from linear algebra

More information

Jim Lambers MAT 419/519 Summer Session Lecture 13 Notes

Jim Lambers MAT 419/519 Summer Session Lecture 13 Notes Jim Lambers MAT 419/519 Summer Session 2011-12 Lecture 13 Notes These notes correspond to Section 4.1 in the text. Least Squares Fit One of the most fundamental problems in science and engineering is data

More information

Matrix Secant Methods

Matrix Secant Methods Equation Solving g(x) = 0 Newton-Lie Iterations: x +1 := x J g(x ), where J g (x ). Newton-Lie Iterations: x +1 := x J g(x ), where J g (x ). 3700 years ago the Babylonians used the secant method in 1D:

More information

Numerical solutions of nonlinear systems of equations

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

More information

MAT 610: Numerical Linear Algebra. James V. Lambers

MAT 610: Numerical Linear Algebra. James V. Lambers MAT 610: Numerical Linear Algebra James V Lambers January 16, 2017 2 Contents 1 Matrix Multiplication Problems 7 11 Introduction 7 111 Systems of Linear Equations 7 112 The Eigenvalue Problem 8 12 Basic

More information

13. Nonlinear least squares

13. Nonlinear least squares L. Vandenberghe ECE133A (Fall 2018) 13. Nonlinear least squares definition and examples derivatives and optimality condition Gauss Newton method Levenberg Marquardt method 13.1 Nonlinear least squares

More information

Gaussian Elimination and Back Substitution

Gaussian Elimination and Back Substitution Jim Lambers MAT 610 Summer Session 2009-10 Lecture 4 Notes These notes correspond to Sections 31 and 32 in the text Gaussian Elimination and Back Substitution The basic idea behind methods for solving

More information

MATH 4211/6211 Optimization Quasi-Newton Method

MATH 4211/6211 Optimization Quasi-Newton Method MATH 4211/6211 Optimization Quasi-Newton Method Xiaojing Ye Department of Mathematics & Statistics Georgia State University Xiaojing Ye, Math & Stat, Georgia State University 0 Quasi-Newton Method Motivation:

More information

Solving Nonlinear Equations

Solving Nonlinear Equations Solving Nonlinear Equations Jijian Fan Department of Economics University of California, Santa Cruz Oct 13 2014 Overview NUMERICALLY solving nonlinear equation Four methods Bisection Function iteration

More information

MATH 3795 Lecture 13. Numerical Solution of Nonlinear Equations in R N.

MATH 3795 Lecture 13. Numerical Solution of Nonlinear Equations in R N. MATH 3795 Lecture 13. Numerical Solution of Nonlinear Equations in R N. Dmitriy Leykekhman Fall 2008 Goals Learn about different methods for the solution of F (x) = 0, their advantages and disadvantages.

More information

SECTION: CONTINUOUS OPTIMISATION LECTURE 4: QUASI-NEWTON METHODS

SECTION: CONTINUOUS OPTIMISATION LECTURE 4: QUASI-NEWTON METHODS SECTION: CONTINUOUS OPTIMISATION LECTURE 4: QUASI-NEWTON METHODS HONOUR SCHOOL OF MATHEMATICS, OXFORD UNIVERSITY HILARY TERM 2005, DR RAPHAEL HAUSER 1. The Quasi-Newton Idea. In this lecture we will discuss

More information

Linear Algebra and Matrix Inversion

Linear Algebra and Matrix Inversion Jim Lambers MAT 46/56 Spring Semester 29- Lecture 2 Notes These notes correspond to Section 63 in the text Linear Algebra and Matrix Inversion Vector Spaces and Linear Transformations Matrices are much

More information

APPENDIX : PARTIAL FRACTIONS

APPENDIX : PARTIAL FRACTIONS APPENDIX : PARTIAL FRACTIONS Appendix : Partial Fractions Given the expression x 2 and asked to find its integral, x + you can use work from Section. to give x 2 =ln( x 2) ln( x + )+c x + = ln k x 2 x+

More information

Search Directions for Unconstrained Optimization

Search Directions for Unconstrained Optimization 8 CHAPTER 8 Search Directions for Unconstrained Optimization In this chapter we study the choice of search directions used in our basic updating scheme x +1 = x + t d. for solving P min f(x). x R n All

More information

CS 450 Numerical Analysis. Chapter 5: Nonlinear Equations

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

More information

Jim Lambers MAT 610 Summer Session Lecture 2 Notes

Jim Lambers MAT 610 Summer Session Lecture 2 Notes Jim Lambers MAT 610 Summer Session 2009-10 Lecture 2 Notes These notes correspond to Sections 2.2-2.4 in the text. Vector Norms Given vectors x and y of length one, which are simply scalars x and y, the

More information

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 6

CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 6 CME 302: NUMERICAL LINEAR ALGEBRA FALL 2005/06 LECTURE 6 GENE H GOLUB Issues with Floating-point Arithmetic We conclude our discussion of floating-point arithmetic by highlighting two issues that frequently

More information

Convex Optimization CMU-10725

Convex Optimization CMU-10725 Convex Optimization CMU-10725 Quasi Newton Methods Barnabás Póczos & Ryan Tibshirani Quasi Newton Methods 2 Outline Modified Newton Method Rank one correction of the inverse Rank two correction of the

More information

Quasi-Newton Methods. Zico Kolter (notes by Ryan Tibshirani, Javier Peña, Zico Kolter) Convex Optimization

Quasi-Newton Methods. Zico Kolter (notes by Ryan Tibshirani, Javier Peña, Zico Kolter) Convex Optimization Quasi-Newton Methods Zico Kolter (notes by Ryan Tibshirani, Javier Peña, Zico Kolter) Convex Optimization 10-725 Last time: primal-dual interior-point methods Given the problem min x f(x) subject to h(x)

More information

Computational Methods. Least Squares Approximation/Optimization

Computational Methods. Least Squares Approximation/Optimization Computational Methods Least Squares Approximation/Optimization Manfred Huber 2011 1 Least Squares Least squares methods are aimed at finding approximate solutions when no precise solution exists Find the

More information

Tangent Planes, Linear Approximations and Differentiability

Tangent Planes, Linear Approximations and Differentiability Jim Lambers MAT 80 Spring Semester 009-10 Lecture 5 Notes These notes correspond to Section 114 in Stewart and Section 3 in Marsden and Tromba Tangent Planes, Linear Approximations and Differentiability

More information

Programming, numerics and optimization

Programming, numerics and optimization Programming, numerics and optimization Lecture C-3: Unconstrained optimization II Łukasz Jankowski ljank@ippt.pan.pl Institute of Fundamental Technological Research Room 4.32, Phone +22.8261281 ext. 428

More information

Math 408A: Non-Linear Optimization

Math 408A: Non-Linear Optimization February 12 Broyden Updates Given g : R n R n solve g(x) = 0. Algorithm: Broyden s Method Initialization: x 0 R n, B 0 R n n Having (x k, B k ) compute (x k+1, B x+1 ) as follows: Solve B k s k = g(x

More information

Scientific Computing: An Introductory Survey

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

More information

AM 205: lecture 19. Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods

AM 205: lecture 19. Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods AM 205: lecture 19 Last time: Conditions for optimality, Newton s method for optimization Today: survey of optimization methods Quasi-Newton Methods General form of quasi-newton methods: x k+1 = x k α

More information

1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0

1. Nonlinear Equations. This lecture note excerpted parts from Michael Heath and Max Gunzburger. f(x) = 0 Numerical Analysis 1 1. Nonlinear Equations This lecture note excerpted parts from Michael Heath and Max Gunzburger. Given function f, we seek value x for which where f : D R n R n is nonlinear. f(x) =

More information

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 5. Nonlinear Equations

Lecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 5. Nonlinear Equations Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T Heath Chapter 5 Nonlinear Equations Copyright c 2001 Reproduction permitted only for noncommercial, educational

More information

Unconstrained Geometric Programming

Unconstrained Geometric Programming Jim Lambers MAT 49/59 Summer Session 20-2 Lecture 8 Notes These notes correspond to Section 2.5 in the text. Unconstrained Geometric Programming Previously, we learned how to use the A-G Inequality to

More information

1. Search Directions In this chapter we again focus on the unconstrained optimization problem. lim sup ν

1. Search Directions In this chapter we again focus on the unconstrained optimization problem. lim sup ν 1 Search Directions In this chapter we again focus on the unconstrained optimization problem P min f(x), x R n where f : R n R is assumed to be twice continuously differentiable, and consider the selection

More information

Projections and Least Square Solutions. Recall that given an inner product space V with subspace W and orthogonal basis for

Projections and Least Square Solutions. Recall that given an inner product space V with subspace W and orthogonal basis for Math 57 Spring 18 Projections and Least Square Solutions Recall that given an inner product space V with subspace W and orthogonal basis for W, B {v 1, v,..., v k }, the orthogonal projection of V onto

More information

Quasi-Newton methods: Symmetric rank 1 (SR1) Broyden Fletcher Goldfarb Shanno February 6, / 25 (BFG. Limited memory BFGS (L-BFGS)

Quasi-Newton methods: Symmetric rank 1 (SR1) Broyden Fletcher Goldfarb Shanno February 6, / 25 (BFG. Limited memory BFGS (L-BFGS) Quasi-Newton methods: Symmetric rank 1 (SR1) Broyden Fletcher Goldfarb Shanno (BFGS) Limited memory BFGS (L-BFGS) February 6, 2014 Quasi-Newton methods: Symmetric rank 1 (SR1) Broyden Fletcher Goldfarb

More information

Numerical Methods in Informatics

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

More information

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

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

More information

Math 411 Preliminaries

Math 411 Preliminaries Math 411 Preliminaries Provide a list of preliminary vocabulary and concepts Preliminary Basic Netwon s method, Taylor series expansion (for single and multiple variables), Eigenvalue, Eigenvector, Vector

More information

AM 205: lecture 19. Last time: Conditions for optimality Today: Newton s method for optimization, survey of optimization methods

AM 205: lecture 19. Last time: Conditions for optimality Today: Newton s method for optimization, survey of optimization methods AM 205: lecture 19 Last time: Conditions for optimality Today: Newton s method for optimization, survey of optimization methods Optimality Conditions: Equality Constrained Case As another example of equality

More information

Simple Iteration, cont d

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

More information

Functions of Several Variables

Functions of Several Variables Jim Lambers MAT 419/519 Summer Session 2011-12 Lecture 2 Notes These notes correspond to Section 1.2 in the text. Functions of Several Variables We now generalize the results from the previous section,

More information

Jim Lambers MAT 460 Fall Semester Lecture 2 Notes

Jim Lambers MAT 460 Fall Semester Lecture 2 Notes Jim Lambers MAT 460 Fall Semester 2009-10 Lecture 2 Notes These notes correspond to Section 1.1 in the text. Review of Calculus Among the mathematical problems that can be solved using techniques from

More information

Math 344 Lecture # Linear Systems

Math 344 Lecture # Linear Systems Math 344 Lecture #12 2.7 Linear Systems Through a choice of bases S and T for finite dimensional vector spaces V (with dimension n) and W (with dimension m), a linear equation L(v) = w becomes the linear

More information

Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane.

Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane. Queens College, CUNY, Department of Computer Science Numerical Methods CSCI 361 / 761 Spring 2018 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 2018 3 Lecture 3 3.1 General remarks March 4, 2018 This

More information

MA 123 September 8, 2016

MA 123 September 8, 2016 Instantaneous velocity and its Today we first revisit the notion of instantaneous velocity, and then we discuss how we use its to compute it. Learning Catalytics session: We start with a question about

More information

17 Solution of Nonlinear Systems

17 Solution of Nonlinear Systems 17 Solution of Nonlinear Systems We now discuss the solution of systems of nonlinear equations. An important ingredient will be the multivariate Taylor theorem. Theorem 17.1 Let D = {x 1, x 2,..., x m

More information

Algorithms for Constrained Optimization

Algorithms for Constrained Optimization 1 / 42 Algorithms for Constrained Optimization ME598/494 Lecture Max Yi Ren Department of Mechanical Engineering, Arizona State University April 19, 2015 2 / 42 Outline 1. Convergence 2. Sequential quadratic

More information

5. Introduction to limit

5. Introduction to limit 5. 5.1. The main idea in calculus is that of finding a desired quantity by pushing to the limit the process of taking ever better approximations (see 0 Introduction). In the implementation, a real number

More information

March 8, 2010 MATH 408 FINAL EXAM SAMPLE

March 8, 2010 MATH 408 FINAL EXAM SAMPLE March 8, 200 MATH 408 FINAL EXAM SAMPLE EXAM OUTLINE The final exam for this course takes place in the regular course classroom (MEB 238) on Monday, March 2, 8:30-0:20 am. You may bring two-sided 8 page

More information

Applications of the Maximum Principle

Applications of the Maximum Principle Jim Lambers MAT 606 Spring Semester 2015-16 Lecture 26 Notes These notes correspond to Sections 7.4-7.6 in the text. Applications of the Maximum Principle The maximum principle for Laplace s equation is

More information

Statistics 580 Optimization Methods

Statistics 580 Optimization Methods Statistics 580 Optimization Methods Introduction Let fx be a given real-valued function on R p. The general optimization problem is to find an x ɛ R p at which fx attain a maximum or a minimum. It is of

More information

MATH 4211/6211 Optimization Basics of Optimization Problems

MATH 4211/6211 Optimization Basics of Optimization Problems MATH 4211/6211 Optimization Basics of Optimization Problems Xiaojing Ye Department of Mathematics & Statistics Georgia State University Xiaojing Ye, Math & Stat, Georgia State University 0 A standard minimization

More information

1. Background: The SVD and the best basis (questions selected from Ch. 6- Can you fill in the exercises?)

1. Background: The SVD and the best basis (questions selected from Ch. 6- Can you fill in the exercises?) Math 35 Exam Review SOLUTIONS Overview In this third of the course we focused on linear learning algorithms to model data. summarize: To. Background: The SVD and the best basis (questions selected from

More information

Math 4329: Numerical Analysis Chapter 03: Fixed Point Iteration and Ill behaving problems. Natasha S. Sharma, PhD

Math 4329: Numerical Analysis Chapter 03: Fixed Point Iteration and Ill behaving problems. Natasha S. Sharma, PhD Why another root finding technique? iteration gives us the freedom to design our own root finding algorithm. The design of such algorithms is motivated by the need to improve the speed and accuracy of

More information

Quasi-Newton Methods. Javier Peña Convex Optimization /36-725

Quasi-Newton Methods. Javier Peña Convex Optimization /36-725 Quasi-Newton Methods Javier Peña Convex Optimization 10-725/36-725 Last time: primal-dual interior-point methods Consider the problem min x subject to f(x) Ax = b h(x) 0 Assume f, h 1,..., h m are convex

More information

5 Finding roots of equations

5 Finding roots of equations Lecture notes for Numerical Analysis 5 Finding roots of equations Topics:. Problem statement. Bisection Method 3. Newton s Method 4. Fixed Point Iterations 5. Systems of equations 6. Notes and further

More information

CS 323: Numerical Analysis and Computing

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.

More information

Numerical Optimization

Numerical Optimization Numerical Optimization Unit 2: Multivariable optimization problems Che-Rung Lee Scribe: February 28, 2011 (UNIT 2) Numerical Optimization February 28, 2011 1 / 17 Partial derivative of a two variable function

More information

Constrained Optimization

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

More information

Matrices. Chapter What is a Matrix? We review the basic matrix operations. An array of numbers a a 1n A = a m1...

Matrices. Chapter What is a Matrix? We review the basic matrix operations. An array of numbers a a 1n A = a m1... Chapter Matrices We review the basic matrix operations What is a Matrix? An array of numbers a a n A = a m a mn with m rows and n columns is a m n matrix Element a ij in located in position (i, j The elements

More information

Notes on Householder QR Factorization

Notes on Householder QR Factorization Notes on Householder QR Factorization Robert A van de Geijn Department of Computer Science he University of exas at Austin Austin, X 7872 rvdg@csutexasedu September 2, 24 Motivation A fundamental problem

More information

Maria Cameron. f(x) = 1 n

Maria Cameron. f(x) = 1 n Maria Cameron 1. Local algorithms for solving nonlinear equations Here we discuss local methods for nonlinear equations r(x) =. These methods are Newton, inexact Newton and quasi-newton. We will show that

More information

Numerical Methods for Large-Scale Nonlinear Systems

Numerical Methods for Large-Scale Nonlinear Systems Numerical Methods for Large-Scale Nonlinear Systems Handouts by Ronald H.W. Hoppe following the monograph P. Deuflhard Newton Methods for Nonlinear Problems Springer, Berlin-Heidelberg-New York, 2004 Num.

More information

Review of matrices. Let m, n IN. A rectangle of numbers written like A =

Review of matrices. Let m, n IN. A rectangle of numbers written like A = Review of matrices Let m, n IN. A rectangle of numbers written like a 11 a 12... a 1n a 21 a 22... a 2n A =...... a m1 a m2... a mn where each a ij IR is called a matrix with m rows and n columns or an

More information

Roundoff Analysis of Gaussian Elimination

Roundoff Analysis of Gaussian Elimination Jim Lambers MAT 60 Summer Session 2009-0 Lecture 5 Notes These notes correspond to Sections 33 and 34 in the text Roundoff Analysis of Gaussian Elimination In this section, we will perform a detailed error

More information

Introduction to gradient descent

Introduction to gradient descent 6-1: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction to gradient descent Derivation and intuitions Hessian 6-2: Introduction to gradient descent Prof. J.C. Kao, UCLA Introduction Our

More information

CS 542G: Robustifying Newton, Constraints, Nonlinear Least Squares

CS 542G: Robustifying Newton, Constraints, Nonlinear Least Squares CS 542G: Robustifying Newton, Constraints, Nonlinear Least Squares Robert Bridson October 29, 2008 1 Hessian Problems in Newton Last time we fixed one of plain Newton s problems by introducing line search

More information

COMP 558 lecture 18 Nov. 15, 2010

COMP 558 lecture 18 Nov. 15, 2010 Least squares We have seen several least squares problems thus far, and we will see more in the upcoming lectures. For this reason it is good to have a more general picture of these problems and how to

More information

Optimization and Root Finding. Kurt Hornik

Optimization and Root Finding. Kurt Hornik Optimization and Root Finding Kurt Hornik Basics Root finding and unconstrained smooth optimization are closely related: Solving ƒ () = 0 can be accomplished via minimizing ƒ () 2 Slide 2 Basics Root finding

More information

Matrix Derivatives and Descent Optimization Methods

Matrix Derivatives and Descent Optimization Methods Matrix Derivatives and Descent Optimization Methods 1 Qiang Ning Department of Electrical and Computer Engineering Beckman Institute for Advanced Science and Techonology University of Illinois at Urbana-Champaign

More information

Section 19 Integral domains

Section 19 Integral domains Section 19 Integral domains Instructor: Yifan Yang Spring 2007 Observation and motivation There are rings in which ab = 0 implies a = 0 or b = 0 For examples, Z, Q, R, C, and Z[x] are all such rings There

More information

Two improved classes of Broyden s methods for solving nonlinear systems of equations

Two improved classes of Broyden s methods for solving nonlinear systems of equations Available online at www.isr-publications.com/jmcs J. Math. Computer Sci., 17 (2017), 22 31 Research Article Journal Homepage: www.tjmcs.com - www.isr-publications.com/jmcs Two improved classes of Broyden

More information

Lecture 6. Regularized least-squares and minimum-norm methods 6 1

Lecture 6. Regularized least-squares and minimum-norm methods 6 1 Regularized least-squares and minimum-norm methods 6 1 Lecture 6 Regularized least-squares and minimum-norm methods EE263 Autumn 2004 multi-objective least-squares regularized least-squares nonlinear least-squares

More information

Root Finding (and Optimisation)

Root Finding (and Optimisation) Root Finding (and Optimisation) M.Sc. in Mathematical Modelling & Scientific Computing, Practical Numerical Analysis Michaelmas Term 2018, Lecture 4 Root Finding The idea of root finding is simple we want

More information

8.3 Householder QR factorization

8.3 Householder QR factorization 83 ouseholder QR factorization 23 83 ouseholder QR factorization A fundamental problem to avoid in numerical codes is the situation where one starts with large values and one ends up with small values

More information

A New Approach for Solving Dual Fuzzy Nonlinear Equations Using Broyden's and Newton's Methods

A New Approach for Solving Dual Fuzzy Nonlinear Equations Using Broyden's and Newton's Methods From the SelectedWorks of Dr. Mohamed Waziri Yusuf August 24, 22 A New Approach for Solving Dual Fuzzy Nonlinear Equations Using Broyden's and Newton's Methods Mohammed Waziri Yusuf, Dr. Available at:

More information

Lecture 26. Quadratic Equations

Lecture 26. Quadratic Equations Lecture 26 Quadratic Equations Quadratic polynomials....................................................... 2 Quadratic polynomials....................................................... 3 Quadratic equations

More information

5 Quasi-Newton Methods

5 Quasi-Newton Methods Unconstrained Convex Optimization 26 5 Quasi-Newton Methods If the Hessian is unavailable... Notation: H = Hessian matrix. B is the approximation of H. C is the approximation of H 1. Problem: Solve min

More information

, b = 0. (2) 1 2 The eigenvectors of A corresponding to the eigenvalues λ 1 = 1, λ 2 = 3 are

, b = 0. (2) 1 2 The eigenvectors of A corresponding to the eigenvalues λ 1 = 1, λ 2 = 3 are Quadratic forms We consider the quadratic function f : R 2 R defined by f(x) = 2 xt Ax b T x with x = (x, x 2 ) T, () where A R 2 2 is symmetric and b R 2. We will see that, depending on the eigenvalues

More information

Math 308 Practice Final Exam Page and vector y =

Math 308 Practice Final Exam Page and vector y = Math 308 Practice Final Exam Page Problem : Solving a linear equation 2 0 2 5 Given matrix A = 3 7 0 0 and vector y = 8. 4 0 0 9 (a) Solve Ax = y (if the equation is consistent) and write the general solution

More information

Math and Numerical Methods Review

Math and Numerical Methods Review Math and Numerical Methods Review Michael Caracotsios, Ph.D. Clinical Associate Professor Chemical Engineering Department University of Illinois at Chicago Introduction In the study of chemical engineering

More information

3.4 Elementary Matrices and Matrix Inverse

3.4 Elementary Matrices and Matrix Inverse Math 220: Summer 2015 3.4 Elementary Matrices and Matrix Inverse A n n elementary matrix is a matrix which is obtained from the n n identity matrix I n n by a single elementary row operation. Elementary

More information

PART I Lecture Notes on Numerical Solution of Root Finding Problems MATH 435

PART I Lecture Notes on Numerical Solution of Root Finding Problems MATH 435 PART I Lecture Notes on Numerical Solution of Root Finding Problems MATH 435 Professor Biswa Nath Datta Department of Mathematical Sciences Northern Illinois University DeKalb, IL. 60115 USA E mail: dattab@math.niu.edu

More information

Next topics: Solving systems of linear equations

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:

More information

Sherman-Morrison-Woodbury

Sherman-Morrison-Woodbury Week 5: Wednesday, Sep 23 Sherman-Mrison-Woodbury The Sherman-Mrison fmula describes the solution of A+uv T when there is already a factization f A. An easy way to derive the fmula is through block Gaussian

More information

Review Questions REVIEW QUESTIONS 71

Review Questions REVIEW QUESTIONS 71 REVIEW QUESTIONS 71 MATLAB, is [42]. For a comprehensive treatment of error analysis and perturbation theory for linear systems and many other problems in linear algebra, see [126, 241]. An overview of

More information

Game Engineering: 2D

Game Engineering: 2D Game Engineering: 2D CS420-2010F-06 2D Math David Galles Department of Computer Science University of San Francisco 06-0: Back to Basics A Vector is a displacement Vector has both direction and length

More information

The Kalman filter is arguably one of the most notable algorithms

The Kalman filter is arguably one of the most notable algorithms LECTURE E NOTES «Kalman Filtering with Newton s Method JEFFREY HUMPHERYS and JEREMY WEST The Kalman filter is arguably one of the most notable algorithms of the 0th century [1]. In this article, we derive

More information

Math 240 Calculus III

Math 240 Calculus III DE Higher Order Calculus III Summer 2015, Session II Tuesday, July 28, 2015 Agenda DE 1. of order n An example 2. constant-coefficient linear Introduction DE We now turn our attention to solving linear

More information

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

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:

More information

Matrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A =

Matrices and Vectors. Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A = 30 MATHEMATICS REVIEW G A.1.1 Matrices and Vectors Definition of Matrix. An MxN matrix A is a two-dimensional array of numbers A = a 11 a 12... a 1N a 21 a 22... a 2N...... a M1 a M2... a MN A matrix can

More information

Lecture 18: November Review on Primal-dual interior-poit methods

Lecture 18: November Review on Primal-dual interior-poit methods 10-725/36-725: Convex Optimization Fall 2016 Lecturer: Lecturer: Javier Pena Lecture 18: November 2 Scribes: Scribes: Yizhu Lin, Pan Liu Note: LaTeX template courtesy of UC Berkeley EECS dept. Disclaimer:

More information

Numerical differentiation

Numerical differentiation Numerical differentiation Paul Seidel 1801 Lecture Notes Fall 011 Suppose that we have a function f(x) which is not given by a formula but as a result of some measurement or simulation (computer experiment)

More information

Equations in Quadratic Form

Equations in Quadratic Form Equations in Quadratic Form MATH 101 College Algebra J. Robert Buchanan Department of Mathematics Summer 2012 Objectives In this lesson we will learn to: make substitutions that allow equations to be written

More information

Quasi-Newton methods for minimization

Quasi-Newton methods for minimization Quasi-Newton methods for minimization Lectures for PHD course on Numerical optimization Enrico Bertolazzi DIMS Universitá di Trento November 21 December 14, 2011 Quasi-Newton methods for minimization 1

More information

Systems of Nonlinear Equations

Systems of Nonlinear Equations Systems of Nonlinear Equations L. T. Biegler Chemical Engineering Department Carnegie Mellon University Pittsburgh, PA 15213 Biegler@cmu.edu http://dynopt.cheme.cmu.edu Obtaining Derivatives Broyden vs.

More information

MecE 390 Final examination, Winter 2014

MecE 390 Final examination, Winter 2014 MecE 390 Final examination, Winter 2014 Directions: (i) a double-sided 8.5 11 formula sheet is permitted, (ii) no calculators are permitted, (iii) the exam is 80 minutes in duration; please turn your paper

More information

SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS BISECTION METHOD

SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS BISECTION METHOD BISECTION METHOD If a function f(x) is continuous between a and b, and f(a) and f(b) are of opposite signs, then there exists at least one root between a and b. It is shown graphically as, Let f a be negative

More information

Numerical Methods I Solving Nonlinear Equations

Numerical Methods I Solving Nonlinear Equations Numerical Methods I Solving Nonlinear Equations Aleksandar Donev Courant Institute, NYU 1 donev@courant.nyu.edu 1 MATH-GA 2011.003 / CSCI-GA 2945.003, Fall 2014 October 16th, 2014 A. Donev (Courant Institute)

More information

Quasi-Newton Methods

Quasi-Newton Methods Quasi-Newton Methods Werner C. Rheinboldt These are excerpts of material relating to the boos [OR00 and [Rhe98 and of write-ups prepared for courses held at the University of Pittsburgh. Some further references

More information

Usually, when we first formulate a problem in mathematics, we use the most familiar

Usually, when we first formulate a problem in mathematics, we use the most familiar Change of basis Usually, when we first formulate a problem in mathematics, we use the most familiar coordinates. In R, this means using the Cartesian coordinates x, y, and z. In vector terms, this is equivalent

More information

CS 323: Numerical Analysis and Computing

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.

More information

Jim Lambers MAT 460/560 Fall Semester Practice Final Exam

Jim Lambers MAT 460/560 Fall Semester Practice Final Exam Jim Lambers MAT 460/560 Fall Semester 2009-10 Practice Final Exam 1. Let f(x) = sin 2x + cos 2x. (a) Write down the 2nd Taylor polynomial P 2 (x) of f(x) centered around x 0 = 0. (b) Write down the corresponding

More information

AM 205: lecture 18. Last time: optimization methods Today: conditions for optimality

AM 205: lecture 18. Last time: optimization methods Today: conditions for optimality AM 205: lecture 18 Last time: optimization methods Today: conditions for optimality Existence of Global Minimum For example: f (x, y) = x 2 + y 2 is coercive on R 2 (global min. at (0, 0)) f (x) = x 3

More information