Topic 4a Introduction to Root Finding & Bracketing Methods
|
|
- Rosalyn Foster
- 6 years ago
- Views:
Transcription
1 /8/18 Couse Instucto D. Raymond C. Rumpf Office: A 337 Phone: (915) E Mail: cumpf@utep.edu Topic 4a Intoduction to Root Finding & Backeting Methods EE 4386/531 Computational Methods in EE Outline Intoduction Backeting Methods The Bisection Method False Position Method Multiple Roots 1
2 /8/18 Intoduction 3 What is Root Finding? What values of does f() =? et f a b c We can figue this out algebaically b b 4ac a But what if f e This cannot be solved analytically. We need to use a numeical method. 4
3 /8/18 Root Finding Methods Backeting Methods Finding a single oot that falls within a known ange. Vey obust Must know something ahead of time. Open Methods Tial and eo iteative methods Do not need bounds, only an initial guess. Moe efficient than backeting methods Can be unstable and not find a solution Roots of Polynomials Algoithm specific to polynomials Physics of you poblem must be fit to a polynomial Able to find all oots. Requies initial bounds U Requies an initial guess 1 Requies multiple points, y,, y,,, y 1 1 N N 5 Genealizing Root Finding Algoithms Root finding algoithms find all values of such that f() =. What if we wish to find all values of such that f() = a? Genealization f a f a et g f a Now pefom standad oot finding on g(). 6 3
4 /8/18 Peliminay Root ocation The basic oot finding algoithms all equie that a oot be oughly located. The oot finding algoithm only efines the ough location. This means we must detemine at least appoimate locations of the oots. This infomation can come fom one of seveal things. 1. Something is known about the physics of the poblem being solved that gives you infomation about whee the oots should be.. If nothing else is known, plot the function to identify the location of the oots and feed each of those ough locations into a ootfinding algoithm. You may have to geneate a numbe of plots and play with ais scaling to find the oots. Slide 7 Need fo Optimized Algoithms In many cicumstances, a single computation of f() may take hous, days, o weeks! In these cases it is highly desied to minimize the total numbe of computations of f(). It is often woth the investment of a few hous to wite an awesome code that uns in an hou than waiting days o weeks fo the answe to come fom a simple code you wote in a few minutes. 8 4
5 /8/18 The Bisection Method 9 Step 1 Pick a lowe and uppe bound, and U that is known to contain a single oot between them f(). -. Root we wish to find. f() U
6 /8/18 Step Calculate the midpoint between and U as the fist guess fo the oot f(). -. Fist guess U f() U Step 3 Calculate the function at the midpoint. f f() U 1 6
7 /8/18 Step 4 Adjust the bounds. f() U 13 Step 5 Calculate the midpoint Second guess U f().4 f() U
8 /8/18 Step 6 Calculate the function at the midpoint f(). -. U f() f Step 7 Adjust the bounds f() U f()
9 /8/18 Step 8 Calculate a midpoint f(). U f() U 17 Step 9 Calculate the function at the midpoint f(). U f() f
10 /8/18 Step 1 Adjust the bounds f() U f() Step 11 Calculate a midpoint f(). U f() U 1
11 /8/18 Step 1 Calculate the function as the midpoint. 1. U f(). f f() Step 13 and Beyond And so on Iteation 1 Iteation f() f() Iteation Iteation f() f() Iteation Iteation f() f() Iteation Iteation f() f()
12 /8/18 Adjusting the Bounds (1 of ) Be caeful about signs when adjusting the bounds. f() Move uppe bound Hee the function is positive and we move the uppe bound. f() Move uppe bound Hee the function is negative and we still move the uppe bound. 3 Adjusting the Bounds ( of ) If f f If f f then thee is a sign change between and. This means the oot is close to than U. Move U to. then the sign change must be between U and. This means the oot is close to U than. Move to. If f f then we ae done because we have the oot eactly. % Adjust the Bounds if f*f< U = ; fu = f; elseif f*f> = ; f = f; else beak; end %oot towad f %oot towad fu 4 1
13 /8/18 When is the Algoithm Finished? i. Calculate the amount has moved fom one iteation to the net. old old ii. At the end of each iteation, check if is less than some theshold. toleance Rule of thumb: If you want some numbe of digits of pecision, ensue is at least an ode of magnitude less than the desied pecision. WARNING! Do not set you convegence condition to U <toleance because this will fail when the same bounday is being adjusted each iteation. 5 Block Diagam of Bisection Method Stat Choose and. Evaluate f() at endpoints. f f f f U U Calculate midpoint. U Move lowe bound. f f Evaluate f() at midpoint. f f f f Done! Calculate. U Move uppe bound. f f U U Calculate change. old no tol yes Done! 6 13
14 /8/18 Algoithm fo Bisection Method 1. Choose lowe and uppe bounds, and U so that they suound a oot.. Evaluate the function at the endpoints, f( ) and f( U ). 3. Calculate midpoint. U 4. Iteate until conveged a) Evaluate the function at the midpoint f( ). b) Adjust the bounds. If f f, then U If f f, then If f, then DONE! c) Update the midpoint. U d) Detemine if conveged i. Calculate step size old ii. Algoithm is conveged if < toleance. 7 Notes on Bisection Method Most obust oot finde east efficient oot finde Guaanteed to find a oot as long as the bounds span a cossing Sometimes good to check sign change of bounds No sign change o even numbe of oots. Sign change odd numbe of oots. 8 14
15 /8/18 False Position Method 9 Moe Intelligent Midpoint False-Position Method U f f f f U U U f U U f Bisection Method U Since f f H, it is a good assumption that the oot is close to than it is to H. 3 15
16 /8/18 Step 1 Pick a lowe and uppe bound, and U that is known to contain a single oot f(). -. Root we wish to find. f() U Step Calculate a bette estimate of the position of the oot using a linea appoimation f(). -. Bette estimate f() U
17 /8/18 Step 3 Calculate the function at the estimate fo f(). f() f U Step 4 Adjust the bounds f() f() -.6 U
18 /8/18 Step 5 Estimate the position of the oot by linea appoimation Second guess.6.4 f(). f() U Deivation of Estimate (1 of 3) The equation of a line given a point (, y ) and slope m is y y m Assuming the function connecting the bounds is close to linea, the slope is appoimately m f f U U 36 18
19 /8/18 Deivation of Estimate ( of 3) et s choose the lowe bound to be the point in ou line equation. y f U f f y f U m We now estimate the position of the oot by setting y = and solving fo. f U f f U f U f U f 37 Deivation of Estimate (3 of 3) et s choose the uppe bound to be the point in ou line equation. f U U f y f U U y f U U We now estimate the position of the oot by setting y = and solving fo. f U f f U U U f U U U f U f m 38 19
20 /8/18 Intepetation of Estimate (1 of ) We have two possible equations to estimate the position of the oot. f U f U f f U U U f U f et s aveage these equations. f f U U U U f U f f U f f f U U f f U U 39 Intepetation of Estimate ( of ) U U f f U f U f Typical bisection method equation A coection tem to give a bette estimate. 4
21 /8/18 Block Diagam of False Position Method Stat Choose and. Evaluate f() at endpoints. f f f f U U Calculate midpoint. U Move lowe bound. f f Evaluate f() at midpoint. f f f f Done! Calculate. f f f f U U U U Move uppe bound. f f U U Calculate change. old This is the only diffeence fom bisection method. no tol yes Done! 41 When False Position Fails The false position method can fail o ehibit etemely slow convegence when the function is highly nonlinea between the bounds. This happens because the estimated oot is a vey poo guess. Slide 4 1
22 /8/18 Notes on False Position Method Vey simila to bisection method Calculates a moe intelligent midpoint. Conveges much faste fo nea linea functions. Conveges slowe fo functions with abupt cuves 43 Multiple Roots 44
23 /8/18 Recognizing Numbe of Roots Single Root Sign changes on eithe side of oot. f f Slope at oot is not zeo. f Tiple Root Sign changes on eithe side of oot. f f Slope at oot is zeo. f Cuvatue changes on eithe side of oot. f f Double Root Sign is same on eithe side of oot. f f Slope at oot is zeo. f Cuvatue is same on eithe side of oot. f f Quaduple Root Sign is same on eithe side of oot. f f Slope at oot is zeo. f Cuvatue is same on eithe side of oot. f f Cuvatue is boad and flat. 45 Poblem with Multiple Roots Backeting methods equie a sign change on eithe side of the oot. This means they only wok fo odd multiple oots. double oot tiple oot WI NOT WORK Slide 46 3
24 /8/18 A Useful Popety We can define an auiliay function u() that will have the same oots as f() but will always change sign on eithe side of any multiple oot. f single oot u f f double oot tiple oot Slide 47 The Fi If f() has even multiple oots, pefom oot finding on the auiliay function u() instead. Slide 48 4
x 1 b 1 Consider the midpoint x 0 = 1 2
1 chapte 2 : oot-finding def : Given a function f(), a oot is a numbe satisfying f() = 0. e : f() = 2 3 = ± 3 question : How can we find the oots of a geneal function f()? 2.1 bisection method idea : Find
More informationNumerical Integration
MCEN 473/573 Chapte 0 Numeical Integation Fall, 2006 Textbook, 0.4 and 0.5 Isopaametic Fomula Numeical Integation [] e [ ] T k = h B [ D][ B] e B Jdsdt In pactice, the element stiffness is calculated numeically.
More informationA Crash Course in (2 2) Matrices
A Cash Couse in ( ) Matices Seveal weeks woth of matix algeba in an hou (Relax, we will only stuy the simplest case, that of matices) Review topics: What is a matix (pl matices)? A matix is a ectangula
More informationThe Substring Search Problem
The Substing Seach Poblem One algoithm which is used in a vaiety of applications is the family of substing seach algoithms. These algoithms allow a use to detemine if, given two chaacte stings, one is
More informationMarkscheme May 2017 Calculus Higher level Paper 3
M7/5/MATHL/HP3/ENG/TZ0/SE/M Makscheme May 07 Calculus Highe level Pape 3 pages M7/5/MATHL/HP3/ENG/TZ0/SE/M This makscheme is the popety of the Intenational Baccalaueate and must not be epoduced o distibuted
More informationyou of a spring. The potential energy for a spring is given by the parabola U( x)
Small oscillations The theoy of small oscillations is an extemely impotant topic in mechanics. Conside a system that has a potential enegy diagam as below: U B C A x Thee ae thee points of stable equilibium,
More informationProblem 1. Part b. Part a. Wayne Witzke ProblemSet #1 PHY 361. Calculate x, the expected value of x, defined by
Poblem Pat a The nomal distibution Gaussian distibution o bell cuve has the fom f Ce µ Calculate the nomalization facto C by equiing the distibution to be nomalized f Substituting in f, defined above,
More informationPhysics 2B Chapter 22 Notes - Magnetic Field Spring 2018
Physics B Chapte Notes - Magnetic Field Sping 018 Magnetic Field fom a Long Staight Cuent-Caying Wie In Chapte 11 we looked at Isaac Newton s Law of Gavitation, which established that a gavitational field
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department Physics 8.07: Electromagnetism II September 15, 2012 Prof. Alan Guth PROBLEM SET 2
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Depatment Physics 8.07: Electomagnetism II Septembe 5, 202 Pof. Alan Guth PROBLEM SET 2 DUE DATE: Monday, Septembe 24, 202. Eithe hand it in at the lectue,
More informationAuchmuty High School Mathematics Department Advanced Higher Notes Teacher Version
The Binomial Theoem Factoials Auchmuty High School Mathematics Depatment The calculations,, 6 etc. often appea in mathematics. They ae called factoials and have been given the notation n!. e.g. 6! 6!!!!!
More informationInternet Appendix for A Bayesian Approach to Real Options: The Case of Distinguishing Between Temporary and Permanent Shocks
Intenet Appendix fo A Bayesian Appoach to Real Options: The Case of Distinguishing Between Tempoay and Pemanent Shocks Steven R. Genadie Gaduate School of Business, Stanfod Univesity Andey Malenko Gaduate
More informationFE FORMULATIONS FOR PLASTICITY
G These slides ae designed based on the book: Finite Elements in Plasticity Theoy and Pactice, D.R.J. Owen and E. Hinton, 970, Pineidge Pess Ltd., Swansea, UK. Couse Content: A INTRODUCTION AND OVERVIEW
More informationASTR415: Problem Set #6
ASTR45: Poblem Set #6 Cuan D. Muhlbege Univesity of Mayland (Dated: May 7, 27) Using existing implementations of the leapfog and Runge-Kutta methods fo solving coupled odinay diffeential equations, seveal
More informationMethod for Approximating Irrational Numbers
Method fo Appoximating Iational Numbes Eic Reichwein Depatment of Physics Univesity of Califonia, Santa Cuz June 6, 0 Abstact I will put foth an algoithm fo poducing inceasingly accuate ational appoximations
More information3.1 Random variables
3 Chapte III Random Vaiables 3 Random vaiables A sample space S may be difficult to descibe if the elements of S ae not numbes discuss how we can use a ule by which an element s of S may be associated
More informationHOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS?
6th INTERNATIONAL MULTIDISCIPLINARY CONFERENCE HOW TO TEACH THE FUNDAMENTALS OF INFORMATION SCIENCE, CODING, DECODING AND NUMBER SYSTEMS? Cecília Sitkuné Göömbei College of Nyíegyháza Hungay Abstact: The
More informationMAGNETIC FIELD AROUND TWO SEPARATED MAGNETIZING COILS
The 8 th Intenational Confeence of the Slovenian Society fo Non-Destuctive Testing»pplication of Contempoay Non-Destuctive Testing in Engineeing«Septembe 1-3, 5, Potoož, Slovenia, pp. 17-1 MGNETIC FIELD
More informationComputational Methods of Solid Mechanics. Project report
Computational Methods of Solid Mechanics Poject epot Due on Dec. 6, 25 Pof. Allan F. Bowe Weilin Deng Simulation of adhesive contact with molecula potential Poject desciption In the poject, we will investigate
More informationNon-Linear Dynamics Homework Solutions Week 2
Non-Linea Dynamics Homewok Solutions Week Chis Small Mach, 7 Please email me at smach9@evegeen.edu with any questions o concens eguading these solutions. Fo the ececises fom section., we sketch all qualitatively
More informationPhysics 521. Math Review SCIENTIFIC NOTATION SIGNIFICANT FIGURES. Rules for Significant Figures
Physics 51 Math Review SCIENIFIC NOAION Scientific Notation is based on exponential notation (whee decimal places ae expessed as a powe of 10). he numeical pat of the measuement is expessed as a numbe
More informationF-IF Logistic Growth Model, Abstract Version
F-IF Logistic Gowth Model, Abstact Vesion Alignments to Content Standads: F-IFB4 Task An impotant example of a model often used in biology o ecology to model population gowth is called the logistic gowth
More informationConjugate Gradient Methods. Michael Bader. Summer term 2012
Gadient Methods Outlines Pat I: Quadatic Foms and Steepest Descent Pat II: Gadients Pat III: Summe tem 2012 Pat I: Quadatic Foms and Steepest Descent Outlines Pat I: Quadatic Foms and Steepest Descent
More informationResearch Design - - Topic 17 Multiple Regression & Multiple Correlation: Two Predictors 2009 R.C. Gardner, Ph.D.
Reseach Design - - Topic 7 Multiple Regession & Multiple Coelation: Two Pedictos 009 R.C. Gadne, Ph.D. Geneal Rationale and Basic Aithmetic fo two pedictos Patial and semipatial coelation Regession coefficients
More informationΔt The textbook chooses to say that the average velocity is
1-D Motion Basic I Definitions: One dimensional motion (staight line) is a special case of motion whee all but one vecto component is zeo We will aange ou coodinate axis so that the x-axis lies along the
More informationRandom Variables and Probability Distribution Random Variable
Random Vaiables and Pobability Distibution Random Vaiable Random vaiable: If S is the sample space P(S) is the powe set of the sample space, P is the pobability of the function then (S, P(S), P) is called
More informationof the contestants play as Falco, and 1 6
JHMT 05 Algeba Test Solutions 4 Febuay 05. In a Supe Smash Bothes tounament, of the contestants play as Fox, 3 of the contestants play as Falco, and 6 of the contestants play as Peach. Given that thee
More informationLight Time Delay and Apparent Position
Light Time Delay and ppaent Position nalytical Gaphics, Inc. www.agi.com info@agi.com 610.981.8000 800.220.4785 Contents Intoduction... 3 Computing Light Time Delay... 3 Tansmission fom to... 4 Reception
More information7.2. Coulomb s Law. The Electric Force
Coulomb s aw Recall that chaged objects attact some objects and epel othes at a distance, without making any contact with those objects Electic foce,, o the foce acting between two chaged objects, is somewhat
More informationCBE Transport Phenomena I Final Exam. December 19, 2013
CBE 30355 Tanspot Phenomena I Final Exam Decembe 9, 203 Closed Books and Notes Poblem. (20 points) Scaling analysis of bounday laye flows. A popula method fo measuing instantaneous wall shea stesses in
More informationTo Feel a Force Chapter 7 Static equilibrium - torque and friction
To eel a oce Chapte 7 Chapte 7: Static fiction, toque and static equilibium A. Review of foce vectos Between the eath and a small mass, gavitational foces of equal magnitude and opposite diection act on
More informationA Two-Dimensional Bisection Envelope Algorithm for Fixed Points
A Two-imensional Bisection Envelope Algoithm fo Fied Points Kis Siosi and Spence Shellman Fom pulished Jounal of Compleity 8, 64-659(00 Intoduction How we solve fo two-dimensional f ( domain: [0, ]X[0,
More informationA Relativistic Electron in a Coulomb Potential
A Relativistic Electon in a Coulomb Potential Alfed Whitehead Physics 518, Fall 009 The Poblem Solve the Diac Equation fo an electon in a Coulomb potential. Identify the conseved quantum numbes. Specify
More informationGoodness-of-fit for composite hypotheses.
Section 11 Goodness-of-fit fo composite hypotheses. Example. Let us conside a Matlab example. Let us geneate 50 obsevations fom N(1, 2): X=nomnd(1,2,50,1); Then, unning a chi-squaed goodness-of-fit test
More informationMULTILAYER PERCEPTRONS
Last updated: Nov 26, 2012 MULTILAYER PERCEPTRONS Outline 2 Combining Linea Classifies Leaning Paametes Outline 3 Combining Linea Classifies Leaning Paametes Implementing Logical Relations 4 AND and OR
More informationCALCULUS II Vectors. Paul Dawkins
CALCULUS II Vectos Paul Dawkins Table of Contents Peface... ii Vectos... 3 Intoduction... 3 Vectos The Basics... 4 Vecto Aithmetic... 8 Dot Poduct... 13 Coss Poduct... 21 2007 Paul Dawkins i http://tutoial.math.lama.edu/tems.aspx
More information1 Similarity Analysis
ME43A/538A/538B Axisymmetic Tubulent Jet 9 Novembe 28 Similaity Analysis. Intoduction Conside the sketch of an axisymmetic, tubulent jet in Figue. Assume that measuements of the downsteam aveage axial
More informationPhysics 161 Fall 2011 Extra Credit 2 Investigating Black Holes - Solutions The Following is Worth 50 Points!!!
Physics 161 Fall 011 Exta Cedit Investigating Black Holes - olutions The Following is Woth 50 Points!!! This exta cedit assignment will investigate vaious popeties of black holes that we didn t have time
More informationSolutions to Problem Set 8
Massachusetts Institute of Technology 6.042J/18.062J, Fall 05: Mathematics fo Compute Science Novembe 21 Pof. Albet R. Meye and Pof. Ronitt Rubinfeld evised Novembe 27, 2005, 858 minutes Solutions to Poblem
More information15.081J/6.251J Introduction to Mathematical Programming. Lecture 6: The Simplex Method II
15081J/6251J Intoduction to Mathematical Pogamming ectue 6: The Simplex Method II 1 Outline Revised Simplex method Slide 1 The full tableau implementation Anticycling 2 Revised Simplex Initial data: A,
More informationParticle Systems. University of Texas at Austin CS384G - Computer Graphics Fall 2010 Don Fussell
Paticle Systems Univesity of Texas at Austin CS384G - Compute Gaphics Fall 2010 Don Fussell Reading Requied: Witkin, Paticle System Dynamics, SIGGRAPH 97 couse notes on Physically Based Modeling. Witkin
More informationExploration of the three-person duel
Exploation of the thee-peson duel Andy Paish 15 August 2006 1 The duel Pictue a duel: two shootes facing one anothe, taking tuns fiing at one anothe, each with a fixed pobability of hitting his opponent.
More informationLINEAR AND NONLINEAR ANALYSES OF A WIND-TUNNEL BALANCE
LINEAR AND NONLINEAR ANALYSES O A WIND-TUNNEL INTRODUCTION BALANCE R. Kakehabadi and R. D. Rhew NASA LaRC, Hampton, VA The NASA Langley Reseach Cente (LaRC) has been designing stain-gauge balances fo utilization
More informationContact impedance of grounded and capacitive electrodes
Abstact Contact impedance of gounded and capacitive electodes Andeas Hödt Institut fü Geophysik und extateestische Physik, TU Baunschweig The contact impedance of electodes detemines how much cuent can
More information5.61 Physical Chemistry Lecture #23 page 1 MANY ELECTRON ATOMS
5.6 Physical Chemisty Lectue #3 page MAY ELECTRO ATOMS At this point, we see that quantum mechanics allows us to undestand the helium atom, at least qualitatively. What about atoms with moe than two electons,
More informationBoundary Layers and Singular Perturbation Lectures 16 and 17 Boundary Layers and Singular Perturbation. x% 0 Ω0æ + Kx% 1 Ω0æ + ` : 0. (9.
Lectues 16 and 17 Bounday Layes and Singula Petubation A Regula Petubation In some physical poblems, the solution is dependent on a paamete K. When the paamete K is vey small, it is natual to expect that
More information6 PROBABILITY GENERATING FUNCTIONS
6 PROBABILITY GENERATING FUNCTIONS Cetain deivations pesented in this couse have been somewhat heavy on algeba. Fo example, detemining the expectation of the Binomial distibution (page 5.1 tuned out to
More informationMath 301: The Erdős-Stone-Simonovitz Theorem and Extremal Numbers for Bipartite Graphs
Math 30: The Edős-Stone-Simonovitz Theoem and Extemal Numbes fo Bipatite Gaphs May Radcliffe The Edős-Stone-Simonovitz Theoem Recall, in class we poved Tuán s Gaph Theoem, namely Theoem Tuán s Theoem Let
More informationReading Assignment. Problem Description for Homework #9. Read Chapters 29 and 30.
Reading Assignment Read Chaptes 29 and 30. Poblem Desciption fo Homewok #9 In this homewok, you will solve the inhomogeneous Laplace s equation to calculate the electic scala potential that exists between
More informationBerkeley Math Circle AIME Preparation March 5, 2013
Algeba Toolkit Rules of Thumb. Make sue that you can pove all fomulas you use. This is even bette than memoizing the fomulas. Although it is best to memoize, as well. Stive fo elegant, economical methods.
More informationMEI Structured Mathematics. Module Summary Sheets. Numerical Methods (Version B reference to new book)
MEI Matematics in Education and Industy MEI Stuctued Matematics Module Summay Seets (Vesion B efeence to new book) Topic : Appoximations Topic : Te solution of equations Topic : Numeical integation Topic
More informationEasy. P4.2 Since the car is moving with constant speed and in a straight line, the. resultant force on it must be regardless of whether it is moving
Chapte 4 Homewok Solutions Easy P4. Since the ca is moving with constant speed and in a staight line, the zeo esultant foce on it must be egadless of whethe it is moving (a) towad the ight o the left.
More informationEEO 401 Digital Signal Processing Prof. Mark Fowler
EEO 41 Digital Signal Pocessing Pof. Mak Fowle Note Set #31 Linea Phase FIR Design Optimum Equiipple (Paks-McClellan) Reading: Sect. 1.2.4 1.2.6 of Poakis & Manolakis 1/2 Motivation The window method and
More informationPhysics 2112 Unit 14
Physics 2112 Unit 14 Today s Concept: What Causes Magnetic Fields d 0I ds ˆ 2 4 Unit 14, Slide 1 You Comments Can you give a summay fo eveything we use the ight hand ule fo? Wasn't too clea on this topic.
More informationWhat to Expect on the Placement Exam
What to Epect on the Placement Eam Placement into: MTH o MTH 44 05 05 The ACCUPLACER placement eam is an adaptive test ceated by the College Boad Educational Testing Sevice. This document was ceated to
More informationC/CS/Phys C191 Shor s order (period) finding algorithm and factoring 11/12/14 Fall 2014 Lecture 22
C/CS/Phys C9 Sho s ode (peiod) finding algoithm and factoing /2/4 Fall 204 Lectue 22 With a fast algoithm fo the uantum Fouie Tansfom in hand, it is clea that many useful applications should be possible.
More informationAbsolute Specifications: A typical absolute specification of a lowpass filter is shown in figure 1 where:
FIR FILTER DESIGN The design of an digital filte is caied out in thee steps: ) Specification: Befoe we can design a filte we must have some specifications. These ae detemined by the application. ) Appoximations
More informationMAC Module 12 Eigenvalues and Eigenvectors
MAC 23 Module 2 Eigenvalues and Eigenvectos Leaning Objectives Upon completing this module, you should be able to:. Solve the eigenvalue poblem by finding the eigenvalues and the coesponding eigenvectos
More informationIntroduction to Vectors and Frames
Intoduction to Vectos and Fames CIS - 600.445 Russell Taylo Saah Gaham Infomation Flow Diagam Model the Patient Plan the Pocedue Eecute the Plan Real Wold Coodinate Fame Tansfomation F = [ R, p] 0 F y
More informationPHYS 1441 Section 002. Lecture #3
PHYS 1441 Section 00 Chapte 1 Lectue #3 Wednesday, Sept. 6, 017 Coulomb s Law The Electic Field & Field Lines Electic Fields and Conductos Motion of a Chaged Paticle in an Electic Field Electic Dipoles
More information18.06 Problem Set 4 Solution
8.6 Poblem Set 4 Solution Total: points Section 3.5. Poblem 2: (Recommended) Find the lagest possible numbe of independent vectos among ) ) ) v = v 4 = v 5 = v 6 = v 2 = v 3 =. Solution (4 points): Since
More informationPhysics 2020, Spring 2005 Lab 5 page 1 of 8. Lab 5. Magnetism
Physics 2020, Sping 2005 Lab 5 page 1 of 8 Lab 5. Magnetism PART I: INTRODUCTION TO MAGNETS This week we will begin wok with magnets and the foces that they poduce. By now you ae an expet on setting up
More informationI. CONSTRUCTION OF THE GREEN S FUNCTION
I. CONSTRUCTION OF THE GREEN S FUNCTION The Helmohltz equation in 4 dimensions is 4 + k G 4 x, x = δ 4 x x. In this equation, G is the Geen s function and 4 efes to the dimensionality. In the vey end,
More informationSolution to HW 3, Ma 1a Fall 2016
Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.
More informationQIP Course 10: Quantum Factorization Algorithm (Part 3)
QIP Couse 10: Quantum Factoization Algoithm (Pat 3 Ryutaoh Matsumoto Nagoya Univesity, Japan Send you comments to yutaoh.matsumoto@nagoya-u.jp Septembe 2018 @ Tokyo Tech. Matsumoto (Nagoya U. QIP Couse
More informationMultiple Experts with Binary Features
Multiple Expets with Binay Featues Ye Jin & Lingen Zhang Decembe 9, 2010 1 Intoduction Ou intuition fo the poect comes fom the pape Supevised Leaning fom Multiple Expets: Whom to tust when eveyone lies
More informationSection 8.2 Polar Coordinates
Section 8. Pola Coodinates 467 Section 8. Pola Coodinates The coodinate system we ae most familia with is called the Catesian coodinate system, a ectangula plane divided into fou quadants by the hoizontal
More informationPROBLEM SET #1 SOLUTIONS by Robert A. DiStasio Jr.
POBLM S # SOLUIONS by obet A. DiStasio J. Q. he Bon-Oppenheime appoximation is the standad way of appoximating the gound state of a molecula system. Wite down the conditions that detemine the tonic and
More informationTemporal-Difference Learning
.997 Decision-Making in Lage-Scale Systems Mach 17 MIT, Sping 004 Handout #17 Lectue Note 13 1 Tempoal-Diffeence Leaning We now conside the poblem of computing an appopiate paamete, so that, given an appoximation
More informationChapter Eight Notes N P U1C8S4-6
Chapte Eight Notes N P UC8S-6 Name Peiod Section 8.: Tigonometic Identities An identit is, b definition, an equation that is alwas tue thoughout its domain. B tue thoughout its domain, that is to sa that
More informationMulti-Objective Optimization Algorithms for Finite Element Model Updating
Multi-Objective Optimization Algoithms fo Finite Element Model Updating E. Ntotsios, C. Papadimitiou Univesity of Thessaly Geece Outline STRUCTURAL IDENTIFICATION USING MEASURED MODAL DATA Weighted Modal
More informationNumerical Modeling in Biomedical Systems
Numeial Modeling in Biomedial Systems BME 15:35 Letue 7 9/6/17 Nonlinea Systems Dunn Chapte 5 Nonlinea equations Root finding Baketing methods Open methods Gaphial Bisetion False Position Newton s method
More informationMATH 415, WEEK 3: Parameter-Dependence and Bifurcations
MATH 415, WEEK 3: Paamete-Dependence and Bifucations 1 A Note on Paamete Dependence We should pause to make a bief note about the ole played in the study of dynamical systems by the system s paametes.
More informationCurrent Balance Warm Up
PHYSICS EXPERIMENTS 133 Cuent Balance-1 Cuent Balance Wam Up 1. Foce between cuent-caying wies Wie 1 has a length L (whee L is "long") and caies a cuent I 0. What is the magnitude of the magnetic field
More informationWhen two numbers are written as the product of their prime factors, they are in factored form.
10 1 Study Guide Pages 420 425 Factos Because 3 4 12, we say that 3 and 4 ae factos of 12. In othe wods, factos ae the numbes you multiply to get a poduct. Since 2 6 12, 2 and 6 ae also factos of 12. The
More information1D2G - Numerical solution of the neutron diffusion equation
DG - Numeical solution of the neuton diffusion equation Y. Danon Daft: /6/09 Oveview A simple numeical solution of the neuton diffusion equation in one dimension and two enegy goups was implemented. Both
More informationVanishing lines in generalized Adams spectral sequences are generic
ISSN 364-0380 (on line) 465-3060 (pinted) 55 Geomety & Topology Volume 3 (999) 55 65 Published: 2 July 999 G G G G T T T G T T T G T G T GG TT G G G G GG T T T TT Vanishing lines in genealized Adams spectal
More informationMath Notes on Kepler s first law 1. r(t) kp(t)
Math 7 - Notes on Keple s fist law Planetay motion and Keple s Laws We conside the motion of a single planet about the sun; fo simplicity, we assign coodinates in R 3 so that the position of the sun is
More informationA Power Method for Computing Square Roots of Complex Matrices
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 13, 39345 1997 ARTICLE NO. AY975517 A Powe Method fo Computing Squae Roots of Complex Matices Mohammed A. Hasan Depatment of Electical Engineeing, Coloado
More informationAn Exact Solution of Navier Stokes Equation
An Exact Solution of Navie Stokes Equation A. Salih Depatment of Aeospace Engineeing Indian Institute of Space Science and Technology, Thiuvananthapuam, Keala, India. July 20 The pincipal difficulty in
More informationNuclear and Particle Physics - Lecture 20 The shell model
1 Intoduction Nuclea and Paticle Physics - Lectue 0 The shell model It is appaent that the semi-empiical mass fomula does a good job of descibing tends but not the non-smooth behaviou of the binding enegy.
More informationInformation Retrieval (Relevance Feedback & Query Expansion)
Infomation Retieval (Relevance Feedback & Quey Epansion) Fabio Aiolli http://www.math.unipd.it/~aiolli Dipatimento di Matematica Univesità di Padova Anno Accademico 1 Relevance feedback and quey epansion
More informationUniform Circular Motion
Unifom Cicula Motion Intoduction Ealie we defined acceleation as being the change in velocity with time: a = v t Until now we have only talked about changes in the magnitude of the acceleation: the speeding
More informationCOORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT
COORDINATE TRANSFORMATIONS - THE JACOBIAN DETERMINANT Link to: phsicspages home page. To leave a comment o epot an eo, please use the auilia blog. Refeence: d Inveno, Ra, Intoducing Einstein s Relativit
More informationChapter 5 Linear Equations: Basic Theory and Practice
Chapte 5 inea Equations: Basic Theoy and actice In this chapte and the next, we ae inteested in the linea algebaic equation AX = b, (5-1) whee A is an m n matix, X is an n 1 vecto to be solved fo, and
More information3.6 Applied Optimization
.6 Applied Optimization Section.6 Notes Page In this section we will be looking at wod poblems whee it asks us to maimize o minimize something. Fo all the poblems in this section you will be taking the
More informationCSCE 478/878 Lecture 4: Experimental Design and Analysis. Stephen Scott. 3 Building a tree on the training set Introduction. Outline.
In Homewok, you ae (supposedly) Choosing a data set 2 Extacting a test set of size > 3 3 Building a tee on the taining set 4 Testing on the test set 5 Repoting the accuacy (Adapted fom Ethem Alpaydin and
More information2 x 8 2 x 2 SKILLS Determine whether the given value is a solution of the. equation. (a) x 2 (b) x 4. (a) x 2 (b) x 4 (a) x 4 (b) x 8
5 CHAPTER Fundamentals When solving equations that involve absolute values, we usually take cases. EXAMPLE An Absolute Value Equation Solve the equation 0 x 5 0 3. SOLUTION By the definition of absolute
More informationTransfer Matrix Method
9/6/17 Instucto D. Ramond Rumpf (915) 747 6958 cumpf@utep.edu 5337 Computational lectomagnetics Lectue #4 Tansfe Mati Method Lectue 4These notes ma contain copighted mateial obtained unde fai use ules.
More information10/04/18. P [P(x)] 1 negl(n).
Mastemath, Sping 208 Into to Lattice lgs & Cypto Lectue 0 0/04/8 Lectues: D. Dadush, L. Ducas Scibe: K. de Boe Intoduction In this lectue, we will teat two main pats. Duing the fist pat we continue the
More informationExplosive Contagion in Networks (Supplementary Information)
Eplosive Contagion in Netwoks (Supplementay Infomation) Jesús Gómez-Gadeñes,, Laua Loteo, Segei N. Taaskin, and Fancisco J. Péez-Reche Institute fo Biocomputation and Physics of Comple Systems (BIFI),
More informationThe Laws of Motion ( ) N SOLUTIONS TO PROBLEMS ! F = ( 6.00) 2 + ( 15.0) 2 N = 16.2 N. Section 4.4. Newton s Second Law The Particle Under a Net Force
SOLUTIONS TO PROBLEMS The Laws of Motion Section 4.3 Mass P4. Since the ca is moving with constant speed and in a staight line, the esultant foce on it must be zeo egadless of whethe it is moving (a) towad
More informationradians). Figure 2.1 Figure 2.2 (a) quadrant I angle (b) quadrant II angle is in standard position Terminal side Terminal side Terminal side
. TRIGONOMETRIC FUNCTIONS OF GENERAL ANGLES In ode to etend the definitions of the si tigonometic functions to geneal angles, we shall make use of the following ideas: In a Catesian coodinate sstem, an
More informationHopefully Helpful Hints for Gauss s Law
Hopefully Helpful Hints fo Gauss s Law As befoe, thee ae things you need to know about Gauss s Law. In no paticula ode, they ae: a.) In the context of Gauss s Law, at a diffeential level, the electic flux
More informationVectors, Vector Calculus, and Coordinate Systems
Apil 5, 997 A Quick Intoduction to Vectos, Vecto Calculus, and Coodinate Systems David A. Randall Depatment of Atmospheic Science Coloado State Univesity Fot Collins, Coloado 80523. Scalas and vectos Any
More informationGalilean Transformation vs E&M y. Historical Perspective. Chapter 2 Lecture 2 PHYS Special Relativity. Sep. 1, y K K O.
PHYS-2402 Chapte 2 Lectue 2 Special Relativity 1. Basic Ideas Sep. 1, 2016 Galilean Tansfomation vs E&M y K O z z y K In 1873, Maxwell fomulated Equations of Electomagnetism. v Maxwell s equations descibe
More informationCentripetal Force OBJECTIVE INTRODUCTION APPARATUS THEORY
Centipetal Foce OBJECTIVE To veify that a mass moving in cicula motion expeiences a foce diected towad the cente of its cicula path. To detemine how the mass, velocity, and adius affect a paticle's centipetal
More informationMotithang Higher Secondary School Thimphu Thromde Mid Term Examination 2016 Subject: Mathematics Full Marks: 100
Motithang Highe Seconday School Thimphu Thomde Mid Tem Examination 016 Subject: Mathematics Full Maks: 100 Class: IX Witing Time: 3 Hous Read the following instuctions caefully In this pape, thee ae thee
More informationImplementation of RCWA
Instucto D. Ramond Rumpf (915) 747 6958 cumpf@utep.edu EE 5337 Computational Electomagnetics Lectue # Implementation of RCWA Lectue These notes ma contain copighted mateial obtained unde fai use ules.
More informationPhysics 181. Assignment 4
Physics 181 Assignment 4 Solutions 1. A sphee has within it a gavitational field given by g = g, whee g is constant and is the position vecto of the field point elative to the cente of the sphee. This
More informationON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION. 1. Introduction. 1 r r. r k for every set E A, E \ {0},
ON INDEPENDENT SETS IN PURELY ATOMIC PROBABILITY SPACES WITH GEOMETRIC DISTRIBUTION E. J. IONASCU and A. A. STANCU Abstact. We ae inteested in constucting concete independent events in puely atomic pobability
More information