Concept of Curve Fitting Difference with Interpolation


 Blaise Baldwin White
 1 years ago
 Views:
Transcription
1 Curve Fitting
2 Content Concept of Curve Fitting Difference with Interpolation Estimation of Linear Parameters by Least Squares Curve Fitting by Polynomial Least Squares Estimation of Nonlinear Parameters by Least Squares Curve Fitting Using Excel
3 Concept of Curve Fitting Difference with Interpolation One of the most extensively used techniques in numerical methods is the estimation of parameters by the principle of least squares. This technique is employed to drive information about the functional relation between xx and yy, assuming such a relation exists, from a set of data pairs xx ii, yy ii (ii = 00, nn). The estimation of parameters by least squares causes an smoothing to a given set of data and eliminates, to some degree, errors in observation, measurement, recording, transmission and conversion, as well as other types of random errors which may have been introduced in the data. This is one of the most important functions of the principle of least squares, and one which distinguishes it from the interpolation. (Recall that an interpolating polynomial exactly fills all data points used, such that any error in the data will be held in the interpolation).
4 Concept of Curve Fitting Difference with Interpolation There are two distinct but related categories of techniques based on the principle of least squares: 1. The estimation of linear parameter by least squares 2. The estimation of nonlinear parameter by least squares
5 Estimation of Linear Parameters by Least Squares Given a set of data pairs xx ii, yy ii (ii = 00, nn), which can be interpreted as the measured coordinates of the coordinates of the points on the graph of yy = ff(xx) values, let us assume that the unknown function ff(xx) can be approximated by a linear combination of suitably chosen functions ff 00 xx, ff 11 xx,, ff mm (xx) of the form FF xx = aa 00 ff 00 xx + aa 11 ff 11 xx + + aa mm ff mm xx where the unknown coefficients aa 00, aa 11, aa 22,, aa mm are independent parameters to be determined, and mm < nn. The difference between the approximating function value FF(xx ii ) and the corresponding data values yy ii is called a residual rr ii and is defined by the relation rr ii = FF xx ii yy ii (ii = 00, nn) We have then a residual rr ii for each data pair xx ii, yy ii (ii = 00, nn).
6 Estimation of Linear Parameters by Least Squares The function FF(xx) that best approximates the given set of data in a least squares sense is the linear combination aa 00 ff 00 xx + aa 11 ff 11 xx + + aa mm ff mm (xx) of functions ff kk (xx) that produces the minimum value of the sum QQ of the squared residuals QQ = rr 22 ii ii ii FF xx ii yy ii 22 Rewriting the above equation in its expanded form, we get QQ = ii aa 00 ff 00 xx + aa 11 ff 11 xx + + aa mm ff mm xx yy ii 22 Considering the parameters aa 00, aa 11, aa 22,, aa mm independent variables of the function QQ, minimizing the sum, that is, differentiating and equating to zero, we obtain QQ aa kk 22 ii FF xx ii yy ii xx ii aa kk = 00 (kk = 00, mm)
7 Estimation of Linear Parameters by Least Squares The constrains imposed by this equation form a system of mm + 11 independent algebraic equations (called normal equations) which are linear in mm + 11 parameters aa kk (kk = 00, mm). The solution (aa 00, aa 11, aa 22,, aa mm ) of this system of normal equations is that set of parameters aa kk which produces the minimum sum of squared residuals. The normal equations can be reduced to a form suitable for computation by the following steps: First, substitute the relation FF(xx ii ) = ff aa kk (xx ii ) kk and express FF(xx ii ) equation in its expanded form QQ 22 aa aa 00 ff 00 xx + aa 11 ff 11 xx + + aa mm ff mm xx yy ii ff kk (xx ii ) = 00 (kk = 00, mm) kk ii
8 Estimation of Linear Parameters by Least Squares We can rewrite the normal equations in the form
9 Estimation of Linear Parameters by Least Squares The mm + 11 normal equations obtained from the above equation, when evaluated for kk = 00, kk = 11, kk = 22,, kk = mm, can be written as a single matrix equation of the form where all summations are over ii (ii = 00, nn). The solution (aa 00, aa 11, aa 22,, aa mm ) of the matrix of the normal equations is the set of parameters aa kk (kk = 00, mm) that minimizes the sum QQ of the squared residuals.
10 Curve Fitting by Polynomial Least Squares Let us now consider the special case of the leastsquares estimation of linear parameters in which the functions ff kk xx = xx kk (kk = 00, mm) so the FF(xx) equation becomes an mthdegree polynomial, mm < nn, denoted PP mm (xx) of the form PP mm xx = aa 00 xx 00 + aa 11 xx 11 + aa 22 xx aa mm xx mm That is, we will approximate function yy = ff(xx) by an mthdegree polynomial PP mm (xx) over the range of data pairs xx ii, yy ii (ii = 00, nn). The parameters aa 00, aa 11,, aa mm are then determined such that QQ = rr 22 ii ff ii PP mm xx ii yy ii 22 is a minimum. That is, we will fit an mthdegree polynomial curve to the data in a leastsquares sense, as defined earlier. This special case of the estimation of linear parameters is commonly referred to as polynomial curvesmoothing by least squares.
11 Curve Fitting by Polynomial Least Squares The normal equations that determine aa 00, aa 11,, aa mm for this special case can be obtained directly by substituting xx ii kk (i.e., xx ii to the kth power) for ff kk (xx ii ) obtained in the above matrix equation. This substitution gives us These normal equations for the leastsquares polynomial can then be written as
12
13 Estimation of Nonlinear Parameters by Least Squares You can use the method previously seen with nonlinear functions, linearizing the function. For example if we have the exponential function We can linearize the function as follows:
14 Estimation of Nonlinear Parameters by Least Squares For this case, the matrix representation of nominal equations is as follows:
15
16 Curve Fitting Using Excel
17 Curve Fitting Using Excel
18 Homework 8 (Individual) 1. Given the following data set, fit a quadratic leastsquares polynomial (degree 2): x y Given the following data set, fit to an exponential function of the form yy = aaee bb by least squares: x y Consider for both problems 6 digits of precision.
19 Computer Program 7 (by team) Submit a computer program that compute the curve fitting of a set of data by the following methods: a) Polynomial Least Squares b) Nonlinear Parameters by Least Squares, exponential fit Hand over: Computational algorithm (printed) Source Code (printed and file) Executable (file)
20 Curve Fitting
Section 6: Quadratic Equations and Functions Part 2
Section 6: Quadratic Equations and Functions Part 2 Topic 1: Observations from a Graph of a Quadratic Function... 147 Topic 2: Nature of the Solutions of Quadratic Equations and Functions... 150 Topic
More informationLesson 19: Translating Functions
Student Outcomes Students recognize and use parent functions for linear, absolute value, quadratic, square root, and cube root functions to perform vertical and horizontal translations. They identify how
More informationSection 7: Exponential Functions
Topic 1: Geometric Sequences... 175 Topic 2: Exponential Functions... 178 Topic 3: Graphs of Exponential Functions  Part 1... 182 Topic 4: Graphs of Exponential Functions  Part 2... 185 Topic 5: Growth
More informationLesson 10. Homework Problem Set Sample Solutions. then Print True else Print False End if. False False True False False False
Homework Problem Set Sample Solutions 1. Perform the instructions in the following programming code as if you were a computer and your paper were the computer screen. Declare xx integer For all xx from
More information1. Answer: x or x. Explanation Set up the two equations, then solve each equation. x. Check
Thinkwell s Placement Test 5 Answer Key If you answered 7 or more Test 5 questions correctly, we recommend Thinkwell's Algebra. If you answered fewer than 7 Test 5 questions correctly, we recommend Thinkwell's
More information3 Nonlinear Regression
CSC 4 / CSC D / CSC C 3 Sometimes linear models are not sufficient to capture the realworld phenomena, and thus nonlinear models are necessary. In regression, all such models will have the same basic
More informationLinear Programming with Bounds
Chapter 481 Linear Programming with Bounds Introduction Linear programming maximizes (or minimizes) a linear objective function subject to one or more constraints. The technique finds broad use in operations
More informationComputational Physics PHYS 420
Computational Physics PHYS 420 Dr Richard H. Cyburt Assistant Professor of Physics My office: 402c in the Science Building My phone: (304) 3846006 My email: rcyburt@concord.edu My webpage: www.concord.edu/rcyburt
More informationTABLE OF CONTENTS. 3 Intro. 4 Foursquare Logo. 6 Foursquare Icon. 9 Colors. 10 Copy & Tone Of Voice. 11 Typography. 13 Crown Usage.
BRANDBOOK TABLE OF CONTENTS 3 Intro 4 Foursquare Logo 6 Foursquare Icon 9 Colors 10 Copy & Tone Of Voice 11 Typography 13 Crown Usage 14 Badge Usage 15 Iconography 16 Trademark Guidelines 2011 FOURSQUARE
More information9.1: GRAPHING QUADRATICS ALGEBRA 1
9.1: GRAPHING QUADRATICS ALGEBRA 1 OBJECTIVES I will be able to graph quadratics: Given in Standard Form Given in Vertex Form Given in Intercept Form What does the graph of a quadratic look like? https://www.desmos.com/calculator
More informationRadical Functions. Attendance Problems. Identify the domain and range of each function.
Page 1 of 12 Radical Functions Attendance Problems. Identify the domain and range of each function. 1. f ( x) = x 2 + 2 2. f ( x) = 3x 3 Use the description to write the quadratic function g based on the
More informationSee the course website for important information about collaboration and late policies, as well as where and when to turn in assignments.
COS Homework # Due Tuesday, February rd See the course website for important information about collaboration and late policies, as well as where and when to turn in assignments. Data files The questions
More informationAn Adaptive Stencil Linear Deviation Method for Wave Equations
211 An Adaptive Stencil Linear Deviation Method for Wave Equations Kelly Hasler Faculty Sponsor: Robert H. Hoar, Department of Mathematics ABSTRACT Wave Equations are partial differential equations (PDEs)
More informationBRANDING AND STYLE GUIDELINES
BRANDING AND STYLE GUIDELINES INTRODUCTION The Dodd family brand is designed for clarity of communication and consistency within departments. Bold colors and photographs are set on simple and clean backdrops
More information3x 2 + 7x + 2. A 86 Factor. Step 1. Step 3 Step 4. Step 2. Step 1 Step 2 Step 3 Step 4
A 86 Factor. Step 1 3x 2 + 7x + 2 Step 2 Step 3 Step 4 3x 2 + 7x + 2 3x 2 + 7x + 2 Step 1 Step 2 Step 3 Step 4 Factor. 1. 3x 2 + 4x +1 = 2. 3x 2 +10x + 3 = 3. 3x 2 +13x + 4 = A 86 Name BDFM? Why? Factor.
More information[1] CURVE FITTING WITH EXCEL
1 Lecture 04 February 9, 2010 Tuesday Today is our third Excel lecture. Our two central themes are: (1) curvefitting, and (2) linear algebra (matrices). We will have a 4 th lecture on Excel to further
More informationPropositional Calculus: Boolean Algebra and Simplification. CS 270: Mathematical Foundations of Computer Science Jeremy Johnson
Propositional Calculus: Boolean Algebra and Simplification CS 270: Mathematical Foundations of Computer Science Jeremy Johnson Propositional Calculus Topics Motivation: Simplifying Conditional Expressions
More informationThe ABC s of Web Site Evaluation
Aa Bb Cc Dd Ee Ff Gg Hh Ii Jj Kk Ll Mm Nn Oo Pp Qq Rr Ss Tt Uu Vv Ww Xx Yy Zz The ABC s of Web Site Evaluation by Kathy Schrock Digital Literacy by Paul Gilster Digital literacy is the ability to understand
More informationLecture 5. If, as shown in figure, we form a right triangle With P1 and P2 as vertices, then length of the horizontal
Distance; Circles; Equations of the form Lecture 5 y = ax + bx + c In this lecture we shall derive a formula for the distance between two points in a coordinate plane, and we shall use that formula to
More informationSketching graphs of polynomials
Sketching graphs of polynomials We want to draw the graphs of polynomial functions y = f(x). The degree of a polynomial in one variable x is the highest power of x that remains after terms have been collected.
More informationZeroInflated Poisson Regression
Chapter 329 ZeroInflated Poisson Regression Introduction The zeroinflated Poisson (ZIP) regression is used for count data that exhibit overdispersion and excess zeros. The data distribution combines
More informationREGULAR GRAPHS OF GIVEN GIRTH. Contents
REGULAR GRAPHS OF GIVEN GIRTH BROOKE ULLERY Contents 1. Introduction This paper gives an introduction to the area of graph theory dealing with properties of regular graphs of given girth. A large portion
More informationAn introduction to interpolation and splines
An introduction to interpolation and splines Kenneth H. Carpenter, EECE KSU November 22, 1999 revised November 20, 2001, April 24, 2002, April 14, 2004 1 Introduction Suppose one wishes to draw a curve
More informationMath 96Radicals #1 Simplify; Combinepage 1
Simplify; Combinepage 1 Part A Number Systems a. Whole Numbers = {0, 1, 2, 3,...} b. Integers = whole numbers and their opposites = {..., 3, 2, 1, 0, 1, 2, 3,...} c. Rational Numbers = quotient of integers
More informationQuadratic Functions CHAPTER. 1.1 Lots and Projectiles Introduction to Quadratic Functions p. 31
CHAPTER Quadratic Functions Arches are used to support the weight of walls and ceilings in buildings. Arches were first used in architecture by the Mesopotamians over 4000 years ago. Later, the Romans
More informationCOMP Logic for Computer Scientists. Lecture 23
COMP 1002 Logic for Computer cientists Lecture 23 B 5 2 J Admin stuff Assignment 3 extension Because of the power outage, assignment 3 now due on Tuesday, March 14 (also 7pm) Assignment 4 to be posted
More informationChapter 5. Radicals. Lesson 1: More Exponent Practice. Lesson 2: Square Root Functions. Lesson 3: Solving Radical Equations
Chapter 5 Radicals Lesson 1: More Exponent Practice Lesson 2: Square Root Functions Lesson 3: Solving Radical Equations Lesson 4: Simplifying Radicals Lesson 5: Simplifying Cube Roots This assignment is
More informationA Short SVM (Support Vector Machine) Tutorial
A Short SVM (Support Vector Machine) Tutorial j.p.lewis CGIT Lab / IMSC U. Southern California version 0.zz dec 004 This tutorial assumes you are familiar with linear algebra and equalityconstrained optimization/lagrange
More informationCIS 580, Machine Perception, Spring 2014: Assignment 4 Due: Wednesday, April 10th, 10:30am (use turnin)
CIS 580, Machine Perception, Spring 2014: Assignment 4 Due: Wednesday, April 10th, 10:30am (use turnin) Solutions (hand calculations, plots) have to be submitted electronically as a single pdf file using
More informationFinite Math  Jterm Homework. Section Inverse of a Square Matrix
Section.577, 78, 79, 80 Finite Math  Jterm 017 Lecture Notes  1/19/017 Homework Section.69, 1, 1, 15, 17, 18, 1, 6, 9, 3, 37, 39, 1,, 5, 6, 55 Section 5.19, 11, 1, 13, 1, 17, 9, 30 Section.5  Inverse
More informationPolynomial Functions Graphing Investigation Unit 3 Part B Day 1. Graph 1: y = (x 1) Graph 2: y = (x 1)(x + 2) Graph 3: y =(x 1)(x + 2)(x 3)
Part I: Polynomial Functions when a = 1 Directions: Polynomial Functions Graphing Investigation Unit 3 Part B Day 1 1. For each set of factors, graph the zeros first, then use your calculator to determine
More information7 Fractions. Number Sense and Numeration Measurement Geometry and Spatial Sense Patterning and Algebra Data Management and Probability
7 Fractions GRADE 7 FRACTIONS continue to develop proficiency by using fractions in mental strategies and in selecting and justifying use; develop proficiency in adding and subtracting simple fractions;
More informationPerspective Mappings. Contents
Perspective Mappings David Eberly, Geometric Tools, Redmond WA 98052 https://www.geometrictools.com/ This work is licensed under the Creative Commons Attribution 4.0 International License. To view a copy
More informationHw 4 Due Feb 22. D(fg) x y z (
Hw 4 Due Feb 22 2.2 Exercise 7,8,10,12,15,18,28,35,36,46 2.3 Exercise 3,11,39,40,47(b) 2.4 Exercise 6,7 Use both the direct method and product rule to calculate where f(x, y, z) = 3x, g(x, y, z) = ( 1
More informationProject 2: How Parentheses and the Order of Operations Impose Structure on Expressions
MAT 51 Wladis Project 2: How Parentheses and the Order of Operations Impose Structure on Expressions Parentheses show us how things should be grouped together. The sole purpose of parentheses in algebraic
More informationPSY 9556B (Feb 5) Latent Growth Modeling
PSY 9556B (Feb 5) Latent Growth Modeling Fixed and random word confusion Simplest LGM knowing how to calculate dfs How many time points needed? Power, sample size Nonlinear growth quadratic Nonlinear growth
More informationI.D. GUIDE Kentucky Campus Version 1.0
I.D. GUIDE 20082009 Kentucky Campus Version 1.0 introduction to the identity guidelines Summer 2008 Dear Asbury College community, As we continue our mission of academic excellence and spiritual vitality
More information8/27/2016. ECE 120: Introduction to Computing. Graphical Illustration of Modular Arithmetic. Representations Must be Unambiguous
University of Illinois at UrbanaChampaign Dept. of Electrical and Computer Engineering ECE 120: Introduction to Computing Signed Integers and 2 s Complement Strategy: Use Common Hardware for Two Representations
More informationCurves and Surfaces Computer Graphics I Lecture 9
15462 Computer Graphics I Lecture 9 Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.110.6] February 19, 2002 Frank Pfenning Carnegie
More informationThe PreImage Problem in Kernel Methods
The PreImage Problem in Kernel Methods James Kwok Ivor Tsang Department of Computer Science Hong Kong University of Science and Technology Hong Kong The PreImage Problem in Kernel Methods ICML2003 1
More informationAlgebra 45 Study Guide: Direct Variation (pp ) Page! 1 of! 9
Page! 1 of! 9 Attendance Problems. Solve for y. 1. 3 + y = 2x 2. 6x = 3y 3. Write an equation that describes the relationship. Solve for x. 3 4.! 5.! 5 = x 6 15 2 = 1.5 x I can identify, write, and graph
More informationAlgebra. Chapter 4: FUNCTIONS. Name: Teacher: Pd:
Algebra Chapter 4: FUNCTIONS Name: Teacher: Pd: Table of Contents Day1: Chapter 41: Relations SWBAT: (1) Identify the domain and range of relations and functions (2) Match simple graphs with situations
More informationLinear and Quadratic Least Squares
Linear and Quadratic Least Squares Prepared by Stephanie Quintal, graduate student Dept. of Mathematical Sciences, UMass Lowell in collaboration with Marvin Stick Dept. of Mathematical Sciences, UMass
More informationIntroduction to Rational Functions Group Activity 5 STEM Project Week #8. AC, where D = dosage for a child, A = dosage for an
MLC at Boise State 013 Defining a Rational Function Introduction to Rational Functions Group Activity 5 STEM Project Week #8 f x A rational function is a function of the form, where f x and g x are polynomials
More information102 Circles. Warm Up Lesson Presentation Lesson Quiz. Holt Algebra2 2
102 Circles Warm Up Lesson Presentation Lesson Quiz Holt Algebra2 2 Warm Up Find the slope of the line that connects each pair of points. 1. (5, 7) and ( 1, 6) 1 6 2. (3, 4) and ( 4, 3) 1 Warm Up Find
More informationIn the Name of God. Lecture 17: ANFIS Adaptive NetworkBased Fuzzy Inference System
In the Name of God Lecture 17: ANFIS Adaptive NetworkBased Fuzzy Inference System Outline ANFIS Architecture Hybrid Learning Algorithm Learning Methods that CrossFertilize ANFIS and RBFN ANFIS as a universal
More informationTranont Mission Statement. Tranont Vision Statement. Change the world s economy, one household at a time.
STYLE GUIDE Tranont Mission Statement Change the world s economy, one household at a time. Tranont Vision Statement We offer individuals world class financial education and training, financial management
More informationSummer Review for Students Entering PreCalculus with Trigonometry. TI84 Plus Graphing Calculator is required for this course.
1. Using Function Notation and Identifying Domain and Range 2. Multiplying Polynomials and Solving Quadratics 3. Solving with Trig Ratios and Pythagorean Theorem 4. Multiplying and Dividing Rational Expressions
More informationTwoStage Least Squares
Chapter 316 TwoStage Least Squares Introduction This procedure calculates the twostage least squares (2SLS) estimate. This method is used fit models that include instrumental variables. 2SLS includes
More informationMAT Business Calculus  Quick Notes
MAT 136  Business Calculus  Quick Notes Last Updated: 4/3/16 Chapter 2 Applications of Differentiation Section 2.1 Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs THE FIRSTDERIVATIVE
More informationLEAP 2025 Algebra I Practice Test Answer Key
LEAP 2025 Algebra I Practice Test Answer This document contains the answer keys and rubrics for the LEAP 2025 Algebra I Practice Test. # Type Value (Points) Session 1a 1 I 1 A1: AAPR.B.3 *order does not
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Lecture  36
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Lecture  36 In last class, we have derived element equations for two d elasticity problems
More informationConvert Local Coordinate Systems to Standard Coordinate Systems
BENTLEY SYSTEMS, INC. Convert Local Coordinate Systems to Standard Coordinate Systems Using 2D Conformal Transformation in MicroStation V8i and Bentley Map V8i Jim McCoy P.E. and Alain Robert 4/18/2012
More informationThe Vertex Cover Problem. Shangqi Wu Presentation of CS 525 March 11 th, 2016
The Vertex Cover Problem Shangqi Wu Presentation of CS 525 March 11 th, 2016 VertexCover Problem Definition If G is an undirected graph, a vertex cover of G is a subset of nodes where every edge of G
More information1.2 Roundoff Errors and Computer Arithmetic
1.2 Roundoff Errors and Computer Arithmetic 1 In a computer model, a memory storage unit word is used to store a number. A word has only a finite number of bits. These facts imply: 1. Only a small set
More informationx = 12 x = 12 1x = 16
2.2  The Inverse of a Matrix We've seen how to add matrices, multiply them by scalars, subtract them, and multiply one matrix by another. The question naturally arises: Can we divide one matrix by another?
More informationModule 1: Introduction to Finite Difference Method and Fundamentals of CFD Lecture 6:
file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_1.htm 1 of 1 6/20/2012 12:24 PM The Lecture deals with: ADI Method file:///d:/chitra/nptel_phase2/mechanical/cfd/lecture6/6_2.htm 1 of 2 6/20/2012
More informationf xx (x, y) = 6 + 6x f xy (x, y) = 0 f yy (x, y) = y In general, the quantity that we re interested in is
1. Let f(x, y) = 5 + 3x 2 + 3y 2 + 2y 3 + x 3. (a) Final all critical points of f. (b) Use the second derivatives test to classify the critical points you found in (a) as a local maximum, local minimum,
More informationFirst of all, we need to know what it means for a parameterize curve to be differentiable. FACT:
CALCULUS WITH PARAMETERIZED CURVES In calculus I we learned how to differentiate and integrate functions. In the chapter covering the applications of the integral, we learned how to find the length of
More information[100] 091 News, Tutorial by Dec. 10, 2012 =======================================
[100] 091 revised on 2012.12.10 cemmath The Simple is the Best News Dec. 10, 2012 ======================================= Cemmath 2.22 (a new name of Msharpmath) is newly upgraded. indefinite integrals
More informationMarcy Mathworks Answers Punchline Radical Expressions
Marcy Mathworks Answers Punchline Free PDF ebook Download: Marcy Mathworks Answers Punchline Download or Read Online ebook marcy mathworks answers punchline radical expressions in PDF Format From The Best
More informationComputer Graphics. Curves and Surfaces. Hermite/Bezier Curves, (B)Splines, and NURBS. By Ulf Assarsson
Computer Graphics Curves and Surfaces Hermite/Bezier Curves, (B)Splines, and NURBS By Ulf Assarsson Most of the material is originally made by Edward Angel and is adapted to this course by Ulf Assarsson.
More informationPrecalculus Notes: Unit 7 Systems of Equations and Matrices
Date: 7.1, 7. Solving Systems of Equations: Graphing, Substitution, Elimination Syllabus Objectives: 8.1 The student will solve a given system of equations or system of inequalities. Solution of a System
More informationFinal Exam Review Algebra Semester 1
Final Exam Review Algebra 015016 Semester 1 Name: Module 1 Find the inverse of each function. 1. f x 10 4x. g x 15x 10 Use compositions to check if the two functions are inverses. 3. s x 7 x and t(x)
More informationLacunary Interpolation Using Quartic BSpline
General Letters in Mathematic, Vol. 2, No. 3, June 2017, pp. 129137 eissn 25199277, pissn 25199269 Available online at http:\\ www.refaad.com Lacunary Interpolation Using Quartic BSpline 1 Karwan
More informationMultiplying and Dividing Rational Expressions
Page 1 of 14 Multiplying and Dividing Rational Expressions Attendance Problems. Simplify each expression. Assume all variables are nonzero. x 6 y 2 1. x 5 x 2 2. y 3 y 3 3. 4. x 2 y 5 Factor each expression.
More informationCS Data Structures and Algorithm Analysis
CS 483  Data Structures and Algorithm Analysis Lecture VI: Chapter 5, part 2; Chapter 6, part 1 R. Paul Wiegand George Mason University, Department of Computer Science March 8, 2006 Outline 1 Topological
More informationMIDI CPU Firmware V User Manual
MIDI CPU Firmware V..2 MIDI CPU Firmware Version.2 User Manual Updated 2353 Additional documentation available at: http://highlyliquid.com/support/ 23 Sonarcana LLC Page / 55 MIDI CPU Firmware V..2 Table
More informationOptimizations and Lagrange Multiplier Method
Introduction Applications Goal and Objectives Reflection Questions Once an objective of any real world application is well specified as a function of its control variables, which may subject to a certain
More informationMathematics. Jaehyun Park. CS 97SI Stanford University. June 29, 2015
Mathematics Jaehyun Park CS 97SI Stanford University June 29, 2015 Outline Algebra Number Theory Combinatorics Geometry Algebra 2 Sum of Powers n k=1 k 3 k 2 = 1 n(n + 1)(2n + 1) 6 = ( k ) 2 = ( 1 2 n(n
More informationCopy Material. Geometry Unit 1. Congruence, Proof, and Constructions. Eureka Math. Eureka Math
Copy Material Geometry Unit 1 Congruence, Proof, and Constructions Eureka Math Eureka Math Lesson 1 Lesson 1: Construct an Equilateral Triangle We saw two different scenarios where we used the construction
More informationRobust Linear Regression (Passing Bablok MedianSlope)
Chapter 314 Robust Linear Regression (Passing Bablok MedianSlope) Introduction This procedure performs robust linear regression estimation using the PassingBablok (1988) medianslope algorithm. Their
More informationGrade 9 Math Terminology
Unit 1 Basic Skills Review BEDMAS a way of remembering order of operations: Brackets, Exponents, Division, Multiplication, Addition, Subtraction Collect like terms gather all like terms and simplify as
More informationAn interesting related problem is Buffon s Needle which was first proposed in the mid1700 s.
Using Monte Carlo to Estimate π using Buffon s Needle Problem An interesting related problem is Buffon s Needle which was first proposed in the mid1700 s. Here s the problem (in a simplified form). Suppose
More informationHomework #6 Brief Solutions 2011
Homework #6 Brief Solutions %page 95 problem 4 data=[,;,;,;4,] data =   4 xk=data(:,);yk=data(:,);s=csfit(xk,yk,,) %Using the program to find the coefficients S =.456 .456 .. .5.9 .5484. .58.87.
More informationAssessment of different model selection criteria by generated experimental data
Assessment of different model selection criteria by generated experimental RADOSLAV MAVREVSKI * Department of Electrical Engineering, Electronics and Automatics SouthWest University "Neofit Rilski" 66
More informationUnifi 45 Projector Retrofit Kit for SMART Board 580 and 560 Interactive Whiteboards
Unifi 45 Projector Retrofit Kit for SMRT oard 580 and 560 Interactive Whiteboards 72 (182.9 cm) 60 (152.4 cm) S580 S560 Cautions, warnings and other important product information are contained in document
More informationTherefore, after becoming familiar with the Matrix Method, you will be able to solve a system of two linear equations in four different ways.
Grade 9 IGCSE A1: Chapter 9 Matrices and Transformations Materials Needed: Straightedge, Graph Paper Exercise 1: Matrix Operations Matrices are used in Linear Algebra to solve systems of linear equations.
More informationChapter 3: Rate Laws Excel Tutorial on Fitting logarithmic data
Chapter 3: Rate Laws Excel Tutorial on Fitting logarithmic data The following table shows the raw data which you need to fit to an appropriate equation k (s 1 ) T (K) 0.00043 312.5 0.00103 318.47 0.0018
More informationCourse of study Algebra Introduction: Algebra 12 is a course offered in the Mathematics Department. The course will be primarily taken by
Course of study Algebra 12 1. Introduction: Algebra 12 is a course offered in the Mathematics Department. The course will be primarily taken by students in Grades 9 and 10, but since all students must
More informationCHAPTER 3 A TIMEDEPENDENT kshortest PATH ALGORITHM FOR ATIS APPLICATIONS
CHAPTER 3 A TIMEDEPENDENT kshortest PATH ALGORITHM FOR ATIS APPLICATIONS 3.1. Extension of a Static ksp Algorithm to the TimeDependent Case Kaufman and Smith [1993] showed that under the consistency
More informationBezier Curves, BSplines, NURBS
Bezier Curves, BSplines, NURBS Example Application: Font Design and Display Curved objects are everywhere There is always need for: mathematical fidelity high precision artistic freedom and flexibility
More informationMath 113 Exam 1 Practice
Math Exam Practice January 6, 00 Exam will cover sections 6.6.5 and 7.7.5 This sheet has three sections. The first section will remind you about techniques and formulas that you should know. The second
More informationCSC Design and Analysis of Algorithms. Lecture 8. Transform and Conquer II Algorithm Design Technique. Transform and Conquer
CSC 301 Design and Analysis of Algorithms Lecture Transform and Conquer II Algorithm Design Technique Transform and Conquer This group of techniques solves a problem by a transformation to a simpler/more
More informationGeneralized Additive Model
Generalized Additive Model by Huimin Liu Department of Mathematics and Statistics University of Minnesota Duluth, Duluth, MN 55812 December 2008 Table of Contents Abstract... 2 Chapter 1 Introduction 1.1
More informationOperations and Properties
. Operations and Properties. OBJECTIVES. Represent the four arithmetic operations using variables. Evaluate expressions using the order of operations. Recognize and apply the properties of addition 4.
More informationEureka Math. Grade 7, Module 6. Teacher Edition
A Story of Units Eureka Math Grade 7, Module 6 Teacher Edition Published by the nonprofit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, sold, or commercialized, in whole
More informationTutorial Four: Linear Regression
Tutorial Four: Linear Regression Imad Pasha Chris Agostino February 25, 2015 1 Introduction When looking at the results of experiments, it is critically important to be able to fit curves to scattered
More informationThe Piecewise Regression Model as a Response Modeling Tool
NESUG 7 The Piecewise Regression Model as a Response Modeling Tool Eugene Brusilovskiy University of Pennsylvania Philadelphia, PA Abstract The general problem in response modeling is to identify a response
More informationBinary Adders: Half Adders and Full Adders
Binary Adders: Half Adders and Full Adders In this set of slides, we present the two basic types of adders: 1. Half adders, and 2. Full adders. Each type of adder functions to add two binary bits. In order
More information3.2  Interpolation and Lagrange Polynomials
3.  Interpolation and Lagrange Polynomials. Polynomial Interpolation: Problem: Givenn pairs of data points x i, y i,wherey i fx i, i 0,,...,n for some function fx, we want to find a polynomial P x of
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationTRIGONOMETRY. T.1 Angles and Degree Measure
1 TRIGONOMETRY Trigonometry is the branch of mathematics that studies the relations between the sides and angles of triangles. The word trigonometry comes from the Greek trigōnon (triangle) and metron
More informationAssignment 1 (concept): Solutions
CS10b Data Structures and Algorithms Due: Thursday, January 0th Assignment 1 (concept): Solutions Note, throughout Exercises 1 to 4, n denotes the input size of a problem. 1. (10%) Rank the following functions
More informationMethod of analysis. Bởi: Sy Hien Dinh
Method of analysis Bởi: Sy Hien Dinh INTRODUCTION Having understood the fundamental laws of circuit theory (Ohm s law and Kirchhhoff s laws), we are now prepared to apply to develop two powerful techniques
More informationLesson 1: Analyzing Quadratic Functions
UNIT QUADRATIC FUNCTIONS AND MODELING Lesson 1: Analyzing Quadratic Functions Common Core State Standards F IF.7 F IF.8 Essential Questions Graph functions expressed symbolically and show key features
More informationPre and PostProcessing for Video Compression
Whitepaper submitted to Mozilla Research Pre and PostProcessing for Video Compression Aggelos K. Katsaggelos AT&T Professor Department of Electrical Engineering and Computer Science Northwestern University
More informationXL2B: Excel2013: Model Trendline Multi 1/24/2018 V0M. Process Advice.
XL2B: Excel2013: Model Trendline Multi 1/24/2018 V0M 1 Model Using Trendline Multiple Models in Excel 2013 by Milo Schield Member: International Statistical Institute US Rep: International Statistical
More informationNumerical Algorithms
Chapter 10 Slide 464 Numerical Algorithms Slide 465 Numerical Algorithms In textbook do: Matrix multiplication Solving a system of linear equations Slide 466 Matrices A Review An n m matrix Column a 0,0
More informationEdge and local feature detection  2. Importance of edge detection in computer vision
Edge and local feature detection Gradient based edge detection Edge detection by function fitting Second derivative edge detectors Edge linking and the construction of the chain graph Edge and local feature
More information