Some notes on Chapter 8: Polynomial and Piecewise-polynomial Interpolation
|
|
- Moris Collins
- 5 years ago
- Views:
Transcription
1 Some notes on Chapter 8: Polynomial and Piecewise-polynomial Interpolation See your notes. 1. Lagrange Interpolation (8.2) 1
2 2. Newton Interpolation (8.3) different form of the same polynomial as Lagrange (!!! the polynomial is unique!) The set of polynomials (1) 1, x x 0, (x x 0 )(x x 1 ),..., (x x 0 )... (x x n 1 ) form a basis for the set of all polynomials of degree n: linearly indep., any polyn. of deg. n can be written as linear combination: (2) P n (x) = a 0 + a 1 (x x 0 ) + a 2 (x x 0 )(x x 1 ) + + a n (x x 0 )... (x x n 1 ) where a 0, a 1,..., a n are constants to be determined from p n (x i ) = f(x i ) at x 0 : a 0 = P n (x 0 ) = f(x 0 ) at x 1 : f(x 0 ) + a 1 (x x 0 ) = P n (x 1 ) = f(x 1 ) a 1 = f(x 1) f(x 0 ) = f(x 1) a 0 x 1 x 0 x 1 x 0 at x 2 : a 2 = f(x 2) a 0 a 1 (x 2 x 0 ) (x 2 x 0 )(x 2 x 1 ) Notice: a 2 = a 2 (x 0, x 1, x 2, a 0, a 1, f(x 2 )), adding a new point x 3 does not change the previously calculated a i Divided differences 0-th: f[x i ] = f(x i ) 1-st: f[x i, x j ] = f[x j] f[x i ] x j x i f (ξ), ξ [x i, x j ] 2-nd: f[x i, x j, x k ] = f[x j, x k ] f[x i, x j ] x k x i f (ξ), ξ [x i, x k ] k-th: f[x i,..., x i+k ] = f[x i+1,..., x i+k ] f[x i,..., x i+k 1 ] x i+k x i f (k) (ξ), ξ [x i, x i+k ] Difference analog for Tailor polynomial f(x) = f(ξ) + (x ξ)f (ξ) + 1 2! (x ξ)2 f (ξ) +... In formula for P n (x): a k = f[x 0,..., x k ] Newton s form of interpolation polynomial (N. dd.): (3) P n (x) = f[x 0 ] + n f[x 0,..., x k ](x x 0 )... (x x k 1 ) k=1 2
3 Lagrange: nodes are fixed, interpolate many diff. functions Newton: interpolate the same functions, increas # of nodes 3. Interpolation Error (8.4) r n (x) = f(x) P n (x) - error of interpolation/remainder Theorem. Let f(x) C n+1 [a, b], x k [a, b], k = 0,..., n. Then x [a, b] ξ(x) [a, b]: Alternative form. thus we have r n (x) = f (n+1) (ξ(x)) (n + 1)! ω(x), ω(x) = r n (x) = f[x 0,..., x n, x] ω(x), ω(x) = f[x 0,..., x n, x] = f (n+1) (ξ(x)) (n + 1)! n (x x k ) k=0 n (x x k ) k=0 Examples. f(x) = 1 x, nodes x 0 = 2, x 1 = 2.75, x 2 = 4 2nd Lagrange polynomial P (x) = 1 64 (x 2.75)(x 4) (x 2)(x 4) (x 2)(x 2.75) = 1 22 x x Determine the error form for this polynomial and the maximum error when the polynomial is used to approximate f(x) for x [2, 4]. Solution: Because f(x) = x 1, we have f (x) = x 2, f (x) = 2x 3, andf (x) = 6x 4. As a consequence, the second Lagrange polynomial has the error form f (ξ(x)) (x x 0 )(x x 1 )(x x 2 ) = (ξ(x)) 4 (x 2)(x 2.75)(x 4), 3! The maximum value of ξ(x) 4 (4 2) 2 = 1/16. We now need to deterthe maximum value on this interval of the absolute value of the polynomial Then critical points: ( 7, 25 ) and ( 7, Max. error: g(x) = (x 2)(x 2.75)(x 4) = x x x ) g (x) = 1 (3x 7)(2x 7) 2 f (ξ(x)) (x x 0 )(x x 1 )(x x 2 ) 1 3! =
4 4. Piecewise-polynomial interpolation: Cubic Splines? Is the interpolation project convergent? Namely, if # of nodes n, can we expect the interpolation error r n (x) = f(x) p n (x) 0? Answer: NO Why? High-degree polynomials (i.e. interpolations on large number of nodes) can oscillate erratically (they are too smooth), that is, a minor fluctuation over a small portion of the interval can induce large fluctuations over the entire range. (depends on the function and choice of nodes) What can we do? Reduce the smoothness of interpolant, i.e. consider piecewise-polynomial approximation. Namely, we divide the approximation interval into a collection of subintervals and construct a (generally) different approximating polynomial on each subinterval. I. Easiest: piecewise-linear, joining a set of data points (x i, f i ), i = 0,..., n with f i = f(x i ) by a series of straight lines (that s how Matlab creates continuous smooth graphs). : sharp endpoints, no differentiability, interpolating func. is not smooth (contradiction with physics) II. piecewise-quadratic interpolant - fitting one quadratic polynomial between each successive pair of nodes: interpolant is continious, its derivative is not 4
5 Figure 1. linear-piecewise inerpolation III. Cubic spline interpolation - piecewise cubic polynomial with two cont. derivatives. What the strange name spline is all about? Mathematical splines originated in the CAD software developed by the aircraft and automobile design industry in the late 1950s and early 1960s and were named after a special wooden or metal drafting tool used in the manual design of ship hulls: a spline. Those were used to manually interpolate the curve using the known points. 5
6 Figure 2. linear (red) and cubic (magenta) spline interpolation with 5 nodes Grid (set of nodes) on [a, b] a = x 0 < x 1 < < x n = b. Cubic spline s(x) corresponding to f(x) and x i : (1) s(x) is cubic polynomial, denoted s i (x) = a i + b i (x x i ) + c i (x x i ) 2 + d i (x x i ) 3 on each [x i 1, x i ], i = 1,..., n [4n unknowns] (2) s(x i ) = f(x i ) - condition of interpolation: s i (x i 1 ) = f(x i 1 ); s i (x i ) = f(x i ) i = 1,..., n [2n eq s] (3) s(x), s (x), s (x) are continuous on [a, b]: s i (x i ) = s i+1 (x i ), (follows from 2) s i(x i ) = s i+1(x i ), i = 1,..., n 1 [2n 2 eq s] s i (x i ) = s i+1(x i ), i = 1,..., n 1 [2n 2 eq s] Total: 4n unknowns vs. 4n 2 eq s: leaves 2 degrees of freedom for conditions on the endpoints: s (x 0 ) = s (x n ) = 0 (natural (or free) boundary) (zero curvature at end, thin rod); s (x 0 ) = f (x 0 ) and s (x n ) = f (x n ) (clamped boundary). Constructing spline: s i (x) = a i + b i (x x i ) + c i 2 (x x i ) 2 + d i (x x 6 i) 3 on [x i, x i+1 ], i = 0,..., n 1 why a i, b i, c i, d i : (4) (5) (6) thus s i(x) = b i + c i (x x i ) + d i 2 (x x i) 2 s i (x) = c i + d i (x x i ) s i (x) = d i a i = s i (x i ), b i = s i(x i ), c i = s i (x i ), d i = s i (x i ) 6
7 Interpolation conditions s(x i ) = f(x i ): a i = f(x i ) = f i, i = 0,..., n Continiuity of s(x): s i (x i ) = s i+1 (x i ), i = 1,..., n 1: a i = a i+1 + b i+1 (x i x i+1 ) + c i+1 2 (x i x i+1 ) 2 + d i+1 6 (x i x i+1 ) 3 Denoting h i = x i x i 1 h i b i h2 i 2 c i + h3 i 6 d i = f i f i 1, i = 1,..., n Continiuity of s (x): s i(x i ) = s i+1(x i ), i = 1,..., n 1: c i h i d i 2 h2 i = b i b i 1, i = 2,..., n Continiuity of s (x): s i (x i ) = s i+1(x i ), i = 1,..., n 1: d i h i = c i c i 1, i = 2,..., n Rewriting we get the three-diagonal system of linear equations: b i = h i 2 c i h2 i 6 d i f i f i 1 h i ; d i = c i c i 1 h i and ( fi+1 f i h i c i 1 + 2(h i + h i+1 )c i + h i+1 c i+1 = 6 h i+1 f i f i 1 h i i = 1,..., n 1, ), c 0 = c n = 0 [= s (x 0 ); = s (x n )= 0 for natural splines] 7
8 FINALLY: We get cubic spline: on each [x i, x i+1 ], i = 0,..., n 1 For h i = x i x i 1 s i (x) = a i + b i (x x i ) + c i 2 (x x i) 2 + d i 6 (x x i) 3 a i = f i, b i = h i 2 c i h2 i 6 d i f i f i 1 h i ; d i = c i c i 1 h i and where α 1 β β 1 α 2 β β n β n 2 α n 1 c 1 c 2. c n 1 = z 1 z 2. z n 1 ( fi+1 f i α i = 2(h i + h i+1 ); β i = h i+1, z i = 6 f ) i f i 1 h i+1 h i 8
9 Special case for uniformly distributed nodes and some adjustements for MATLAB The cubic spline s i (x) = a i + b i (x x i ) + c i 2 (x x i) 2 + d i 6 (x x i) 3 MATLAB: for k=1:n-1 x1 = x((x<=nodes(k+1))&(x>=nodes(k)))-nodes(k+1); Spline((x<=Nodes(k+1))&(x>=Nodes(k)))=a(k)+b(k)*x1+c(k)./2*x1. 2+ d(k)./6*x1. 3; end For h i = x i x i 1 (mesh step) we have a i = f i+1, b i = h ( c i h ) 2 3 d i + f i+1 f i ; d i = c i c i 1 h i h and linear system for c 1, c 2,..., c n 1 (remember, c 0 = c n = 0) 4h h c 1 h 4h h c h h 4h c n 1 = z 1 z 2. z n 1 where z i = 6 h (f i 2f i+1 + f i+2 ) 9
10 10
Cubic Splines MATH 375. J. Robert Buchanan. Fall Department of Mathematics. J. Robert Buchanan Cubic Splines
Cubic Splines MATH 375 J. Robert Buchanan Department of Mathematics Fall 2006 Introduction Given data {(x 0, f(x 0 )), (x 1, f(x 1 )),...,(x n, f(x n ))} which we wish to interpolate using a polynomial...
More information3.1 Interpolation and the Lagrange Polynomial
MATH 4073 Chapter 3 Interpolation and Polynomial Approximation Fall 2003 1 Consider a sample x x 0 x 1 x n y y 0 y 1 y n. Can we get a function out of discrete data above that gives a reasonable estimate
More informationSPLINE INTERPOLATION
Spline Background SPLINE INTERPOLATION Problem: high degree interpolating polynomials often have extra oscillations. Example: Runge function f(x = 1 1+4x 2, x [ 1, 1]. 1 1/(1+4x 2 and P 8 (x and P 16 (x
More informationJim 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 informationLecture Note 3: Interpolation and Polynomial Approximation. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 3: Interpolation and Polynomial Approximation Xiaoqun Zhang Shanghai Jiao Tong University Last updated: October 10, 2015 2 Contents 1.1 Introduction................................ 3 1.1.1
More informationCurve Fitting and Interpolation
Chapter 5 Curve Fitting and Interpolation 5.1 Basic Concepts Consider a set of (x, y) data pairs (points) collected during an experiment, Curve fitting: is a procedure to develop or evaluate mathematical
More informationInterpolation Theory
Numerical Analysis Massoud Malek Interpolation Theory The concept of interpolation is to select a function P (x) from a given class of functions in such a way that the graph of y P (x) passes through the
More information1 Piecewise Cubic Interpolation
Piecewise Cubic Interpolation Typically the problem with piecewise linear interpolation is the interpolant is not differentiable as the interpolation points (it has a kinks at every interpolation point)
More informationLecture Note 3: Polynomial Interpolation. Xiaoqun Zhang Shanghai Jiao Tong University
Lecture Note 3: Polynomial Interpolation Xiaoqun Zhang Shanghai Jiao Tong University Last updated: October 24, 2013 1.1 Introduction We first look at some examples. Lookup table for f(x) = 2 π x 0 e x2
More informationEmpirical Models Interpolation Polynomial Models
Mathematical Modeling Lia Vas Empirical Models Interpolation Polynomial Models Lagrange Polynomial. Recall that two points (x 1, y 1 ) and (x 2, y 2 ) determine a unique line y = ax + b passing them (obtained
More informationData Analysis-I. Interpolation. Soon-Hyung Yook. December 4, Soon-Hyung Yook Data Analysis-I December 4, / 1
Data Analysis-I Interpolation Soon-Hyung Yook December 4, 2015 Soon-Hyung Yook Data Analysis-I December 4, 2015 1 / 1 Table of Contents Soon-Hyung Yook Data Analysis-I December 4, 2015 2 / 1 Introduction
More informationScientific Computing
2301678 Scientific Computing Chapter 2 Interpolation and Approximation Paisan Nakmahachalasint Paisan.N@chula.ac.th Chapter 2 Interpolation and Approximation p. 1/66 Contents 1. Polynomial interpolation
More informationMaria Cameron Theoretical foundations. Let. be a partition of the interval [a, b].
Maria Cameron 1 Interpolation by spline functions Spline functions yield smooth interpolation curves that are less likely to exhibit the large oscillations characteristic for high degree polynomials Splines
More informationLectures 9-10: Polynomial and piecewise polynomial interpolation
Lectures 9-1: Polynomial and piecewise polynomial interpolation Let f be a function, which is only known at the nodes x 1, x,, x n, ie, all we know about the function f are its values y j = f(x j ), j
More informationCHAPTER 4. Interpolation
CHAPTER 4 Interpolation 4.1. Introduction We will cover sections 4.1 through 4.12 in the book. Read section 4.1 in the book on your own. The basic problem of one-dimensional interpolation is this: Given
More informationPolynomial Review Problems
Polynomial Review Problems 1. Find polynomial function formulas that could fit each of these graphs. Remember that you will need to determine the value of the leading coefficient. The point (0,-3) is on
More informationExam 2. Average: 85.6 Median: 87.0 Maximum: Minimum: 55.0 Standard Deviation: Numerical Methods Fall 2011 Lecture 20
Exam 2 Average: 85.6 Median: 87.0 Maximum: 100.0 Minimum: 55.0 Standard Deviation: 10.42 Fall 2011 1 Today s class Multiple Variable Linear Regression Polynomial Interpolation Lagrange Interpolation Newton
More informationQ 0 x if x 0 x x 1. S 1 x if x 1 x x 2. i 0,1,...,n 1, and L x L n 1 x if x n 1 x x n
. - Piecewise Linear-Quadratic Interpolation Piecewise-polynomial Approximation: Problem: Givenn pairs of data points x i, y i, i,,...,n, find a piecewise-polynomial Sx S x if x x x Sx S x if x x x 2 :
More informationChapter 4: Interpolation and Approximation. October 28, 2005
Chapter 4: Interpolation and Approximation October 28, 2005 Outline 1 2.4 Linear Interpolation 2 4.1 Lagrange Interpolation 3 4.2 Newton Interpolation and Divided Differences 4 4.3 Interpolation Error
More informationMA2501 Numerical Methods Spring 2015
Norwegian University of Science and Technology Department of Mathematics MA5 Numerical Methods Spring 5 Solutions to exercise set 9 Find approximate values of the following integrals using the adaptive
More informationInterpolation. Escuela de Ingeniería Informática de Oviedo. (Dpto. de Matemáticas-UniOvi) Numerical Computation Interpolation 1 / 34
Interpolation Escuela de Ingeniería Informática de Oviedo (Dpto. de Matemáticas-UniOvi) Numerical Computation Interpolation 1 / 34 Outline 1 Introduction 2 Lagrange interpolation 3 Piecewise polynomial
More informationINTERPOLATION. and y i = cos x i, i = 0, 1, 2 This gives us the three points. Now find a quadratic polynomial. p(x) = a 0 + a 1 x + a 2 x 2.
INTERPOLATION Interpolation is a process of finding a formula (often a polynomial) whose graph will pass through a given set of points (x, y). As an example, consider defining and x 0 = 0, x 1 = π/4, x
More informationLecture 10 Polynomial interpolation
Lecture 10 Polynomial interpolation Weinan E 1,2 and Tiejun Li 2 1 Department of Mathematics, Princeton University, weinan@princeton.edu 2 School of Mathematical Sciences, Peking University, tieli@pku.edu.cn
More informationNumerical interpolation, extrapolation and fi tting of data
Chapter 6 Numerical interpolation, extrapolation and fi tting of data 6.1 Introduction Numerical interpolation and extrapolation is perhaps one of the most used tools in numerical applications to physics.
More informationChapter 3 Interpolation and Polynomial Approximation
Chapter 3 Interpolation and Polynomial Approximation Per-Olof Persson persson@berkeley.edu Department of Mathematics University of California, Berkeley Math 128A Numerical Analysis Polynomial Interpolation
More informationMATH ASSIGNMENT 07 SOLUTIONS. 8.1 Following is census data showing the population of the US between 1900 and 2000:
MATH4414.01 ASSIGNMENT 07 SOLUTIONS 8.1 Following is census data showing the population of the US between 1900 and 2000: Years after 1900 Population in millions 0 76.0 20 105.7 40 131.7 60 179.3 80 226.5
More informationIn practice one often meets a situation where the function of interest, f(x), is only represented by a discrete set of tabulated points,
1 Interpolation 11 Introduction In practice one often meets a situation where the function of interest, f(x), is only represented by a discrete set of tabulated points, {x i, y i = f(x i ) i = 1 n, obtained,
More informationCubic Splines; Bézier Curves
Cubic Splines; Bézier Curves 1 Cubic Splines piecewise approximation with cubic polynomials conditions on the coefficients of the splines 2 Bézier Curves computer-aided design and manufacturing MCS 471
More informationChapter 1 Numerical approximation of data : interpolation, least squares method
Chapter 1 Numerical approximation of data : interpolation, least squares method I. Motivation 1 Approximation of functions Evaluation of a function Which functions (f : R R) can be effectively evaluated
More informationApplied Numerical Analysis Quiz #2
Applied Numerical Analysis Quiz #2 Modules 3 and 4 Name: Student number: DO NOT OPEN UNTIL ASKED Instructions: Make sure you have a machine-readable answer form. Write your name and student number in the
More information(0, 0), (1, ), (2, ), (3, ), (4, ), (5, ), (6, ).
1 Interpolation: The method of constructing new data points within the range of a finite set of known data points That is if (x i, y i ), i = 1, N are known, with y i the dependent variable and x i [x
More informationIntroductory Numerical Analysis
Introductory Numerical Analysis Lecture Notes December 16, 017 Contents 1 Introduction to 1 11 Floating Point Numbers 1 1 Computational Errors 13 Algorithm 3 14 Calculus Review 3 Root Finding 5 1 Bisection
More informationPowerPoints organized by Dr. Michael R. Gustafson II, Duke University
Part 5 Chapter 17 Numerical Integration Formulas PowerPoints organized by Dr. Michael R. Gustafson II, Duke University All images copyright The McGraw-Hill Companies, Inc. Permission required for reproduction
More informationMath Numerical Analysis Mid-Term Test Solutions
Math 400 - Numerical Analysis Mid-Term Test Solutions. Short Answers (a) A sufficient and necessary condition for the bisection method to find a root of f(x) on the interval [a,b] is f(a)f(b) < 0 or f(a)
More information1 Review of Interpolation using Cubic Splines
cs412: introduction to numerical analysis 10/10/06 Lecture 12: Instructor: Professor Amos Ron Cubic Hermite Spline Interpolation Scribes: Yunpeng Li, Mark Cowlishaw 1 Review of Interpolation using Cubic
More informationComputational Physics
Interpolation, Extrapolation & Polynomial Approximation Lectures based on course notes by Pablo Laguna and Kostas Kokkotas revamped by Deirdre Shoemaker Spring 2014 Introduction In many cases, a function
More information2
1 Notes for Numerical Analysis Math 54 by S. Adjerid Virginia Polytechnic Institute and State University (A Rough Draft) 2 Contents 1 Polynomial Interpolation 5 1.1 Review...............................
More informationThere is a unique function s(x) that has the required properties. It turns out to also satisfy
Numerical Analysis Grinshpan Natural Cubic Spline Let,, n be given nodes (strictly increasing) and let y,, y n be given values (arbitrary) Our goal is to produce a function s() with the following properties:
More informationPolynomial Interpolation Part II
Polynomial Interpolation Part II Prof. Dr. Florian Rupp German University of Technology in Oman (GUtech) Introduction to Numerical Methods for ENG & CS (Mathematics IV) Spring Term 2016 Exercise Session
More informationInterpolation and Polynomial Approximation I
Interpolation and Polynomial Approximation I If f (n) (x), n are available, Taylor polynomial is an approximation: f (x) = f (x 0 )+f (x 0 )(x x 0 )+ 1 2! f (x 0 )(x x 0 ) 2 + Example: e x = 1 + x 1! +
More informationNumerical techniques to solve equations
Programming for Applications in Geomatics, Physical Geography and Ecosystem Science (NGEN13) Numerical techniques to solve equations vaughan.phillips@nateko.lu.se Vaughan Phillips Associate Professor,
More informationOutline. 1 Interpolation. 2 Polynomial Interpolation. 3 Piecewise Polynomial Interpolation
Outline Interpolation 1 Interpolation 2 3 Michael T. Heath Scientific Computing 2 / 56 Interpolation Motivation Choosing Interpolant Existence and Uniqueness Basic interpolation problem: for given data
More informationQ1. Discuss, compare and contrast various curve fitting and interpolation methods
Q1. Discuss, compare and contrast various curve fitting and interpolation methods McMaster University 1 Curve Fitting Problem statement: Given a set of (n + 1) point-pairs {x i,y i }, i = 0,1,... n, find
More informationInterpolation. Chapter Interpolation. 7.2 Existence, Uniqueness and conditioning
76 Chapter 7 Interpolation 7.1 Interpolation Definition 7.1.1. Interpolation of a given function f defined on an interval [a,b] by a polynomial p: Given a set of specified points {(t i,y i } n with {t
More informationMidterm Review. Igor Yanovsky (Math 151A TA)
Midterm Review Igor Yanovsky (Math 5A TA) Root-Finding Methods Rootfinding methods are designed to find a zero of a function f, that is, to find a value of x such that f(x) =0 Bisection Method To apply
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc
More informationA first order divided difference
A first order divided difference For a given function f (x) and two distinct points x 0 and x 1, define f [x 0, x 1 ] = f (x 1) f (x 0 ) x 1 x 0 This is called the first order divided difference of f (x).
More informationWe consider the problem of finding a polynomial that interpolates a given set of values:
Chapter 5 Interpolation 5. Polynomial Interpolation We consider the problem of finding a polynomial that interpolates a given set of values: x x 0 x... x n y y 0 y... y n where the x i are all distinct.
More informationChapter 2 Interpolation
Chapter 2 Interpolation Experiments usually produce a discrete set of data points (x i, f i ) which represent the value of a function f (x) for a finite set of arguments {x 0...x n }. If additional data
More informationLagrange Interpolation and Neville s Algorithm. Ron Goldman Department of Computer Science Rice University
Lagrange Interpolation and Neville s Algorithm Ron Goldman Department of Computer Science Rice University Tension between Mathematics and Engineering 1. How do Mathematicians actually represent curves
More informationInterpolation. 1. Judd, K. Numerical Methods in Economics, Cambridge: MIT Press. Chapter
Key References: Interpolation 1. Judd, K. Numerical Methods in Economics, Cambridge: MIT Press. Chapter 6. 2. Press, W. et. al. Numerical Recipes in C, Cambridge: Cambridge University Press. Chapter 3
More informationApplied Math 205. Full office hour schedule:
Applied Math 205 Full office hour schedule: Rui: Monday 3pm 4:30pm in the IACS lounge Martin: Monday 4:30pm 6pm in the IACS lounge Chris: Tuesday 1pm 3pm in Pierce Hall, Room 305 Nao: Tuesday 3pm 4:30pm
More informationNUMERICAL METHODS. x n+1 = 2x n x 2 n. In particular: which of them gives faster convergence, and why? [Work to four decimal places.
NUMERICAL METHODS 1. Rearranging the equation x 3 =.5 gives the iterative formula x n+1 = g(x n ), where g(x) = (2x 2 ) 1. (a) Starting with x = 1, compute the x n up to n = 6, and describe what is happening.
More informationINTERPOLATION Background Polynomial Approximation Problem:
INTERPOLATION Background Polynomial Approximation Problem: given f(x) C[a, b], find P n (x) = a 0 + a 1 x + a 2 x 2 + + a n x n with P n (x) close to f(x) for x [a, b]. Motivations: f(x) might be difficult
More informationNumerical integration and differentiation. Unit IV. Numerical Integration and Differentiation. Plan of attack. Numerical integration.
Unit IV Numerical Integration and Differentiation Numerical integration and differentiation quadrature classical formulas for equally spaced nodes improper integrals Gaussian quadrature and orthogonal
More informationCurve Fitting. Objectives
Curve Fitting Objectives Understanding the difference between regression and interpolation. Knowing how to fit curve of discrete with least-squares regression. Knowing how to compute and understand the
More informationPiecewise Polynomial Interpolation
Piecewise Polynomial Interpolation 1 Piecewise linear interpolation Suppose we have data point (x k,y k ), k =0, 1,...N. A piecewise linear polynomial that interpolates these points is given by p(x) =p
More information19.4 Spline Interpolation
c9-b.qxd 9/6/5 6:4 PM Page 8 8 CHAP. 9 Numerics in General 9.4 Spline Interpolation Given data (function values, points in the xy-plane) (x, ƒ ), (x, ƒ ),, (x n, ƒ n ) can be interpolated by a polynomial
More informationUniversity of Houston, Department of Mathematics Numerical Analysis, Fall 2005
4 Interpolation 4.1 Polynomial interpolation Problem: LetP n (I), n ln, I := [a,b] lr, be the linear space of polynomials of degree n on I, P n (I) := { p n : I lr p n (x) = n i=0 a i x i, a i lr, 0 i
More informationSection 6.6 Gaussian Quadrature
Section 6.6 Gaussian Quadrature Key Terms: Method of undetermined coefficients Nonlinear systems Gaussian quadrature Error Legendre polynomials Inner product Adapted from http://pathfinder.scar.utoronto.ca/~dyer/csca57/book_p/node44.html
More informationInterpolation and extrapolation
Interpolation and extrapolation Alexander Khanov PHYS6260: Experimental Methods is HEP Oklahoma State University October 30, 207 Interpolation/extrapolation vs fitting Formulation of the problem: there
More informationMath 578: Assignment 2
Math 578: Assignment 2 13. Determine whether the natural cubic spline that interpolates the table is or is not the x 0 1 2 3 y 1 1 0 10 function 1 + x x 3 x [0, 1] f(x) = 1 2(x 1) 3(x 1) 2 + 4(x 1) 3 x
More informationInput: A set (x i -yy i ) data. Output: Function value at arbitrary point x. What for x = 1.2?
Applied Numerical Analysis Interpolation Lecturer: Emad Fatemizadeh Interpolation Input: A set (x i -yy i ) data. Output: Function value at arbitrary point x. 0 1 4 1-3 3 9 What for x = 1.? Interpolation
More informationFixed point iteration and root finding
Fixed point iteration and root finding The sign function is defined as x > 0 sign(x) = 0 x = 0 x < 0. It can be evaluated via an iteration which is useful for some problems. One such iteration is given
More informationNumerical Methods of Approximation
Contents 31 Numerical Methods of Approximation 31.1 Polynomial Approximations 2 31.2 Numerical Integration 28 31.3 Numerical Differentiation 58 31.4 Nonlinear Equations 67 Learning outcomes In this Workbook
More informationLecture 20: Lagrange Interpolation and Neville s Algorithm. for I will pass through thee, saith the LORD. Amos 5:17
Lecture 20: Lagrange Interpolation and Neville s Algorithm for I will pass through thee, saith the LORD. Amos 5:17 1. Introduction Perhaps the easiest way to describe a shape is to select some points on
More informationCurve Fitting. 1 Interpolation. 2 Composite Fitting. 1.1 Fitting f(x) 1.2 Hermite interpolation. 2.1 Parabolic and Cubic Splines
Curve Fitting Why do we want to curve fit? In general, we fit data points to produce a smooth representation of the system whose response generated the data points We do this for a variety of reasons 1
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 7 Interpolation Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign Copyright c 2002. Reproduction permitted
More informationCS 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 informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximating a function f(x, wose values at a set of distinct points x, x, x 2,,x n are known, by a polynomial P (x
More informationCS 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 informationSection 5.3 The Newton Form of the Interpolating Polynomial
Section 5.3 The Newton Form of the Interpolating Polynomial Key terms Divided Difference basis Add-on feature Divided Difference Table The Lagrange form is not very efficient if we have to evaluate the
More information2-4 Zeros of Polynomial Functions
Write a polynomial function of least degree with real coefficients in standard form that has the given zeros. 33. 2, 4, 3, 5 Using the Linear Factorization Theorem and the zeros 2, 4, 3, and 5, write f
More informationLösning: Tenta Numerical Analysis för D, L. FMN011,
Lösning: Tenta Numerical Analysis för D, L. FMN011, 090527 This exam starts at 8:00 and ends at 12:00. To get a passing grade for the course you need 35 points in this exam and an accumulated total (this
More informationAdditional exercises with Numerieke Analyse
Additional exercises with Numerieke Analyse March 10, 017 1. (a) Given different points x 0, x 1, x [a, b] and scalars y 0, y 1, y, z 1, show that there exists at most one polynomial p P 3 with p(x i )
More informationCS412: Introduction to Numerical Methods
CS412: Introduction to Numerical Methods MIDTERM #1 2:30PM - 3:45PM, Tuesday, 03/10/2015 Instructions: This exam is a closed book and closed notes exam, i.e., you are not allowed to consult any textbook,
More informationCHAPTER 3 Further properties of splines and B-splines
CHAPTER 3 Further properties of splines and B-splines In Chapter 2 we established some of the most elementary properties of B-splines. In this chapter our focus is on the question What kind of functions
More informationNumerical Analysis: Interpolation Part 1
Numerical Analysis: Interpolation Part 1 Computer Science, Ben-Gurion University (slides based mostly on Prof. Ben-Shahar s notes) 2018/2019, Fall Semester BGU CS Interpolation (ver. 1.00) AY 2018/2019,
More informationGENG2140, S2, 2012 Week 7: Curve fitting
GENG2140, S2, 2012 Week 7: Curve fitting Curve fitting is the process of constructing a curve, or mathematical function, f(x) that has the best fit to a series of data points Involves fitting lines and
More informationPreliminary Examination in Numerical Analysis
Department of Applied Mathematics Preliminary Examination in Numerical Analysis August 7, 06, 0 am pm. Submit solutions to four (and no more) of the following six problems. Show all your work, and justify
More informationCubic Spline. s(x) = s j (x) = a j + b j (x x j ) + c j (x x j ) 2 + d j (x x j ) 3, j = 0,... n 1 (1)
Cubic Spline Suppose we are given a set of interpolating points (x i, y i ) for i = 0, 1, 2, n We seek to construct a piecewise cubic function s(x) that has the for x [[x j, x j+1 ] we have: s(x) = s j
More informationCubic Splines. Antony Jameson. Department of Aeronautics and Astronautics, Stanford University, Stanford, California, 94305
Cubic Splines Antony Jameson Department of Aeronautics and Astronautics, Stanford University, Stanford, California, 94305 1 References on splines 1. J. H. Ahlberg, E. N. Nilson, J. H. Walsh. Theory of
More informationPolynomial Functions and Their Graphs
Polynomial Functions and Their Graphs Definition of a Polynomial Function Let n be a nonnegative integer and let a n, a n- 1,, a 2, a 1, a 0, be real numbers with a n 0. The function defined by f (x) a
More informationConvergence rates of derivatives of a family of barycentric rational interpolants
Convergence rates of derivatives of a family of barycentric rational interpolants J.-P. Berrut, M. S. Floater and G. Klein University of Fribourg (Switzerland) CMA / IFI, University of Oslo jean-paul.berrut@unifr.ch
More informationTo find an approximate value of the integral, the idea is to replace, on each subinterval
Module 6 : Definition of Integral Lecture 18 : Approximating Integral : Trapezoidal Rule [Section 181] Objectives In this section you will learn the following : Mid point and the Trapezoidal methods for
More informationCHAPTER 2 POLYNOMIALS KEY POINTS
CHAPTER POLYNOMIALS KEY POINTS 1. Polynomials of degrees 1, and 3 are called linear, quadratic and cubic polynomials respectively.. A quadratic polynomial in x with real coefficient is of the form a x
More informationUniversity of British Columbia Math 307, Final
1 University of British Columbia Math 307, Final April 23, 2012 3.30-6.00pm Name: Student Number: Signature: Instructor: Instructions: 1. No notes, books or calculators are allowed. A MATLAB/Octave formula
More informationn 1 f n 1 c 1 n+1 = c 1 n $ c 1 n 1. After taking logs, this becomes
Root finding: 1 a The points {x n+1, }, {x n, f n }, {x n 1, f n 1 } should be co-linear Say they lie on the line x + y = This gives the relations x n+1 + = x n +f n = x n 1 +f n 1 = Eliminating α and
More informationIntro Polynomial Piecewise Cubic Spline Software Summary. Interpolation. Sanzheng Qiao. Department of Computing and Software McMaster University
Interpolation Sanzheng Qiao Department of Computing and Software McMaster University January, 2014 Outline 1 Introduction 2 Polynomial Interpolation 3 Piecewise Polynomial Interpolation 4 Natural Cubic
More informationBSM510 Numerical Analysis
BSM510 Numerical Analysis Polynomial Interpolation Prof. Manar Mohaisen Department of EEC Engineering Review of Precedent Lecture Polynomial Regression Multiple Linear Regression Nonlinear Regression Lecture
More informationIntegration, differentiation, and root finding. Phys 420/580 Lecture 7
Integration, differentiation, and root finding Phys 420/580 Lecture 7 Numerical integration Compute an approximation to the definite integral I = b Find area under the curve in the interval Trapezoid Rule:
More informationMTH301 Calculus II Glossary For Final Term Exam Preparation
MTH301 Calculus II Glossary For Final Term Exam Preparation Glossary Absolute maximum : The output value of the highest point on a graph over a given input interval or over all possible input values. An
More informationUNIT-II INTERPOLATION & APPROXIMATION
UNIT-II INTERPOLATION & APPROXIMATION LAGRANGE POLYNAMIAL 1. Find the polynomial by using Lagrange s formula and hence find for : 0 1 2 5 : 2 3 12 147 : 0 1 2 5 : 0 3 12 147 Lagrange s interpolation formula,
More informationNumerical Integration
Numerical Integration Sanzheng Qiao Department of Computing and Software McMaster University February, 2014 Outline 1 Introduction 2 Rectangle Rule 3 Trapezoid Rule 4 Error Estimates 5 Simpson s Rule 6
More informationIntro Polynomial Piecewise Cubic Spline Software Summary. Interpolation. Sanzheng Qiao. Department of Computing and Software McMaster University
Interpolation Sanzheng Qiao Department of Computing and Software McMaster University July, 2012 Outline 1 Introduction 2 Polynomial Interpolation 3 Piecewise Polynomial Interpolation 4 Natural Cubic Spline
More informationExtrapolation in Numerical Integration. Romberg Integration
Extrapolation in Numerical Integration Romberg Integration Matthew Battaglia Joshua Berge Sara Case Yoobin Ji Jimu Ryoo Noah Wichrowski Introduction Extrapolation: the process of estimating beyond the
More informationyou expect to encounter difficulties when trying to solve A x = b? 4. A composite quadrature rule has error associated with it in the following form
Qualifying exam for numerical analysis (Spring 2017) Show your work for full credit. If you are unable to solve some part, attempt the subsequent parts. 1. Consider the following finite difference: f (0)
More information9.5. Polynomial and Rational Inequalities. Objectives. Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater.
Chapter 9 Section 5 9.5 Polynomial and Rational Inequalities Objectives 1 3 Solve quadratic inequalities. Solve polynomial inequalities of degree 3 or greater. Solve rational inequalities. Objective 1
More informationChapter 3: Root Finding. September 26, 2005
Chapter 3: Root Finding September 26, 2005 Outline 1 Root Finding 2 3.1 The Bisection Method 3 3.2 Newton s Method: Derivation and Examples 4 3.3 How To Stop Newton s Method 5 3.4 Application: Division
More informationNumerical Integration (Quadrature) Another application for our interpolation tools!
Numerical Integration (Quadrature) Another application for our interpolation tools! Integration: Area under a curve Curve = data or function Integrating data Finite number of data points spacing specified
More information