DOCSLIB.ORG

ECE 3040 Lecture 17: Polynomial Interpolation © Prof

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

ECE 3040 Lecture 17: Polynomial Interpolation © Prof. Mohamad Hassoun This lecture covers the following topics: Introduction Interpolation using a single polynomial Newton’s interpolation polynomials Matlab built-in polynomial interpolation: polyfit The curse of high-dimensional polynomials Cubic spline interpolation Matlab built-in cubic spline interpolation: spline Interpolation using rational functions Introduction Polynomial interpolation is a procedure for modeling a set of precise data points using a polynomial function, 푝(푥), that fits the data exactly (passes through all provided data points). The data points are normally obtained from a complicated mathematical model, 푓(푥), of an engineering or scientific system derived from physical principles. Once an interpolation polynomial is computed, it can be used to replace the complicated mathematical model for the purpose of analysis and design. For instance, the interpolating polynomial can be used to estimate the value of the function at a new point 푥′, as 푓(푥′) ≅ 푝(푥′). The solution for the coefficients of the interpolation polynomial 푝(푥) can be determined by solving an associated system of linear equations, or can be computed using formulas. The data points used for interpolation can also be a set of very accurately measured experimental values. In practice, if the set of data points is large, different polynomials are used to construct a piece-wise interpolation function; this procedure is known as spline interpolation. Rational functions may also be used for interpolation. This lecture will also introduces two built-in Matlab polynomial-based interpolation functions: polyfit and spline. Interpolation Using a Single Polynomial A straight line can be completely defined using two points on the straight line. The slope, 푎1, and 푦-intercept, 푎0, coefficients in the representation 푝(푥) = 푎1푥 + 푎0 are sufficient to define the straight line. The coefficients of the straight line that passes through the points (푥1, 푓(푥1)) and (푥2, 푓(푥2)) are unique and are the solution of the system of two linear equations: 푓(푥1) = 푎1푥1 + 푎0 푓(푥2) = 푎1푥2 + 푎0 or, in matrix form, 푥 1 푎 푓(푥 ) [ 1 ] [ 1] = [ 1 ] 푥2 1 푎0 푓(푥2) Similarly, three points are required to compute the coefficients of a parabola, 2 푝(푥) = 푎2푥 + 푎1푥 + 푎0. For example, the three coefficients of the parabola that passes through the distinct (non-collinear) points (푥1, 푓(푥1)), (푥2, 푓(푥2)) and (푥3, 푓(푥3)) are unique and are the solution of the system of three linear equations: 2 푓(푥1) = 푎2푥1 + 푎1푥1 + 푎0 2 푓(푥2) = 푎2푥2 + 푎1푥2 + 푎0 2 푓(푥3) = 푎2푥3 + 푎1푥3 + 푎0 or, in matrix form, 2 푥1 푥1 1 푎2 푓(푥1) 2 [푥2 푥2 1] [푎1 ] = [푓(푥2)] 2 푎 푥3 푥3 1 0 푓(푥3) In fact, fitting a polynomial to a set of given data points is not new to us. Recall that we had encountered this problem in the context of parabolic interpolation-based optimization in Lecture 14. The following is an example of second-order polynomial interpolation. Example. Use parabolic interpolation to approximate the function 푥2 푓(푥) = − 2sin(푥) 10 from three of its points at 푥1 = 0, 푥2 = 1 and 푥3 = 2. Then, employ the quadratic model to predict the value of 푓(푥), at 푥 = 1.5. Also, compute the true error. Solution. Substituting the values of 푥 in the function 푓(푥) leads to the three interpolation points: (0,0), (1, −1.583) and (2, −1.419). The coefficients of the parabola can then be obtained (employing the above formulation) as the solution of 0 0 1 푎2 0 [1 1 1] [푎1 ] = [−1.583] 4 2 1 푎0 −1.419 The (green) plot of the interpolation parabola 푝(푥) = 0.874푥2 − 2.457푥 and the (blue) plot of 푓(푥), are shown below, within the interval [-0.5 3]. The interpolated value at 푥 = 1.5 is 푝(1.5) = −1.719, giving a true error of 푓(1.5) − 푝(1.5) = −1.770 − (−1.719) = −0.051. The above formulation can be extended to interpolating the 푛 points { (푥1, 푓(푥1)), (푥2, 푓(푥2)), … , (푥푛, 푓(푥푛))} Using the an (푛 − 1)-order polynomial 푛−1 2 푝(푥) = 푎푛−1푥 + ⋯ + 푎2푥 + 푎1푥 + 푎0 which results in the following 푛x푛 system of linear equation: 푎 푥푛−1 … 푥2 푥 1 푛−1 푓(푥 ) 1 1 1 ⋮ 1 푛−1 2 푥 ⋯ 푥 푥2 1 푓(푥2) 2 2 푎2 = [ ] ⋮ ⋮ ⋮ ⋮ ⋮ 푎 ⋮ 푛−1 2 1 [푥 ⋯ 푥 푥 1] 푓(푥푛) 푛 푛 푛 [ 푎0 ] Coefficient matrices of this form are referred to as Vandermonde matrices. Matlab can generate the Vandermonde matrix using the built-in function vander(x), where x is a vector of the 푥푖 data points. The following is an example. The Vandermonde matrix can become ill-conditioned (especially for large 푛), and therefore the solution for the above system formulation is very sensitive to round- off errors. The Matlab instruction cond(A) can be used to return the condition number of a matrix A; a high valued condition number indicates a nearly singular matrix. For example, The next section presents an alternative, numerically robust formulation for the polynomial interpolation problem. Newton’s Interpolation Polynomials There are a variety of alternative forms for expressing an interpolation polynomial beyond the previous formulation. Newton’s interpolation polynomial is among the most popular. The simplest form of interpolation is to start with two points (푥1, 푓(푥1)) and (푥2, 푓(푥2)) and connect them with a straight line. The equation for a straight line that passes through two points can be easily derived and is expressed as 푓(푥2) − 푓(푥1) 푝푁1(푥) = (푥 − 푥1) + 푓(푥1) 푥2 − 푥1 We will refer to 푝푁1(푥) as Newton’s linear-interpolation formula. It allows us to approximate the value of 푓(푥) at some point 푥 inside [푥1 푥2]. The notation 푝푁1 designates that this is a first-order Newton interpolating polynomial. The nominal form for 푝푁1 is given by 푝푁1(푥) = 푏1 + 푏2(푥 − 푥1) where the coefficients are given by 푓(푥2) − 푓(푥1) 푏1 = 푓(푥1), 푏2 = 푥2 − 푥1 It is obvious here that the smaller the interval between the data points, the better the approximation. Alternatively, one can increase the order of the polynomial for increased accuracy if more data points are available. For three data points, the second-order Newton’s interpolation polynomial, 푝푁2(푥), is 푝푁2(푥) = 푏1 + 푏2(푥 − 푥1) + 푏3(푥 − 푥1)(푥 − 푥2) with coefficients 푓(푥 ) − 푓(푥 ) 푓(푥 ) − 푓(푥 ) 3 2 − 2 1 푓(푥2) − 푓(푥1) 푥3 − 푥2 푥2 − 푥1 푏1 = 푓(푥1), 푏2 = , 푏3 = 푥2 − 푥1 푥3 − 푥1 Notice that the coefficients 푏1 and 푏2 are the same in the second and first-order Newton’s polynomials. This means that if two points are given and a first order Newton’s polynomial is fitted to pass through these points and then a third point is added, the polynomial 푝푁2 would then be computed as 푝푁2 = 푝푁1 + 푏3(푥 − 푥1)(푥 − 푥2) and only one new coefficient (푏3) would need to be computed. This observation is also true for higher order Newton polynomials. It is interesting to note that 푏3 is the ′′ finite-difference, second-derivative approximation of 푓 (푥3). The general form for Newton’s (푛 − 1)-order interpolating polynomial on the 푛 data points {(푥1, 푓(푥1)), (푥2, 푓(푥2)), … , (푥푛, 푓(푥푛))}, is given by 푝푁(푛−1)(푥) = 푏1 + 푏2(푥 − 푥1) + 푏3(푥 − 푥1)(푥 − 푥2) + ⋯ + 푏푛(푥 − 푥1)(푥 − 푥2) … (푥 − 푥푛−1) where the 푏푖 constants are evaluated directly (and efficiently) from the data points, using the formulas: 푏1 = 푓(푥1) 푓(푥2) − 푏1 푏2 = 푥2 − 푥1 푓(푥3) − 푓(푥2) − 푏2 푥3 − 푥2 푏3 = 푥3 − 푥1 푓(푥 ) − 푓(푥 ) 푓(푥 ) − 푓(푥 ) 4 3 − 3 2 푥4 − 푥3 푥3 − 푥2 − 푏3 푥4 − 푥2 푏4 = 푥4 − 푥1 and so on. Your turn: Deduce the formula for 푏5. Example. Solve the last example using a second-order Newton’s interpolation polynomial. Again, the three interpolation points are (0,0), (1, −1.583) and (2, −1.419). Solution. The 푏 coefficients are: 푏1 = 푓(푥1) = 푓(0) = 0 푓(푥2) − 푏1 −1.583 − 0 푏2 = = = −1.583 푥2 − 푥1 1 − 0 푓(푥3) − 푓(푥2) −1.419 − (−1.583) − 푏2 − (−1.583) 푥3 − 푥2 2 − 1 푏3 = = = 0.874 푥3 − 푥1 2 − 0 which lead to the interpolation polynomial, 푝푁2(푥) = 0 − 1.583(푥 − 0) + 0.874(푥 − 0)(푥 − 1) = −1.583푥 + 0.874푥(푥 − 1) = 0.874푥2 − 2.457푥 The interpolated value at 푥 = 1.5 is 푝푁2(1.5) = −1.719. These results are consistent with those obtained based on solving the Vandermonde matrix formulation (refer to the previous example). The coefficients of the Newton polynomial can also be computed systematically by solving (using forward substitution utilizing the “\” operator) the system of equations shown below. Your turn: Use it to find {푏1, 푏2, 푏3} for the last example. 1 0 0 0 … 0 푏 푓(푥 ) 1 1 1 (푥 − 푥 ) 0 0 … 0 푏 푓(푥 ) 2 1 2 2 1 (푥 − 푥 ) (푥 − 푥 )(푥 − 푥 ) 0 … 0 푏 푓(푥 ) 3 1 3 1 3 2 3 = 3 . [1 (푥푛 − 푥1) (푥푛 − 푥1)(푥푛 − 푥2) … (푥푛 − 푥1)(푥푛 − 푥2)(… )(푥푛 − 푥푛−1)] [푏푛] [푓(푥푛)] Function newtint is a Matlab implementation of Newton’s interpolation. The following is an application of the function newtint to the solution of the last example: Your turn: Rewrite the above function using the matrix formulation shown at the end of the last page. Email your Matlab function to the grader before the due date for Assignment 11.
Recommended publications
  • Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform

    Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform

    World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:13, No:8, 2019 Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform Lagrange and Newton Curves Taweechai Nuntawisuttiwong, Natasha Dejdumrong Abstract—Newton-Lagrange Interpolations are widely used in quadratic computation time, which is slower than the others. numerical analysis. However, it requires a quadratic computational However, there are some curves in CAGD, Wang-Ball, DP time for their constructions. In computer aided geometric design and Dejdumrong curves with linear complexity. In order to (CAGD), there are some polynomial curves: Wang-Ball, DP and Dejdumrong curves, which have linear time complexity algorithms. reduce their computational time, Bezier´ curve is converted Thus, the computational time for Newton-Lagrange Interpolations into any of Wang-Ball, DP and Dejdumrong curves. Hence, can be reduced by applying the algorithms of Wang-Ball, DP and the computational time for Newton-Lagrange interpolations Dejdumrong curves. In order to use Wang-Ball, DP and Dejdumrong can be reduced by converting them into Wang-Ball, DP and algorithms, first, it is necessary to convert Newton-Lagrange Dejdumrong algorithms. Thus, it is necessary to investigate polynomials into Wang-Ball, DP or Dejdumrong polynomials. In this work, the algorithms for converting from both uniform and the conversion from Newton-Lagrange interpolation into non-uniform Newton-Lagrange polynomials into Wang-Ball, DP and the curves with linear complexity, Wang-Ball, DP and Dejdumrong polynomials are investigated. Thus, the computational Dejdumrong curves. An application of this work is to modify time for representing Newton-Lagrange polynomials can be reduced sketched image in CAD application with the computational into linear complexity.
  • Generalized Barycentric Coordinates on Irregular Polygons

    Generalized Barycentric Coordinates on Irregular Polygons

    Generalized Barycentric Coordinates on Irregular Polygons Mark Meyery Haeyoung Leez Alan Barry Mathieu Desbrunyz Caltech - USC y z Abstract In this paper we present an easy computation of a generalized form of barycentric coordinates for irregular, convex n-sided polygons. Triangular barycentric coordinates have had many classical applications in computer graphics, from texture mapping to ray-tracing. Our new equations preserve many of the familiar properties of the triangular barycentric coordinates with an equally simple calculation, contrary to previous formulations. We illustrate the properties and behavior of these new generalized barycentric coordinates through several example applications. 1 Introduction The classical equations to compute triangular barycentric coordinates have been known by mathematicians for centuries. These equations have been heavily used by the earliest computer graphics researchers and have allowed many useful applications including function interpolation, surface smoothing, simulation and ray intersection tests. Due to their linear accuracy, barycentric coordinates can also be found extensively in the finite element literature [Wac75]. (a) (b) (c) (d) Figure 1: (a) Smooth color blending using barycentric coordinates for regular polygons [LD89], (b) Smooth color blending using our generalization to arbitrary polygons, (c) Smooth parameterization of an arbitrary mesh using our new formula which ensures non-negative coefficients. (d) Smooth position interpolation over an arbitrary convex polygon (S-patch of depth 1). Despite the potential benefits, however, it has not been obvious how to generalize barycentric coordinates from triangles to n-sided poly- gons. Several formulations have been proposed, but most had their own weaknesses. Important properties were lost from the triangular barycentric formulation, which interfered with uses of the previous generalized forms [PP93, Flo97].
  • Mathematical Construction of Interpolation and Extrapolation Function by Taylor Polynomials Arxiv:2002.11438V1 [Math.NA] 26 Fe

    Mathematical Construction of Interpolation and Extrapolation Function by Taylor Polynomials Arxiv:2002.11438V1 [Math.NA] 26 Fe

    Mathematical Construction of Interpolation and Extrapolation Function by Taylor Polynomials Nijat Shukurov Department of Engineering Physics, Ankara University, Ankara, Turkey E-mail: [email protected] , [email protected] Abstract: In this present paper, I propose a derivation of unified interpolation and extrapolation function that predicts new values inside and outside the given range by expanding direct Taylor series on the middle point of given data set. Mathemati- cal construction of experimental model derived in general form. Trigonometric and Power functions adopted as test functions in the development of the vital aspects in numerical experiments. Experimental model was interpolated and extrapolated on data set that generated by test functions. The results of the numerical experiments which predicted by derived model compared with analytical values. KEYWORDS: Polynomial Interpolation, Extrapolation, Taylor Series, Expansion arXiv:2002.11438v1 [math.NA] 26 Feb 2020 1 1 Introduction In scientific experiments or engineering applications, collected data are usually discrete in most cases and physical meaning is likely unpredictable. To estimate the outcomes and to understand the phenomena analytically controllable functions are desirable. In the mathematical field of nu- merical analysis those type of functions are called as interpolation and extrapolation functions. Interpolation serves as the prediction tool within range of given discrete set, unlike interpola- tion, extrapolation functions designed to predict values out of the range of given data set. In this scientific paper, direct Taylor expansion is suggested as a instrument which estimates or approximates a new points inside and outside the range by known individual values. Taylor se- ries is one of most beautiful analogies in mathematics, which make it possible to rewrite every smooth function as a infinite series of Taylor polynomials.
  • On Multivariate Interpolation

    On Multivariate Interpolation

    On Multivariate Interpolation Peter J. Olver† School of Mathematics University of Minnesota Minneapolis, MN 55455 U.S.A. [email protected] http://www.math.umn.edu/∼olver Abstract. A new approach to interpolation theory for functions of several variables is proposed. We develop a multivariate divided difference calculus based on the theory of non-commutative quasi-determinants. In addition, intriguing explicit formulae that connect the classical finite difference interpolation coefficients for univariate curves with multivariate interpolation coefficients for higher dimensional submanifolds are established. † Supported in part by NSF Grant DMS 11–08894. April 6, 2016 1 1. Introduction. Interpolation theory for functions of a single variable has a long and distinguished his- tory, dating back to Newton’s fundamental interpolation formula and the classical calculus of finite differences, [7, 47, 58, 64]. Standard numerical approximations to derivatives and many numerical integration methods for differential equations are based on the finite dif- ference calculus. However, historically, no comparable calculus was developed for functions of more than one variable. If one looks up multivariate interpolation in the classical books, one is essentially restricted to rectangular, or, slightly more generally, separable grids, over which the formulae are a simple adaptation of the univariate divided difference calculus. See [19] for historical details. Starting with G. Birkhoff, [2] (who was, coincidentally, my thesis advisor), recent years have seen a renewed level of interest in multivariate interpolation among both pure and applied researchers; see [18] for a fairly recent survey containing an extensive bibli- ography. De Boor and Ron, [8, 12, 13], and Sauer and Xu, [61, 10, 65], have systemati- cally studied the polynomial case.
  • Easy Way to Find Multivariate Interpolation

    Easy Way to Find Multivariate Interpolation

    IJETST- Vol.||04||Issue||05||Pages 5189-5193||May||ISSN 2348-9480 2017 International Journal of Emerging Trends in Science and Technology IC Value: 76.89 (Index Copernicus) Impact Factor: 4.219 DOI: https://dx.doi.org/10.18535/ijetst/v4i5.11 Easy way to Find Multivariate Interpolation Author Yimesgen Mehari Faculty of Natural and Computational Science, Department of Mathematics Adigrat University, Ethiopia Email: [email protected] Abstract We derive explicit interpolation formula using non-singular vandermonde matrix for constructing multi dimensional function which interpolates at a set of distinct abscissas. We also provide examples to show how the formula is used in practices. Introduction Engineers and scientists commonly assume that past and currently. But there is no theoretical relationships between variables in physical difficulty in setting up a frame work for problem can be approximately reproduced from discussing interpolation of multivariate function f data given by the problem. The ultimate goal whose values are known.[1] might be to determine the values at intermediate Interpolation function of more than one variable points, to approximate the integral or to simply has become increasingly important in the past few give a smooth or continuous representation of the years. These days, application ranges over many variables in the problem different field of pure and applied mathematics. Interpolation is the method of estimating unknown For example interpolation finds applications in the values with the help of given set of observations. numerical integrations of differential equations, According to Theile Interpolation is, “The art of topography, and the computer aided geometric reading between the lines of the table” and design of cars, ships, airplanes.[1] According to W.M.
  • Math 541 - Numerical Analysis Interpolation and Polynomial Approximation — Piecewise Polynomial Approximation; Cubic Splines

    Math 541 - Numerical Analysis Interpolation and Polynomial Approximation — Piecewise Polynomial Approximation; Cubic Splines

    Polynomial Interpolation Cubic Splines Cubic Splines... Math 541 - Numerical Analysis Interpolation and Polynomial Approximation — Piecewise Polynomial Approximation; Cubic Splines Joseph M. Mahaffy, [email protected] Department of Mathematics and Statistics Dynamical Systems Group Computational Sciences Research Center San Diego State University San Diego, CA 92182-7720 http://jmahaffy.sdsu.edu Spring 2018 Piecewise Poly. Approx.; Cubic Splines — Joseph M. Mahaffy, [email protected] (1/48) Polynomial Interpolation Cubic Splines Cubic Splines... Outline 1 Polynomial Interpolation Checking the Roadmap Undesirable Side-effects New Ideas... 2 Cubic Splines Introduction Building the Spline Segments Associated Linear Systems 3 Cubic Splines... Error Bound Solving the Linear Systems Piecewise Poly. Approx.; Cubic Splines — Joseph M. Mahaffy, [email protected] (2/48) Polynomial Interpolation Checking the Roadmap Cubic Splines Undesirable Side-effects Cubic Splines... New Ideas... An n-degree polynomial passing through n + 1 points Polynomial Interpolation Construct a polynomial passing through the points (x0,f(x0)), (x1,f(x1)), (x2,f(x2)), ... , (xN ,f(xn)). Define Ln,k(x), the Lagrange coefficients: n x − xi x − x0 x − xk−1 x − xk+1 x − xn Ln,k(x)= = ··· · ··· , Y xk − xi xk − x0 xk − xk−1 xk − xk+1 xk − xn i=0, i=6 k which have the properties Ln,k(xk) = 1; Ln,k(xi)=0, for all i 6= k. Piecewise Poly. Approx.; Cubic Splines — Joseph M. Mahaffy, [email protected] (3/48) Polynomial Interpolation Checking the Roadmap Cubic Splines Undesirable Side-effects Cubic Splines... New Ideas... The nth Lagrange Interpolating Polynomial We use Ln,k(x), k =0,...,n as building blocks for the Lagrange interpolating polynomial: n P (x)= f(x )L (x), X k n,k k=0 which has the property P (xi)= f(xi), for all i =0, .
  • CHAPTER 6 Parametric Spline Curves

    CHAPTER 6 Parametric Spline Curves

    CHAPTER 6 Parametric Spline Curves When we introduced splines in Chapter 1 we focused on spline curves, or more precisely, vector valued spline functions. In Chapters 2 and 4 we then established the basic theory of spline functions and B-splines, and in Chapter 5 we studied a number of methods for constructing spline functions that approximate given data. In this chapter we return to spline curves and show how the approximation methods in Chapter 5 can be adapted to this more general situation. We start by giving a formal definition of parametric curves in Section 6.1, and introduce parametric spline curves in Section 6.2.1. In the rest of Section 6.2 we then generalise the approximation methods in Chapter 5 to curves. 6.1 Definition of Parametric Curves In Section 1.2 we gave an intuitive introduction to parametric curves and discussed the significance of different parameterisations. In this section we will give a more formal definition of parametric curves, but the reader is encouraged to first go back and reread Section 1.2 in Chapter 1. 6.1.1 Regular parametric representations A parametric curve will be defined in terms of parametric representations. s Definition 6.1. A vector function or mapping f :[a, b] 7→ R of the interval [a, b] into s m R for s ≥ 2 is called a parametric representation of class C for m ≥ 1 if each of the s components of f has continuous derivatives up to order m. If, in addition, the first derivative of f does not vanish in [a, b], Df(t) = f 0(t) 6= 0, for t ∈ [a, b], then f is called a regular parametric representation of class Cm.
  • Spatial Interpolation Methods

    Spatial Interpolation Methods

    Page | 0 of 0 SPATIAL INTERPOLATION METHODS 2018 Page | 1 of 1 1. Introduction Spatial interpolation is the procedure to predict the value of attributes at unobserved points within a study region using existing observations (Waters, 1989). Lam (1983) definition of spatial interpolation is “given a set of spatial data either in the form of discrete points or for subareas, find the function that will best represent the whole surface and that will predict values at points or for other subareas”. Predicting the values of a variable at points outside the region covered by existing observations is called extrapolation (Burrough and McDonnell, 1998). All spatial interpolation methods can be used to generate an extrapolation (Li and Heap 2008). Spatial Interpolation is the process of using points with known values to estimate values at other points. Through Spatial Interpolation, We can estimate the precipitation value at a location with no recorded data by using known precipitation readings at nearby weather stations. Rationale behind spatial interpolation is the observation that points close together in space are more likely to have similar values than points far apart (Tobler’s Law of Geography). Spatial Interpolation covers a variety of method including trend surface models, thiessen polygons, kernel density estimation, inverse distance weighted, splines, and Kriging. Spatial Interpolation requires two basic inputs: · Sample Points · Spatial Interpolation Method Sample Points Sample Points are points with known values. Sample points provide the data necessary for the development of interpolator for spatial interpolation. The number and distribution of sample points can greatly influence the accuracy of spatial interpolation.
  • Mials P

    Mials P

    Euclid's Algorithm 171 Euclid's Algorithm Horner’s method is a special case of Euclid's Algorithm which constructs, for given polyno- mials p and h =6 0, (unique) polynomials q and r with deg r<deg h so that p = hq + r: For variety, here is a nonstandard discussion of this algorithm, in terms of elimination. Assume that d h(t)=a0 + a1t + ···+ adt ;ad =06 ; and n p(t)=b0 + b1t + ···+ bnt : Then we seek a polynomial n−d q(t)=c0 + c1t + ···+ cn−dt for which r := p − hq has degree <d. This amounts to the square upper triangular linear system adc0 + ad−1c1 + ···+ a0cd = bd adc1 + ad−1c2 + ···+ a0cd+1 = bd+1 . adcn−d−1 + ad−1cn−d = bn−1 adcn−d = bn for the unknown coefficients c0;:::;cn−d which can be uniquely solved by back substitution since its diagonal entries all equal ad =0.6 19aug02 c 2002 Carl de Boor 172 18. Index Rough index for these notes 1-1:-5,2,8,40 cartesian product: 2 1-norm: 79 Cauchy(-Bunyakovski-Schwarz) 2-norm: 79 Inequality: 69 A-invariance: 125 Cauchy-Binet formula: -9, 166 A-invariant: 113 Cayley-Hamilton Theorem: 133 absolute value: 167 CBS Inequality: 69 absolutely homogeneous: 70, 79 Chaikin algorithm: 139 additive: 20 chain rule: 153 adjugate: 164 change of basis: -6 affine: 151 characteristic function: 7 affine combination: 148, 150 characteristic polynomial: -8, 130, 132, 134 affine hull: 150 circulant: 140 affine map: 149 codimension: 50, 53 affine polynomial: 152 coefficient vector: 21 affine space: 149 cofactor: 163 affinely independent: 151 column map: -6, 23 agrees with y at Λt:59 column space: 29 algebraic dual: 95 column
  • CS321-001 Introduction to Numerical Methods

    CS321-001 Introduction to Numerical Methods

    CS321-001 Introduction to Numerical Methods Lecture 3 Interpolation and Numerical Differentiation Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506-0633 Polynomial Interpolation Given a set of discrete values, how can we estimate other values between these data The method that we will use is called polynomial interpolation. We assume the data we had are from the evaluation of a smooth function. We may be able to use a polynomial p(x) to approximate this function, at least locally. A condition: the polynomial p(x) takes the given values at the given points (nodes), i.e., p(xi) = yi with 0 ≤ i ≤ n. The polynomial is said to interpolate the table, since we do not know the function. 2 Polynomial Interpolation Note that all the points are passed through by the curve 3 Polynomial Interpolation We do not know the original function, the interpolation may not be accurate 4 Order of Interpolating Polynomial A polynomial of degree 0, a constant function, interpolates one set of data If we have two sets of data, we can have an interpolating polynomial of degree 1, a linear function x x1 x x0 p(x) y0 y1 x0 x1 x1 x0 y1 y0 y0 (x x0 ) x1 x0 Review carefully if the interpolation condition is satisfied Interpolating polynomials can be written in several forms, the most well known ones are the Lagrange form and Newton form. Each has some advantages 5 Lagrange Form For a set of fixed nodes x0, x1, …, xn, the cardinal functions, l0, l1,…, ln, are defined as 0 if i j li (x j ) ij
  • Automatic Constraint-Based Synthesis of Non-Uniform Rational B-Spline Surfaces " (1994)

    Automatic Constraint-Based Synthesis of Non-Uniform Rational B-Spline Surfaces " (1994)

    Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1994 Automatic constraint-based synthesis of non- uniform rational B-spline surfaces Philip Chacko Theruvakattil Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Mechanical Engineering Commons Recommended Citation Theruvakattil, Philip Chacko, "Automatic constraint-based synthesis of non-uniform rational B-spline surfaces " (1994). Retrospective Theses and Dissertations. 10515. https://lib.dr.iastate.edu/rtd/10515 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. It has been accepted for inclusion in Retrospective Theses and Dissertations by an authorized administrator of Iowa State University Digital Repository. For more information, please contact [email protected]. INFORMATION TO USERS This manuscript has been reproduced from the microSlm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand comer and continuing from left to right in equal sections with small overlaps.
  • CS 450 – Numerical Analysis Chapter 7: Interpolation =1[2]

    CS 450 – Numerical Analysis Chapter 7: Interpolation =1[2]

    CS 450 { Numerical Analysis Chapter 7: Interpolation y Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign [email protected] January 28, 2019 yLecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80 2 Interpolation 3 Interpolation I Basic interpolation problem: for given data (t1; y1); (t2; y2);::: (tm; ym) with t1 < t2 < ··· < tm determine function f : R ! R such that f (ti ) = yi ; i = 1;:::; m I f is interpolating function, or interpolant, for given data I Additional data might be prescribed, such as slope of interpolant at given points I Additional constraints might be imposed, such as smoothness, monotonicity, or convexity of interpolant I f could be function of more than one variable, but we will consider only one-dimensional case 4 Purposes for Interpolation I Plotting smooth curve through discrete data points I Reading between lines of table I Differentiating or integrating tabular data I Quick and easy evaluation of mathematical function I Replacing complicated function by simple one 5 Interpolation vs Approximation I By definition, interpolating function fits given data points exactly I Interpolation is inappropriate if data points subject to significant errors I It is usually preferable to smooth noisy data, for example by least squares approximation I Approximation is also more appropriate for special function libraries 6 Issues in Interpolation