INTERPOLATION BY QUARTIC SPLINE
|
|
- Shanna French
- 6 years ago
- Views:
Transcription
1 INTERPOLATION BY QUARTIC SPLINE K.K.Nigam 1, Y.P. Dubey 2, Brajendra Tiwari 3, Anil Shukla 4 1 Mathematics Deptt., LNCT Jabalpur, MP, (India) 2 Sarswati Institution of Engg. & Technology, Jabalpur MP, (India) 3 R.K.D.F University, Bhopal MP, (India) 4 Mathematics Deptt., GGCT, Jabalpur, (India) ABSTRACT In this paper we have obtained a precise estimate of error bounds or deficient quartic spline interpolation. Keywords: Defficient, Error Bound, Quartic Spline, Interpolation I. INTRODUCTION For smooth and more efficient approximation of function cubic and higher degree spline are popular (see Deboor [1]), As in the study of linear spline the maximum error between function and its interpolation can be controlled by mesh spacing such function have corner at the joint two linear pieces and therefore the usually require more data than higher order method to get desired accuracy. In the direction of same higher order method, we refer to Rana & Dubey [5,6], Hall & Mayer [7], the error bounds for quartic spline interpolation obtain by Howell and Verma [ 2 ] and the error bound of spline of degree six obtained by Dubey [ 3 ], (also see Meir and Sharma [9]). The purpose of this paper is to construct a spline method for solving an interpolation problem using piecewise quartic polynomial and obtain error bound. II. EXISTENCE AND UNIQUENESS Let a mesh on [0, 1] be given by P : O = x 0 < x 1 <.x n =1 with h i =x i -x i-1 for i=1, 2 n. Let S(4, p) denote the set of or all algebraic polynomial of degree 4. And s i is the restriction of s over [, ], the class s*(4, p) of deficient quantic spline is defined by S* (4, P) = {s i : s i ε c 1 [0, 1], s i ε s (4, P) for i=1, 2,..n} in S**(4, P) denotes the class of all deficient spline S * (4, P) which satisfies the boundary condition. s(x 0 ) = f(x 0 ), s(x n ) = f(x n ) (2.1) Problem 2.1 : suppose f exist over P then there a exit unique spline interpolation s ε s** (4, P) of f which satisfies interpolation conditions s( i ) = f( i ) s( i ) = f( i ) s 1 ( i ) = f 1 ( i ) (2.2) 103 P a g e
2 Let Q(z) be a quartic polynomial on [0, 1] then it is easy to verify that (2.3) We are now set to answer problem 2.1 in theorem 2.1 Theorem 2.1: There exist a unique deficient quartic spline in S**(4, P) which satisfies interpolatory condition ( 2.1 ) (2.2 ) Proof : Let 1 Then in view of condition (2.1) - (2.2) we now express equation (2.3 ) in terms of restriction s i of s in [x i,x i+1 ] as follows s i (x)=f( i )P 1 (t)+f( i )P 2 (t)+h i f 1 ( i )P 3 (t)+s i (x)p 4 (t)+s i+1 P 5 (t) (2.4) Since sε C 1 [a, b] we have from equation ( 2.4 ) we get Clearly matrix is diagonally dominant and hence invertible. Thus the system of equation has unique solution this completes the proof of theorem 2.1 III. ERROR BOUNDS In this section of paper error bounds e(x) = f(x) s(x) is obtained for the spline interpolation of theorem 2.1 by following approach used by Hall and Meyar. We sell denote by hi [f, x] the unique quartic acquiring with f( i ), f( i ), f ( i ), f(x i ) and f 1 (x i+1 ) and let f ε C 5 [0, 1] Now, consider a first continuously differentiable quartic spline S of theorem 2.1 we have for x i < x < x i+1 f(x) s(x) = f(x) s i (x) < f(x)-l i [f, x] + L i [f,x] - S(x) (3.1) Thus it is clear from (3.1) that in order to get the bounds of e(x) We have to estimate point wise bounds of both the terms on the right hand side of (3.1) by a well know remainder theorem of Cauchy (see Davis [8 ] we see that 104 P a g e
3 (3.2) F = max f (5) (x) we next proceed to obtain bounds for L i [f, x] s i (x) it follows from (2.4 ) that L i [f, x]-s i (x) < e(x i ) P 4 (t) + e(x i+1 ) P 5 (t) (3.3) Thus L i [f, x] s i (x) < e(x i ) P 4 (t) + e(x i+1 ) P 5 (t) (3.4) Since P 4 (t) = [1-9t + 29t 2 39t t 4 ] and 0 < t < 1 = K(t) (say) (3.5) Now using (3.5) in (3.4) we have L i [f, x] s i (x) < max { e(x i ), e(x i+1 ) } k(t) Setting e(x j ) = max e(x i ) (3.6) j=1, 2,.n-1 and h = max (h i ) (3.7) i =1, 2, n-1 we see that (3.6) may be written as L i [f, x]-s(x) < e(x j ) k(t) (3.8) First we have to find upper bound of e(x j ) Replacing s(x j ) by e(x j ) in (3.8) we get Now (3.9) (3.10) Since E(f) is a linear functional which is zero for polynomials of degree 4 or less and applying the peano theorem (see Davis [8 ]) we have (3.11) Now from (3.11) it follows that (3.12) 105 P a g e
4 We rewrite the expression (3.9) in the following symmetric form of x j Thus j < y < x j+1 j < y < j x j < y < j x j-1 < y < j-1 = y j-1 j-1 < y < j-1 From above expression it is follows that j-1 < y < x j E [(x-y) + 4 ] is non negative in x j-1 < y < x j+1 Thus we see that (3.13) Thus we have following from (3.12) when we apply to (3.13) (3.14) Combining (3.7) (3.9) (3.12) with (3.14) we have max e(x j ) = e j j=1, 2,..n-1 Now making use of equation (3.2) and (3.7) in (3.1) and then using (3.15) along with (3.8) we see that (3.15) (3.16) (3.17) Thus we prove the following Theorem 3.1 : Suppose S(x) is the quartic spline interpolation of theorem 2.1 and f ε c 5 [0, 1] then (3.18) k = max c(t) given by (3.17) also we have 0 < x < 1 (3.19) Equation on (3.15) and (3.17) resp. prove the inequality (3.18) and (3.19) of Theorem 3.1 we shall prove the inequality (3.18) is best possible in the limit. we can easily seen that by cauchy formula given in [4 ] that 106 P a g e
5 (3.20) More over for equally spaced knots, we have from (3.9) that (3.21) Consider for a moment (3.22) We have from ( 3.6 ) (3.23) Combine (3.20) and (3.23) we have (3.24) From (3.24) it is clearly observed that (3.18) is best possible provide that. We could prove that (3.25) In fact (3.25) is attained only in the limit. The difficulty will take place in the boundary condition e(x 0 ) = e(x n ) = 0 However it can be shown is that as we move many submitted away from the boundary for that we shall apply (3.21) in inductively to move away from the end condition e(x 0 ) = e(x n )=0 first step in this direction is to show that e(x j )> for j=0,..n Which can be prove by contradiction assumption. Let e(x j )<0 for some i=1, 2, n-1. Now a making use of (3.19) 4>77 this is a contradiction. Hence e(x j )>0 for j=0,.n Now from equation (3.21) Similarly e j > 0 from j=1, 2,..n-1 (3.26) 107 P a g e
6 Again using ( 3.26) in (3.21) We have, Repeated use of (3.21) following that (3.27) Now it can be easily see that r, h, s of And hence in the limiting case (3.28) Which verifies (3.18) inequality thus corresponding to the function (3.29) in the limit for equally spaced knot this complete the proof of theorem (3.1) REFERENCES Journal Papers [1] Deboor, C.A. Practical Guide to splines,applied Mathematical Science, vol.27 Springer,Verlage, Newyork,1979. [2] Howell,G and Verma,A.K Best Error Bound of Quartic Spline Interpolation, J.Approximation Theory [3] 58(1989), [4] Dubey, Y.P. Best error bounds of spline of degree six Int.Journal of Mathematical Analysis vol5 (2011),pp [5] Gemling,R.H.J.and Meyling,G. Interpolation by Bivariate Quintic Spline of class construction of theory of function 87(ed) Sendor et al (1987) [6] Rana,S.S and Dubey,Y.P.Dubey. Best Error bounds of Quinticspline interpolation.j.pure and Applied mathematics 28(10) (1997). [7] Rana,S.S and Dubey, Y.P. Best error bounds of quartic spline interpolation Indian Journal of pure and applied mathematics30(4)(1999) Books [1] Hall, C.A. and Meyar,W.W, J.Approximation Theory 16(1976),pp [2] Davis,P.J. Interpolation and approximation,new York,1969. [3] Meir, A and Sharma,A..Convergence of a class of interpolatory spline J.Approx. Theory (1968) pp P a g e
Deficient Quartic Spline Interpolation
International Journal of Computational Science and Mathematics. ISSN 0974-3189 Volume 3, Number 2 (2011), pp. 227-236 International Research Publication House http://www.irphouse.com Deficient Quartic
More informationConvergence of C 2 Deficient Quartic Spline Interpolation
Advances in Computational Sciences and Technology ISSN 0973-6107 Volume 10, Number 4 (2017) pp. 519-527 Research India Publications http://www.ripublication.com Convergence of C 2 Deficient Quartic Spline
More informationLacunary Interpolation Using Quartic B-Spline
General Letters in Mathematic, Vol. 2, No. 3, June 2017, pp. 129-137 e-issn 2519-9277, p-issn 2519-9269 Available online at http:\\ www.refaad.com Lacunary Interpolation Using Quartic B-Spline 1 Karwan
More informationGeneralised Mean Averaging Interpolation by Discrete Cubic Splines
Publ. RIMS, Kyoto Univ. 30 (1994), 89-95 Generalised Mean Averaging Interpolation by Discrete Cubic Splines By Manjulata SHRIVASTAVA* Abstract The aim of this work is to introduce for a discrete function,
More informationDiscrete Cubic Interpolatory Splines
Publ RIMS, Kyoto Univ. 28 (1992), 825-832 Discrete Cubic Interpolatory Splines By Manjulata SHRIVASTAVA* Abstract In the present paper, existence, uniqueness and convergence properties of a discrete cubic
More informationFebruary 2017 (1/20) 2 Piecewise Polynomial Interpolation 2.2 (Natural) Cubic Splines. MA378/531 Numerical Analysis II ( NA2 )
f f f f f (/2).9.8.7.6.5.4.3.2. S Knots.7.6.5.4.3.2. 5 5.2.8.6.4.2 S Knots.2 5 5.9.8.7.6.5.4.3.2..9.8.7.6.5.4.3.2. S Knots 5 5 S Knots 5 5 5 5.35.3.25.2.5..5 5 5.6.5.4.3.2. 5 5 4 x 3 3.5 3 2.5 2.5.5 5
More informationNatural Quartic Spline
Natural Quartic Spline Rafael E Banchs INTRODUCTION This report describes the natural quartic spline algorithm developed for the enhanced solution of the Time Harmonic Field Electric Logging problem As
More informationPolynomials tend to oscillate (wiggle) a lot, even when our true function does not.
AMSC/CMSC 460 Computational Methods, Fall 2007 UNIT 2: Spline Approximations Dianne P O Leary c 2001, 2002, 2007 Piecewise polynomial interpolation Piecewise polynomial interpolation Read: Chapter 3 Skip:
More informationInterpolation & Polynomial Approximation. Cubic Spline Interpolation II
Interpolation & Polynomial Approximation Cubic Spline Interpolation II Numerical Analysis (9th Edition) R L Burden & J D Faires Beamer Presentation Slides prepared by John Carroll Dublin City University
More informationPositivity Preserving Interpolation of Positive Data by Rational Quadratic Trigonometric Spline
IOSR Journal of Mathematics (IOSR-JM) e-issn: 2278-5728, p-issn:2319-765x. Volume 10, Issue 2 Ver. IV (Mar-Apr. 2014), PP 42-47 Positivity Preserving Interpolation of Positive Data by Rational Quadratic
More informationSolve Non-Linear Parabolic Partial Differential Equation by Spline Collocation Method
Solve Non-Linear Parabolic Partial Differential Equation by Spline Collocation Method P.B. Choksi 1 and A.K. Pathak 2 1 Research Scholar, Rai University,Ahemdabad. Email:pinalchoksey@gmail.com 2 H.O.D.
More informationParameterization. Michael S. Floater. November 10, 2011
Parameterization Michael S. Floater November 10, 2011 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to generate from point
More informationMar. 20 Math 2335 sec 001 Spring 2014
Mar. 20 Math 2335 sec 001 Spring 2014 Chebyshev Polynomials Definition: For an integer n 0 define the function ( ) T n (x) = cos n cos 1 (x), 1 x 1. It can be shown that T n is a polynomial of degree n.
More informationOn the deviation of a parametric cubic spline interpolant from its data polygon
Computer Aided Geometric Design 25 (2008) 148 156 wwwelseviercom/locate/cagd On the deviation of a parametric cubic spline interpolant from its data polygon Michael S Floater Department of Computer Science,
More informationLecture VIII. Global Approximation Methods: I
Lecture VIII Global Approximation Methods: I Gianluca Violante New York University Quantitative Macroeconomics G. Violante, Global Methods p. 1 /29 Global function approximation Global methods: function
More informationCS321 Introduction To Numerical Methods
CS3 Introduction To Numerical Methods Fuhua (Frank) Cheng Department of Computer Science University of Kentucky Lexington KY 456-46 - - Table of Contents Errors and Number Representations 3 Error Types
More informationspecified or may be difficult to handle, we often have a tabulated data
Interpolation Introduction In many practical situations, for a function which either may not be explicitly specified or may be difficult to handle, we often have a tabulated data where and for In such
More informationImportant Properties of B-spline Basis Functions
Important Properties of B-spline Basis Functions P2.1 N i,p (u) = 0 if u is outside the interval [u i, u i+p+1 ) (local support property). For example, note that N 1,3 is a combination of N 1,0, N 2,0,
More informationLinear Interpolating Splines
Jim Lambers MAT 772 Fall Semester 2010-11 Lecture 17 Notes Tese notes correspond to Sections 112, 11, and 114 in te text Linear Interpolating Splines We ave seen tat ig-degree polynomial interpolation
More informationInterpolation and Splines
Interpolation and Splines Anna Gryboś October 23, 27 1 Problem setting Many of physical phenomenona are described by the functions that we don t know exactly. Often we can calculate or measure the values
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 informationAPPM/MATH Problem Set 4 Solutions
APPM/MATH 465 Problem Set 4 Solutions This assignment is due by 4pm on Wednesday, October 16th. You may either turn it in to me in class on Monday or in the box outside my office door (ECOT 35). Minimal
More informationInterpolation by Spline Functions
Interpolation by Spline Functions Com S 477/577 Sep 0 007 High-degree polynomials tend to have large oscillations which are not the characteristics of the original data. To yield smooth interpolating curves
More informationFall CSCI 420: Computer Graphics. 4.2 Splines. Hao Li.
Fall 2014 CSCI 420: Computer Graphics 4.2 Splines Hao Li http://cs420.hao-li.com 1 Roller coaster Next programming assignment involves creating a 3D roller coaster animation We must model the 3D curve
More informationResearch Article Data Visualization Using Rational Trigonometric Spline
Applied Mathematics Volume Article ID 97 pages http://dx.doi.org/.//97 Research Article Data Visualization Using Rational Trigonometric Spline Uzma Bashir and Jamaludin Md. Ali School of Mathematical Sciences
More informationConsider functions such that then satisfies these properties: So is represented by the cubic polynomials on on and on.
1 of 9 3/1/2006 2:28 PM ne previo Next: Trigonometric Interpolation Up: Spline Interpolation Previous: Piecewise Linear Case Cubic Splines A piece-wise technique which is very popular. Recall the philosophy
More informationQuasi-Quartic Trigonometric Bézier Curves and Surfaces with Shape Parameters
Quasi-Quartic Trigonometric Bézier Curves and Surfaces with Shape Parameters Reenu Sharma Assistant Professor, Department of Mathematics, Mata Gujri Mahila Mahavidyalaya, Jabalpur, Madhya Pradesh, India
More informationVideo 11.1 Vijay Kumar. Property of University of Pennsylvania, Vijay Kumar
Video 11.1 Vijay Kumar 1 Smooth three dimensional trajectories START INT. POSITION INT. POSITION GOAL Applications Trajectory generation in robotics Planning trajectories for quad rotors 2 Motion Planning
More information8 Piecewise Polynomial Interpolation
Applied Math Notes by R. J. LeVeque 8 Piecewise Polynomial Interpolation 8. Pitfalls of high order interpolation Suppose we know the value of a function at several points on an interval and we wish to
More informationSplines. Parameterization of a Curve. Curve Representations. Roller coaster. What Do We Need From Curves in Computer Graphics? Modeling Complex Shapes
CSCI 420 Computer Graphics Lecture 8 Splines Jernej Barbic University of Southern California Hermite Splines Bezier Splines Catmull-Rom Splines Other Cubic Splines [Angel Ch 12.4-12.12] Roller coaster
More informationarxiv: v1 [math.gt] 11 May 2018
TRIPLE CROSSING NUMBER AND DOUBLE CROSSING BRAID INDEX DAISHIRO NISHIDA arxiv:1805.04428v1 [math.gt] 11 May 2018 Abstract. Traditionally, knot theorists have considered projections of knots where there
More informationParameterization of triangular meshes
Parameterization of triangular meshes Michael S. Floater November 10, 2009 Triangular meshes are often used to represent surfaces, at least initially, one reason being that meshes are relatively easy to
More informationA C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions
A C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions Nira Dyn School of Mathematical Sciences Tel Aviv University Michael S. Floater Department of Informatics University of
More informationLower estimate of the square-to-linear ratio for regular Peano curves
DOI 10.1515/dma-2014-0012 Discrete Math. Appl. 2014; 24 (3):123 12 Konstantin E. Bauman Lower estimate of the square-to-linear ratio for regular Peano curves Abstract: We prove that the square-to-linear
More informationSome bounds on chromatic number of NI graphs
International Journal of Mathematics and Soft Computing Vol.2, No.2. (2012), 79 83. ISSN 2249 3328 Some bounds on chromatic number of NI graphs Selvam Avadayappan Department of Mathematics, V.H.N.S.N.College,
More informationModule 4 : Solving Linear Algebraic Equations Section 11 Appendix C: Steepest Descent / Gradient Search Method
Module 4 : Solving Linear Algebraic Equations Section 11 Appendix C: Steepest Descent / Gradient Search Method 11 Appendix C: Steepest Descent / Gradient Search Method In the module on Problem Discretization
More informationA Modified Spline Interpolation Method for Function Reconstruction from Its Zero-Crossings
Scientific Papers, University of Latvia, 2010. Vol. 756 Computer Science and Information Technologies 207 220 P. A Modified Spline Interpolation Method for Function Reconstruction from Its Zero-Crossings
More informationLecture 9: Introduction to Spline Curves
Lecture 9: Introduction to Spline Curves Splines are used in graphics to represent smooth curves and surfaces. They use a small set of control points (knots) and a function that generates a curve through
More informationChapter 18 out of 37 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal.
Chapter 8 out of 7 from Discrete Mathematics for Neophytes: Number Theory, Probability, Algorithms, and Other Stuff by J. M. Cargal 8 Matrices Definitions and Basic Operations Matrix algebra is also known
More informationAn Improved Measurement Placement Algorithm for Network Observability
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 819 An Improved Measurement Placement Algorithm for Network Observability Bei Gou and Ali Abur, Senior Member, IEEE Abstract This paper
More informationTable for Third-Degree Spline Interpolation Using Equi-Spaced Knots. By W. D. Hoskins
MATHEMATICS OF COMPUTATION, VOLUME 25, NUMBER 116, OCTOBER, 1971 Table for Third-Degree Spline Interpolation Using Equi-Spaced Knots By W. D. Hoskins Abstract. A table is given for the calculation of the
More information(Refer Slide Time: 00:02:24 min)
CAD / CAM Prof. Dr. P. V. Madhusudhan Rao Department of Mechanical Engineering Indian Institute of Technology, Delhi Lecture No. # 9 Parametric Surfaces II So these days, we are discussing the subject
More informationA new 8-node quadrilateral spline finite element
Journal of Computational and Applied Mathematics 195 (2006) 54 65 www.elsevier.com/locate/cam A new 8-node quadrilateral spline finite element Chong-Jun Li, Ren-Hong Wang Institute of Mathematical Sciences,
More informationFriday, 11 January 13. Interpolation
Interpolation Interpolation Interpolation is not a branch of mathematic but a collection of techniques useful for solving computer graphics problems Basically an interpolant is a way of changing one number
More informationBilinear Programming
Bilinear Programming Artyom G. Nahapetyan Center for Applied Optimization Industrial and Systems Engineering Department University of Florida Gainesville, Florida 32611-6595 Email address: artyom@ufl.edu
More informationSAT-CNF Is N P-complete
SAT-CNF Is N P-complete Rod Howell Kansas State University November 9, 2000 The purpose of this paper is to give a detailed presentation of an N P- completeness proof using the definition of N P given
More informationCurve fitting using linear models
Curve fitting using linear models Rasmus Waagepetersen Department of Mathematics Aalborg University Denmark September 28, 2012 1 / 12 Outline for today linear models and basis functions polynomial regression
More informationCubic Spline Interpolation with New Conditions on M 0 and M n
Cubic Spline Interpolation with New Conditions on M 0 and M n 1 Parcha Kalyani and P.S. Rama Chandra Rao 2 1,2 Kakatiya Institute of Technology and Sciences, Warangal-506009-India Abstract In this communication,
More informationOn the Dimension of the Bivariate Spline Space S 1 3( )
On the Dimension of the Bivariate Spline Space S 1 3( ) Gašper Jaklič Institute of Mathematics, Physics and Mechanics University of Ljubljana Jadranska 19, 1000 Ljubljana, Slovenia Gasper.Jaklic@fmf.uni-lj.si
More informationPiecewise Polynomial Interpolation, cont d
Jim Lambers MAT 460/560 Fall Semester 2009-0 Lecture 2 Notes Tese notes correspond to Section 4 in te text Piecewise Polynomial Interpolation, cont d Constructing Cubic Splines, cont d Having determined
More informationExtremal Graph Theory: Turán s Theorem
Bridgewater State University Virtual Commons - Bridgewater State University Honors Program Theses and Projects Undergraduate Honors Program 5-9-07 Extremal Graph Theory: Turán s Theorem Vincent Vascimini
More informationAdvanced Operations Research Techniques IE316. Quiz 1 Review. Dr. Ted Ralphs
Advanced Operations Research Techniques IE316 Quiz 1 Review Dr. Ted Ralphs IE316 Quiz 1 Review 1 Reading for The Quiz Material covered in detail in lecture. 1.1, 1.4, 2.1-2.6, 3.1-3.3, 3.5 Background material
More informationTILING RECTANGLES SIMON RUBINSTEIN-SALZEDO
TILING RECTANGLES SIMON RUBINSTEIN-SALZEDO. A classic tiling problem Question.. Suppose we tile a (large) rectangle with small rectangles, so that each small rectangle has at least one pair of sides with
More informationMath 485, Graph Theory: Homework #3
Math 485, Graph Theory: Homework #3 Stephen G Simpson Due Monday, October 26, 2009 The assignment consists of Exercises 2129, 2135, 2137, 2218, 238, 2310, 2313, 2314, 2315 in the West textbook, plus the
More informationSEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION
CHAPTER 5 SEQUENCES, MATHEMATICAL INDUCTION, AND RECURSION Alessandro Artale UniBZ - http://www.inf.unibz.it/ artale/ SECTION 5.5 Application: Correctness of Algorithms Copyright Cengage Learning. All
More informationLocal Approximation by Splines with Displacement of Nodes
ISSN 1055-1344, Siberian Advances in Mathematics, 013, Vol. 3, No. 1, pp. 69 75. c Allerton Press, Inc., 013. Original Russian Text c Yu. S. Volkov, E. V. Strelkova, and V. T. Shevaldin, 011, published
More informationCPS 102: Discrete Mathematics. Quiz 3 Date: Wednesday November 30, Instructor: Bruce Maggs NAME: Prob # Score. Total 60
CPS 102: Discrete Mathematics Instructor: Bruce Maggs Quiz 3 Date: Wednesday November 30, 2011 NAME: Prob # Score Max Score 1 10 2 10 3 10 4 10 5 10 6 10 Total 60 1 Problem 1 [10 points] Find a minimum-cost
More informationBlock-based Thiele-like blending rational interpolation
Journal of Computational and Applied Mathematics 195 (2006) 312 325 www.elsevier.com/locate/cam Block-based Thiele-like blending rational interpolation Qian-Jin Zhao a, Jieqing Tan b, a School of Computer
More information(i, j)-almost Continuity and (i, j)-weakly Continuity in Fuzzy Bitopological Spaces
International Journal of Scientific and Innovative Mathematical Research (IJSIMR) Volume 4, Issue 2, February 2016, PP 89-98 ISSN 2347-307X (Print) & ISSN 2347-3142 (Online) www.arcjournals.org (i, j)-almost
More informationA C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions
A C 2 Four-Point Subdivision Scheme with Fourth Order Accuracy and its Extensions Nira Dyn Michael S. Floater Kai Hormann Abstract. We present a new four-point subdivision scheme that generates C 2 curves.
More informationA New Class of Quasi-Cubic Trigonometric Bezier Curve and Surfaces
A New Class of Quasi-Cubic Trigonometric Bezier Curve and Surfaces Mridula Dube 1, Urvashi Mishra 2 1 Department of Mathematics and Computer Science, R.D. University, Jabalpur, Madhya Pradesh, India 2
More informationBézier Splines. B-Splines. B-Splines. CS 475 / CS 675 Computer Graphics. Lecture 14 : Modelling Curves 3 B-Splines. n i t i 1 t n i. J n,i.
Bézier Splines CS 475 / CS 675 Computer Graphics Lecture 14 : Modelling Curves 3 n P t = B i J n,i t with 0 t 1 J n, i t = i=0 n i t i 1 t n i No local control. Degree restricted by the control polygon.
More informationRational Bezier Surface
Rational Bezier Surface The perspective projection of a 4-dimensional polynomial Bezier surface, S w n ( u, v) B i n i 0 m j 0, u ( ) B j m, v ( ) P w ij ME525x NURBS Curve and Surface Modeling Page 97
More informationLecture 2 September 3
EE 381V: Large Scale Optimization Fall 2012 Lecture 2 September 3 Lecturer: Caramanis & Sanghavi Scribe: Hongbo Si, Qiaoyang Ye 2.1 Overview of the last Lecture The focus of the last lecture was to give
More informationBounds on the signed domination number of a graph.
Bounds on the signed domination number of a graph. Ruth Haas and Thomas B. Wexler September 7, 00 Abstract Let G = (V, E) be a simple graph on vertex set V and define a function f : V {, }. The function
More informationON COMMON FIXED POINT THEOREMS FOR SEMI-COMPATIBLE AND OCCASIONALLY WEAKLY COMPATIBLE MAPPINGS IN MENGER SPACE
www.arpapress.com/volumes/vol14issue3/ijrras_14_3_23.pdf ON COMMON FIXED POINT THEOREMS FOR SEMI-COMPATIBLE AND OCCASIONALLY WEAKLY COMPATIBLE MAPPINGS IN MENGER SPACE Arihant Jain 1 & Basant Chaudhary
More informationand the crooked shall be made straight, and the rough ways shall be made smooth; Luke 3:5
ecture 8: Knot Insertion Algorithms for B-Spline Curves and Surfaces and the crooked shall be made straight, and the rough ways shall be made smooth; uke 3:5. Motivation B-spline methods have several advantages
More informationA Quintic Spline method for fourth order singularly perturbed Boundary Value Problem of Reaction-Diffusion type
A Quintic Spline method for fourth order singularly perturbed Boundary Value Problem of Reaction-Diffusion type Jigisha U. Pandya and Harish D. Doctor Asst. Prof., Department of Mathematics, Sarvajanik
More informationThe 4/5 Upper Bound on the Game Total Domination Number
The 4/ Upper Bound on the Game Total Domination Number Michael A. Henning a Sandi Klavžar b,c,d Douglas F. Rall e a Department of Mathematics, University of Johannesburg, South Africa mahenning@uj.ac.za
More information1, 2
A QUARTIC LEGENDRE SPLINE COLLOCATION METHOD TO SOLVE FREDHOLM INTEGRO DIFFERENTIAL EQUATION B. M. Pya 1, D. C. Joshi 2 1 Asst. Prof., Dept.of Applied Mathematics, Sardar Vallabhbhai Patel Institute of
More informationA note on isolate domination
Electronic Journal of Graph Theory and Applications 4 (1) (016), 94 100 A note on isolate domination I. Sahul Hamid a, S. Balamurugan b, A. Navaneethakrishnan c a Department of Mathematics, The Madura
More informationLet v be a vertex primed by v i (s). Then the number f(v) of neighbours of v which have
Let v be a vertex primed by v i (s). Then the number f(v) of neighbours of v which have been red in the sequence up to and including v i (s) is deg(v)? s(v), and by the induction hypothesis this sequence
More informationRational Bezier Curves
Rational Bezier Curves Use of homogeneous coordinates Rational spline curve: define a curve in one higher dimension space, project it down on the homogenizing variable Mathematical formulation: n P(u)
More informationA Cumulative Averaging Method for Piecewise Polynomial Approximation to Discrete Data
Applied Mathematical Sciences, Vol. 1, 16, no. 7, 331-343 HIKARI Ltd, www.m-hiari.com http://dx.doi.org/1.1988/ams.16.5177 A Cumulative Averaging Method for Piecewise Polynomial Approximation to Discrete
More informationConstructing Cubic Splines on the Sphere
Constructing Cubic Splines on the Sphere Except where reference is made to the work of others, the work described in this thesis is my own or was done in collaboration with my advisory committee. This
More informationQuasilinear First-Order PDEs
MODULE 2: FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 16 Lecture 3 Quasilinear First-Order PDEs A first order quasilinear PDE is of the form a(x, y, z) + b(x, y, z) x y = c(x, y, z). (1) Such equations
More informationInteractive Graphics. Lecture 9: Introduction to Spline Curves. Interactive Graphics Lecture 9: Slide 1
Interactive Graphics Lecture 9: Introduction to Spline Curves Interactive Graphics Lecture 9: Slide 1 Interactive Graphics Lecture 13: Slide 2 Splines The word spline comes from the ship building trade
More informationLecture 25: Bezier Subdivision. And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10
Lecture 25: Bezier Subdivision And he took unto him all these, and divided them in the midst, and laid each piece one against another: Genesis 15:10 1. Divide and Conquer If we are going to build useful
More informationTHE CONNECTED COMPLEMENT DOMINATION IN GRAPHS V.MOHANASELVI 1. Assistant Professor of Mathematics, Nehru Memorial College, Puthanampatti,
THE CONNECTED COMPLEMENT DOMINATION IN GRAPHS V.MOHANASELVI 1 Assistant Professor of Mathematics, Nehru Memorial College, Puthanampatti, Tiruchirappalli-621 00 S.DHIVYAKANNU 2 Assistant Professor of Mathematics,
More informationParameterization for curve interpolation
Working title: Topics in Multivariate Approximation and Interpolation 101 K. Jetter et al., Editors c 2005 Elsevier B.V. All rights reserved Parameterization for curve interpolation Michael S. Floater
More informationCS 475 / CS Computer Graphics. Modelling Curves 3 - B-Splines
CS 475 / CS 675 - Computer Graphics Modelling Curves 3 - Bézier Splines n P t = i=0 No local control. B i J n,i t with 0 t 1 J n,i t = n i t i 1 t n i Degree restricted by the control polygon. http://www.cs.mtu.edu/~shene/courses/cs3621/notes/spline/bezier/bezier-move-ct-pt.html
More informationCLASSES OF VERY STRONGLY PERFECT GRAPHS. Ganesh R. Gandal 1, R. Mary Jeya Jothi 2. 1 Department of Mathematics. Sathyabama University Chennai, INDIA
Inter national Journal of Pure and Applied Mathematics Volume 113 No. 10 2017, 334 342 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu ijpam.eu Abstract: CLASSES
More informationMATH 890 HOMEWORK 2 DAVID MEREDITH
MATH 890 HOMEWORK 2 DAVID MEREDITH (1) Suppose P and Q are polyhedra. Then P Q is a polyhedron. Moreover if P and Q are polytopes then P Q is a polytope. The facets of P Q are either F Q where F is a facet
More informationUnit 7 Number System and Bases. 7.1 Number System. 7.2 Binary Numbers. 7.3 Adding and Subtracting Binary Numbers. 7.4 Multiplying Binary Numbers
Contents STRAND B: Number Theory Unit 7 Number System and Bases Student Text Contents Section 7. Number System 7.2 Binary Numbers 7.3 Adding and Subtracting Binary Numbers 7.4 Multiplying Binary Numbers
More informationWinning Positions in Simplicial Nim
Winning Positions in Simplicial Nim David Horrocks Department of Mathematics and Statistics University of Prince Edward Island Charlottetown, Prince Edward Island, Canada, C1A 4P3 dhorrocks@upei.ca Submitted:
More informationRigid Tilings of Quadrants by L-Shaped n-ominoes and Notched Rectangles
Rigid Tilings of Quadrants by L-Shaped n-ominoes and Notched Rectangles Aaron Calderon a, Samantha Fairchild b, Michael Muir c, Viorel Nitica c, Samuel Simon d a Department of Mathematics, The University
More informationMATH 423 Linear Algebra II Lecture 17: Reduced row echelon form (continued). Determinant of a matrix.
MATH 423 Linear Algebra II Lecture 17: Reduced row echelon form (continued). Determinant of a matrix. Row echelon form A matrix is said to be in the row echelon form if the leading entries shift to the
More informationNURBS: Non-Uniform Rational B-Splines AUI Course Denbigh Starkey
NURBS: Non-Uniform Rational B-Splines AUI Course Denbigh Starkey 1. Background 2 2. Definitions 3 3. Using NURBS to define a circle 4 4. Homogeneous coordinates & control points at infinity 9 5. Constructing
More informationPS Geometric Modeling Homework Assignment Sheet I (Due 20-Oct-2017)
Homework Assignment Sheet I (Due 20-Oct-2017) Assignment 1 Let n N and A be a finite set of cardinality n = A. By definition, a permutation of A is a bijective function from A to A. Prove that there exist
More informationABSTRACT. Keywords: Continuity; interpolation; monotonicity; rational Bernstein-Bézier ABSTRAK
Sains Malaysiana 40(10)(2011): 1173 1178 Improved Sufficient Conditions for Monotonic Piecewise Rational Quartic Interpolation (Syarat Cukup yang Lebih Baik untuk Interpolasi Kuartik Nisbah Cebis Demi
More informationNonparametric regression using kernel and spline methods
Nonparametric regression using kernel and spline methods Jean D. Opsomer F. Jay Breidt March 3, 016 1 The statistical model When applying nonparametric regression methods, the researcher is interested
More informationLemma (x, y, z) is a Pythagorean triple iff (y, x, z) is a Pythagorean triple.
Chapter Pythagorean Triples.1 Introduction. The Pythagorean triples have been known since the time of Euclid and can be found in the third century work Arithmetica by Diophantus [9]. An ancient Babylonian
More informationMultiple-Choice Test Spline Method Interpolation COMPLETE SOLUTION SET
Multiple-Choice Test Spline Method Interpolation COMPLETE SOLUTION SET 1. The ollowing n data points, ( x ), ( x ),.. ( x, ) 1, y 1, y n y n quadratic spline interpolation the x-data needs to be (A) equally
More informationComponent Connectivity of Generalized Petersen Graphs
March 11, 01 International Journal of Computer Mathematics FeHa0 11 01 To appear in the International Journal of Computer Mathematics Vol. 00, No. 00, Month 01X, 1 1 Component Connectivity of Generalized
More informationFixed points of Kannan mappings in metric spaces endowed with a graph
An. Şt. Univ. Ovidius Constanţa Vol. 20(1), 2012, 31 40 Fixed points of Kannan mappings in metric spaces endowed with a graph Florin Bojor Abstract Let (X, d) be a metric space endowed with a graph G such
More informationINTERNATIONAL JOURNAL OF ELECTRONICS AND COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET)
INTERNATIONAL JOURNAL OF ELECTRONICS AND COMMUNICATION ENGINEERING & TECHNOLOGY (IJECET) International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 ISSN 0976 6464(Print)
More informationPAIRED-DOMINATION. S. Fitzpatrick. Dalhousie University, Halifax, Canada, B3H 3J5. and B. Hartnell. Saint Mary s University, Halifax, Canada, B3H 3C3
Discussiones Mathematicae Graph Theory 18 (1998 ) 63 72 PAIRED-DOMINATION S. Fitzpatrick Dalhousie University, Halifax, Canada, B3H 3J5 and B. Hartnell Saint Mary s University, Halifax, Canada, B3H 3C3
More informationAlgebraic Graph Theory- Adjacency Matrix and Spectrum
Algebraic Graph Theory- Adjacency Matrix and Spectrum Michael Levet December 24, 2013 Introduction This tutorial will introduce the adjacency matrix, as well as spectral graph theory. For those familiar
More informationA New Algorithm for Developing Block Methods for Solving Fourth Order Ordinary Differential Equations
Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 12, Number 2 (2016), pp. 1465-1471 Research India Publications http://www.ripublication.com A New Algorithm for Developing Block Methods
More information3.7. Vertex and tangent
3.7. Vertex and tangent Example 1. At the right we have drawn the graph of the cubic polynomial f(x) = x 2 (3 x). Notice how the structure of the graph matches the form of the algebraic expression. The
More information