Easy way to Find Multivariate Interpolation

Transcription

1 International Journal of Emerging Trends in Science Technology IC Value: 7689 (Index Copernicus) Impact Factor: 4219 DOI: Easy way to Find Multivariate Interpolation Author Yimesgen Mehari Faculty of Natural Computational Science, Department of Mathematics Adigrat University, Ethiopia Abstract We derive explicit interpolation formula using non-singular vermonde 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 scientists commonly assume that relationships between variables in physical problem can be approximately reproduced from data given by the problem The ultimate goal might be to determine the values at intermediate points, to approximate the integral or to simply give a smooth or continuous representation of the variables in the problem Interpolation is the method of estimating unknown values with the help of given set of observations According to Theile Interpolation is, The art of reading between the lines of the table According to WM Harper: Interpolation consists in reading a value which lies between two extreme points [3] Also interpolation means insertion or filling up intermediate terms of the series It is the technique of obtaining the value of a function for any intermediate values of the independent variables ie, of argument when the values of the function corresponding to number of values of argument are given The most elementary type of interpolation consists of fitting a polynomial to a collection of data points called polynomial interpolation Multivariate interpolation is an interpolation of function more than one variable The problem of finding smooth interpolate for several variable is a difficult one that has much attention both in the past currently But there is no theoretical difficulty in setting up a frame work for discussing interpolation of multivariate function f whose values are known [1] Interpolation function of more than one variable has become increasingly important in the past few years These days, application ranges over many different field of pure applied mathematics For example interpolation finds applications in the numerical integrations of differential equations, topography, the computer aided geometric design of cars, ships, airplanes [1] 1 Polynomial interpolation Given the values of function values of x say, polynomial say It implies, for us write in matrix form: distinct we can find a Yimesgen Mehari wwwijetstin Page 5189

2 This system of equation has unique solution if the matrix functions in the linear space of all continuous mapping from to, the real numbers Thus none of the functions can be expressed as a linear combination of the others Finally, let denote the span of That is the set of all linear combination of let denote the function that is not in Then we can obtain a unique solution of the interpolating problem of determining if only if the matrix called vermonde matrix, is non-singular If then the matrix V is non-singular But the determinant of V is given by the product is taken over all i j such that Since the abscissas are all distinct it is clear from (3) that Thus, the vermonde matrix V is non-singular the system of linear equation (1) has a unique solution So, our target is how to determine the coefficients To find the values of those coefficients we can use matrix inverse method as follow;, is non-singular As special case, let us take choose the as first N monomials then is simply the set of polynomial of degree at most Then the above matrix A is vermonde matrix whose form is given in (2) It implies that Note: If the set of data points are large in number we can use MATH Lab Software to find inverse of the matrix Example: Suppose we are given three data points (0, 0, 1), (0, 1, 2), (1, 1, 3) that lie on Then the interpolating polynomial that passes through these points is of the form Finally, replace the corresponding values of We obtain interpolation polynomial of degree at most n 2 Multivariate case denote N distinct abscissas in denote N linearly independent in Yimesgen Mehari wwwijetstin Page 5190

3 It implies 3 Multivariate interpolation using determinant of matrix be two set of abscissas, then the Lagrange multivariate interpolation on is given by Thus, Therefore, the interpolation polynomial is; Example 2: Suppose we are given four data points (-1, -1, 1), (-1, 1, 5),(1, -1, -5) (1,1 3) that lie on Then the interpolating polynomial that passes through these points is of the form It implies With the property zero all other points on Equation (31) is called two variable Lagrange interpolations [1] This interpolation formula works only when the data points are taken from a rectangular region (or the point Now we can find interpolation of multivariate function on a set of distinct abscissas (the points are not necessarily taken from ) using determinant of matrix as follow be multinomial function, the we need to find multivariate interpolation polynomial for Suppose for be the interpolation points, then we have the following system of equations Thus, Therefore, the interpolation polynomial is; Yimesgen Mehari wwwijetstin Page 5191

4 In matrix form; Therefore, the formula, is called Lagrange Multivariate interpolation formula Example 1: Suppose we are given three data points (0, 0, 1), (0, 1, 2), (1, 1, 3) that lie on then find the interpolating polynomial Solution: Since we have given three data points the interpolating polynomial can be defined uniquely as be a matrix obtained by replacing the row of the above matrix by the monomials in beginning from the second column From (33) From (34) From the interpolation formula, (36) we have Note: From the matrix, if the rows are the same (have same position) then, If rows are different (have different position), then the matrix has different rows with equal entries It implies, Thus, with the properties is an interpolation polynomial Example 2: Suppose we are given four data points (-1, -1, 1), (-1, 1, 5),(1, -1, -5) (1,1 3) that lie on Then the interpolating polynomial that passes through these points is of the form Yimesgen Mehari wwwijetstin Page 5192

5 From (33) From (34) Specifically, we showed how to interpolate function of two independent variables at distinct points give examples to show how the formula is used in practice Two dimensional interpolations are implemented through double blending process by applying one dimensional interpolation operation on one coordinate at a time while keeping the values of the other coordinate make fixed As the same manner we can extend these interpolation method to three, four,,d dimensions From the interpolation formula, (36) we have Reference 1 GEORGE MPHILLIPS, Interpolations Approximation by polynomials, Canadian mathematical society SCHOICHIRONAKAMORA, Numerical analysis Graphic visualization with MATLA the ohw state university, AKJaiswal, Text Book of computer based numerical statistical techniques, New age international publishers 4 JF STEFFENSEN, Interpolation Dover Publications, Inc, New York, second edition, 2006 polynomial is an interpolation Conclusion We derived function of two variables interpolation formula using determinant inverse of nonsingular matrix If that matrix is singular we can t represent the interpolation polynomial uniquely, in case all the given interpolant points are distinct non-collinear the matrix that we used in this method is non-singular The points that results in a singular matrix deserves further investigation Yimesgen Mehari wwwijetstin Page 5193