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Easy Way to Find Multivariate Interpolation

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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. Harper: “Interpolation consists in reading a value which lies between two 1. Polynomial interpolation extreme points”.[3] Also interpolation means Given the values of function distinct insertion or filling up intermediate terms of the values of x say, we can find a series. It is the technique of obtaining the value of polynomial say a function for any intermediate values of the such that for independent variables i.e., of argument when the It implies, values of the function corresponding to number of values of argument are given. The most elementary type of interpolation consists of fitting Let us write in matrix form: 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 Yimesgen Mehari www.ijetst.in Page 5189 IJETST- Vol.||04||Issue||05||Pages 5189-5193||May||ISSN 2348-9480 2017 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 and let denote the function that is not in Then we can obtain a unique solution of the This system of equation has unique solution if the interpolating problem of determining such that matrix if and only if the matrix called vandermonde matrix, is non-singular. If then the matrix V is non-singular. But the determinant of V is given by is non-singular. As special case, let us take choose the Where the product is taken over all i and j such as first N monomials and then is simply the set of polynomial of degree at most . that Then the above matrix A is vandermonde Since the abscissas are all distinct it is clear from (3) that matrix whose form is given in Thus, the vandermonde matrix V is non-singular (2). and the system of linear equation (1) has a unique It implies that solution. So, our target is how to determine the coefficients To find the values of those coefficients we can use Where matrix inverse method as follow; Note: If the set of data points are large in number Where, 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), and (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 in By inserting the given data points we obtain the We obtain interpolation polynomial of degree at following system of equations. most n. 2. Multivariate case This system is equivalent to; Let denote N distinct abscissas in and denote N linearly independent Yimesgen Mehari www.ijetst.in Page 5190 IJETST- Vol.||04||Issue||05||Pages 5189-5193||May||ISSN 2348-9480 2017 3. Multivariate interpolation using determinant of matrix Let be two Where set of abscissas, then the Lagrange multivariate interpolation on is and given by It implies Thus, . Therefore, the interpolation polynomial is; Example 2: Suppose we are given four data points (-1, -1, 1), (-1, 1, 5),(1, -1, -5) and (1,1 3) that lie on Then the interpolating polynomial that passes through these points is of the form . By inserting the given data points we obtain the following system of equations. With the property and zero all other points on Equation (3.1) is called two variable Lagrange interpolations [1]. This system is equivalent to; 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 Where are not necessarily taken from ) using determinant of matrix as follow. and Let be multinomial function, the we It implies need to find multivariate interpolation polynomial such that for Suppose for be the interpolation points, then we have the following system of equations. Thus, . Therefore, the interpolation polynomial is; Yimesgen Mehari www.ijetst.in Page 5191 IJETST- Vol.||04||Issue||05||Pages 5189-5193||May||ISSN 2348-9480 2017 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), and (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 By inserting the given data points we obtain the following system of equations. From (3.3) Let be a matrix obtained by replacing the row of the above matrix by the monomials in and beginning from the second column. From (3.4) Let and From the interpolation formula, (3.6) we have is an interpolation polynomial. Example 2: Suppose we are given four data Note: From the matrix, if the and points (-1, -1, 1), (-1, 1, 5),(1, -1, -5) and (1,1 3) rows are the same (have same position) then, that lie on Then the interpolating polynomial that passes through these points is of If rows are different (have the form different position), then the matrix has different . rows with equal entries. It implies, Thus, with the properties By inserting the given data points we obtain the following system of equations. Yimesgen Mehari www.ijetst.in Page 5192 IJETST- Vol.||04||Issue||05||Pages 5189-5193||May||ISSN 2348-9480 2017 Specifically, we showed how to interpolate This system is equivalent to; function of two independent variables at distinct points and give examples to show how the formula is used in practice. From (3.3) 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 From (3.4) the other coordinate make fixed. As the same manner we can extend these interpolation method to three, four,…,d dimensions. Reference 1. GEORGE M.PHILLIPS, Interpolations and Approximation by polynomials, Canadian mathematical society 2003. 2. SCHOICHIRO.NAKAMORA, Numerical analysis and Graphic visualization with MATLA the ohw state university, 1996. 3. A.K.Jaiswal, Text Book of computer based From the interpolation formula, (3.6) we have numerical and statistical techniques, New age international publishers. 4. J.F. STEFFENSEN, Interpolation. Dover Publications, Inc., New York, second edition, 2006. is an interpolation polynomial. Conclusion We derived function of two variables interpolation formula using determinant and inverse of non- singular matrix. If that matrix is singular we can’t represent the interpolation polynomial uniquely, in case all the given interpolant points are distinct and 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 www.ijetst.in Page 5193 .
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