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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
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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 that the set of polynomial of degree at most . 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
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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;
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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.
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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.
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