Alex Lewis Final Project
Multivariate Lagrange Interpolation
Abstract. Explain how the standard linear Lagrange interpolation can be generalized to construct a formula that interpolates a set of points in . We will also provide examples to show how the formula is used in practice.
1 Introduction
Interpolation is a fundamental topic in Numerical Analysis. It is the practice of creating a function that fits a finite sampled data set. These sampled values can be used to construct an interpolant, which must fit the interpolated function at each date point.
2 Polynomial Interpolation “The Problem of determining a polynomial of degree one that passes through the distinct points and is the same as approximating a function for which and by means of a first degree polynomial interpolating, or agreeing with, the values of f at the given points. We define the functions
and , And then define
We can generalize this concept of linear interpolation and consider a polynomial of degree n that passes through n+1 points.
In this case we can construct, for each , a function with the property that when and . To satisfy for each requires that the numerator of contains the term
To satisfy , the denominator of must be equal to this term evaluated at . Thus,
This is called the Th Lagrange Interpolating Polynomial The polynomial is given once again as,
Example 1 If and , then is the polynomial that agrees with at , and ; that is,
3 Multivariate Interpolation First we define a function be an -variable multinomial function of degree . In other words, the , could, in our case be ; in this case it would be a 3 variable function of some degree . Since there are number of terms, it is a necessary condition that we have number of distinct points for to be uniquely defined. In other words,
Where the ’s are all coefficients in , and is the -tuple of independent variables of .
Following Lagrange method, we wish to write in the form , where is a multinomial function in the independent variables with the property that when is equal to to the data value, or 1, ,…, , , then ℓx =1 and ℓ x =0 ≠ . To do this, let us consider the system of linear equations where . From this system we can construct the a matrix
We must assume , that is M must be non-singular. If is singular then it means that the coefficients of are not uniquely determined, in which case would not be unique.
Let . Now we make substitutions in ; this produces the following matrix :
Now, let ). Now make the substitutions in . This gives the following matrix,
Note that the appears twice in . That means . Thus when then . By construction, , Hence
And thus
Looks pretty similar to our Lagrange Polynomial, right?
Example 2
Suppose we are given three data points and that lie on . These points define uniquely a linear function of two variables, so , for some coefficients . Hence the coefficients must satisfy
Thus it follows that
We get F:\NUMERICAL ANALYSIS\MutiVariate.mw
Example 3 Suppose we are given the following data points, and , that lie on . These points define uniquely a degree two function of two variables, so , , for some coefficients . Hence the coefficients must satisfy
And the determinants of each matrix are as follows. (done using Maple)
F:\NUMERICAL ANALYSIS\MultiVariate2.mw
4 Conclusion We have given a brief review of Lagrange interpolation and shown how to apply this to construct a multivariate Lagrange interpolant. We have also given examples for both cases to show both methods in practice.
5 References
Saniee, Kamron. "A Simple Expression for Multivariate Lagrange Interpolation." SIAM (2007). Web. 16 May 2010.
Burden, Richard L., and J. Douglas. Faires. Numerical Analysis. Boston: PWS- Kent Pub., 1993. Print.