**Comparison of Lagrangian and Multivariate Interpolation**

This project compared the "efficiency" between Lagrangian Polynomial Interpolation and Multivariate Polynomial Interpolation. The results determined what one should use to attain an equation that most resembles a set of points. Additionally, an autonomous process of interpolation was constructed for both methods using Python. The modules Numpy, Sympy, and Scipy were integrated into some of the mathematical processes that had to be implemented into the code. These included the following: Rectification, the Vandermonde Matrix, intricate integration, substitution of variables for both univariate and bivariate instances, and the determination of the roots of a polynomial. For every equation that received the Lagrangian Interpolation treatment, the same initial equation was used and received the Multivariate Interpolation treatment. The variable "t," which denotes the amount of times the function had to be further partitioned and interpolated, was compared amongst both procedures. There were many interpolated equations that did not equate to the original function. To represent these equations, the y-values of the functions at an arbitrarily high x-value were acquired because continuity and expectedness of each equation at this value can be assumed. Additionally, the percent error of the most accurate y-values for each equation type for both methods and the y-values of every original equation were analyzed through a matched pairs t-test. For the purpose of this investigation, the research question should be constructed as follows: Which Method of Polynomial Interpolation is More Efficient/Accurate: Lagrangian Interpolation or Multivariate Interpolation?

Keywords: Polynomial, Lagrangian, Multivariate

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