Numerical analysis is a branch of mathematics that deals with developing and implementing algorithms for solving mathematical problems that cannot be solved analytically. It involves using numerical methods to obtain approximate solutions to mathematical problems.
An interpolating polynomial is a polynomial function that passes through a given set of data points. It is used to approximate a function that is not known or difficult to evaluate at certain points.
An interpolating polynomial can be calculated using various methods such as Lagrange interpolation, Newton's divided differences interpolation, or spline interpolation. These methods involve using the given data points to construct a polynomial function that passes through them.
The main purpose of interpolating polynomials is to approximate a function at points where the function is not known or difficult to evaluate. It is also used to fill in missing data points in a set of data, and to create smooth curves and surfaces from a set of scattered data points.
Interpolating polynomials can sometimes produce inaccurate results, especially when the data points are widely spaced or when the degree of the polynomial is high. They can also lead to overfitting, where the polynomial closely matches the given data points but does not accurately represent the true underlying function.