In class today we began discussing methods of numerical integration.  Doing research upon methods of approximating definite integrals, I came across the Monte Carlo integration, and Monte Carlo methods in general. 
Monte Carlo methods utilize random numbers to randomly sample from the domain in order to (accurately) compute the desired result.  In Monte Carlo integration, […]

When dealing with polynomial interpolations, it seems intuitively clear that the higher the degree of the polynomial, the more accurate the approximation.  However, Runge’s phenomenon shows that this is not the case.  In 1901, German mathematician Carl Runge discovered that sometimes using polynomial interpolation for polynomials of high degrees can cause very large errors.  For […]

When discussing Newton’s algorithm for polynomial interpolation in class, I was reminded of some other work that Newton had done with polynomials, known as Newton’s sums.  If a_0x^n+a_1x^{n-1}+…+a_n is a polynomial of degree n with roots r_1, r_2, … , r_n, Newton’s sums gives us an iterative method to compute r_1^k+r_2^k+…+r_n^k for k>=1 without explicitly calculating […]