MATH 105B: NUMERICAL ANALYSIS II

In this course, we study approximation methods for important problems in Mathematics. Interpolation is a method of constructing a polynomial that coincides with given data points. We learn polynomial approximation using Lagrange interpolation, Hermite interpolation, or Cubic spline interpolation. We also focus on discrete methods for differentiation and integration in Calculus. We use numerical differentiation formulas based on Taylor series, and we see Richardson’s extrapolation used to generate high-accuracy results while using low-order formulas. The basic method involved in approximating the definite integral is called numerical quadrature. We learn basic quadrature rules, such as Trapezoidal and Simpson’s rules, as well as Composite and Gaussian quadrature rule. Adaptive quadrature methods are also covered. Furthermore, when we determine the best approximate polynomial for a function, the method of least squares is often used. We deal with the least squares approximation in both discrete and continuous ways. Some important bases of polynomials are introduced with a proper inner product: Orthogonal polynomials, Chebyshev polynomials, and Trigonometric polynomials. We also look at the fast Fourier transform (FFT) algorithm.