## Math 373 Lecture Notes

• Lecture 1: (Brief review of Calculus. Intro to the solution of nonlinear equations.)

• Lecture 2: (Bisection, false position, secant method, Newton's method.)

• Lecture 3: (Newton's method for systems of equations, fixed point iteration.)

• Lecture 4: (Local convergence results, order of convergence.)

• Lecture 5: (Weierstrass approximation theorem, Lagrange form of the interpolating polynomial.)

• Lecture 6: (Divided differences, Newton form of the interpolating polynomial, interpolation error.)

• Lecture 7: (Hermite interpolation, divided differences for repeated points, Runge example.)

• Lecture 8: (Piecewise polynomial approximation, error estimates and basis functions for continuous piecewise linear and piecewise cubic Hermite approximation.)

• Lecture 9: (Cubic spline approximation.)

• Lecture 10: (Approximation of parametric curves; Bezier curves.)

• Lecture 11: (Approximation of derivatives.)

• Lecture 12: (Approximation of integrals; basic and composite integration rules)

• Lecture 13: (Iterative approaches to the approximation of integrals; Richardson extrapolation and Romberg integration)

• Lecture 14: (Gaussian quadrature)

• Lecture 16: (Numerical solution of ODEs: theoretical background; Euler's method)

• Lecture 17: (Taylor series methods)

• Lecture 18: (Runge-Kutta methods)

• Lecture 19: (Estimation of local error and adaptive methods for ODEs)

• Lecture 20: (Linear multistep methods: derivation, consistency, local truncation error)

• Lecture 21: (Consistency, zero-stability, and convergence of linear multistep methods)

• Lecture 22: (Numerical stability: strong, weak, absolute, relative)

• Lecture 23: (Predictor-Corrector methods; generalizations to systems)