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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 15:
(Adaptive 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)