Syllabus
Nonlinear systems of equations, existence theorems (Banach, Brouwer, Zarantonello).
Convergence speed, orders of convergence.
Scalar equations, basic methods (bisection, fixed point iteration, regula falsi).
Newton and secant methods, local convergence, types of failure, difference approximation.
Sophisticated and hybrid algorithms (Muller's method, inverse quadratic interpolation, Brent's method).
Systems of equations, properties, Ostrowski theorem.
Newton's method for systems, local convergence, quasi-Newton approaches.
Global convergence, continuation methods.
Annotation
The course deals with theoretical and practical aspects of the numerical solution of nonlinear equations and their systems. The emphasis is on Newton's method and its modifications.
Students will also test the algorithms practically.