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This thesis offers an overview of the relatively new family of Rosenbrock-Krylov numerical methods for ODEs. These methods are a further development of Rosenbrock methods, using a lower-dimension approximation of the Jacobian. The thesis gives the mathematical background to Rosenbrock-Krylov methods by first presenting Runge-Kutta methods and Rosenbrock methods, followed by the concept of Krylov s
