Analysis of Fixed-Stepsize Methods

Implementing Adams Methods with Preassigned Stepsize Ratios

Runge-Kutta and Adams methods are the most popular codes to solve numerically nonstiff ODEs. The Adams methods are useful to reduce the number of function calls, but they usually require more CPU time than the Runge-Kutta methods. In this work we develop a numerical study of a variable step length Adams implementation, which can only take preassigned step-size ratios. Our aim is the reduction o...

Stepsize Restrictions for Boundedness and Monotonicity of Multistep Methods

In this paper nonlinear monotonicity and boundedness properties are analyzed for linear multistep methods. We focus on methods which satisfy a weaker boundedness condition than strict monotonicity for arbitrary starting values. In this way, many linear multistep methods of practical interest are included in the theory. Moreover, it will be shown that for such methods monotonicity can still be v...

Analysis of variable-stepsize linear multistep methods with special emphasis on symmetric ones

In this paper we deal with several issues concerning variablestepsize linear multistep methods. First, we prove their stability when their fixed-stepsize counterparts are stable and under mild conditions on the stepsizes and the variable coefficients. Then we prove asymptotic expansions on the considered tolerance for the global error committed. Using them, we study the growth of error with tim...

Fixed Point Methods in Nonlinear Analysis

In this paper we present a selection of fixed point theorems with applications in nonlinear analysis. We begin with the Banach fixed point theorem, which we use to prove the inverse and implicit mapping theorems and the Picard-Lindelöf theorem for Banach spaces. We then prove in succession the fixed point theorems of Brouwer, Schauder, and Schaeffer, after which we conclude with two example app...

Modified Two-Point Stepsize Gradient Methods for Unconstrained Optimization