Results in Numerical Analysis, obtained at the Institute

Finite element methods

Some results regarding constructive aspects in solving initial and boundary value problems for partial differential equations were obtained

Spectral methods

Convergence orders of sequences

Connecting different definitions of (classical) C-orders, Q- and R-orders were obtained, together with a survey of the convergence orders of the basic iterative methods (Newton, secant, successive approximations):

Numerical optimization

Newton and Newton-Krylov methods for nonlinear systems in Rn

The high convergence orders of the Newton methods have been characterized, while considering all sources of errors; the Newton methods with large number of unknowns were studied when the linear systems are solved by Krylov methods, results regarding convergence, monotony and asymptotical behavior being obtained.

Solving of nonlinear equations by Newton, secant, Chebyshev, Steffensen or Aitken methods

Local and semilocal convergence results were obtained:

For a series of papers in this field, I. Păvăloiu was awarded the “Gheorghe Lazăr” prize of the Romanian Academy, in 1970.

Monotone sequences for approximating the solutions of nonlinear equations

Some classes of Steffensen, Aitken and Aitken-Steffensen methods were introduced and studied, leading to sequences approximating bilateraly the solutions of nonlinear equations:

Iterative methods of interpolatory type, with high efficiency index

Among certain classes of iterative methods of interpolatory type, the methods with high efficiency index were determined:

Krylov methods for numerical computing of large linear systems in Rn

Connections between the residuals and the backward errors of the approximative solutions of certain Krylov methods were found, as well as some results regarding relations satisfied by the errors of these approximative solutions.

Iterative methods for numerical solving of eigenvalues/eigenvectors

Simpler convergence conditions were obtained for different methods (Newton, Chebyshev, chord and Steffensen method) for the case when the system of nonlinear equations has as solutions the eigenvalues and eigenvectors of a linear operator.