**Newton type iterative methods and Newton-Krylov methods for numerical solving of nonlinear systems in R ^{n}.**

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.

- E. Cătinaş,
*Inexact perturbed Newton methods, and applications to a class of Krylov solvers*, J. Optim. Theory Appl., vol. 108 (2001) no. 3, pp. 543-570. - E. Cătinaş,
*The inexact, inexact perturbed and quasi-Newton methods are equivalent models*, Math. Comp., 74 (2005) no. 249, pp. 291-301. - E. Cătinaş,
*Methods of Newton and Newton-Krylov type*, Risoprint, Cluj-Napoca, 2007, ISBN 978-973-751-533-9.

The papers have been cited in reputed journals (such as SIAM journals), by ISI Highly Cited authors.

**Solving of nonlinear equations by Newton, Chebyshev, Steffensen, Aitken and Aitken-Steffensen type methods.**

Local and semilocal convergence results were obtained:

- I. Păvăloiu,
*Rezolvarea ecuaţiilor prin interpolare*, Ed. Dacia, 1981, 190 pp . - I. Păvăloiu,
*Sur une generalisation de la methode de Steffensen*, Rev. Anal. Numer. Theor. Approx., v. 21 (1992) no. 1, pp. 59-67. - E. Cătinaş,
*On some iterative methods for solving nonlinear equations*, Rev. Anal. Numer. Theor. Approx., 23 (1994) no. 1, pp. 47-53.

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:

- I. Păvăloiu,
*Approximation of the roots of equations by Aitken-Steffensen-type monotonic sequences*, Calcolo, v. 32 (1995) nos 1-2, pp. 69-82. - I. Păvăloiu and E. Cătinaş,
*On a Steffensen-Hermite method of order three*, Applied Mathematics and Computation, v. 215 (2009) no. 7, pp. 2663-2672.

**Finite element and spectral methods.**

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

- C.I. Gheorghiu,
*A Constructive Introduction to Finite Elements Method*, Editura Quo-Vadis, Cluj-Napoca, 1999, 170 pp., ISBN 973-99137-0-9 - C.I. Gheorghiu,
*Spectral Methods for Differential Problems*, Casa Cărţii de Stiintă, Cluj-Napoca, 2007, X+154 pp., ISBN 978-973-133-099-0. - C.I. Gheorghiu and I.S. Pop,
*A modified Chebyshev-Tau method for a hydrodynamic stability problem*, Proceedings of ICAOR 1996, v. II, pp. 119-126. - C.I. Gheorghiu, M.E. Hochstenbach, B. Plestenjak, J. Rommes, Spectral collocation solutions to multiparameter Mathieu’s system, system, Appl. Math. Comput. (2012)
- C.I. Gheorghiu, Spectral Methods for Non-Standard Eigenvalue Problems, Springer Briefs in Mathematics, 2014, pp. 120+X, print ISBN 978-3-319-06229-7, electronic ISBN 978-3-319-06230-3. (Preface, Content)

**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:

- I. Păvăloiu,
*On computational complexity in solving equations by interpolation methods*, Rev. Anal. Numer. Theor. Approx., 24 (1995) no. 1, 201-214. - I. Păvăloiu,
*Optimal efficiency indexes for iterative methods of interpolatory-type*, Computer Science Journal of Moldova, 5 (1997) no. 1(13), 20-43.

**Krylov methods for numerical computing of large linear systems in R ^{n}.**

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.

- E. Cătinaş,
*Inexact perturbed Newton methods, and applications to a class of Krylov solvers*, J. Optim. Theory Appl., vol. 108 (2001) no. 3, pp. 543-570. - E. Cătinaş,
*On the high convergence orders of the Newton-GMBACK methods*, Rev. Anal. Numer. Theor. Approx., 28 (1999) no. 2, pp. 125-132. - E. Cătinaş,
*Methods of Newton and Newton-Krylov type*, Risoprint, Cluj-Napoca, 2007, ISBN 978-973-751-533-9.

**Spline functions applied to boundary value problems for ordinary differential equations.**

Results regarding the derivative-interpolatory splines, applied to bilocal linear problems, and to singulary perturbed bilocal problems.

- C. Mustăţa, C. Iancu,
*Error estimation in the approximation of function by interpolation cubic splines*, Mathematica (Cluj) 29 (52) (1987) no. 1, 33-39 - C. Mustăţa,
*On a problem of B.A. Karpilovskaya*, Rev. Anal. Numer. Theor. Approx., 28 (1999) no. 2, pp. 179-189.

**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.

- I. Păvăloiu, E. Cătinaş,
*Remarks on some Newton and Chebyshev-type methods for approximating the eigenvalues and eigenvectors of matrices*, Computer Science Journal of Moldova, 7 (1999) no. 1(19), 3-17. - I. Păvăloiu, E. Cătinaş,
*On approximating the eigenvalues and eigenvectors of linear continuous operators*, Rev. Anal. Numer. Theor. Approx., 26 (1997) nos. 1-2, 19-28.