Numerical Analysis

(in Romanian )

Newton type iterative methods and Newton-Krylov methods for numerical solving of nonlinear systems in Rⁿ.

The high convergence orders of the Newton methods in presence of all sources of errors has been characterized; the Newton methods with large number of unknowns were studied when the linear systems are solved by Krylov methods, and some results regarding convergence, monotony and asymptotical behavior were 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 (impact factor 2009: 0.996) (citations in ISI journals, citations in other journals).
• E. Cătinaş, The inexact, inexact perturbed and quasi-Newton methods are equivalent models, Math. Comp., 74 (2005) no. 249, pp. 291-301 (impact factor 2009: 1.598). (citations in ISI journals, citations in other journals).
• 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, 190pp (citations in ISI journals, citations in other journals).
• I. Păvăloiu, Sur une generalisation de la methode de Steffensen, Rev. Anal. Numer. Theor. Approx., v. 21 (1992) no. 1, pp. 59-67 (citations in ISI journals, citations in other journals).
• E. Cătinaş, On some iterative methods for solving nonlinear equations, Rev. Anal. Numer. Theor. Approx., 23 (1994) no. 1, pp. 47-53 (citations in ISI journals, citations in other journals).

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 were determined and studied, which lead 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. (impact factor 2009: 1.124)

Finite element method and spectral methods.

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

• T. Petrila, C.I. Gheorghiu, Metoda elementului finit şi aplicaţii, Editura Academiei Romane, 1987, 299 pp.
• 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 (citations in ISI journals, citations in other journals).
• 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. (citations in ISI journals, citations in other journals).

For the first of the cited books, C.I. Gheorghiu has obtained in 1988 the "Gheorghe Lazăr" prize of the Romanian Academy.

Interpolatory iterative methods, with highest efficiency index.

Among certain classes of interpolatory iterative methods, the methods with highest 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 solving 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.

• 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 (impact factor 2009: 0.996) (citations in ISI journals, citations in other journals).
• 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.

Certain results were obtained regarding the derivative-interpolatory splines, applied to bilocal linear problems, and to singulary perturbed bilocal problems.

• Mustăţa, C., Iancu, C., Error estimation in the approximation of function by interpolation cubic splines, Mathematica (Cluj) 29 (52) (1987) no. 1, 33-39
• 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.