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The present work is a mathematical analysis of two algorithms, namely
the Roothaan and the level-shifting algorithms, commonly used in
practice to solve the Hartree-Fock equations. The level-shifting
algorithm is proved to be well-posed and to converge provided the shift
parameter is large enough. On the contrary, cases when the Roothaan
algorithm is not well defined or fails in converging are
exhibited. These mathematical results are confronted to numerical
experiments performed by chemists.
Si discretizza il problema dell'ostacolo parabolico con differenze all'indietro nel tempo ed elementi finiti lineari nello spazio e si dimostrano stime dell'errore per la frontiera libera discreta.
The backward Euler algorithm for the multidimensional nonhomogeneous heat equation is analyzed, based on the finite element method. The existence and uniqueness of the numerical solution is investigated. Also, the convergence of the numerical solutions is studied.
We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.
In this article we consider elliptic partial differential equations with random coefficients and/or random forcing terms. In the current treatment of such problems by stochastic Galerkin methods it is standard to assume that the random diffusion coefficient is bounded by positive deterministic constants or modeled as lognormal random field. In contrast, we make the significantly weaker assumption that the non-negative random coefficients can be bounded strictly away from zero and infinity by random...
We introduce a new idea of recurrent functions to provide a new semilocal convergence analysis for two-step Newton-type methods of high efficiency index. It turns out that our sufficient convergence conditions are weaker, and the error bounds are tighter than in earlier studies in many interesting cases. Applications and numerical examples, involving a nonlinear integral equation of Chandrasekhar type, and a differential equation containing a Green's kernel are also provided.
Using systematically a tricky idea of N.V. Krylov, we obtain general results on the rate of convergence of a certain class of monotone approximation schemes for stationary Hamilton-Jacobi-Bellman equations with variable coefficients. This result applies in particular to control schemes based on the dynamic programming principle and to finite difference schemes despite, here, we are not able to treat the most general case. General results have been obtained earlier by Krylov for finite difference...
Using systematically a tricky idea of N.V. Krylov, we obtain
general results on the rate of convergence of a certain class of
monotone approximation schemes for stationary
Hamilton-Jacobi-Bellman equations with variable coefficients.
This result applies in particular to control schemes based on the
dynamic programming principle and to finite difference schemes
despite, here, we are not able to treat the most general case.
General results have been obtained earlier by Krylov for
finite...
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