On some recently developed architectures, a numerical program may give different answersdepending on the execution hardware and the compilation. These discrepancies of the resultscome from the fact that each floating-point computation is calculated with different precisions.The goal of this thesis is to formally prove properties about numerical programs while takingthe architect...

If each off-diagonal entry and the sum of each row of a diagonally dominant M -matrix are known to certain relative accuracy, then its smallest eigenvalue and the entries of its inverse are known to the same order relative accuracy independent of any condition numbers. In this paper, we devise algorithms that compute these quantities with relative errors in the magnitude of the machine precisio...

The need to determine a few eigenvalues of a large sparse generalised eigenvalue problem Ax = Bx with semi-positive deenite B arises in many physical situations, for example, in a stability analysis of the discretised Navier-Stokes equation. A common technique is to apply Arnoldi's method to the shift-invert transformation, but this can suuer from numerical instabilities as is illustrated by a ...

We define several abstract semantics for the static analysis of finite precision computations, that bound not only the ranges of values taken by numerical variables of a program, but also the difference with the result of the same sequence of operations in an idealized real number semantics. These domains point out with more or less detail (control point, block, function for instance) sources o...

The study of the battery charge process as the only power storage agent in off-grid systems is of significant importance. The battery charge process has different modes, and the battery in these modes is dependent on the amount of charge. In order to charge the battery in off-grid systems, two charge controllers including Pulse Width Modulation (PWM) and Maximum Power Point Tracker (MPPT) are c...

An iterative algorithm is presented for solving the extended Sylvester-conjugate matrix equation. By this iterativemethod, the solvability of thematrix equation can be determined automatically. When the matrix equation is consistent, a solution can be obtained within finite iteration steps for any initial values in the absence of round-off errors. The algorithm is also generalized to solve a mo...

An efficient iterative algorithm is applied to seek analytical solutions of higher dimensional initial boundary value problems (IBVPS) with variable coefficients. The proposed scheme finds the solution without any discretization, perturbation, transformation or restrictive assumptions and avoids the round off errors. Numerical results clearly reveal the complete reliability and efficiency of th...

We introduce a concrete semantics for floating-point operations which describes the propagation of roundoff errors throughout a calculation. This semantics is used to assert the correctness of a static analysis which can be straightforwardly derived from it. In our model, every elementary operation introduces a new first order error term, which is later propagated and combined with other error ...

The vast use of computers on scientific numerical computation makes the awareness of the limited precision that these machines are able to provide us an essential matter. A limited and insufficient precision allied to the truncation and rounding errors may induce the user to incorrect interpretation of his/hers answer. In this work, we have developed a computational package to minimize this kin...

Because of the importance of special functions, several books and a large collection of papers have been devoted to the numerical computation of these functions, the most well-known being the Abramowitz and Stegun handbook [1]. But up to this date, no environment offers routines for the provable correct evaluation of these special functions. We point out how series and limit-periodic continued ...