Fayyaz Ahmad | University of Barcelona (original) (raw)

Papers by Fayyaz Ahmad

Algorithms, 2017

A modification to an existing iterative method for computing zeros with unknown multiplicities of... more A modification to an existing iterative method for computing zeros with unknown multiplicities of nonlinear equations or a system of nonlinear equations is presented. We introduce preconditioners to nonlinear equations or a system of nonlinear equations and their corresponding Jacobians. The inclusion of preconditioners provides numerical stability and accuracy. The different selection of preconditioner offers a family of iterative methods. We modified an existing method in a way that we do not alter its inherited quadratic convergence. Numerical simulations confirm the quadratic convergence of the preconditioned iterative method. The influence of preconditioners is clearly reflected in the numerically achieved accuracy of computed solutions.

Numerical Algorithms

Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion ... more Bogoya, Böttcher, Grudsky, and Maximenko have recently obtained the precise asymptotic expansion for the eigenvalues of a sequence of Toeplitz matrices {T n (f)}, under suitable assumptions on the associated generating function f. In this paper, we provide numerical evidence that some of these assumptions can be relaxed and extended to the case of a sequence of preconditioned Toeplitz matrices {T −1 n (g)T n (f)}, for f trigonometric polynomial, g nonnegative, not identically zero trigonometric polynomial, r = f/g, and where the ratio r plays the same role as f

A multi-step frozen Jacobian iterative scheme for solving system of nonlinear equations associate... more A multi-step frozen Jacobian iterative scheme for solving system of nonlinear equations associated with IVPs (initial value problems) and BVPs (boundary value problems) is constructed. The multi-step iterative schemes consist of two parts, namely base method and a multi-step part. The proposed iterative scheme uses higher order Fréchet derivatives in the base method part and offers high convergence order (CO) 3s + 1, here s is the number of steps. The increment in the CO per step is three, and we solve three upper and lower triangles systems per step in the multi-step part. A single inversion of the is not working in latexfrozen Jacobian is required and in fact, we avoid the direct inversion of the frozen Jacobian by computing the LU factors. The LU-factors are utilized in the multi-step part to solve upper and lower triangular systems repeatedly that makes the iterative scheme computationally efficient. We solve a set of IVPs and BVPs to show the validity, accuracy and efficiency of our proposed iterative scheme.

In this note, we proposed multi-step iterative methods to solve systems of weakly nonlinear equat... more In this note, we proposed multi-step iterative methods to solve systems of weakly nonlinear equations associated with PDEs and ODEs.

It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the sol... more It is well-known that the solution of Hamilton-Jacobi equation may have singularity i.e., the solution is non-smooth or nearly non-smooth. We construct a frozen Jacobian multi-step iterative method for solving Hamilton-Jacobi equation under the assumption that the solution is nearly singular. The frozen Jacobian iterative methods are computationally very efficient because a single instance of the iterative method uses a single inversion (in the scene of LU factorization) of the frozen Jacobian. The multi-step part enhances the convergence order by solving lower and upper triangular systems. The convergence order of our proposed iterative method is 3(m − 1) for m ≥ 3. For attaining good numerical accuracy in the solution, we use Chebyshev pseudo-spectral collocation method. Some Hamilton-Jacobi equations are solved, and numerically obtained results show high accuracy.

Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equa... more Frozen Jacobian iterative methods are of practical interest to solve the system of nonlinear equations. A frozen Jacobian multi-step iterative method is presented. We divide the multi-step iterative method into two parts namely base method and multi-step part. The convergence order of the constructed frozen Jacobian iterative method is three, and we design the base method in a way that we can maximize the convergence order in the multi-step part. In the multi-step part, we utilize a single evaluation of the function, solve four systems of lower and upper triangular systems and a second frozen Jacobian. The attained convergence order per multi-step is four. Hence, the general formula for the convergence order is 3 + 4(m − 2) for m ≥ 2 and m is the number of multi-steps. In a single instance of the iterative method, we employ only single inversion of the Jacobian in the form of LU factors that makes the method computationally cheaper because the LU factors are used to solve four system of lower and upper triangular systems repeatedly. The claimed convergence order is verified by computing the computational order of convergence for a system of nonlinear equations. The efficiency and validity of the proposed iterative method are narrated by solving many nonlinear initial and boundary value problems. c 2016 All rights reserved. Keywords: Frozen Jacobian iterative methods, multi-step iterative methods, systems of nonlinear equations, nonlinear initial value problems, nonlinear boundary value problems. 2010 MSC: 65H10, 65N22.

A systematic way is presented for the construction of multi-step iterative method with frozen Jac... more A systematic way is presented for the construction of multi-step iterative method with frozen Jacobian. The inclusion of an auxiliary function is discussed. The presented analysis shows that how to incorporate auxiliary function in a way that we can keep the order of convergence and computational cost of Newton multi-step method. The auxiliary function provides us the way to overcome the singularity and ill-conditioning of the Jacobian. The order of convergence of proposed p-step iterative method is p + 1. Only one Jacobian inversion in the form of LU-factorization is required for a single iteration of the iterative method and in this way, it offers an efficient scheme. For the construction of our proposed iterative method, we used a decomposition technique that naturally provides different iterative schemes. We also computed the computational convergence order that confirms the claimed theoretical order of convergence. The developed iterative scheme is applied to large scale problems, and numerical results show that our iterative scheme is promising.

Boundary conditions are y(0) = 0, y(1) = 0.

The quasilinearization iterative method is famous to solve boundary value problems with quadratic... more The quasilinearization iterative method is famous to solve boundary value problems with quadratic convergence or in other words it is equivalent to classical Newton-Raphson iterative method with second-order convergence for system of nonlinear equations. The Chebyshev-Halley method is also designed to solve system of nonlinear equations with third-order convergence but as far as we know, it is never discussed in the context of ordinary differential equations (ODEs) or partial differential equations (PDEs). In this short note, we proposed a general procedure to design higherorder iterative methods for ODEs and PDEs. In fact quasilinearization and Chebyshev-Hally method are the particular cases of our proposal.

Use of iterative schemes (for nonlinear algebraic equations ) for ODE's

Algorithms, 2017

Numerical Algorithms

Boundary conditions are y(0) = 0, y(1) = 0.

Use of iterative schemes (for nonlinear algebraic equations ) for ODE's