The principle of majorization-minimization (MM) provides a general framework for eliciting effective algorithms to solve optimization problems. However, the resulting methods often suffer from slow ...convergence, especially in large-scale and high-dimensional data settings. This has motivated several acceleration schemes tailored for MM algorithms, but many existing approaches are either problem-specific, or rely on approximations and heuristics loosely inspired by the optimization literature. We propose a novel quasi-Newton method for accelerating any valid MM algorithm, cast as seeking a fixed point of the MM algorithm map. The method does not require specific information or computation from the objective function or its gradient, and enjoys a limited-memory variant amenable to efficient computation in high-dimensional settings. By rigorously connecting our approach to Broyden's classical root-finding methods, we establish convergence guarantees and identify conditions for linear and super-linear convergence. These results are validated numerically and compared to peer methods in a thorough empirical study, showing that it achieves state-of-the-art performance across a diverse range of problems.
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The calculation of volume roots from cubic equation of state, for given temperature and pressure, is still an important operation both in industry and academic field. It is proposed the use of ...Ferrari's formula to calculate the pressure range containing three real positive roots (for pressure and temperature below the critical). In addition, a modification in the Cardano‐Tartáglia's formula is proposed to provide the roots in the ascending order. For low values of reduced temperatures, it is proposed a simple and efficient approximation method to avoid the round‐off errors presented by Cardano‐Tartáglia's formula. It is also proposed a method for calculating the saturation pressure from cubic equations of state. Examples of the methods proposed are presented for the Van der Waals, Soave‐Redlich‐Kwong e Peng‐Robinson equations of state in order to show the quality and reliability of the methods proposed.
Direction of arrival (DOA) estimation using coprime array is studied, and an extended-aperture unitary root multiple signal classification (MUSIC)-based method is proposed. The geometry of prototype ...coprime array is modified through some translations, which enable one subarray of the coprime array to achieve aperture extension after unitary transformation. Then, the aperture of the other subarray can also be extended based on the rotational invariance of the extended subarray, and 2-D parameter estimations from the two extended subarrays can be achieved in succession via 1-D root MUSIC-based technique. Finally, unique DOA is determined from the intersection of the two automatically paired and coprime estimations. In contrast to the partial search MUSIC method, the estimation of signal parameters via rotational invariance technique based method, the root MUSIC-based method, and the real-valued cross covariance matrix based method, the proposed method gives better DOA estimation results and manages more sources. Furthermore, it has low complexity for the real-valued decomposition. Simulation results verify the improvement of the proposed approach.
Zeroing neural network (ZNN, or termed Zhang neural network after its inventors) is an effective approach to dynamic matrix square root (DMSR) finding arising in numerous fields of science and ...engineering. The conventional ZNN models can obtain the theoretical DMSR in infinitely long time or in finite time. However, in some applications especially the ones that require to fulfill hard time constraints, these ZNN models may be not competent to guarantee a timely convergence. Hence, for solving DMSR, a ZNN model with explicitly and antecedently definable convergence time is more preferable. Being robust to external noises is very significant for a neural network model. Unfortunately, the existing ZNN models exhibit limited noise-tolerance capability and the corresponding steady-state residual errors would be theoretically bounded when the ZNN models are perturbed by dynamic bounded non-vanishing noises. To enhance the existing ZNN models, by using two novel activation functions, this paper for the first time enables the ZNN model to be predefined-time convergent with improved noise-tolerance capability. The convergence time of the accelerated ZNN model can be explicitly defined as a prior constant parameter. More importantly, such a predefined-time convergent ZNN (PTZNN) is capable of theoretically and completely enduring dynamic bounded vanishing and non-vanishing noises. For handling constant noises such as large constant model-implementation errors, the PTZNN can achieve improved noise-tolerance performance as compared with the existing ZNN models. Comparative simulation results demonstrate that the proposed PTZNN delivers superior convergence and robustness performance for solving DMSR in comparison with the existing ZNN models.
A root‐finding absorbing boundary condition (RFABC) for scalar‐wave propagation problems in infinite anisotropic media was developed. Although the phase velocity and the group velocity in isotropic ...media have the same sign, the sign of the latter can differ from that of the former in anisotropic media. Therefore, a RFABC for anisotropic scalar waves consistent with the group velocity of the considered media is developed, as the velocity is closely related to the direction of energy propagation. The developed boundary condition is shown to satisfy a criterion for an “enough accurate” boundary condition. The well‐posedness of the boundary condition is proven at the continuous level. Its finite‐element formulation, which ensures well‐posedness at a discrete level, is derived, after which it is demonstrated that accurate and stable solutions to the problem of antiplane shear‐wave propagation in an anisotropic elastic waveguide, an example of anisotropic scalar‐wave propagation, can be obtained using the proposed numerical approach.
We present an algorithm, based on Newton’s method, for finding all roots of univariate complex polynomials so that the observed complexity is linear in the degree, up to logarithmic factors. Unlike ...the usual Newton method, which finds at most one root at a time, it is global in the sense that it attempts to find all roots of polynomials simultaneously.
We demonstrate the feasibility of this algorithm by employing it to find all roots of several families of polynomials of degrees up to more than one billion (109). In all cases, the observed (empirical) complexity for finding all roots of a polynomial of degree d – measured either as the number of Newton iterations or computing time – was between O(dlnd) and O(dln3d), with small constants.
•Survey of fixed point iteration processes.•Extension of the pseudo-Newton method idea to other root finding methods.•Modifications of the MMP-methods and their convergence ...visualizations.•Non-trivial and intriguing polynomiographs have been obtained.
In this paper, an iteration process, referred to in short as MMP, will be considered. This iteration is related to finding the maximum modulus of a complex polynomial over a unit disc on the complex plane creating intriguing images. Kalantari calls these images polynomiographs independently from whether they are generated by the root finding or maximum modulus finding process applied to any polynomial. We show that the images can be easily modified using different MMP methods (pseudo-Newton, MMP-Householder, methods from the MMP-Basic, MMP-Parametric Basic or MMP-Euler–Schröder Families of Iterations) with various kinds of non-standard iterations. Such images are interesting from three points of views: scientific, educational and artistic. We present the results of experiments showing automatically generated non-trivial images obtained for different modifications of root finding MMP-methods. The colouring by iteration reveals the dynamic behaviour of the used root finding process and its speed of convergence. The results of the present paper extend Kalantari’s recent results in finding the maximum modulus of a complex polynomial based on Newton’s process with the Picard iteration to other MMP-processes with various non-standard iterations.
In this study, we propose a new approach for development of absorbing boundary conditions for scalar-wave propagation problems in infinite media based on a root-finding algorithm for the solution of ...the exact wave dispersion relation. We select the Newton–Raphson method as the root-finding algorithm in the present study and assess the accuracy of the newly developed boundary condition by estimating its reflection coefficient. Furthermore, we evaluate and verify the stability of the boundary condition. We apply our development to various scalar-wave propagation problems and demonstrate that the proposed approach leads to accurate and stable computations.
•New boundary conditions are proposed for unbounded domains to absorb scalar waves.•A root-finding algorithm for the solution of exact dispersion relation is utilized.•In the present study, we select the Newton–Raphson method as the algorithm.•We assess the accuracy of the new boundary condition and verify its stability.•We demonstrate that the proposed approach leads to accurate and stable computations.
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