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Quadratically convergent algorithm for computing real root of non-linear transcendental equations.

ABSTRACT: OBJECTIVES:The present paper describes a new algorithm to find a root of non-linear transcendental equations. It is found that Regula-Falsi method always gives guaranteed result but slow convergence. However, Newton-Raphson method does not give guaranteed result but faster than Regula-Falsi method. Therefore, the present paper used these two ideas and developed a new algorithm which has better convergence than Regula-Falsi and guaranteed result. One of the major issue in Newton-Raphson method is, it fails when first derivative is zero or approximately zero. RESULTS:The proposed method implemented the failure condition of Newton-Raphson method with better convergence. Error calculation has been discussed for certain real life examples using Bisection, Regula-Falsi, Newton-Raphson method and new proposed method. The computed results show that the new proposed quadratically convergent method provides better convergence than other methods.

SUBMITTER: Thota S

PROVIDER: S-EPMC6302413 | biostudies-literature | 2018 Dec

REPOSITORIES: biostudies-literature

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Quadratically convergent algorithm for computing real root of non-linear transcendental equations.

Thota Srinivasarao S Srivastav Vivek Kumar VK

BMC research notes 20181220 1

<h4>Objectives</h4>The present paper describes a new algorithm to find a root of non-linear transcendental equations. It is found that Regula-Falsi method always gives guaranteed result but slow convergence. However, Newton-Raphson method does not give guaranteed result but faster than Regula-Falsi method. Therefore, the present paper used these two ideas and developed a new algorithm which has better convergence than Regula-Falsi and guaranteed result. One of the major issue in Newton-Raphson m ...[more]

PMID: 30572943

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Project description:BackgroundReverse engineering gene networks and identifying regulatory interactions are integral to understanding cellular decision making processes. Advancement in high throughput experimental techniques has initiated innovative data driven analysis of gene regulatory networks. However, inherent noise associated with biological systems requires numerous experimental replicates for reliable conclusions. Furthermore, evidence of robust algorithms directly exploiting basic biological traits are few. Such algorithms are expected to be efficient in their performance and robust in their prediction.ResultsWe have developed a network identification algorithm to accurately infer both the topology and strength of regulatory interactions from time series gene expression data in the presence of significant experimental noise and non-linear behavior. In this novel formulism, we have addressed data variability in biological systems by integrating network identification with the bootstrap resampling technique, hence predicting robust interactions from limited experimental replicates subjected to noise. Furthermore, we have incorporated non-linearity in gene dynamics using the S-system formulation. The basic network identification formulation exploits the trait of sparsity of biological interactions. Towards that, the identification algorithm is formulated as an integer-programming problem by introducing binary variables for each network component. The objective function is targeted to minimize the network connections subjected to the constraint of maximal agreement between the experimental and predicted gene dynamics. The developed algorithm is validated using both in silico and experimental data-sets. These studies show that the algorithm can accurately predict the topology and connection strength of the in silico networks, as quantified by high precision and recall, and small discrepancy between the actual and predicted kinetic parameters. Furthermore, in both the in silico and experimental case studies, the predicted gene expression profiles are in very close agreement with the dynamics of the input data.ConclusionsOur integer programming algorithm effectively utilizes bootstrapping to identify robust gene regulatory networks from noisy, non-linear time-series gene expression data. With significant noise and non-linearities being inherent to biological systems, the present formulism, with the incorporation of network sparsity, is extremely relevant to gene regulatory networks, and while the formulation has been validated against in silico and E. Coli data, it can be applied to any biological system.

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Quadratically convergent algorithm for computing real root of non-linear transcendental equations.

Publications

Quadratically convergent algorithm for computing real root of non-linear transcendental equations.

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