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MERICAN Journal of Public Diplomacy and International Studies
www.
 grnjournal.us 
 
AMERICAN Journal of Public Diplomacy and
International Studies 
Volume 01, Issue 06, 2023 ISSN (E): 2993-2157 
 
 
 
Numerical Methods in Mathematics and Informatics: 
Bridging Theory and Computation
 
Turganbaeva Ramiza Tolibaevna, Qa’llibekov U’mitbek O’serbay Uli,
Abatbaev Rasulbek Nietbay Uli, Mnajatdinov Atabek Rafatdin Uli
 
A student of the Faculty of Mathematics of KSU 
 
 
Abstract: 
Numerical methods play a pivotal role in the fields of mathematics and informatics by 
providing powerful tools for solving complex mathematical problems that are often intractable 
through analytical methods alone. This article explores the fundamental concepts, applications, 
and advancements in numerical methods, highlighting their significance in both theoretical 
research and practical computational tasks. We delve into various numerical techniques, their 
underlying principles, and their applications across different branches of mathematics and 
informatics.
Keywords: 
Numerical methods, mathematics, informatics, root finding, optimization, numerical 
integration, numerical differentiation, differential equations, linear algebra, machine learning, 
simulation, modeling.
 
Numerical methods encompass a wide range of techniques used to approximate solutions to 
mathematical problems, where analytical solutions may be elusive or computationally expensive. 
These methods bridge the gap between mathematical theory and computational implementation, 
enabling researchers and practitioners to tackle problems that arise in diverse fields such as 
physics, engineering, economics, and computer science. Numerical methods for root finding, 
such as the Newton-Raphson method and the bisection method, play a crucial role in finding 
solutions to equations that cannot be solved explicitly. Optimization techniques like gradient 
descent and simulated annealing are fundamental tools for finding optimal solutions in a wide 
range of applications. Numerical integration techniques, including the trapezoidal rule and 
Simpson's rule, enable the approximation of definite integrals. These methods are essential for 
computing areas, volumes, and probabilities in various mathematical and scientific contexts. 
Similarly, numerical differentiation methods help estimate derivatives of functions, which are 
vital in fields like physics and engineering. [1.82] 
Numerical methods offer robust solutions to ordinary and partial differential equations. 
Techniques like Euler's method and the finite difference method provide approximations to 
differential equations that describe real-world phenomena, from fluid dynamics to population 
growth. In linear algebra, numerical methods are indispensable for solving large systems of 
linear equations and eigenvalue problems. Methods like Gaussian elimination and iterative 
techniques such as the Jacobi method and Gauss-Seidel method underpin simulations and 
modeling in engineering and physics. Numerical methods are at the core of machine learning 
algorithms, facilitating the optimization of model parameters through techniques like stochastic 

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