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Advanced Numerical Methods for Equations, Systems Equations and Optimization

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Numerical Methods for Energy Applications

Part of the book series: Power Systems ((POWSYS))

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Abstract

In this chapter an overview of advanced numerical methods is presented. After errors are defined, the readers are initiated with the principles of approximating functions, numerical methods for solving equations and systems equations and optimization methods. The theoretical basics of polynomial interpolation, numerical differentiation and numerical integration are presented, and, as a natural sequel, for each method some examples are given. Numerical methods for linear and nonlinear equations and system equations, numerical methods for computing eigenvalues and eigenvectors are discussed and examples are given. The main goal of the optimization methods is to find the maximum or minimum of the objective function by using linear or nonlinear programming as shown in this chapter. Finally, an introduction in Matlab is done to put in evidence the easy-to-use and attractiveness of this popular software. Each method is accompanied by examples to help understanding.

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Abbreviations

A. :

Acronyms

Matlab :

Matrix Laboratory

OS :

Operating Systems

Mac OS X :

Macintosh Operating System

B. :

Symbols/Parameters

\(\tilde{x}\) :

The approximate value

x :

The true (unknown) quantity

\(\varepsilon\) :

The error

\(\left| \varepsilon \right|\) :

The absolute error

\(\varepsilon_{r}\) :

The relative error

f(x) :

The function of real variable x

A:

The Vandermonde matrix

tk :

The k-moment of time

p:

The p-step of numerical algorithm

xr :

The exact solution

xo :

The first solution

\(\Re\) :

The real set

T:

The matrix of the coefficient

X:

The vector the unknowns

G:

The column vector of free terms

\({\text{det}}{{T}}\) :

The determinant of matrix T

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Author information

Authors and Affiliations

  1. Doctoral School of Engineering Sciences, University Valahia, Targoviste, Romania

    Horia Andrei

  2. Numerical Methods Research Center, Technical University of Cluj-Napoca, Cluj-Napoca, Romania

    Dan D. Micu

  3. Department of Control Systems and Electrical Engineering, University Dunarea de Jos Galati, Galati, Romania

    Marian Gaiceanu

  4. Department of Electrical Engineering, University Politehnica Bucharest, Bucharest, Romania

    Marilena Stanculescu & Paul Cristian Andrei

Authors
  1. Horia Andrei
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  2. Dan D. Micu
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  3. Marian Gaiceanu
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  4. Marilena Stanculescu
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  5. Paul Cristian Andrei
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Corresponding author

Correspondence to Horia Andrei .

Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, Seraj Higher Education Institute, Tabriz, Iran

    Naser Mahdavi Tabatabaei

  2. Faculty of Electronics, Communications and Computers, University of Pitesti, Pitesti, Romania

    Nicu Bizon

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Andrei, H., Micu, D.D., Gaiceanu, M., Stanculescu, M., Andrei, P.C. (2021). Advanced Numerical Methods for Equations, Systems Equations and Optimization. In: Mahdavi Tabatabaei, N., Bizon, N. (eds) Numerical Methods for Energy Applications. Power Systems. Springer, Cham. https://doi.org/10.1007/978-3-030-62191-9_1

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  • DOI: https://doi.org/10.1007/978-3-030-62191-9_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-62190-2

  • Online ISBN: 978-3-030-62191-9

  • eBook Packages: EnergyEnergy (R0)

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