Fundamental Numerical Methods for Electrical Engineering

Contents

About the Author

Introduction

Methods for Numerical Solution of Linear Equations

1.1 Direct Methods

1.1.1 The Gauss Elimination Method

1.1.2 The Gauss–Jordan Elimination Method

1.1.3 The LU Matrix Decomposition Method

1.1.4 The Method of Inverse Matrix

1.2 Indirect or Iterative Methods

1.2.1 The Direct Iteration Method

1.2.2 Jacobi and Gauss–Seidel Methods

1.3 Examples of Applications in Electrical Engineering

References

Methods for Numerical Solving the Single Nonlinear Equations

2.1 Determination of the Complex Roots of Polynomial Equations by Using the Lin’s and Bairstow’s Methods

2.1.1 Lin’s Method

2.1.2 Bairstow’s Method

2.1.3 Laguerre Method

2.2 Iterative Methods Used for Solving Transcendental Equations

2.2.1 Bisection Method of Bolzano

2.2.2 The Secant Method

2.2.3 Method of Tangents (Newton–Raphson)

2.3 Optimization Methods

2.4 Examples of Applications

References

Methods for Numerical Solution of Nonlinear Equations

3.1 The Method of Direct Iterations

3.2 The Iterative Parameter Perturbation Procedure

3.3 The Newton Iterative Method

3.4 The Equivalent Optimization Strategies

3.5 Examples of Applications in the Microwave Technique

References

Methods for the Interpolation and Approximation of One Variable Function

4.1 Fundamental Interpolation Methods

4.1.1 The Piecewise Linear Interpolation

4.1.2 The Lagrange Interpolating Polynomial

4.1.3 The Aitken Interpolation Method

4.1.4 The Newton–Gregory Interpolating Polynomial

4.1.5 Interpolation by Cubic Spline Functions

4.1.6 Interpolation by a Linear Combination of Chebyshev Polynomials of the First Kind

4.2 Fundamental Approximation Methods for One Variable Functions

4.2.1 The Equal Ripple (Chebyshev) Approximation

4.2.2 The Maximally Flat (Butterworth) Approximation

4.2.3 Approximation (Curve Fitting) by the Method of Least Squares

4.2.4 Approximation of Periodical Functions by Fourier Series

4.3 Examples of the Application of Chebyshev Polynomials in Synthesis of Radiation Patterns of the In- Phase Linear Array Antenna

References

Methods for Numerical Integration of One and Two Variable Functions

5.1 Integration of Definite Integrals by Expanding the Integrand Function in Finite Series of Analytically Integrable Functions

5.2 Fundamental Methods for Numerical Integration of One Variable Functions

5.2.1 Rectangular and Trapezoidal Methods of Integration

5.2.2 The Romberg Integration Rule

5.2.3 The Simpson Method of Integration

5.2.4 The Newton–Cotes Method of Integration

5.2.5 The Cubic Spline Function Quadrature

5.2.6 The Gauss and Chebyshev Quadratures

5.3 Methods for Numerical Integration of Two Variable Functions

5.3.1 The Method of Small (Elementary) Cells

5.3.2 The Simpson Cubature Formula

5.4 An Example of Applications

References

Numerical Differentiation of One and Two Variable Functions

6.1 Approximating the Derivatives of One Variable Functions

6.2 Calculating the Derivatives of One Variable Function by Differentiation of the Corresponding Interpolating Polynomial

6.2.1 Differentiation of the Newton–Gregory Polynomial and Cubic Spline Functions

6.3 Formulas for Numerical Differentiation of Two Variable Functions

6.4 An Example of the Two-Dimensional Optimization Problem and its Solution by Using the Gradient Minimization Technique

References

Methods for Numerical Integration of Ordinary Differential Equations

7.1 The Initial Value Problem and Related Solution Methods

7.2 The One-Step Methods

7.2.1 The Euler Method and its Modified Version

7.2.2 The Heun Method

7.2.3 The Runge–Kutta Method (RK 4)

7.2.4 The Runge–Kutta–Fehlberg Method (RKF 45)

7.3 The Multi-step Predictor –Corrector Methods

7.3.1 The Adams–Bashforth–Moulthon Method

7.3.2 The Milne–Simpson Method

7.3.3 The Hamming Method

7.4 Examples of Using the RK 4 Method for Integration of Differential Equations Formulated for Some Electrical Rectifier Devices

7.4.1 The Unsymmetrical Voltage Doubler

7.4.2 The Full-Wave Rectifier Integrated with the Three-Element Low- Pass Filter

7.4.3 The Quadruple Symmetrical Voltage Multiplier

7.5 An Example of Solution of Riccati Equation Formulated for a Nonhomogenous Transmission Line Segment

7.6 An Example of Application of the Finite Difference Method for Solving the Linear Boundary Value Problem

References

The Finite Difference Method Adopted for Solving Laplace Boundary Value Problems

8.1 The Interior and External Laplace Boundary Value Problems

8.2 The Algorithm for Numerical Solving of Two-Dimensional Laplace Boundary Problems by Using the Finite Difference Method

8.2.1 The Liebmann Computational Procedure

8.2.2 The Successive Over-Relaxation Method (SOR)

8.3 Difference Formulas for Numerical Calculation of a Normal Component of an Electric Field Vector at Good Conducting Planes

8.4 Examples of Computation of the Characteristic Impedance and Attenuation Coefficient for Some TEM Transmission Lines

8.4.1 The Shielded Triplate Stripline

8.4.2 The Square Coaxial Line

8.4.3 The Triplate Stripline

8.4.4 The Shielded Inverted Microstrip Line

8.4.5 The Shielded Slab Line

8.4.6 Shielded Edge Coupled Triplate Striplines

References

A Equation of a Plane in Three-Dimensional Space

B The Inverse of the Given Nonsingular Square Matrix

C The Fast Elimination Method

D The Doolittle Formulas Making Possible Presentation of a Nonsingular Square Matrix in the form of the Product of Two Triangular Matrices

E Difference Formula for Calculation of the Electric Potential at Points Lying on the Border Between two Looseless Dielectric Media Without Electrical Charges

F Complete Elliptic Integrals of the First Kind

Subject Index