## Fundamental Numerical Methods for Electrical EngineeringStormy development of electronic computation techniques (computer systems and software), observed during the last decades, has made possible automation of data processing in many important human activity areas, such as science, technology, economics and labor organization. In a broadly understood technology area, this developmentledtoseparationofspecializedformsofusingcomputersforthedesign and manufacturing processes, that is: – computer-aided design (CAD) – computer-aided manufacture (CAM) In order to show the role of computer in the rst of the two applications m- tioned above, let us consider basic stages of the design process for a standard piece of electronic system, or equipment: – formulation of requirements concerning user properties (characteristics, para- ters) of the designed equipment, – elaboration of the initial, possibly general electric structure, – determination of mathematical model of the system on the basis of the adopted electric structure, – determination of basic responses (frequency- or time-domain) of the system, on the base of previously established mathematical model, – repeated modi cation of the adopted diagram (changing its structure or element values) in case, when it does not satisfy the adopted requirements, – preparation of design and technological documentation, – manufacturing of model (prototype) series, according to the prepared docum- tation, – testing the prototype under the aspect of its electric properties, mechanical du- bility and sensitivity to environment conditions, – modi cation of prototype documentation, if necessary, and handing over the documentation to series production. The most important stages of the process under discussion are illustrated in Fig. I. 1. xi xii Introduction Fig. I. |

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### Contents

1 | |

112 The GaussJordan Elimination Method | 9 |

113 The LU Matrix Decomposition Method | 11 |

114 The Method of Inverse Matrix | 14 |

12 Indirect or Iterative Methods | 17 |

122 Jacobi and GaussSeidel Methods | 18 |

13 Examples of Applications in Electrical Engineering | 23 |

References | 27 |

532 The Simpson Cubature Formula | 148 |

54 An Example of Applications | 151 |

References | 154 |

Numerical Differentiation of One and Two Variable Functions | 155 |

61 Approximating the Derivatives of One Variable Functions | 157 |

62 Calculating the Derivatives of One Variable Function by Differentiation of the Corresponding Interpolating Polynomial | 163 |

63 Formulas for Numerical Differentiation of Two Variable Functions | 168 |

64 An Example of the TwoDimensional Optimization Problem and its Solution by Using the Gradient Minimization Technique | 172 |

Methods for Numerical Solving the Single Nonlinear Equations | 29 |

21 Determination of the Complex Roots of Polynomial Equations by Using the Lins and Bairstows Methods | 30 |

212 Bairstows Method | 32 |

213 Laguerre Method | 35 |

22 Iterative Methods Used for Solving Transcendental Equations | 36 |

221 Bisection Method of Bolzano | 37 |

222 The Secant Method | 38 |

223 Method of Tangents NewtonRaphson | 40 |

23 Optimization Methods | 42 |

24 Examples of Applications | 44 |

References | 47 |

Methods for Numerical Solution of Nonlinear Equations | 49 |

32 The Iterative Parameter Perturbation Procedure | 51 |

33 The Newton Iterative Method | 52 |

34 The Equivalent Optimization Strategies | 56 |

35 Examples of Applications in the Microwave Technique | 58 |

References | 68 |

Methods for the Interpolation and Approximation of One Variable Function | 69 |

41 Fundamental Interpolation Methods | 72 |

412 The Lagrange Interpolating Polynomial | 73 |

413 The Aitken Interpolation Method | 76 |

414 The NewtonGregory Interpolating Polynomial | 77 |

415 Interpolation by Cubic Spline Functions | 82 |

416 Interpolation by a Linear Combination of Chebyshev Polynomials of the First Kind | 86 |

42 Fundamental Approximation Methods for One Variable Functions | 89 |

422 The Maximally Flat Butterworth Approximation | 94 |

423 Approximation Curve Fitting by the Method of Least Squares | 97 |

424 Approximation of Periodical Functions by Fourier Series | 102 |

43 Examples of the Application of Chebyshev Polynomials in Synthesis of Radiation Patterns of the InPhase Linear Array Antenna | 111 |

References | 120 |

Methods for Numerical Integration of One and Two Variable Functions | 121 |

51 Integration of Deﬁnite Integrals by Expanding the Integrand Function in Finite Series of Analytically Integrable Functions | 123 |

52 Fundamental Methods for Numerical Integration of One Variable Functions | 125 |

522 The Romberg Integration Rule | 130 |

523 The Simpson Method of Integration | 132 |

524 The NewtonCotes Method of Integration | 136 |

525 The Cubic Spline Function Quadrature | 138 |

526 The Gauss and Chebyshev Quadratures | 140 |

53 Methods for Numerical Integration of Two Variable Functions | 147 |

References | 177 |

Methods for Numerical Integration of Ordinary Differential Equations | 178 |

72 The OneStep Methods | 180 |

722 The Heun Method | 182 |

723 The RungeKutta Method RK 4 | 184 |

724 The RungeKuttaFehlberg Method RKF 45 | 186 |

73 The Multistep PredictorCorrector Methods | 189 |

731 The AdamsBashforthMoulthon Method | 193 |

732 The MilneSimpson Method | 194 |

733 The Hamming Method | 197 |

74 Examples of Using the RK 4 Method for Integration of Differential Equations Formulated for Some Electrical Rectiﬁer Devices | 199 |

742 The FullWave Rectiﬁer Integrated with the ThreeElement LowPass Filter | 204 |

743 The Quadruple Symmetrical Voltage Multiplier | 208 |

75 An Example of Solution of Riccati Equation Formulated for a Nonhomogenous Transmission Line Segment | 215 |

76 An Example of Application of the Finite Difference Method for Solving the Linear Boundary Value Problem | 219 |

References | 221 |

The Finite Difference Method Adopted for Solving Laplace Boundary Value Problems | 223 |

81 The Interior and External Laplace Boundary Value Problems | 226 |

82 The Algorithm for Numerical Solving of TwoDimensional Laplace Boundary Problems by Using the Finite Difference Method | 228 |

821 The Liebmann Computational Procedure | 231 |

822 The Successive OverRelaxation Method SOR | 238 |

83 Difference Formulas for Numerical Calculation of a Normal Component of an Electric Field Vector at Good Conducting Planes | 242 |

Impedance and Attenuation Coefﬁcient for Some TEM Transmission Lines | 245 |

841 The Shielded Triplate Stripline | 246 |

842 The Square Coaxial Line | 249 |

843 The Triplate Stripline | 251 |

844 The Shielded Inverted Microstrip Line | 253 |

845 The Shielded Slab Line | 258 |

846 Shielded Edge Coupled Triplate Striplines | 263 |

References | 268 |

Equation of a Plane in ThreeDimensional Space | 269 |

The Inverse of the Given Nonsingular Square Matrix | 271 |

The Fast Elimination Method | 273 |

The Doolittle Formulas Making Possible Presentation of a Nonsingular Square Matrix in the form of the Product of Two Triangular Matrices | 275 |

Difference Formula for Calculation of the Electric Potential at Points Lying on the Border Between two Looseless Dielectric Media Without Electrical... | 277 |

Complete Elliptic Integrals of the First Kind | 279 |

281 | |

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Fundamental Numerical Methods for Electrical Engineering Stanislaw Rosloniec No preview available - 2010 |

Fundamental Numerical Methods for Electrical Engineering Stanislaw Rosloniec No preview available - 2011 |

Fundamental Numerical Methods for Electrical Engineering Stanislaw Rosloniec No preview available - 2008 |

### Common terms and phrases

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