Numerical Methods for Scientists and Engineers

The “chain of spheres” (COS) algorithm, as part of the RIJCOSX SCF procedure, approximates the exchange term by performing analytic integration with respect to the coordinates of only one of the two electrons, whereas for the remaining coordinates, integration is carried out numerically. In the present work, we attempt to enhance the efficiency of the method by minimizing numerical errors in the COS procedure. The main idea is based on the work of Friesner and consists of finding a fitting matrix, Q, which leads the numerical and analytically evaluated overlap matrices to coincide. Using Q, the evaluation of exchange integrals can indeed be improved. Improved results and timings are obtained with the present default grid setup for both single point calculations and geometry optimizations. The fitting procedure results in a reduction of grid sizes necessary for achieving chemical accuracy. We demonstrate this by testing a number of grids and comparing results to the fully analytic and the earlier COS approximations. This turns out to be favourable for total and reaction energies, for which chemical accuracy can now be reached with a corresponding ∼30% speedup over the original RIJCOSX procedure for single point energies. Results are slightly less favourable for the accuracy of geometry optimizations, but the procedure is still shown to yield geometries with errors well below the method inherent errors of the employed theoretical framework.

An overlap fitted chain of spheres exchange method.

This paper proposes a spatially common sparsity based adaptive channel estimation and feedback scheme for frequency division duplex based massive multi-input multi-output (MIMO) systems, which adapts training overhead and pilot design to reliably estimate and feed back the downlink channel state information (CSI) with significantly reduced overhead. Specifically, a nonorthogonal downlink pilot design is first proposed, which is very different from standard orthogonal pilots. By exploiting the spatially common sparsity of massive MIMO channels, a compressive sensing (CS) based adaptive CSI acquisition scheme is proposed, where the consumed time slot overhead only adaptively depends on the sparsity level of the channels. In addition, a distributed sparsity adaptive matching pursuit algorithm is proposed to jointly estimate the channels of multiple subcarriers. Furthermore, by exploiting the temporal channel correlation, a closed-loop channel tracking scheme is provided, which adaptively designs the nonorthogonal pilot according to the previous channel estimation to achieve an enhanced CSI acquisition. Finally, we generalize the results of the multiple-measurement-vectors case in CS and derive the Cramer–Rao lower bound of the proposed scheme, which enlightens us to design the nonorthogonal pilot signals for the improved performance. Simulation results demonstrate that the proposed scheme outperforms its counterparts, and it is capable of approaching the performance bound.

Spatially Common Sparsity Based Adaptive Channel Estimation and Feedback for FDD Massive MIMO

Key generation from the randomness of wireless channels is a promising alternative to public key cryptography for the establishment of cryptographic keys between any two users. This paper reviews the current techniques for wireless key generation. The principles, performance metrics and key generation procedure are comprehensively surveyed. Methods for optimizing the performance of key generation are also discussed. Key generation applications in various environments are then introduced along with the challenges of applying the approach in each scenario. The paper concludes with some suggestions for future studies.

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Key Generation From Wireless Channels: A Review

A modified ABC algorithm based fast satellite image segmentation has been presented.ABC, PSO and GA methods are compared with this proposed method.The experimental results demonstrate better performance of MABC based technique.The proposed MABC based approach is much faster (CPU time is less).The validity of the proposed technique is reported both qualitatively and quantitatively. In this paper, a modified artificial bee colony (MABC) algorithm based satellite image segmentation using different objective function has been presented to find the optimal multilevel thresholds. Three different methods are compared with this proposed method such as ABC, particle swarm optimization (PSO) and genetic algorithm (GA) using Kapur's, Otsu and Tsallis objective function for optimal multilevel thresholding. The experimental results demonstrate that the proposed MABC algorithm based segmentation can efficiently and accurately search multilevel thresholds, which are very close to optimal ones examined by the exhaustive search method. In MABC algorithm, an improved solution search equation is used which is based on the bee's search only around the best solution of previous iteration to improve exploitation. In addition, to improve global convergence when generating initial population, both chaotic system and opposition-based learning method are employed. Compared to other thresholding methods, segmentation results of the proposed MABC algorithm is most promising, and the computational time is also minimized.

Modified artificial bee colony based computationally efficient multilevel thresholding for satellite image segmentation using Kapur's, Otsu and Tsallis functions

MIMO-OFDM is a key technology for next-generation cellular communications (3GPP-LTE, Mobile WiMAX, IMT-Advanced) as well as wireless LAN (IEEE 802.11a, IEEE 802.11n), wireless PAN (MB-OFDM), and broadcasting (DAB, DVB, DMB). In MIMO-OFDM Wireless Communications with MATLAB, the authors provide a comprehensive introduction to the theory and practice of wireless channel modeling, OFDM, and MIMO, using MATLAB programs to simulate the various techniques on MIMO-OFDM systems. One of the only books in the area dedicated to explaining simulation aspects Covers implementation to help cement the key concepts Uses materials that have been classroom-tested in numerous universities Provides the analytic solutions and practical examples with downloadable MATLAB codes Simulation examples based on actual industry and research projects Presentation slides with key equations and figures for instructor use MIMO-OFDM Wireless Communications with MATLAB is a key text for graduate students in wireless communications. Professionals and technicians in wireless communication fields, graduate students in signal processing, as well as senior undergraduates majoring in wireless communications will find this book a practical introduction to the MIMO-OFDM techniques. Instructor materials and MATLAB code examples available for download at www.wiley.com/go/chomimo

MIMO-OFDM Wireless Communications with MATLAB

Preface. 1. MATLAB Usage and Computational Errors. 1.1 Basic Operations of MATLAB. 1.1.1 Input/Output of Data from MATLAB Command Window. 1.1.2 Input/Output of Data Through Files. 1.1.3 Input/Output of Data Using Keyboard. 1.1.4 2-D Graphic Input/Output. 1.1.5 3-D Graphic Output. 1.1.6 Mathematical Functions. 1.1.7 Operations on Vectors and Matrices. 1.1.8 Random Number Generators. 1.1.9 Flow Control. 1.2 Computer Errors Versus Human Mistakes. 1.2.1 IEEE 64-bit Floating-Point Number Representation. 1.2.2 Various Kinds of Computing Errors. 1.2.3 Absolute/Relative Computing Errors. 1.2.4 Error Propagation. 1.2.5 Tips for Avoiding Large Errors. 1.3 Toward Good Program. 1.3.1 Nested Computing for Computational Efficiency. 1.3.2 Vector Operation Versus Loop Iteration. 1.3.3 Iterative Routine Versus Nested Routine. 1.3.4 To Avoid Runtime Error. 1.3.5 Parameter Sharing via Global Variables. 1.3.6 Parameter Passing Through Varargin. 1.3.7 Adaptive Input Argument List. Problems. 2. System of Linear Equations. 2.1 Solution for a System of Linear Equations. 2.1.1 The Nonsingular Case (M = N). 2.1.2 The Underdetermined Case (M N): Least-Squares Error Solution. 2.1.4 RLSE (Recursive Least-Squares Estimation). 2.2 Solving a System of Linear Equations. 2.2.1 Gauss Elimination. 2.2.2 Partial Pivoting. 2.2.3 Gauss-Jordan Elimination. 2.3 Inverse Matrix. 2.4 Decomposition (Factorization). 2.4.1 LU Decomposition (Factorization): Triangularization. 2.4.2 Other Decomposition (Factorization): Cholesky, QR, and SVD. 2.5 Iterative Methods to Solve Equations. 2.5.1 Jacobi Iteration. 2.5.2 Gauss-Seidel Iteration. 2.5.3 The Convergence of Jacobi and Gauss-Seidel Iterations. Problems. 3. Interpolation and Curve Fitting. 3.1 Interpolation by Lagrange Polynomial. 3.2 Interpolation by Newton Polynomial. 3.3 Approximation by Chebyshev Polynomial. 3.4 Pade Approximation by Rational Function. 3.5 Interpolation by Cubic Spline. 3.6 Hermite Interpolating Polynomial. 3.7 Two-dimensional Interpolation. 3.8 Curve Fitting. 3.8.1 Straight Line Fit: A Polynomial Function of First Degree. 3.8.2 Polynomial Curve Fit: A Polynomial Function of Higher Degree. 3.8.3 Exponential Curve Fit and Other Functions. 3.9 Fourier Transform. 3.9.1 FFT Versus DFT. 3.9.2 Physical Meaning of DFT. 3.9.3 Interpolation by Using DFS. Problems. 4. Nonlinear Equations. 4.1 Iterative Method Toward Fixed Point. 4.2 Bisection Method. 4.3 False Position or Regula Falsi Method. 4.4 Newton(-Raphson) Method. 4.5 Secant Method. 4.6 Newton Method for a System of Nonlinear Equations. 4.7 Symbolic Solution for Equations. 4.8 A Real-World Problem. Problems. 5. Numerical Differentiation/Integration. 5.1 Difference Approximation for First Derivative. 5.2 Approximation Error of First Derivative. 5.3 Difference Approximation for Second and Higher Derivative. 5.4 Interpolating Polynomial and Numerical Differential. 5.5 Numerical Integration and Quadrature. 5.6 Trapezoidal Method and Simpson Method. 5.7 Recursive Rule and Romberg Integration. 5.8 Adaptive Quadrature. 5.9 Gauss Quadrature. 5.9.1 Gauss-Legendre Integration. 5.9.2 Gauss-Hermite Integration. 5.9.3 Gauss-Laguerre Integration. 5.9.4 Gauss-Chebyshev Integration. 5.10 Double Integral. Problems. 6. Ordinary Differential Equations. 6.1 Euler's Method. 6.2 Heun's Method: Trapezoidal Method. 6.3 Runge-Kutta Method. 6.4 Predictor-Corrector Method. 6.4.1 Adams-Bashforth-Moulton Method. 6.4.2 Hamming Method. 6.4.3 Comparison of Methods. 6.5 Vector Differential Equations. 6.5.1 State Equation. 6.5.2 Discretization of LTI State Equation. 6.5.3 High-Order Differential Equation to State Equation. 6.5.4 Stiff Equation. 6.6 Boundary Value Problem (BVP). 6.6.1 Shooting Method. 6.6.2 Finite Difference Method. Problems. 7. Optimization. 7.1 Unconstrained Optimization [L-2, Chapter 7]. 7.1.1 Golden Search Method. 7.1.2 Quadratic Approximation Method. 7.1.3 Nelder-Mead Method [W-8]. 7.1.4 Steepest Descent Method. 7.1.5 Newton Method. 7.1.6 Conjugate Gradient Method. 7.1.7 Simulated Annealing Method [W-7]. 7.1.8 Genetic Algorithm [W-7]. 7.2 Constrained Optimization [L-2, Chapter 10]. 7.2.1 Lagrange Multiplier Method. 7.2.2 Penalty Function Method. 7.3 MATLAB Built-In Routines for Optimization. 7.3.1 Unconstrained Optimization. 7.3.2 Constrained Optimization. 7.3.3 Linear Programming (LP). Problems. 8. Matrices and Eigenvalues. 8.1 Eigenvalues and Eigenvectors. 8.2 Similarity Transformation and Diagonalization. 8.3 Power Method. 8.3.1 Scaled Power Method. 8.3.2 Inverse Power Method. 8.3.3 Shifted Inverse Power Method. 8.4 Jacobi Method. 8.5 Physical Meaning of Eigenvalues/Eigenvectors. 8.6 Eigenvalue Equations. Problems. 9. Partial Differential Equations. 9.1 Elliptic PDE. 9.2 Parabolic PDE. 9.2.1 The Explicit Forward Euler Method. 9.2.2 The Implicit Backward Euler Method. 9.2.3 The Crank-Nicholson Method. 9.2.4 Two-Dimensional Parabolic PDE. 9.3 Hyperbolic PDE. 9.3.1 The Explicit Central Difference Method. 9.3.2 Two-Dimensional Hyperbolic PDE. 9.4 Finite Element Method (FEM) for solving PDE. 9.5 GUI of MATLAB for Solving PDEs: PDETOOL. 9.5.1 Basic PDEs Solvable by PDETOOL. 9.5.2 The Usage of PDETOOL. 9.5.3 Examples of Using PDETOOL to Solve PDEs. Problems. Appendix A: Mean Value Theorem. Appendix B: Matrix Operations/Properties. Appendix C: Differentiation with Respect to a Vector. Appendix D: Laplace Transform. Appendix E: Fourier Transform. Appendix F: Useful Formulas. Appendix G: Symbolic Computation. Appendix H: Sparse Matrices. Appendix I: MATLAB. References. Subject Index. Index for MATLAB Routines. Index for Tables.

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Applied Numerical Methods Using MATLAB

Applied Numerical Methods Using MATLAB®: Yang/Applied Numerical MATLAB

This book is primarily intended for junior-level students who take the courses on signals and systems. It may be useful as a reference text for practicing engineers and scientists who want to acquire some of the concepts required for signal proce- ing. The readers are assumed to know the basics about linear algebra, calculus (on complex numbers, differentiation, and integration), differential equations, Laplace R transform, and MATLAB . Some knowledge about circuit systems will be helpful. Knowledge in signals and systems is crucial to students majoring in Electrical Engineering. The main objective of this book is to make the readers prepared for studying advanced subjects on signal processing, communication, and control by covering from the basic concepts of signals and systems to manual-like introduc- R R tions of how to use the MATLAB and Simulink tools for signal analysis and lter design. The features of this book can be summarized as follows: 1. It not only introduces the four Fourier analysis tools, CTFS (continuous-time Fourier series), CTFT (continuous-time Fourier transform), DFT (discrete-time Fourier transform), and DTFS (discrete-time Fourier series), but also illuminates the relationship among them so that the readers can realize why only the DFT of the four tools is used for practical spectral analysis and why/how it differs from the other ones, and further, think about how to reduce the difference to get better information about the spectral characteristics of signals from the DFT analysis.

Signals and Systems with MATLAB

This chapter presents the basic principle of OFDM (orthogonal frequency division multiplexing) transmission techniques and their performance, including the ready-to-run MATLAB? programs for the simulation. It also introduces the concept of orthogonal frequency division multiple access (OFDMA), which is the multiple access technique associated with OFDM.

Won Young Yang

Papers

MIMO-OFDM Wireless Communications with MATLAB

Applied Numerical Methods Using MATLAB

Applied Numerical Methods Using MATLAB®: Yang/Applied Numerical MATLAB

Signals and Systems with MATLAB

Introduction to OFDM