Sigam Series in Applied Mathematics Volume 7 Rizwan Butt Introduction to Applied Linear Algebra with MATLAB Heldermann Verlag
Contents Number Systems and Errors 1 1.1 Introduction 1 1.2 Number Representation and Base of Numbers 1 1.2.1 Normalized Floating-point Representation 3 1.2.2 Rounding and Chopping 5 1.3 Error 6 1.4 Sources of Errors 7 1.4.1 Human Error 7 1.4.2 Truncation Error 8 1.4.3 Round-off Error 8 1.5 Effect of Round-off Errors in Arithmetic Operation 9 1.5.1 Rounding off Errors in Addition and Subtraction 9 1.5.2 Rounding off Errors in Multiplication 10 1.5.3 Rounding off Errors in Division 12 1.5.4 Rounding off Errors in Powers and roots 14 1.6 Summary 15 1.7 Problems 17 Systems of Linear Equations 19 2.1 Introduction 19 2.1.1 Linear System in Matrix Notation 23 2.2 Properties of Matrices and Determinant 25 2.2.1 Introduction of Matrices 25 2.2.2 Some Special Matrix Forms 29 2.2.3 The Determinant of Matrix 40 2.3 Numerical Methods for Linear Systems 45 2.4 Direct Methods for Linear Systems 46 2.4.1 Cramer's Rule 46 2.4.2 Gaussian Elimination Method 49 2.4.3 Pivoting Strategies 61 2.4.4 Gauss-Jordan Method 66 2.4.5 LU Decomposition Method 71 2.4.6 Tridiagonal Systems of linear equations 90
VIII Contents 2.5 Applications 93 2.5.1 Curve Fitting, Electric Networks and Traffic Flow 93 2.5.2 Heat Conduction 100 2.5.3 Chemical Solutions and Balancing Chemical Equations... 102 2.5.4 Manufacturing, Social, and Financial Issues 105 2.5.5 Allocation of Resources 109 2.6 Summary 110 2.7 Problems 112 3 Conditioning of Linear Systems 123 3.1 Introduction 123 3.2 Norms of Vectors and Matrices 123 3.3 Errors in Solving Linear Systems 127 3.4 Summary 138 3.5 Problems 139 4 Iterative Methods for Linear Systems 143 4.1 Introduction 143 4.2 Jacobi Iterative Method 144 4.3 Gauss-Seidel Iterative Method 149 4.4 Convergence Criteria 153 4.5 Eigenvalues and Eigenvectors 158 4.6 Successive Over-Relaxation Method 165 4.7 Conjugate Gradient Method 171 4.8 Iterative Refinement 175 4.9 Summary 177 4.10 Problems 178 5 The Eigenvalue problems 183 5.1 Introduction 183 5.2 Linear Algebra and Eigenvalues Problems 193 5.3 Diagonalization of Matrices 197 5.4 Basic Properties of Eigenvalue Problems 209 5.5 Some Results of Eigenvalues Problems 224 5.6 Applications of Eigenvalue Problems 227 5.6.1 System of Differential Equations 227 5.6.2 Difference Equations 233 5.7 Summary 237 5.8 Problems 238 6 Numerical Computation of Eigenvalues 243 6.1 Introduction 243 6.2 Vector Iterative Methods for Eigenvalues 244 6.2.1 Power Method 244
Contents IX 6.2.2 Inverse Power Method 248 6.2.3 Shifted Inverse Power Method 251 6.3 Location of the Eigenvalues 255 6.3.1 Gerschgorin Circles Theorem 255 6.3.2 Rayleigh Quotient 256 6.4 Intermediate Eigenvalues 258 6.5 Eigenvalues of Symmetric Matrices 261 6.5.1 Jacobi Method 262 6.5.2 Sturm Sequence Iteration 268 6.5.3 Given's Method 271 6.5.4 Householder's Method 275 6.6 Matrix Decomposition Methods 279 6.6.1 QR Method 279 6.6.2 LR Method 283 6.6.3 Upper Hessenberg Form 285 6.6.4 Singular Value Decomposition 292 6.7 Summary 299 6.8 Problems 300 7 Approximating Functions 307 7.1 Introduction 307 7.2 Polynomial Approximation 308 7.2.1 Aitken's Method 308 7.3 Least Squares Approximation 312 7.3.1 Linear Least Squares 313 7.3.2 Polynomial Least Squares 317 7.3.3 Nonlinear Least Squares 321 7.3.4 Least Squares Plane 328 7.3.5 Least Squares Solution of a Overdetermined System 330 7.3.6 Least Squares with QR Decomposition 334 7.3.7 Least Squares with Singular Value Decomposition 339 7.4 Summary 344 7.5 Problems 345 8 Linear Programming 351 8.1 Introduction 351 8.2 General Formulation 352 8.2.1 Terminology 353 8.3 Linear programming Problems 353 8.4 Graphical Solution of LP Models 356 8.4.1 Reversed Inequality Constraints 362 8.4.2 Equality Constraints 362 8.4.3 Minimum Value of a Function 362 8.4.4 Linear Program in Canonical Form 368
X Contents 8.4.5 Linear Program in Standard Form 369 8.4.6 Some Important Definitions 372 8.5 The Simplex Method 373 8.5.1 Unrestricted-in-Sign Variables 381 8.6 Finding a Feasible Basis 383 8.6.1 Big M Simplex Method 385 8.6.2 Two-Phase Simplex Method 387 8.7 Duality 391 8.7.1 Comparison of Primal and Dual Problems 393 8.7.2 Primal-Dual Problems in Standard Form 395 8.8 Sensitivity Analysis in Linear Programming 400 8.9 Summary 402 8.10 Problems 403 Appendices 413 A Complex Numbers and Inner Products 413 A.I Complex Numbers 413 A. 1.1 Geometric Representation of Complex Number 414 A. 1.2 Operations on Complex Numbers 414 A.1.3 Polar Form of Complex Number 416 A. 1.4 Matrices with Complex Entries 419 A. 1.5 Solving Systems with Complex Entries 420 A.1.6 Determinants of Complex Numbers 420 A.I.7 Complex Eigenvalues and Eigenvectors 421 A.2 Inner Product Space 421 A.2.1 Properties of Inner Products 422 A.2.2 Complex Inner Products 425 B Introduction of MATLAB 427 B.I MATLAB Built-in-Functions 463 B.2 Symbolic Computation 465 B.3 Symbolic Math Toolbox Functions 481 B.4 Index of MATLAB Programs 482 C Answers to Selected Problems 485 Bibliography 508 Index 509