Introduction to Applied Linear Algebra with MATLAB

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1 Sigam Series in Applied Mathematics Volume 7 Rizwan Butt Introduction to Applied Linear Algebra with MATLAB Heldermann Verlag

2 Contents Number Systems and Errors Introduction Number Representation and Base of Numbers Normalized Floating-point Representation Rounding and Chopping Error Sources of Errors Human Error Truncation Error Round-off Error Effect of Round-off Errors in Arithmetic Operation Rounding off Errors in Addition and Subtraction Rounding off Errors in Multiplication Rounding off Errors in Division Rounding off Errors in Powers and roots Summary Problems 17 Systems of Linear Equations Introduction Linear System in Matrix Notation Properties of Matrices and Determinant Introduction of Matrices Some Special Matrix Forms The Determinant of Matrix Numerical Methods for Linear Systems Direct Methods for Linear Systems Cramer's Rule Gaussian Elimination Method Pivoting Strategies Gauss-Jordan Method LU Decomposition Method Tridiagonal Systems of linear equations 90

3 VIII Contents 2.5 Applications Curve Fitting, Electric Networks and Traffic Flow Heat Conduction Chemical Solutions and Balancing Chemical Equations Manufacturing, Social, and Financial Issues Allocation of Resources Summary Problems Conditioning of Linear Systems Introduction Norms of Vectors and Matrices Errors in Solving Linear Systems Summary Problems Iterative Methods for Linear Systems Introduction Jacobi Iterative Method Gauss-Seidel Iterative Method Convergence Criteria Eigenvalues and Eigenvectors Successive Over-Relaxation Method Conjugate Gradient Method Iterative Refinement Summary Problems The Eigenvalue problems Introduction Linear Algebra and Eigenvalues Problems Diagonalization of Matrices Basic Properties of Eigenvalue Problems Some Results of Eigenvalues Problems Applications of Eigenvalue Problems System of Differential Equations Difference Equations Summary Problems Numerical Computation of Eigenvalues Introduction Vector Iterative Methods for Eigenvalues Power Method 244

4 Contents IX Inverse Power Method Shifted Inverse Power Method Location of the Eigenvalues Gerschgorin Circles Theorem Rayleigh Quotient Intermediate Eigenvalues Eigenvalues of Symmetric Matrices Jacobi Method Sturm Sequence Iteration Given's Method Householder's Method Matrix Decomposition Methods QR Method LR Method Upper Hessenberg Form Singular Value Decomposition Summary Problems Approximating Functions Introduction Polynomial Approximation Aitken's Method Least Squares Approximation Linear Least Squares Polynomial Least Squares Nonlinear Least Squares Least Squares Plane Least Squares Solution of a Overdetermined System Least Squares with QR Decomposition Least Squares with Singular Value Decomposition Summary Problems Linear Programming Introduction General Formulation Terminology Linear programming Problems Graphical Solution of LP Models Reversed Inequality Constraints Equality Constraints Minimum Value of a Function Linear Program in Canonical Form 368

5 X Contents Linear Program in Standard Form Some Important Definitions The Simplex Method Unrestricted-in-Sign Variables Finding a Feasible Basis Big M Simplex Method Two-Phase Simplex Method Duality Comparison of Primal and Dual Problems Primal-Dual Problems in Standard Form Sensitivity Analysis in Linear Programming Summary 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

1 Number Systems and Errors 1

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