Table Of Contents:
Preface xv
Acknowledgments xix

1 Matrices and Linear Systems 1(242)

1.1 Introduction 1(9)

1.1.1 Linear Systems in Matrix Notation 7(3)

1.2 Properties of Matrices and Determinants 10(64)

1.2.1 Introduction to Matrices 10(5)

1.2.2 Some Special Matrix Forms 15(15)

1.2.3 Solutions of Linear Systems of Equations 30(8)

1.2.4 The Determinant of a Matrix 38(24)

1.2.5 Homogeneous Linear Systems 62(6)

1.2.6 Matrix Inversion Method 68(3)

1.2.7 Elementary Matrices 71(3)

1.3 Numerical Methods for Linear Systems 74(1)

1.4 Direct Methods for Linear Systems 74(87)

1.4.1 Cramer's Rule 75(4)

1.4.2 Gaussian Elimination Method 79(20)

1.4.3 Pivoting Strategies 99(7)

1.4.4 Gauss-Jordan Method 106(5)

1.4.5 LU Decomposition Method 111(46)

1.4.6 Tridiagonal Systems of Linear Equations 157(4)

1.5 Conditioning of Linear Systems 161(19)

1.5.1 Norms of Vectors and Matrices 162(5)

1.5.2 Errors in Solving Linear Systems 167(13)

1.6 Applications 180(22)

1.6.1 Curve Fitting, Electric Networks, and Traffic Flow 180(9)

1.6.2 Heat Conduction 189(3)

1.6.3 Chemical Solutions and Balancing Chemical Equations 192(3)

1.6.4 Manufacturing, Social, and Financial Issues 195(6)

1.6.5 Allocation of Resources 201(1)

1.7 Summary 202(2)

1.8 Problems 204(39)

2 Iterative Methods for Linear Systems 243(84)

2.1 Introduction 243(2)

2.2 Jacobi Iterative Method 245(7)

2.3 Gauss-Seidel Iterative Method 252(18)

2.4 Convergence Criteria 270(10)

2.5 Eigenvalues and Eigenvectors 280(14)

2.6 Successive Over-Relaxation Method 294(14)

2.7 Conjugate Gradient Method 308(5)

2.8 Iterative Refinement 313(2)

2.9 Summary 315(1)

2.10 Problems 316(11)

3 The Eigenvalue Problems 327(90)

3.1 Introduction 327(30)

Linear Algebra and Eigenvalues Problems 348(9)

3.3 Diagonalization of Matrices 357(17)

3.4 Basic Properties of Eigenvalue Problems 374(19)

3.5 Some Results of Eigenvalues Problems 393(4)

3.6 Applications of Eigenvalue Problems 397(12)

3.6.1 System of Differential Equations 397(8)

3.6.2 Difference Equations 405(4)

3.7 Summary 409(1)

3.8 Problems 410(7)

4 Numerical Computation of Eigenvalues 417(94)

4.1 Introduction 417(1)

4.2 Vector Iterative Methods for Eigenvalues 418(20)

4.2.1 Power Method 419(10)

4.2.2 Inverse Power Method 429(4)

4.2.3 Shifted Inverse Power Method 433(5)

4.3 Location of the Eigenvalues 438(4)

4.3.1 Gerschgorin Circles Theorem 438(2)

4.3.2 Rayleigh Quotient 440(2)

4.4 Intermediate Eigenvalues 442(4)

4.5 Eigenvalues of Symmetric Matrices 446(27)

4.5.1 Jacobi Method 448(7)

4.5.2 Sturm Sequence Iteration 455(5)

4.5.3 Given's Method 460(5)

4.5.4 Householder's Method 465(8)

4.6 Matrix Decomposition Methods 473(26)

4.6.1 QR Method 473(6)

4.6.2 LR Method 479(3)

4.6.3 Upper Hessenberg Form 482(9)

4.6.4 Singular Value Decomposition 491(8)

4.7 Summary 499(1)

4.8 Problems 500(11)

5 Interpolation and Approximation 511(142)

5.1 Introduction 511(2)

5.2 Polynomial Approximation 513(61)

5.2.1 Lagrange Interpolating Polynomials 514(16)

5.2.2 Newton's General Interpolating Formula 530(22)

5.2.3 Aitken's Method 552(5)

5.2.4 Chebyshev Polynomials 557(17)

5.3 Least Squares Approximation 574(60)

5.3.1 Linear Least Squares 575(6)

5.3.2 Polynomial Least Squares 581(4)

5.3.3 Nonlinear Least Squares 585(16)

5.3.4 Least Squares Plane 601(3)

5.3.5 Trigonometric Least Squares Polynomial 604(4)

5.3.6 Least Squares Solution of an Overdetermined System 608(5)

5.3.7 Least Squares Solution of an Underdetermined System 613(6)

5.3.8 The Pseudoinverse of a Matrix 619(3)

5.3.9 Least Squares with QR Decomposition 622(6)

5.3.10 Least Squares with Singular Value Decomposition 628(6)

5.4 Summary 634(1)

5.5 Problems 635(18)

6 Linear Programming 653(82)

6.1 Introduction 653(2)

6.2 General Formulation 655(1)

6.3 Terminology 656(1)

6.4 Linear Programming Problems 656(4)

6.4.1 Formulation of Mathematical Model 657(2)

6.4.2 Formulation of Mathematical Model 659(1)

6.5 Graphical Solution of LP Models 660(23)

6.5.1 Reversed Inequality Constraints 668(1)

6.5.2 Equality Constraints 668(1)

6.5.3 Minimum Value of a Function 669(7)

6.5.4 LP Problem in Canonical Form 676(1)

6.5.5 LP Problem in Standard Form 677(5)

6.5.6 Some Important Definitions 682(1)

6.6 The Simplex Method 683(10)

6.6.1 Basic and Nonbasic Variables 683(1)

6.6.2 The Simplex Algorithm 684(6)

6.6.3 Simplex Method for Minimization Problem 690(3)

6.7 Unrestricted in Sign Variables 693(2)

6.8 Finding a Feasible Basis 695(2)

6.8.1 By Trial and Error 695(1)

6.8.2 Use of Artificial Variables 696(1)

6.9 Big M Simplex Method 697(4)

6.10 Two-Phase Simplex Method 701(5)

6.11 Duality 706(11)

6.11.1 Comparison of Primal and Dual Problems 708(3)

6.11.2 Primal-Dual Problems in Standard Form 711(6)

6.12 Sensitivity Analysis in Linear Programming 717(4)

6.13 Summary 721(1)

6.14 Problems 722(13)

7 Nonlinear Programming 735(182)

7.1 Introduction 735(1)

7.2 Review of Differential Calculus 736(38)

7.2.1 Limits of Functions 736(2)

7.2.2 Continuity of a Function 738(1)

7.2.3 Derivative of a Function 739(3)

7.2.4 Local Extrema of a Function 742(10)

7.2.5 Directional Derivatives and the Gradient Vector 752(5)

7.2.6 Hessian Matrix 757(5)

7.2.7 Taylor's Series Expansion 762(6)

7.2.8 Quadratic Forms 768(6)

7.3 Nonlinear Equations and Systems 774(28)

7.3.1 Bisection Method 775(6)

7.3.2 Fixed-Point Method 781(5)

7.3.3 Newton's Method 786(3)

7.3.4 System of Nonlinear Equations 789(13)

7.4 Convex and Concave Functions 802(16)

7.5 Standard Form of a Nonlinear Programming Problem 818(1)

7.6 One-Dimensional Unconstrained Optimization 819(16)

7.6.1 Golden-Section Search 819(6)

7.6.2 Quadratic Interpolation 825(6)

7.6.3 Newton's Method 831(4)

7.7 Multidimensional Unconstrained Optimization 835(20)

7.7.1 Gradient Methods 840(10)

7.7.2 Newton's Method 850(5)

7.8 Constrained Optimization 855(26)

7.8.1 Lagrange Multipliers 855(13)

7.8.2 The Kuhn-Tucker Conditions 868(2)

7.8.3 Karush-Kuhn-Tucker Conditions 870(11)

7.9 Generalized Reduced-Gradient Method 881(9)

7.10 Separable Programming 890(5)

7.11 Quadratic Programming 895(5)

7.12 Summary 900(1)

7.13 Problems 901(16)

Appendices 917(212)

A Number Representations and Errors 917(24)

A.1 Introduction 917(1)

A.2 Number Representations and the Base of Numbers 918(3)

A.2.1 Normalized Floating-Point Representations 921(3)

A.2.2 Rounding and Chopping 924(1)

A.3 Error 925(2)

A.4 Sources of Errors 927(1)

A.4.1 Human Errors 927(1)

A.4.2 Truncation Errors 927(1)

A.4.3 Round-off Errors 928(1)

A.5 Effect of Round-off Errors in Arithmetic Operations 929(1)

A.5.1 Round-off Errors in Addition and Subtraction 929(2)

A.5.2 Round-off Errors in Multiplication 931(2)

A.5.3 Round-off Errors in Division 933(2)

A.5.4 Round-off Errors in Powers and Roots 935(2)

A.6 Summary 937(1)

A.7 Problems 938(3)

B Mathematical Preliminaries 941(66)

B.1 The Vector Space 941(1)

B.1.1 Vectors in Two Dimensions 942(5)

B.1.2 Vectors in Three Dimensions 947(17)

B.1.3 Lines and Planes in Space 964(12)

B.2 Complex Numbers 976(1)

B.2.1 Geometric Representation of Complex Numbers 977(1)

B.2.2 Operations on Complex Numbers 978(2)

B.2.3 Polar Forms of Complex Numbers 980(3)

B.2.4 Matrices with Complex Entries 983(1)

B.2.5 Solving Systems with Complex Entries 984(1)

B.2.6 Determinants of Complex Numbers 984(1)

B.2.7 Complex Eigenvalues and Eigenvectors 985(1)

B.3 Inner Product Spaces 986(1)

B.3.1 Properties of Inner Products 987(3)

B.3.2 Complex Inner Products 990(2)

B.4 Problems 992(15)

C Introduction to MATLAB 1007(90)

C.1 Introduction 1007(1)

C.2 Some Basic MATLAB Operations 1008(2)

C.2.1 MATLAB Numbers and Numeric Formats 1010(2)

C.2.2 Arithmetic Operations 1012(2)

C.2.3 MATLAB Mathematical Functions 1014(1)

C.2.4 Scalar Variables 1015(1)

C.2.5 Vectors 1016(4)

C.2.6 Matrices 1020(4)

C.2.7 Creating Special Matrices 1024(8)

C.2.8 Matrix Operations 1032(3)

C.2.9 Strings and Printing 1035(2)

C.2.10 Solving Linear Systems 1037(7)

C.2.11 Graphing in MATLAB 1044(7)

C.3 Programming in MATLAB 1051(1)

C.3.1 Statements for Control Flow 1051(1)

C.3.2 For Loop 1052(1)

C.3.3 While Loop 1052(1)

C.3.4 Nested for Loops 1053(1)

C.3.5 Structure 1054(2)

C.4 Defining Functions 1056(3)

C.5 MATLAB Built-in Functions 1059(2)

C.6 Symbolic Computation 1061(3)

C.6.1 Some Important Symbolic Commands 1064(5)

C.6.2 Solving Equations Symbolically 1069(2)

C.6.3 Calculus 1071(6)

C.6.4 Symbolic Ordinary Differential Equations 1077(2)

C.6.5 Linear Algebra 1079(1)

C.6.6 Eigenvalues and Eigenvectors 1080(1)

C.6.7 Plotting Symbolic Expressions 1081(2)

C.7 Symbolic Math Toolbox Functions 1083(3)

C.8 Index of MATLAB Programs 1086(3)

C.9 Summary 1089(1)

C.10 Problems 1090(7)

D Answers to Selected Exercises 1097(32)

D.0.1 Chapter 1 1097(10)

D.0.2 Chapter 2 1107(1)

D.0.3 Chapter 3 1108(3)

D.0.4 Chapter 4 1111(4)

D.0.5 Chapter 5 1115(3)

D.0.6 Chapter 6 1118(2)

D.0.7 Chapter 7 1120(2)

D.0.8 Appendix A 1122(1)

D.0.9 Appendix B 1123(3)

D.0.10 Appendix C 1126(3)
Bibliography 1129(16)
Index 1145