Numerical methods for engineers and scientists using MATLAB:
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| Format: | Book |
| Language: | English |
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Boca Raton, FL
CRC Press, Taylor & Francis Group
[2017]
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| Edition: | Second edition |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | xxi, 471 Seiten Diagramme |
| ISBN: | 9781498777421 |
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| 245 | 1 | 0 | |a Numerical methods for engineers and scientists using MATLAB |c Ramin S. Esfandiari, PdD |
| 250 | |a Second edition | ||
| 264 | 1 | |a Boca Raton, FL |b CRC Press, Taylor & Francis Group |c [2017] | |
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| adam_text | Titel: Numerical methods for engineers and scientists using MATLAB
Autor: Esfandiari, Ramin S
Jahr: 2017
Contents
Preface xv
Acknowledgments xix
Author xxi
1. Background and Introduction 1
Part 1: Background 1
1.1 Differential Equations 1
1.1.1 Linear, First-Order ODEs 1
1.1.2 Second-Order ODEs with Constant Coefficients 2
.. 1.1.2.1 Homogeneous Solution 2
1.1.2.2 Particular Solution 3
1.1.3 Method of Undetermined Coefficients 3
1.2 Matrix Analysis 4
1.2.1 Matrix Operations 5
1.2.2 Matrix Transpose 5
1.2.3 Special Matrices 6
1.2.4 Determinant of a Matrix 6
1.2.5 Properties of Determinant 6
1.2.5.1 Cramer s Rule 7
1.2.6 Inverse of a Matrix 8
1.2.7 Properties of Inverse 9
1.2.8 Solving a Linear System of Equations 9
1.3 Matrix Eigenvalue Problem 9
1.3.1 Solving the Eigenvalue Problem 10
1.3.2 Similarity Transformation 11
1.3.3 Matrix Diagonalization 11
1.3.4 Eigenvalue Properties of Matrices 12
Part 2: Introduction to Numerical Methods 12
1.4 Errors and Approximations 12
1.4.1 Sources of Computational Error 12
1.4.2 Binary and Hexadecimal Numbers 13
1.4.3 Floating Point and Rounding Errors 13
1.4.4 Round-Off: Chopping and Rounding 14
1.4.5 Absolute and Relative Errors 15
1.4.6 Error Bound 16
1.4.7 Transmission of Error from a Source to the Final Result 16
1.4.8 Subtraction of Nearly Equal Numbers 17
1.5 Iterative Methods 19
1.5.1 Fundamental Iterative Method 20
1.5.2 Rate of Convergence of an Iterative Method 21
Problem Set (Chapter 1) 22
2. Introduction to MATLAB® 27
2.1 MATLAB Built-in Functions 27
vii
Contents
Vlll
2.1.1 Rounding Commands 27
2.1.2 Relational Operators 28
2.1.3 Format Options 28
2.2 Vectors and Matrices 2^
2.2.1 Linspace 30
2.2.2 Matrices ^
2.2.3 Determinant, Transpose, and Inverse 32
2.2.4 Slash Operators 33
2.2.5 Element-by-Element Operations 33
2.2.6 Diagonal Matrices and Diagonals of a Matrix 34
2.3 Symbolic Math Toolbox 36
2.3.1 Anonymous Functions 38
2.3.2 MATLAB Function 38
2.3.3 Differentiation 39
2.3.4 Partial Derivatives 40
2.3.5 Integration 40
2.4 Program Flow Control 41
2.4.1 for Loop 41
2.4.2 The if Command 42
2.4.3 while Loop 43
2.5 Displaying Formatted Data 43
2.5.1 Differential Equations 44
2.6 Plotting 45
2.6.1 subplot 45
2.6.2 Plotting Analytical Expressions 46
2.6.3 Multiple Plots 46
2.7 User-Defined Functions and Script Files 47
2.7.1 Setting Default Values for Input Variables 49
2.7.2 Creating Script Files 50
Problem Set (Chapter 2) 51
3. Numerical Solution of Equations of a Single Variable 55
3.1 Numerical Solution of Equations 55
3.2 Bisection Method 55
3.2.1 MATLAB Built-in Function fzero 60
3.3 Regula Falsi Method (Method of False Position) 61
3.3.1 Modified Regula Falsi Method 64
3.4 Fixed-Point Method gg
3.4.1 Selection of a Suitable Iteration Function 66
3.4.2 A Note on Convergence 67
3.4.3 Rate of Convergence of the Fixed-Point Iteration 71
3.5 Newton s Method (Newton-Raphson Method) 72
3.5.1 Rate of Convergence of Newton s Method 76
3.5.2 A Few Notes on Newton s Method [ 77
3.5.3 Modified Newton s Method for Roots with Multiplicity
2 or Higher 7o
3.6 Secant Method g^
3.6.1 Rate of Convergence of Secant Method 83
3.6.2 A Few Notes on Secant Method 01
Contents ix
3.7 Equations with Several Roots 83
3.7.1 Finding Roots to the Right of a Specified Point ; 83
3.7.2 Finding Several Roots in an Interval Using f zero 84
Problem Set (Chapter 3) 88
4. Numerical Solution of Systems of Equations 95
4.1 Linear Systems of Equations 95
4.2 Numerical Solution of Linear Systems 96
4.3 Gauss Elimination Method 96
4.3.1 Choosing the Pivot Row: Partial Pivoting with Row Scaling 98
4.3.2 Permutation Matrices 99
4.3.3 Counting the Number of Operations 102
4.3.3.1 Elimination 102
4.3.3.2 Back Substitution 103
4.3.4 Tridiagonal Systems 103
4.3.4.1 Thomas Method 104
4.3.4.2 MATLAB Built-in Function 106
4.4 LU Factorization Methods 107
4.4.1 Doolittle Factorization 107
4.4.2 Finding L and U Using Steps of Gauss Elimination 108
4.4.3 Finding L and U Directly 108
4.4.3.1 Doolittle s Method to Solve a Linear System 110
4.4.3.2 Operations Count 112
4.4.4 Cholesky Factorization 112
4.4.4.1 Cholesky s Method to Solve a Linear System 113
4.4.4.2 Operations Count 115
4.4.4.3 MATLAB Built-in Functions lu and chol 115
4.5 Iterative Solution of Linear Systems 116
4.5.1 Vector Norms 116
4.5.2 Matrix Norms 118
4.5.2.1 Compatibility of Vector and Matrix Norms 119
4.5.3 General Iterative Method 120
4.5.3.1 Convergence of the General Iterative Method 120
4.5.4 Jacobi Iteration Method 121
4.5.4.1 Convergence of the Jacobi Iteration Method 122
4.5.5 Gauss-Seidel Iteration Method 125
4.5.5.1 Convergence of the Gauss-Seidel Iteration Method 127
4.5.6 Indirect Methods versus Direct Methods for Large Systems 130
4.6 Ill-Conditioning and Error Analysis 131
4.6.1 Condition Number 131
4.6.2 Ill-Conditioning 132
4.6.2.1 Indicators of Ill-Conditioning 133
4.6.3 Computational Error 133
4.6.3.1 Consequences of Ill-Conditioning 135
4.6.4 Effects of Parameter Changes on the Solution 136
4.7 Systems of Nonlinear Equations 138
4.7.1 Newton s Method for a System of Nonlinear Equations 138
4.7.1.1 Newton s Method for Solving a System of
Two Nonlinear Equations 138
X
Contents
4.7.1.2 Newton s Method for Solving a System of n Nonlinear
Equations 1^2
4.7.1.3 Convergence of Newton s Method 142
4.7.2 Fixed-Point Iteration Method for a System of Nonlinear Equations.... 143
4.7.2.1 Convergence of the Fixed-Point Iteration Method 143
Problem Set (Chapter 4) ^
5. Curve Fitting and Interpolation 161
5.1 Least-Squares Regression 161
5.2 Linear Regression 162
5.2.1 Deciding a Best Fit Criterion 163
5.2.2 Linear Least-Squares Regression 164
5.3 Linearization of Nonlinear Data 167
5.3.1 Exponential Function 167
5.3.2 Power Function 167
5.3.3 Saturation Function 168
5.4 Polynomial Regression 172
5.4.1 Quadratic Least-Squares Regression 174
5.4.2 Cubic Least-Squares Regression 176
5.4.3 MATLAB Built-in Functions Polyf it and Polyval 178
5.5 Polynomial Interpolation 179
5.5.1 Lagrange Interpolating Polynomials 180
5.5.2 Drawbacks of Lagrange Interpolation 183
. 5.5.3 Newton Divided-Difference Interpolating Polynomials 184
5.5.4 Special Case: Equally-Spaced Data 190
5.5.5 Newton Forward-Difference Interpolating Polynomials 191
5.6 Spline Interpolation 193
5.6.1 Linear Splines 194
5.6.2 Quadratic Splines 195
5.6.2.1 Function Values at the Endpoints (2 Equations) 195
5.6.2.2 Function Values at the Interior Knots (In - 2 Equations) 196
5.6.2.3 First Derivatives at the Interior Knots (n - 1 Equations) 196
5.6.2.4 Second Derivative at the Left Endpoint is Zero (1 Equation).... 196
5.6.3 Cubic Splines 198
5.6.3.1 Clamped Boundary Conditions 199
5.6.3.2 Free Boundary Conditions 199
5.6.4 Construction of Cubic Splines: Clamped Boundary Conditions 199
5.6.5 Construction of Cubic Splines: Free Boundary Conditions 204
5.6.6 MATLAB Built-in Functions interpl and spline 205
5.6.7 Boundary Conditions 207
5.6.8 Interactive Curve Fitting and Interpolation in MATLAB 208
5.7 Fourier Approximation and Interpolation 209
5.7.1 Sinusoidal Curve Fitting 209
5.7.1.1 Fourier Approximation 210
5.7.1.2 Fourier Interpolation 210
5.7.2 Linear Transformation of Data 210
5.7.3 Discrete Fourier Transform 215
5.7.4 Fast Fourier Transform 216
5.7.4.1 Sande-Tukey Algorithm (N = 2v,p = integer) 217
Contents xj
5.7.4.2 Case Study: N = 23 = 8 218
5.7.4.3 Cooley-Tukey Algorithm (N = 2p, p = integer) 219
5.7.5 MATLAB Built-in Function f f t 220
5.7.5.1 Interpolation Using f f t 220
Problem Set (Chapter 5) ..223
6. Numerical Differentiation and Integration 249
6.1 Numerical Differentiation 249
6.2 Finite-Difference Formulas for Numerical Differentiation 249
6.2.1 Finite-Difference Formulas for the First Derivative 250
6.2.1.1 Two-Point Backward Difference Formula 250
6.2.1.2 Two-Point Forward Difference Formula 251
6.2.1.3 Two-Point Central Difference Formula 251
6.2.1.4 Three-Point Backward Difference Formula 252
6.2.1.5 Three-Point Forward Difference Formula 253
6.2.2 Finite-Difference Formulas for the Second Derivative 254
6.2.2.1 Three-Point Backward Difference Formula 254
6.2.2.2 Three-Point Forward Difference Formula 254
6.2.2.3 Three-Point Central Difference Formula 255
6.2.2.4 Summary of Finite-Difference pormulas for First to
Fourth Derivatives 256
6.2.3 Estimate Improvement: Richardson s Extrapolation 256
6.2.4 Richardson s Extrapolation for Discrete Sets of Data 259
6.2.5 Derivative Estimates for Non-Evenly Spaced Data 259
6.2.6 MATLAB Built-in Functions dif f and polyder 260
6.3 Numerical Integration: Newton-Cotes Formulas 261
6.3.1 Newton-Cotes Formulas 262
6.3.2 Rectangular Rule 262
6.3.2.1 Composite Rectangular Rule 262
6.3.3 Error Estimate for Composite Rectangular Rule 264
6.3.4 Trapezoidal Rule 266
6.3.4.1 Composite Trapezoidal Rule 267
6.3.4.2 Error Estimate for Composite Trapezoidal Rule 267
6.3.5 Simpson s Rules 269
6.3.5.1 Simpson s 1/3 Rule 269
6.3.5.2 Composite Simpson s 1/3 Rule 270
6.3.5.3 Error Estimate for Composite Simpson s 1/3 Rule 270
6.3.5.4 Simpson s 3/8 Rule 271
6.3.5.5 Composite Simpson s 3/8 Rule 272
6.3.5.6 Error Estimate for Composite Simpson s 3/8 Rule 273
6.3.6 MATLAB Built-in Functions quad and trapz 273
6.4 Numerical Integration of Analytical Functions: Romberg Integration,
Gaussian Quadrature 275
6.4.1 Romberg Integration 275
6.4.1.1 Richardson s Extrapolation 275
6.4.1.2 Romberg Integration 278
6.4.2 Gaussian Quadrature 280
6.5 Improper Integrals 285
Problem Set (Chapter 6) 286
Contents
xn
, Numerical Solution of Initial-Value Problems 301
7.1 Introduction ^
7.2 One-Step Methods ^01
7.3 Euler s Method 302
7.3.1 Error Analysis for Euler s Method 305
7.3.2 Calculation of Local and Global Truncation Errors 305
7.3.3 Higher-Order Taylor Methods 307
7.4 Runge-Kutta Methods 309
7.4.1 Second-Order Runge-Kutta (RK2) Methods 310
7.4.1.1 Improved Euler s Method 311
7.4.1.2 Heun s Method 311
7.4.1.3 Ralston s Method 312
7.4.1.4 Graphical Representation of Heun s Method 312
7.4.2 Third-Order Runge-Kutta (RK3) Methods 315
7.4.2.1 The Classical RK3 Method 315
7.4.2.2 Heun s RK3 Method 315
7.4.3 Fourth-Order Runge-Kutta (RK4) Methods 316
7.4.3.1 The Classical RK4 Method 317
7.4.4 Higher-Order Runge-Kutta Methods 319
7.4.5 Selection of Optimal Step Size: Runge-Kutta Fehlberg (RKF)
Method 320
7.4.5.1 Adjustment of Step Size 321
7.5 Multistep Methods 322
7.5.1 Adams-Bashforth Method 323
7.5.1.1 Second-Order Adams-Bashforth Formula 324
7.5.1.2 Third-Order Adams-Bashforth Formula 324
7.5.1.3 Fourth-Order Adams-Bashforth Formula 324
7.5.2 Adams-Moulton Method 325
7.5.2.1 Second-Order Adams-Moulton Formula 326
7.5.2.2 Third-Order Adams-Moulton Formula 326
7.5.2.3 Fourth-Order Adams-Moulton Formula 326
7.5.3 Predictor-Corrector Methods 326
7.5.3.1 Heun s Predictor-Corrector Method 327
7.5.3.2 Adams-Bashforth-Moulton (ABM)
Predictor-Corrector Method 327
7.6 Systems of Ordinary Differential Equations 330
7.6.1 Transformation into a System of First-Order ODEs 330
7.6.1.1 State Variables
7.6.1.2 Notation 33q
7.6.1.3 State-Variable Equations
7.6.2 Numerical Solution of a System of First-Order ODEs 332
7.6.2.1 Euler s Method for Systems 332
7.6.2.2 Heun s Method for Systems . 1.335
7.6.2.3 Classical RK4 Method for Systems.. 336
7.7 Stability lllllllllllllllllllllll 340
7.7.1 Euler s Method 041
7.7.2 Euler s Implicit Method 041
7.8 Stiff Differential Equations 343
7.9 MATLAB Built-in Functions for Solving Initial-Value Problems!!! . !!!! !!!!!!!!345
Contents xiii
7.9.1 Non-Stiff Equations 345
7.9.2 A Single First-Order IVP 345
7.9.3 Setting ODE Solver Options 347
7.9.4 A System of First-Order IVPs 348
7.9.5 Stiff Equations 349
Problem Set (Chapter 7) 350
8. Numerical Solution of Boundary-Value Problems 367
8.1 Second-Order BVP 367
8.2 Boundary Conditions 367
8.3 Higher-Order BVP 368
8.4 Shooting Method 368
8.5 Finite-Difference Method 374
8.5.1 Boundary-Value Problems with Mixed Boundary Conditions 379
8.6 MATLAB Built-in Function bvp4c for Boundary-Value Problems 381
8.6.1 Second-Order BVP 382
Problem Set (Chapter 8) 386
9. Matrix Eigenvalue Problem 393
9.1 Matrix Eigenvalue Problem 393
9.2 Power Method: Estimation of the Dominant Eigenvalue 393
9.2.1 Different Cases of Dominant Eigenvalue 395
9.2.2 Algorithm for the Power Method 395
9.3 Inverse Power Method: Estimation of the Smallest Eigenvalue 398
9.4 Shifted Inverse Power Method: Estimation of the Eigenvalue
Nearest a Specified Value 399
9.4.1 Notes on the Shifted Inverse Power Method 400
9.5 Shifted Power Method 401
9.5.1 Strategy to Estimate All Eigenvalues of a Matrix 401
9.6 MATLAB Built-in Function eig 403
9.7 Deflation Methods 403
9.7.1 Wielandt s Deflation Method 404
9.7.2 Deflation Process 405
9.8 Householder Tridiagonalization and QR Factorization Methods 407
9.8.1 Householder s Tridiagonalization Method
(Symmetric Matrices) 408
9.8.2 Determination of Symmetric Orthogonal Pk (k = 1,2,... ,n - 2) 409
9.8.3 QR Factorization Method 411
9.8.4 Determination of Qk and Rk Matrices 412
9.8.5 Structure of Lk (k = 2,3,..., n) 412
9.9 MATLAB Built-in Function qr 413
9.10 A Note on the Terminating Condition Used in HouseholderQR 414
9.11 Transformation to Hessenberg Form (Nonsymmetric Matrices) 417
Problem Set (Chapter 9) 418
10. Numerical Solution of Partial Differential Equations 423
10.1 Introduction 423
10.2 Elliptic Partial Differential Equations 424
10.2.1 Dirichlet Problem 424
xjv Contents
10.2.2 Alternating Direction Implicit (ADI) Methods 428
10.2.2.1 Peaceman-Rachford Alternating Direction
Implicit (PRADI) Method 429
10.2.3 Neumann Problem 433
10.2.3.1 Existence of a Solution for the Neumann Problem 435
10.2.4 Mixed Problem 436
10.2.5 More Complex Regions 437
10.3 Parabolic Partial Differential Equations 440
10.3.1 Finite-Difference Method 440
10.3.1.1 Stability and Convergence of the Finite-Difference
Method 441
10.3.2 Crank-Nicolson Method 443
10.3.2.1 Crank-Nicolson (CN) Method versus Finite-Difference
(FD) Method 446
10.4 Hyperbolic Partial Differential Equations 448
10.4.1 Starting the Procedure 449
Problem Set (Chapter 10) 452
Index 461
|
| any_adam_object | 1 |
| author | Esfandiari, Ramin S. |
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| discipline | Informatik |
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| indexdate | 2024-07-10T07:53:28Z |
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| isbn | 9781498777421 |
| language | English |
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| publisher | CRC Press, Taylor & Francis Group |
| record_format | marc |
| spelling | Esfandiari, Ramin S. (DE-588)1046926926 aut Numerical methods for engineers and scientists using MATLAB Ramin S. Esfandiari, PdD Second edition Boca Raton, FL CRC Press, Taylor & Francis Group [2017] xxi, 471 Seiten Diagramme txt rdacontent n rdamedia nc rdacarrier Engineering mathematics Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf MATLAB (DE-588)4329066-8 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s MATLAB (DE-588)4329066-8 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029857419&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Esfandiari, Ramin S. Numerical methods for engineers and scientists using MATLAB Engineering mathematics Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd MATLAB (DE-588)4329066-8 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4329066-8 |
| title | Numerical methods for engineers and scientists using MATLAB |
| title_auth | Numerical methods for engineers and scientists using MATLAB |
| title_exact_search | Numerical methods for engineers and scientists using MATLAB |
| title_full | Numerical methods for engineers and scientists using MATLAB Ramin S. Esfandiari, PdD |
| title_fullStr | Numerical methods for engineers and scientists using MATLAB Ramin S. Esfandiari, PdD |
| title_full_unstemmed | Numerical methods for engineers and scientists using MATLAB Ramin S. Esfandiari, PdD |
| title_short | Numerical methods for engineers and scientists using MATLAB |
| title_sort | numerical methods for engineers and scientists using matlab |
| topic | Engineering mathematics Numerical analysis Numerische Mathematik (DE-588)4042805-9 gnd MATLAB (DE-588)4329066-8 gnd |
| topic_facet | Engineering mathematics Numerical analysis Numerische Mathematik MATLAB |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029857419&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT esfandiariramins numericalmethodsforengineersandscientistsusingmatlab |