An introduction to numerical computation:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey
World Scientific
[2016]
|
| Subjects: | |
| Online Access: | Klappentext Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xii, 255 Seiten Illustrationen, Diagramme 25 cm |
| ISBN: | 9789814730068 |
Staff View
MARC
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| 245 | 1 | 0 | |a An introduction to numerical computation |c Wen Shen, Penn State University, USA |
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| 264 | 4 | |c 2016 | |
| 300 | |a xii, 255 Seiten |b Illustrationen, Diagramme |c 25 cm | ||
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Record in the Search Index
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| adam_text | Developed during ten years of teaching experience, this book serves as a set of lecture notes for an introductory course on numerical computation, at the senior undergraduate level. These notes contain the material that can be covered in a semester, together with a few optional sections for additional reading. Rather than surveying a large number of algorithms, the book presents the most important computational methods and emphasises the underlying mathematical ideas. In most chapters, graphs and drawings are relied on, to build up intuition.
The notes are written in a rather colloquial style, presenting the subject matter in the same form as it can be explained in a classroom. For instructors, this will minimize the amount of effort required to prepare their blackboard presentations.
As prerequisites, the book only relies on standard calculus, an introductory course on matrices, and some basic computer programming skills. As a new feature, these notes are supplemented by two sets of videos from the author s Youtube channel. These videos contain a complete set of live lectures given in Spring 2015, together with a complete set of short tutorials, from 5 to 15 minutes each.
A set of homework problems is included at the end of each chapter. Homework projects cover a variety of
appIica11ar s, in conriecrion wiifi popelation dynarr11 cs, engineering, met:hanics, image reconstruction, etc. A complete set of solutions is available for instructors,
upon request.
Contents
Preface vii
1. Computer Arithmetic 1
1.1 Introduction........................................................ 1
1.2 Representation of Numbers in Different Bases........................ 1
1.3 Floating Point Representation....................................... 4
1.4 Loss of Significance................................................ 7
1.5 Review of Taylor Series and Taylor Theorem.......................... 8
1.6 Numerical Differentiations and Finite Differences.................. 11
1.7 Homework Problems for Chapter 1.................................... 14
2. Polynomial Interpolation 17
2.1 Introduction....................................................... 17
2.2 Lagrange Interpolation Polynomials................................. 22
2.3 Newton’s Divided Differences....................................... 24
2.4 Errors in Polynomial Interpolation................................. 30
2.5 Properties of the Newton’s Divided Differences (optional) ......... 38
2.6 Convergence of Polynomial Interpolation............................ 40
2.7 Homework Problems for Chapter 2.................................... 42
3. Piecewise Polynomial Interpolation: Splines 47
3.1 Introduction....................................................... 47
3.2 Linear Splines..................................................... 49
3.3 Quadratic Splines (optional)....................................... 51
3.4 Natural Cubic Splines.............................................. 52
3.5 Homework Problems for Chapter 3.................................... 61
4. Numerical Integration 65
4.1 Introduction....................................................... 65
4.2 Trapezoid Rule..................................................... 66
ix
70
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92
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151
An Introduction to Numerical Computation
4.3 Simpson’s Rule...............................................
4.4 Recursive Trapezoid Rule.....................................
4.5 Romberg Algorithm ...........................................
4.6 Adaptive Simpson’s Quadrature Scheme.........................
4.7 Gaussian Quadrature Formulas.................................
4.8 Matlab Simulations...........................................
4.9 A More Abstract Discussion (optional)........................
4.10 Homework Problems for Chapter 4..............................
Numerical Solutions of Non-linear Equations
5.1 Introduction.................................................
5.2 Bisection Method.............................................
5.3 Fixed Point Iterations.......................................
5.4 Newton’s Method..............................................
5.5 Secant Method................................................
5.6 Systems of Non-linear Equations..............................
5.7 Homework Problems for Chapter 5..............................
Direct Methods for Systems of Linear Equations
6.1 Introduction.................................................
6.2 Naive Gaussian Elimination: Simplest Version.................
6.3 Gaussian Elimination with Pivoting (optional)................
6.3.1 Difficulties with Naive Gaussian Elimination..........
6.3.2 Gaussian Elimination with Partial Pivoting............
6.3.3 When Partial Pivoting is not Effective................
6.3.4 Gaussian Elimination with Scaled Partial Pivoting . . . .
6.4 LU-Factorization (optional)..................................
6.5 Matlab Simulations...........................................
6.6 Tridiagonal and Banded Systems...............................
6.7 Review of Linear Algebra.....................................
6.8 Homework Problems for Chapter 6..............................
Fixed Point Iterative Solvers for Linear Systems
7.1 General Introduction to Iterative Solvers ...................
7.2 Jacobi Iterations............................................
7.3 Gauss-Seidel Iterations......................................
7.4 SOR Iterations...............................................
7.5 Matlab Simulations...........................................
7.6 Writing All Three Methods in Matrix-Vector Form..............
7.7 Error Analysis and Convergence...............................
7.8 Homework Problems for Chapter 7..............................
Contents
xi
8. The Method of Least Squares 153
8.1 Problem Description............................................153
8.2 Linear Regression..............................................154
8.3 Method of Least Squares with Quadratic Functions...............157
8.4 Method of Least Squares with General Functions (optional) .... 158
8.5 General Linear Method of Least Squares.........................159
8.6 Mat lab Simulations for Linear Method of Leasts Squares........161
8.7 Non-linear Methods of Least Squares............................162
8.8 Method of Least Squares for Continuous Functions...............164
8.9 Homework Problems for Chapter 8................................169
9. Numerical Solutions of ODEs 173
9.1 Introduction...................................................173
9.2 Taylor Series Methods for ODEs ................................174
9.3 Runge-Kutta Methods............................................182
9.4 An Adaptive Runge-Kutta-Fehlberg Method........................187
9.5 Multi-Step Methods ............................................189
9.6 A Case Study for a Scalar ODE, Solved in Matlab (optional) . . . 193
9.6.1 Euler’s Method..........................................194
9.6.2 Heun’s Method...........................................196
9.6.3 RK4 Method..............................................196
9.6.4 RKF5 Method.............................................197
9.6.5 Comparison .............................................199
9.7 Numerical Solution of Systems of First Order ODEs..............201
9.8 Higher Order Equations and Systems.............................203
9.9 A Case Study for a System of ODEs by Various Methods (optional) 205
9.10 Stiff Systems .................................................208
9.11 Homework Problems for Chapter 9................................214
10. Two-Point Boundary Value Problems 217
10.1 Introduction...................................................217
10.2 Shooting Method................................................218
10.2.1 Linear Shooting.........................................218
10.2.2 Some Extensions of Linear Shooting......................220
10.2.3 Non-linear Shooting.....................................221
10.3 Finite Difference Method.......................................223
10.4 Homework Problems for Chapter 10 229
11. FDM for Partial Differential Equations 231
11.1 Laplace Equation in 2D: Finite Difference Methods..............231
11.2 Some Extensions of the Laplace Equation........................236
11.3 Heat Equation in ID............................................240
w
xii
An Introduction to Numerical Computation
11.3.1 Forward-Euler Scheme for the Heat Equation...........241
11.3.2 Backward Euler Scheme for the Heat Equation...........243
11.3.3 Crank-Nicolson Scheme for the Heat Equation...........245
11.3.4 The 0-scheme for the Heat Equation...................247
11.4 Other Forms of the Heat Equation............................247
11.5 Homework Set for Chapter 11.................................249
Bibliography 251
Index 253
|
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| isbn | 9789814730068 |
| language | English |
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| physical | xii, 255 Seiten Illustrationen, Diagramme 25 cm |
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| publisher | World Scientific |
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| spelling | Shen, Wen Verfasser aut An introduction to numerical computation Wen Shen, Penn State University, USA New Jersey World Scientific [2016] 2016 xii, 255 Seiten Illustrationen, Diagramme 25 cm txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references and index Numerical analysis Textbooks Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Numerische Mathematik (DE-588)4042805-9 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028922991&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Klappentext Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028922991&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Shen, Wen An introduction to numerical computation Numerical analysis Textbooks Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4042805-9 |
| title | An introduction to numerical computation |
| title_auth | An introduction to numerical computation |
| title_exact_search | An introduction to numerical computation |
| title_full | An introduction to numerical computation Wen Shen, Penn State University, USA |
| title_fullStr | An introduction to numerical computation Wen Shen, Penn State University, USA |
| title_full_unstemmed | An introduction to numerical computation Wen Shen, Penn State University, USA |
| title_short | An introduction to numerical computation |
| title_sort | an introduction to numerical computation |
| topic | Numerical analysis Textbooks Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Numerical analysis Textbooks Numerische Mathematik |
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