Elements of scientific computing: with 18 tables
Saved in:
| Format: | Book |
|---|---|
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2010
|
| Series: | Texts in computational science and engineering
7 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XII, 459 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783642112980 |
Staff View
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Record in the Search Index
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IMAGE 1
CONTENTS
1 COMPUTING INTEGRALS 1
1.1 BACKGROUND 1
1.2 THE COMPUTATIONAL PROBLEM 3
1.3 THE TRAPEZOIDAL METHOD 4
1.3.1 APPROXIMATING THE INTEGRAL USING ONE TRAPEZOID 5 1.3.2
APPROXIMATING THE INTEGRAL USING TWO TRAPEZOIDS 7 1.3.3 APPROXIMATING
THE INTEGRAL USING N TRAPEZOIDS 9 1.4 COMPUTATIONAL ANALYSIS 12
1.5 BACK TO THE BAGELS 14
1.6 EXERCISES 15
1.7 PROJECTS 17
1.7.1 SHOW THAT THE ERROR IS O(H 2 ) 17
1.7.2 DERIVE OTHER METHODS FOR NUMERICAL INTEGRATION 22 1.7.3 COMPUTE
THE LENGTH OF A CABLE 26
2 DIFFERENTIAL EQUATIONS: THE FIRST STEPS 31
2.1 MODELING CHANGES 31
2.1.1 THE CULTIVATION OF RABBITS 32
2.1.2 THE SIMPLEST POSSIBLE CASE 32
2.1.3 EXPONENTIAL GROWTH 33
2.1.4 LOGISTIC GROWTH 37
2.2 NUMERICAL SOLUTION 40
2.2.1 THE SIMPLEST POSSIBLE MODEL 41
2.2.2 NUMERICAL APPROXIMATION OF EXPONENTIAL GROWTH 44
2.2.3 NUMERICAL STABILITY 46
2.2.4 AN IMPLICIT SCHEME 48
2.2.5 EXPLICIT AND IMPLICIT SCHEMES 49
2.2.6 NUMERICAL SOLUTION OF THE LOGISTIC EQUATION 55
2.3 EXERCISES 60
2.4 PROJECTS 67
2.4.1 MORE ON STABILITY 67
2.4.2 MORE ON ACCURACY 69
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/100650768X
DIGITALISIERT DURCH
IMAGE 2
VILI CONTENTS
3 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 75
3.1 RABBITS AND FOXES; FISH AND SHARKS 75
3.2 A NUMERICAL METHOD: UNLIMITED RESOURCES 78
3.3 A NUMERICAL METHOD: LIMITED RESOURCES 79
3.4 PHASE PLANE ANALYSIS 80
3.4.1 A SIMPLIFIED MODEL 80
3.4.2 THE PHASE PLANE 81
3.4.3 CIRCLES IN THE ANALYTICAL PHASE PLANE 83
3.4.4 ALTERNATIVE ANALYSIS 84
3.4.5 CIRCLES IN THE NUMERICAL PHASE PLANE 85
3.4.6 MORE ON NUMERICS 88
3.5 EXERCISES 91
3.6 PROJECT: ANALYSIS OF A SIMPLE SYSTEM 95
4 NONLINEAR ALGEBRAIC EQUATIONS 99
4.1 THE BISECTION METHOD 102
4.2 EFFICIENCY CONSIDERATION 106
4.3 NEWTON'S METHOD 107
4.4 THE SECANT METHOD 112
4.5 FIXED-POINT ITERATIONS 115
4.5.1 CONVERGENCE OF FIXED-POINT ITERATIONS 117
4.5.2 SPEED OF CONVERGENCE 120
4.5.3 EXISTENCE AND UNIQUENESS OF A SOLUTION 121
4.5.4 UNIQUENESS 122
4.5.5 EXISTENCE 123
4.6 SYSTEMS OF NONLINEAR EQUATIONS 125
4.6.1 A LINEAR SYSTEM 126
4.6.2 A NONLINEAR SYSTEM 127
4.6.3 NEWTON'S METHOD 129
4.6.4 A NONLINEAR EXAMPLE 132
4.6.5 THE NONLINEAR SYSTEM REVISITED 133
4.7 EXERCISES 134
4.8 PROJECT: CONVERGENCE OF NEWTON'S METHOD 138
5 THE METHOD OF LEAST SQUARES 147
5.1 BUILDING MODELS OF DISCRETE DATA SETS 150
5.1.1 APPROXIMATION BY A CONSTANT 150
5.1.2 APPROXIMATION BY A LINEAR FUNCTION 154
5.1.3 APPROXIMATION BY A QUADRATIC FUNCTION 157
5.1.4 LARGE DATA SETS 161
5.2 APPLICATION TO POPULATION MODELS 165
5.2.1 EXPONENTIAL GROWTH OF THE WORLD POPULATION? 166 5.2.2 LOGISTIC
GROWTH OF THE WORLD POPULATION 170
5.3 LEAST SQUARES APPROXIMATIONS OF FUNCTIONS 173
5.3.1 APPROXIMATING FUNCTIONS BY A CONSTANT 176
IMAGE 3
CONTENTS IX
5.3.2 APPROXIMATION USING LINEAR FUNCTIONS 178
5.3.3 APPROXIMATION USING QUADRATIC FUNCTIONS 180 5.3.4 SUMMARY OF THE
EXAMPLES 183
5.4 EXERCISES 185
5.5 PROJECT: COMPUTING COEFFICIENTS 190
6 ABOUT SCIENTIFIC SOFTWARE 195
6.1 ALGORITHMS EXPRESSED AS PSEUDO CODE 197
6.1.1 BASIC INGREDIENTS OF PSEUDO CODES 197
6.1.2 INTEGRATION 199
6.1.3 OPTIMIZATION OF ALGORITHMS AND IMPLEMENTATIONS 200 6.1.4
DEVELOPING ALGORITHMS FOR SIMPSON'S RULE 203 6.1.5 ADAPTIVE INTEGRATION
RULES 207
6.1.6 ORDINARY DIFFERENTIAL EQUATIONS 211
6.2 ABOUT PROGRAMMING LANGUAGES 212
6.2.1 STATIC TYPING VERSUS DYNAMIC TYPING 213
6.2.2 COMPUTATIONAL EFFICIENCY 214
6.2.3 BUILT-IN HIGH-PERFORMANCE UTILITIES 215
6.2.4 SUPPORT FOR USER-DEFINED OBJECTS 216
6.3 THE TRAPEZOIDAL RULE IN DIFFERENT LANGUAGES 217
6.3.1 CODE STRUCTURE 217
6.3.2 FORTRAN 77 218
6.3.3 C AND C ++ 222
6.3.4 JAVA 224
6.3.5 MATLAB 226
6.3.6 PYTHON 228
6.3.7 VECTORIZATION 230
6.3.8 MAPLE 236
6.3.9 SUMMARY 237
6.4 HEUN'S SCHEME IN DIFFERENT LANGUAGES 239
6.4.1 CODE STRUCTURE 239
6.4.2 FORTRAN 77 239
6.4.3 GRAPHICS 242
6.4.4 C ++ : 244
6.4.5 JAVA 248
6.4.6 MATLAB 249
6.4.7 PYTHON 251
6.4.8 SUMMARY 252
6.5 NUMERICAL SOFTWARE ENGINEERING 254
6.5.1 FUNCTION LIBRARIES 254
6.5.2 MOTIVATION FOR OBJECT-ORIENTED LIBRARIES 259
6.5.3 DESIGN OF A NUMERICAL INTEGRATION HIERARCHY 262 6.5.4 A CLASS
HIERARCHY IN JAVA 263
6.5.5 A CLASS HIERARCHY IN PYTHON 265
6.5.6 OBJECT-ORIENTED PROGRAMMING IN MATLAB 268
IMAGE 4
X CONTENTS
6.6 EXERCISES 269
6.7 PROJECTS 270
6.7.1 COMPUTING THE VOLUME OF A CYLINDRICAL CONTAINER 270 6.7.2 A CLASS
HIERARCHY FOR SCALAR ODE SOLVERS 271
6.7.3 SOFTWARE FOR SYSTEMS OF ODES 272
7 THE DIFFUSION EQUATION 273
7.1 BASICS OF DIFFUSION PROCESSES 274
7.1.1 HEAT CONDUCTION 274
7.1.2 DIFFUSIVE TRANSPORT OF A SUBSTANCE 275
7.1.3 DIFFUSION VERSUS CONVECTION 276
7.1.4 THE RELEVANCE OF ONE-DIMENSIONAL MODELS 277 7.2 THE MATHEMATICAL
MODEL OF DIFFUSION 285
7.2.1 THE DIFFUSION EQUATION 285
7.2.2 INITIAL AND BOUNDARY CONDITIONS 286
7.2.3 THE ONE-DIMENSIONAL INITIAL-BOUNDARY VALUE PROBLEM. .287 7.2.4
THE THREE-DIMENSIONAL DIFFUSION EQUATION 287 7.3 DERIVATION OF DIFFUSION
EQUATIONS 290
7.3.1 DIFFUSION OF A SUBSTANCE 290
7.3.2 HEAT CONDUCTION 295
7.3.3 VISCOUS FLUID FLOW 300
7.3.4 SUMMARIZING THE MODELS 304
7.3.5 SCALING 307
7.4 EXPLICIT NUMERICAL METHODS 316
7.4.1 THE BASICS OF FINITE DIFFERENCE DISCRETIZATIONS 316 7.4.2
INCORPORATING DIRICHLET BOUNDARY CONDITIONS 321 7.4.3 INCORPORATING
NEUMANN BOUNDARY CONDITIONS 321 7.4.4 HOW TO VERIFY A COMPUTER
IMPLEMENTATION 324
7.4.5 INSTABILITY 327
7.4.6 A DISCRETE ALGORITHM DIRECTLY FROM PHYSICS 332 7.4.7 VARIABLE
COEFFICIENTS 334
7.5 IMPLICIT NUMERICAL METHODS 336
7.5.1 THE BACKWARD EULER SCHEME 336
7.5.2 THE LINEAR SYSTEM OF EQUATIONS : 337
7.5.3 SOLUTION OF TRIDIAGONAL LINEAR SYSTEMS 340
7.5.4 COMPARING THE EXPLICIT AND IMPLICIT METHODS 342 7.5.5 THE
CRANK-NICOLSON SCHEME 344
7.5.6 THEO SCHEME 346
7.6 EXERCISES 348
7.7 PROJECTS 352
7.7.1 DIFFUSION OF A JUMP 352
7.7.2 PERIODICAL INJECTION OF POLLUTION 353
7.7.3 ANALYZING DISCRETE SOLUTIONS 355
7.7.4 COMPARE DIFFERENT SCALINGS 356
IMAGE 5
CONTENTS XI
8 ANALYSIS OF THE DIFFUSION EQUATION 359
8.1 PROPERTIES OF THE SOLUTION 360
8.1.1 ENERGY ARGUMENTS 360
8.1.2 A BOUND ON THE DERIVATIVE 362
8.1.3 STABILITY 364
8.1.4 UNIQUENESS 366
8.1.5 MAXIMUM PRINCIPLES 366
8.1.6 PHYSICAL CONSIDERATIONS 367
8.1.7 ANALYTICAL CONSIDERATIONS 367
8.1.8 THE MINIMUM PRINCIPLE 372
8.1.9 SUMMARY 373
8.1.10 UNIQUENESS REVISITED 374
8.2 SEPARATION OF VARIABLES AND FOURIER ANALYSIS 374
8.2.1 SEPARATION OF VARIABLES 375
8.2.2 SUPER-POSITIONING 378
8.2.3 FOURIER SERIES AND THE INITIAL CONDITION 381
8.2.4 SOME SIMPLE EXAMPLES 381
8.2.5 INITIAL CONDITIONS GIVEN BY A SUM OF SINE FUNCTIONS 382 8.2.6
COMPUTING FOURIER SINE SERIES 384
8.2.7 SUMMARY 386
8.2.8 MORE EXAMPLES 387
8.2.9 ANALYSIS OF AN EXPLICIT FINITE DIFFERENCE SCHEME 392 8.2.10
CONSEQUENCES OF THE STABILITY CRITERION 395
8.2.11 EXERCISES 396
8.3 PROJECTS 403
8.3.1 NEUMANN BOUNDARY CONDITIONS 403
8.3.2 VARIABLE COEFFICIENTS 406
9 PARAMETER ESTIMATION AND INVERSE PROBLEMS 411
9. 1 PARAMETER ESTIMATION IN EXPONENTIAL GROWTH 412
9.1.1 A SIMPLER PROBLEM 414
9.2 THE BACKWARD DIFFUSION EQUATION 415
9.3 ESTIMATING THE DIFFUSION COEFFICIENT 418
9.4 EXERCISES 419
10 A GLIMPSE OF PARALLEL COMPUTING 423
10.1 MOTIVATIONS FOR PARALLEL COMPUTING 423
10.1.1 FROM THE PERSPECTIVE OF SPEED 423
10.1.2 FROM THE PERSPECTIVE OF MEMORY 426
10.1.3 PARALLEL COMPUTERS 426
10.2 MORE ABOUT PARALLEL COMPUTING 427
10.2.1 INSPIRATIONS FROM PARALLEL PROCESSING IN REAL LIFE 428 10.2.2
FROM SERIAL COMPUTING TO PARALLEL COMPUTING 428 10.2.3 EXAMPLE 1 OF DATA
PARALLELISM 430
10.2.4 EXAMPLE 2 OF DATA PARALLELISM 432
IMAGE 6
XII CONTENTS
10.2.5 EXAMPLE 3 OF DATA PARALLELISM 434
10.2.6 PERFORMANCE ANALYSIS 438
10.2.7 OVERHEAD RELATED TO PARALLELIZATION 440
10.3 PARALLEL PROGRAMMING 442
10.3.1 OPENMP PROGRAMMING 443
10.3.2 MPI PROGRAMMING 445
10.3.3 CONCLUDING REMARKS 449
10.4 EXERCISES 449
10.5 PROJECT: PARALLEL SOLUTION OF A 2D DIFFUSION EQUATION 451
REFERENCES 455
INDEX 457 |
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| dewey-full | 518 502.85 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis 502 - Miscellany |
| dewey-raw | 518 502.85 |
| dewey-search | 518 502.85 |
| dewey-sort | 3518 |
| dewey-tens | 510 - Mathematics 500 - Natural sciences and mathematics |
| discipline | Allgemeine Naturwissenschaft Mathematik |
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| indexdate | 2024-07-20T10:49:31Z |
| institution | BVB |
| isbn | 9783642112980 |
| language | English |
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| physical | XII, 459 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Springer |
| record_format | marc |
| series | Texts in computational science and engineering |
| series2 | Texts in computational science and engineering |
| spelling | Elements of scientific computing with 18 tables Aslak Tveito ... Berlin [u.a.] Springer 2010 XII, 459 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Texts in computational science and engineering 7 Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Wissenschaftliches Rechnen (DE-588)4338507-2 gnd rswk-swf Wissenschaftliches Rechnen (DE-588)4338507-2 s Numerische Mathematik (DE-588)4042805-9 s DE-604 Tveito, Aslak Sonstige oth Erscheint auch als Online-Ausgabe 978-3-642-11299-7 Texts in computational science and engineering 7 (DE-604)BV016971315 7 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3532010&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020645924&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Elements of scientific computing with 18 tables Texts in computational science and engineering Numerische Mathematik (DE-588)4042805-9 gnd Wissenschaftliches Rechnen (DE-588)4338507-2 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4338507-2 |
| title | Elements of scientific computing with 18 tables |
| title_auth | Elements of scientific computing with 18 tables |
| title_exact_search | Elements of scientific computing with 18 tables |
| title_full | Elements of scientific computing with 18 tables Aslak Tveito ... |
| title_fullStr | Elements of scientific computing with 18 tables Aslak Tveito ... |
| title_full_unstemmed | Elements of scientific computing with 18 tables Aslak Tveito ... |
| title_short | Elements of scientific computing |
| title_sort | elements of scientific computing with 18 tables |
| title_sub | with 18 tables |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd Wissenschaftliches Rechnen (DE-588)4338507-2 gnd |
| topic_facet | Numerische Mathematik Wissenschaftliches Rechnen |
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| volume_link | (DE-604)BV016971315 |
| work_keys_str_mv | AT tveitoaslak elementsofscientificcomputingwith18tables |