Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Boca Raton ; London ; New York
CRC Press, an imprint of Taylor & Francis Group
[2015]
|
| Schriftenreihe: | Computational science series
|
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027763821&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | xvii, 416 Seiten Illustrationen |
| ISBN: | 9781482239867 1482239868 |
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| 245 | 1 | 0 | |a The end of error |b unum computing |c John L. Gustafson |
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Datensatz im Suchindex
| _version_ | 1819384371496353792 |
|---|---|
| adam_text | Titel: The end of error
Autor: Gustafson, John L
Jahr: 2015
Contents
Preface xiii
Acknowledgments xvii
How to read this book xix
Part 1
A New Number Format: The Unum
Chapter 1 Overview 1
1.1 Fewer bits. Better answers 2
1.2 Why better arithmetic can save energy arid power 5
Chapter 2 Building up to the unum format 7
2.1 A graphical view of bit strings: Value and closure plots 8
2.2 Negative numbers 10
2.3 Fixed point format 14
2.4 Floating point format, almost 17
2.5 What about infinity and NaN? Improving on IEEE rules 22
Chapter 3 The original sin of Computer arithmetic 25
3.1 The acceptance of incorrect answers 25
3.2 Almost infinite and beyond infinity 28
3.3 No overflow, no underflow, and no rounding 30
3.4 Visualizing ubit-enabled numbers 33
vi | 77ie End ot Error
Chapter 4 The complete unum format 35
4.1 Overcoming the tyranny of fixed storage size 35
4.2 The IEEE Standard float formats 36
4.3 Unum format: Flexible ränge and precision 38
4.4 How can appending extra bits save storage? 40
4.5 Ludicrous precision? The vast ränge of unums 41
4.6 Changing environment settings within a Computing task 42
4.7 The reference prototype 43
4.8 Special values in a flexible precision environment 47
4.9 Converting exact unums to real numbers 48
4.10 A complete exact unum set for a small utag 50
4.11 Inexact unums 51
4.12 A visualizer for unum strings 53
Chapter 5 Hidden scratchpads and the three layers 55
5.1 The hidden scratchpad 56
5.1.1 The Cray-1 Supercomputer 57
5.1.2 The original definition of the C language 58
5.1.3 Coprocessors also try to be helpful 59
5.1.4 IBM almost gets it right: The fused multiply-add 60
5.1.5 Kulisch gets it right: The exact dot product (EDP) 61
5.2 The unum layer 62
5.2.1 The ubound 62
5.2.2 Processor design for the u-layer 64
5.3 The math layer 65
5.3.1 The general bound, or gbound 65
5.3.2 Who uses NaN? Sun Microsystems gets a nasty surprise 67
5.3.3 An explosion of storage demand in the scratchpad? 68
5.3.4 Traditional interval arithmetic. Watch out 69
5.3.5 Significance arithmetic 72
5.3.6 Why endpoints must be marked open or closed 72
5.4 The human layer 74
5.4.1 Numbers as people perceive them 74
5.4.2 Demands and concessions 75
5.5 Moving between layers 78
5.5.1 Defining the conversion function 78
5.5.2 Testing the conversion function 81
5.6 Summary of conversions between layers in the prototype 82
5.7 Are floats good enough for government work ? 83
Contents vii
Chapter 6 Information per bit 85
6.1 Information as the reciprocal of uncertainty 85
6.2 Unifying a bound to a Single ULP 86
6.3 Unification in the prototype 87
6.3.1 The Option of lossy compression 87
6.3.2 Intelligent unification 88
6.3.3 Much ado about almost nothing and almost infinite 90
6.4 Can ubounds save storage compared with traditional floats? 91
Chapter 7 Fixed-size unum storage 93
7.1 The Warlpiri unums 93
7.2 The Warlpiri ubounds 95
7.3 Hardware for unums: Faster than float hardware? 98
7.3.1 General comments about the cost of handling exceptions 98
7.3.2 Unpacked unum format and the summary bits idea 99
Chapter 8 Comparison Operations 103
8.1 Less than, greaterthan 103
8.1.1 Conceptual definition of ordering forgeneral intervals 103
8.1.2 Hardware design for less than and greater than tests 105
8.2 Equal, nowhere equal, and not nowhere equal 106
8.2.1 Conceptual definition of equal for general intervals 106
8.2.2 Hardware approach for equal and not nowhere equal 109
8.3 Intersection 109
Chapter 9 Add/subtract, and the unbiased rounding myth..lll
9.1 Re-Iearning the addition table... for all real numbers 111
9.1.1 Examples and tests of data motion 114
9.1.2 Three-dimensional visualization of ubound addition 115
9.1.3 Hardware design for unum addition and subtraction 116
9.2 Creeping crud and the myth of unbiased rounding 117
9.3 Automatic accuracy control and a simple unum math test 121
Chapter 10 Multiplication and division 127
10.1 Multiplication requires examining each quadrant 128
10.2 Hardware for unum multiplication 132
10.2.1 Multiplies that fit the Standard hardware approach 132
10.2.2 Extended precision and the complete accumulator 135
10.2.3 Really big multiplies 135
10.3 Division introduces asymmetry in the arguments 136
10.3.1 The bad boy of arithmetic 136
10.3.2 Hardware for unum division 139
viii | The End of Error
Chapter 11 Powers 141
11.1 Square 141
11.2 Square root 143
11.3 Nested Square roots and ULP straddling 145
11.4 Taxing the scratchpad: Integers to integer powers 146
11.5 A practice calculation ofxy at low precision 147
11.6 Practical considerations and the actual working routine 149
11.6.1 Why the power function can be fast 149
11.6.2 The prototype power function 150
11.6.3 A challenge to the defenders of floats 151
11.7 Exp(x) and The Table-Maker s Dilemma 152
11.7.1 Another diiemma caused by the dishonesty of rounding 152
11.7.2 The prototype exponential function 153
Chapter 12 Other important unary Operations 155
12.1 Scope of the prototype 155
12.2 Absolute value 155
12.3 Natural logarithm, and a mention of log base 2 156
12.4 Trig functions: Ending the madness by degrees 156
Chapter 13 Fused Operations (single-use expressions) 159
13.1 Standardizing a set of fused Operations 159
13.2 Fused multiply-add and fused multiply-subtract 160
13.3 Solving the paradox of slow arithmetic for complex numbers 161
13.4 Unum hardware for the complete accumulator 162
13.4.1 Cost within the unpacked unum environment 162
13.4.2 Fused dot product and fused sums in the prototype 164
13.4.3 Consistent results for parallel Computing 165
13.5 Other fused Operations 167
13.5.1 Not every pairing of Operations should be fused 167
13.5.2 Fused product 167
13.5.3 Fused add-multiply 169
13.5.4 Fused product ratio 169
13.5.5 Fused norm, fused root dot product, and fused mean 171
Chapter 14 Trial runs: Unums face challenge calculations...l73
14.1 Floating point II: The wrath of Kahan 173
14.2 Rump s royal pain 179
14.3 The quadratic formula 181
14.4 Bailey s numerical nightmare 184
14.5 Fast Fourier Transforms using unums 188
Part 1 Summary 193
Contents | ix
Part 2
A New Way to Solve: The Ubox
Chapter 15 The other kind of error 195
15.1 Sampling error 195
15.2 The deeply unsatisfying nature of classical error bounds 197
15.3 The ubox approach 199
15.4 Walking the line 201
15.5 A ubox connected-region example: Computing the unit circle area 203
15.6 A definition of answer quality and Computing speed 210
15.7 Another Kahan booby trap: The smooth surprise 212
Chapter 16 Avoiding interval arithmetic pitfalls 215
16.1 Useless error bounds 215
16.2 The wrapping problem 215
16.3 The dependency problem 219
16.4 Intelligent Standard library routines 221
16.5 Polynomial evaluation without the dependency problem 222
16.6 Other fused multiple-use expressions 231
Chapter 17 What does it mean to solve an equation? 233
17.1 Another break from traditional numerical methods 233
17.2 A linear equation in one unknown, solved by inversion 234
17.2.1 Inversion with unums and intervals 236
17.2.2 Ubounds as coefficients 237
17.3 Try everything! Exhaustive search of the number line 241
17.3.1 The quadratic formula revisited 241
17.3.2 To boldly split infinities: Try everything 242
17.3.3 Highly adjustable parallelism 245
17.4 The universal equation solver 245
17.4.1 Methods that usually work 245
17.4.2 The general failure of the inversion method 247
17.4.3 Inequality testing; finding extrema 250
17.4.4 The intractable fallacy 251
17.5 Solvers in more than one dimension 251
17.6 Summary of the ubox solver approach 256
Chapter 18 Permission to guess 257
18.1 Algorithms that work for floats also work for unums 257
18.2 A fixed-point problem 261
18.3 Large Systems of linear equations 270
18.4 The last resort 272
x | The End of Error
Chapter 19 Pendulums done correctly 273
19.1 The introductory physics approach 273
19.2 The usual numerical approach 274
19.3 Space stepping: A new source of massive parallelism 276
19.4 It s not just for pendulums 282
Chapter 20 The two-body problem (and beyond) 287
20.1 A differential equation with multiple dimensions 287
20.1.1 Setting up a traditional Simulation 288
20.1.2 Trying a traditional method out on a Single orbit 292
20.1.3 What about interval arithmetic methods? 294
20.2 Ubox approach: The initial Space Step 294
20.2.1 Solving the chicken-and-egg puzzle 294
20.2.2 First, bound the position 296
20.3 The next starting point, and some State law enforcement 302
20.4 The general Space Step 305
20.5 The three-body problem 308
20.6 The n-body problem and the galaxy colliders 310
Chapter 21 Calculus considered evil: Discrete physics 311
21.1 Continuum versus discrete physics 311
21.2 The discrete version of a vibrating string 313
21.3 The single-atom gas 316
21.4 Structural analysis 321
Chapter 22 The end of error 327
Glossary 333
Contents | xi
Appendix A: Glossary of unum functions 337
A.1 Environment and bit-extraction functions 337
A.2 Constructors 337
A.3 Visualization functions 337
A.4 Conversion functions 338
A.5 Argument validity tests 338
A.6 Helper functions for functions defined above 338
A.7 Comparison Operations 339
A.8 Arithmetic Operations 339
A.9 Fused Operations (single-use expressions) 340
A.10 Some defined data values 341
A.11 Auto-precision functions 342
Appendix B: Glossary of ubox functions 343
B.1 ULP-related manipulations 343
B.2 Neighbor-finding functions 343
B.3 Ubox creation by Splitting 343
B.4 Coalescing ubox sets 344
B.5 Ubox bounds, width, volume 344
B.6 Test Operations on sets 344
B.7 Fused polynomial evaluation and acceleration formula 345
B.8 The try-everything solver 345
B.9 The guessing function 345
Appendix C: Algorithm listings for Part 1 347
C. 1 The set-the-environment function 348
C.2 Type-checking functions 349
C.3 The unum-to-float Converter and supporting functions 351
C.4 The u-layer to general interval conversions 352
C.5 The real-to-unum or x conversion function 353
C.6 Unification functions and supporting functions 356
C.7 The general interval to unum Converter 358
C.8 Comparison tests and ubound intersection 359
C.9 Addition and subtraction functions 362
C.10 Multiplication functions 363
C.11 Division routines 365
C.12 Automatic precision adjustment functions 367
C.13 Fused Operations (single-use expressions) 367
C.14 Square and Square root 372
C.15 The power function xy and exp(x) 373
C.16 Absolute value, logarithm, and trigonometry functions 378
C.17 The unum Fast Fourier Transform 382
xii | The End of Error
Appendix D: Algorithm listings for Part 2 383
D.1 ULP manipulation functions 383
D.2 Neighbor-finding functions 385
D.3 Ubox creation by Splitting 387
D.4 Coaiescing ubox sets 389
D.5 Ubox bounds, width, volumes 392
D.6 Test Operations on sets 393
D.7 Fused polynomial evaluation and acceleration formula 394
D.8 The try-everything solver 398
D.9 The guess function 399
For Further Reading 401
Index 403
|
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| id | DE-604.BV042326995 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T17:08:07Z |
| institution | BVB |
| isbn | 9781482239867 1482239868 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027763821 |
| oclc_num | 908175670 |
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| owner_facet | DE-703 DE-20 DE-706 DE-29T |
| physical | xvii, 416 Seiten Illustrationen |
| publishDate | 2015 |
| publishDateSearch | 2015 |
| publishDateSort | 2015 |
| publisher | CRC Press, an imprint of Taylor & Francis Group |
| record_format | marc |
| series2 | Computational science series |
| spellingShingle | Gustafson, John L. The end of error unum computing Datenverarbeitung (DE-588)4011152-0 gnd Computerarithmetik (DE-588)4135485-0 gnd Kalkulationsverfahren (DE-588)4163111-0 gnd |
| subject_GND | (DE-588)4011152-0 (DE-588)4135485-0 (DE-588)4163111-0 |
| title | The end of error unum computing |
| title_auth | The end of error unum computing |
| title_exact_search | The end of error unum computing |
| title_full | The end of error unum computing John L. Gustafson |
| title_fullStr | The end of error unum computing John L. Gustafson |
| title_full_unstemmed | The end of error unum computing John L. Gustafson |
| title_short | The end of error |
| title_sort | the end of error unum computing |
| title_sub | unum computing |
| topic | Datenverarbeitung (DE-588)4011152-0 gnd Computerarithmetik (DE-588)4135485-0 gnd Kalkulationsverfahren (DE-588)4163111-0 gnd |
| topic_facet | Datenverarbeitung Computerarithmetik Kalkulationsverfahren |
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