A history of algorithms: from the pebble to the microchip
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| Format: | Book |
|---|---|
| Language: | English French |
| Published: |
Berlin [u.a.]
Springer
1999
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | IX, 524 S. Ill., graph. Darst. |
| ISBN: | 3540633693 |
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MARC
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| 130 | 0 | |a Histoire d'algorithmes | |
| 245 | 1 | 0 | |a A history of algorithms |b from the pebble to the microchip |c Jean-Luc Chabert (ed.). [Authors: Evelyne Barbin ...] |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1999 | |
| 300 | |a IX, 524 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
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| adam_text | Contents
Introduction l
1 Algorithms for Arithmetic Operations 7
1.1 Sumerian Division 8
1.2 A Babylonian Algorithm for Calculating Inverses 11
1.3 Egyptian Algorithms for Arithmetic 15
1.4 Tableau Multiplication 20
1.5 Optimising Calculations 28
1.6 Simple Division by Difference on a Counting Board 30
1.7 Division on the Chinese Abacus 35
1.8 Numbers Written as Decimals 37
1.9 Binary Arithmetic 40
1.10 Computer Arithmetic 43
Bibliography 46
2 Magic Squares 49
2.1 Squares with Borders 53
2.2 The Marking Cells Method 58
2.3 Proceeding by 2 and by 3 64
2.4 Arnauld s Borders Method 70
Bibliography 81
3 Methods of False Position 83
3.1 Mesopotamia: a Geometric False Position 86
3.2 Egypt: Problem 26 of the Rhind Papyrus 88
3.3 China: Chapter VII of the Jiuzhang Suanshu 91
3.4 India: Bhaskara and the Rule of Simple False Position 96
3.5 Qusta Ibn Luqa: A Geometric Justification 98
3.6 Ibn al Banna: The Method of the Scales 101
3.7 Fibonacci: the Elchatayn rule 103
3.8 Pellos: The Rule of Three and The Method of Simple False Position 106
3.9 Clavius: Solving a System of Equations 107
Bibliography Ill
VI Contents
4 Euclid s Algorithm 113
4.1 Euclid s Algorithm 113
4.2 Comparing Ratios 118
4.3 Bdzout s Identity 122
4.4 Continued Fractions 126
4.5 The Number of Roots of an Equation 132
Bibliography 136
5 From Measuring the Circle to Calculating n 139
Geometric Approaches 140
5.1 The Circumference of the Circle 140
5.2 The Area of the Circle in the Jiuzhang Suanshu 146
5.3 The Method of Isoperimeters 152
Analytic Approaches 156
5.4 Arithmetic Quadrature 156
5.5 Using Series 161
5.6 Epilogue 164
Bibliography 166
6 Newton s Methods 169
The Tangent Method 170
6.1 Straight Line Approximations 170
6.2 Recurrence Formulas 175
6.3 Initial Conditions 178
6.4 Measure of Convergence 183
6.5 Complex Roots 188
Newton s Polygon 191
6.6 The Ruler and Small Parallelograms 191
Bibliography 196
7 Solving Equations by Successive Approximations 199
Extraction of Square Roots 200
7.1 The Method of Heron of Alexandria 202
7.2 The Method of Theon of Alexandria 203
7.3 Mediaeval Binomial Algorithms 205
Numerical Solutions of Equations 208
7.4 Al TusPs Tables 208
7.5 Viete s Method 213
7.6 Kepler s Equation , 219
7.7 Bernoulli s Method of Recurrent Series 223
7.8 Approximation by Continued Fractions 227
Contents VII
Horner like Transformations of Polynomial Equations 230
7.9 The Ruffini Budan Schema 230
Bibliography 236
8 Algorithms in Arithmetic 239
Factors and Multiples 240
8.1 The Sieve of Eratosthenes 241
8.2 Criteria For Divisibility 243
8.3 Quadratic Residues 248
Tests for Primality 251
8.4 The Converse of Fermat s Theorem 252
8.5 The Lucas Test 256
8.6 Pe pin sTest 260
Factorisation Algorithms 263
8.7 Factorisation by the Difference of Two Squares 264
8.8 Factorisation by Quadratic Residues 267
8.9 Factorisation by Continued Fractions 269
The Pell Fermat Equation 272
8.10 The Arithmetica of Diophantus 273
8.11 The Lagrange Result 275
Bibliography 280
9 Solving Systems of Linear Equations 283
9.1 Cramer s Rule 284
9.2 The Method of Least Squares 287
9.3 The Gauss Pivot Method 291
9.4 A Gauss Iterative Method 296
9.5 Jacobi s Method 300
9.6 Seidel s Method 302
9.7 Nekrasov and the Rate of Convergence 306
9.8 Cholesky s Method 310
9.9 Epilogue 314
Bibliography 315
10 Tables and Interpolation 319
10.1 Ptolemy s Chord Tables 321
10.2 Briggs and Decimal Logarithms 328
10.3 The Gregory Newton Formula 332
10.4 Newton s Interpolation Polynomial 336
10.5 The Lagrange Interpolation Polynomial 340
VIII Contents
10.6 An Error Upper Bound 345
10.7 Neville s Algorithm 347
Bibliography 350
11 Approximate Quadratures 353
11.1 Gregory s Formula 354
11.2 Newton s Three Eighths Rule 356
11.3 The Newton Cotes Formulas 357
11.4 Stirling s Correction Formulas 359
11.5 Simpson s Rule 362
11.6 The Gauss Quadrature Formulas 363
11.7 Chebyshev s Choice 367
11.8 Epilogue 369
Bibliography 370
12 Approximate Solutions of Differential Equations 373
12.1 Euler s Method 374
12.2 The Existence of a Solution 378
12.3 Runge s Methods 381
12.4 Heun s Methods 388
12.5 Kutta s Methods , 392
12.6 John Adams and the Use of Finite Differences 396
12.7 Epilogue 401
Bibliography..... 402
13 Approximation of Functions 405
Uniform Approximation 407
13.1 Taylor s Formula 407
13.2 The Lagrange Remainder 409
13.3 Chebyshev s Polynomial of Best Approximation 412
13.4 Spline Fitting 418
Mean Quadratic Approximation 420
13.5 Fourier Series 422
13.6 The Fast Fourier Transform 424
Bibliography 427
14 Acceleration of Convergence 429
14.1 Stirling s Method for Series 430
14.2 The Euler Maclaurin Summation Formula 434
14.3 The Euler Constant 439
Contents IX
14.4 Aitken s Method 443
14.5 Richardson s Extrapolation Method 447
14.6 Romberg s Integration Method 451
Bibliography 453
15 Towards the Concept of Algorithm 455
Recursive Functions and Computable Functions 458
15.1 The 1931 Definition 458
15.2 General G6del Recursive Functions 460
15.3 Alonzo Church and Effective Calculability 462
15.4 Recursive Functions in the Kleene Sense 466
Machines 468
15.5 The Turing Machine 468
15.6 Post s Machine 474
15.7 Conclusion 479
Bibliography 480
Biographies 481
General Index 517
Index of Names 521
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| institution | BVB |
| isbn | 3540633693 |
| language | English French |
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| spelling | Histoire d'algorithmes A history of algorithms from the pebble to the microchip Jean-Luc Chabert (ed.). [Authors: Evelyne Barbin ...] Berlin [u.a.] Springer 1999 IX, 524 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Algorithmentheorie Geschichte Algorithms -- History Algorithmus (DE-588)4001183-5 gnd rswk-swf Algorithmentheorie (DE-588)4200409-3 gnd rswk-swf Geschichte (DE-588)4020517-4 gnd rswk-swf Algorithmus (DE-588)4001183-5 s Geschichte (DE-588)4020517-4 s DE-604 Algorithmentheorie (DE-588)4200409-3 s Chabert, Jean-Luc Sonstige oth Barbin, Évelyne 1949- Sonstige (DE-588)1016467923 oth HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008118282&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | A history of algorithms from the pebble to the microchip Algorithmentheorie Geschichte Algorithms -- History Algorithmus (DE-588)4001183-5 gnd Algorithmentheorie (DE-588)4200409-3 gnd Geschichte (DE-588)4020517-4 gnd |
| subject_GND | (DE-588)4001183-5 (DE-588)4200409-3 (DE-588)4020517-4 |
| title | A history of algorithms from the pebble to the microchip |
| title_alt | Histoire d'algorithmes |
| title_auth | A history of algorithms from the pebble to the microchip |
| title_exact_search | A history of algorithms from the pebble to the microchip |
| title_full | A history of algorithms from the pebble to the microchip Jean-Luc Chabert (ed.). [Authors: Evelyne Barbin ...] |
| title_fullStr | A history of algorithms from the pebble to the microchip Jean-Luc Chabert (ed.). [Authors: Evelyne Barbin ...] |
| title_full_unstemmed | A history of algorithms from the pebble to the microchip Jean-Luc Chabert (ed.). [Authors: Evelyne Barbin ...] |
| title_short | A history of algorithms |
| title_sort | a history of algorithms from the pebble to the microchip |
| title_sub | from the pebble to the microchip |
| topic | Algorithmentheorie Geschichte Algorithms -- History Algorithmus (DE-588)4001183-5 gnd Algorithmentheorie (DE-588)4200409-3 gnd Geschichte (DE-588)4020517-4 gnd |
| topic_facet | Algorithmentheorie Geschichte Algorithms -- History Algorithmus |
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