ISBN: 3-540-63369-3
TITLE: A History of Algorithms
AUTHOR: Chabert, Jean-Luc (Ed.)
TOC:

Introduction 1 
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 111 
4 Euclid's Algorithm 113 
4.1 Euclid's Algorithm 113 
4.2 Comparing Ratios 118 
4.3 Bzout'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 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-Tusi's Tables 208 
7.5 Vite'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 
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 Ppin's Test 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 
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 
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 Gdel 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 
END
