The Birth of Numerical Analysis

Cover......Page 1
Half Title......Page 2
Copyright......Page 4
1 The limitations of computers......Page 5
3 Sixty years young \back to the roots of the future"......Page 6
4 Extrapolation......Page 7
5 Functional equations......Page 8
6 The importance of software and the in°uence of
hardware......Page 10
7 Approximation and optimization......Page 11
9 And there is more......Page 13
10 Acknowledgements......Page 14
Table of Contents......Page 17
Some pioneers of extrapolation methods......Page 18
What is interpolation?......Page 19
2.1 First contributions......Page 20
2.2 C. Huygens......Page 21
2.3 L.F. Richardson......Page 23
2.4 W. Romberg......Page 26
3 Aitken's process and Steffensen's method......Page 28
3.1 Seki Takakazu......Page 31
3.2 A.C. Aitken......Page 32
3.3 J.F. Steffensen......Page 33
3.4 D. Shanks......Page 34
3.5 P. Wynn......Page 35
4 And now?......Page 36
References......Page 37
Abstract......Page 40
1.1 Software......Page 41
1.2 N-dimensional quadrature rules......Page 42
2.1 One dimension; regular integrand......Page 43
Theorem 2.1.......Page 44
Theorem 2.2. [3, Theorem 2.2]......Page 45
3.1 An N-dimensional example......Page 46
Theorem 3.1.......Page 47
Theorem 3.3.......Page 48
5 Gaussian formulas for singular integrands......Page 49
References......Page 51
Numerical methods for ordinary differential
equations: early days......Page 52
2 Notable events, ideas and people......Page 53
Linear multistep methods......Page 54
Runge-Kutta methods......Page 55
Linear and non-linear stability......Page 56
S. Gill and R. H. Merson......Page 57
T. E. Hull......Page 59
References......Page 60
Abstract......Page 62
Developing a taste for dynamical system theory......Page 66
Retirement......Page 68
Addicted to cycling......Page 69
Abstract......Page 70
2 A survey of numerical methods......Page 71
2.1 Degenerate kernel approximation methods......Page 72
2.2 Projection methods......Page 73
2.3 Nyström methods......Page 75
3 Error analysis and some history......Page 76
3.1 Degenerate kernel methods......Page 77
3.2.1 Kantorovich and Krylov regularization......Page 78
3.2.2 The iterated projection solution......Page 79
3.3 Nyström methods......Page 80
3.3.1 Product integration......Page 83
3.3.2 The eigenvalue problem......Page 84
4 Boundary integral equation methods......Page 85
Acknowledgements......Page 86
References......Page 87
1 Introduction......Page 90
2 Prelude { the pre-NAG days......Page 91
3 Announcement of the ICL 1906A......Page 92
5 Selection of library contents......Page 93
6 Library contents......Page 94
7 Comments on the chapter contents developed......Page 97
8 Chapter subdivisions......Page 98
9 Library contribution......Page 99
10.1 Numerical linear algebra......Page 100
11 Types of Library Software......Page 101
13 Issues of numerical software portability......Page 102
16 NAG Library Manual......Page 103
18 Algorithm testing......Page 104
19 Validation and library assembly......Page 105
22 Operational principles......Page 106
References......Page 107
Abstract......Page 109
1 Introduction......Page 110
2 A short history of supercomputers......Page 111
3 2000-2005: Cluster, Intel processors, and the Earth
Simulator......Page 112
3.1 Explosion of cluster based system......Page 113
3.2 Intel-ization of the processor landscape......Page 115
3.3 The Earth Simulator shock......Page 116
4 2005 and beyond......Page 117
4.1 Dynamic of the market......Page 118
4.2 Consumer and producer......Page 119
4.4 Projections......Page 121
References......Page 123
Abstract......Page 124
1 Historical comments on enforcing nonnegativity......Page 125
Perron-Frobenius theorem:......Page 126
Karush-Kuhn-Tucker conditions:......Page 127
NNLS problem......Page 128
3.2 Numerical approaches and algorithms......Page 129
Lawson and Hanson's algorithm......Page 130
Algorithm PQN-NNLS......Page 133
3.2.3 Other methods:......Page 134
Block principal pivoting algorithm......Page 135
Newton Like method for NNLS......Page 136
4.1 Nonnegative matrix factorization......Page 137
Alternating Least Squares (ALS) algorithms for NMF......Page 138
ALS algorithm for NMF......Page 139
Nonnegative Rank-k Tensor Decomposition Problem......Page 140
Alternating least squares for NTF......Page 141
5.1 Support vector machines......Page 142
5.2 Image processing and computer vision......Page 143
5.3 Text mining......Page 145
5.5 Speech recognition......Page 146
5.6 Spectral unmixing by NMF and NTF......Page 147
References......Page 150
Abstract......Page 155
1 Earlier algorithms......Page 156
2 Two major advances in unconstrained optimization......Page 158
3 Unconstrained objective functions for constrained
problems......Page 162
4 Sequential quadratic programming......Page 165
5 Trust region methods......Page 168
6 Further remarks......Page 171
References......Page 173
Abstract......Page 175
2 Edinburgh: early years......Page 176
3 How I became a numerical analyst......Page 180
4 St Andrews......Page 182
5 Collaboration between St Andrews and Edinburgh......Page 183
6 Dundee......Page 185
7 The evolution of computing facilities in Dundee......Page 187
8 Postscript......Page 189
References......Page 191
Remembering Philip Rabinowitz......Page 192
Philip J. Davis......Page 193
Aviezri S. Fraenkel......Page 197
My early experiences with scientific computation......Page 200
Abstract......Page 206
1 Introduction......Page 207
2 Properties of the Chebyshev polynomials......Page 209
3 Inversion of the Laplace transform......Page 210
4 Solution of the Abel integral equation......Page 211
5 The computation of Laplace, Fourier and Hankel
transforms......Page 212
6 Solution of integral equations of the second kind using
modi¯ed moments......Page 214
7 An extension of Clenshaw-Curtis quadrature to
oscillating and singular integrals......Page 216
References......Page 217
Name Index......Page 219
Subject Index......Page 227