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E-raamat: Scientific Computation

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(Eidgenössische Technische Hochschule Zürich),
  • Formaat: PDF+DRM
  • Ilmumisaeg: 05-Nov-2009
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9780511654343
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Scientific ComputationSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 05-Nov-2009
  • Kirjastus: Cambridge University Press
  • Keel: eng
  • ISBN-13: 9780511654343
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Using real-life applications, this graduate-level textbook introduces different mathematical methods of scientific computation to solve minimization problems using examples ranging from locating an aircraft, finding the best time to replace a computer, analyzing developments on the stock market, and constructing phylogenetic trees. The textbook focuses on several methods, including nonlinear least squares with confidence analysis, singular value decomposition, best basis, dynamic programming, linear programming, and various optimization procedures. Each chapter solves several realistic problems, introducing the modeling optimization techniques and simulation as required. This allows readers to see how the methods are put to use, making it easier to grasp the basic ideas. There are also worked examples, practical notes, and background materials to help the reader understand the topics covered. Interactive exercises are available at www.cambridge.org/9780521849890.

Using real-life applications, this graduate-level textbook introduces different mathematical methods of scientific computation to solve minimization problems.

Arvustused

' this book will help students understand methods in some of their appropriate contexts, in a relatively easy, fresh, and concise way.' E. Vavalis, Computing Reviews ' the text makes an important contribution: it describes the thought process necessary to solve a problem computationally, considers the various possible models, and shows which ones lead to the most precise solutions. I recommend this text for instructors who are interested in problem-based or other interactive learning styles.' Physics Today

Muu info

Using real-life applications, this graduate-level textbook introduces different mathematical methods of scientific computation to solve minimization problems.
Preface xi
List of abbreviations xiii
1 Determination of the accurate location of an aircraft 1
1.1 Introduction
1
1.2 Modelling the problem as a least squares problem
4
1.2.1 How to solve a mixed problem: equation system and minimization
6
1.3 Solving the non linear least squares problem
6
1.4 Error analysis confidence analysis
8
1.4.1 An approximation for a complicated S
10
1.4.2 Interactive exercise
12
Further reading
13
2 When to replace equipment 14
2.1 The replacement paradox
14
2.2 Car replacement
15
2.2.1 Simulation
18
2.2.2 Linear approximations
18
2.2.3 Mathematical background
20
2.3 Computer replacement and Moore's law
22
2.3.1 Piecewise lineal approximation
23
3 Secondary structure prediction using least squares and singular value decomposition 25
3.1 Extremely brief introduction to protein biochemistry
25
3.1.1 The structure Whyseter catadon)– P02185
27
3.1.2 Additional Milo illation about the biochemical structure of α-helices
28
3.2 General introduction to modelling and prediction
30
3.2.1 SSP as modelling pisiblem
32
3.3 Least squares singulat value decomposition
33
3.3.1 The need for weighting
35
3.3.2 Singular value decomposition and eigenvalue decomposition
36
3.3.3 Criteria for discarding singular values
40
3.3.4 A concrete example
43
4 Secondary structure prediction using least squares and best basis 49
4.1 Definition of "best basis"
49
4.1.1 Stepwise regression
49
4.1.2 The difficulty of the problem
50
4.2 Mathematical formulation using the best basis approach
51
4.3 Algorithms for finding local minima
52
4.3.1 Early abort
52
4.3.2 (Discrete) steepest descent
53
4.3.3 Improved EA or SD algorithm (avoiding repeated paths)
54
4.3.4 Exploring the solution space
54
4.4 The neighbor function
55
4.5 The confidence interval
56
4.5.1 Analysis of the variance of the coefficients
57
4.6 Confidence analysis with random columns
58
4.7 Final observations on the least squares methods
59
4.7.1 Refining the predictor function
59
4.7.2 Validation
60
4.7.3 Other models
61
4.8 Linear classification/discrimination
62
4.8.1 Trying to do better
63
4.8.2 Constructing initial solutions using the sigmoid function
64
5 Secondary structure prediction with learning methods (nearest neighbors) 66
5.1 Formulation of the problem
67
5.1.1 The general idea
67
5.1.2 Application to helix prediction
68
5.2 Searching and data structures for NN problems
69
5.2.1 Sequential search
69
5.2.2 Range searching
69
5.2.3 Quad trees
70
5.2.4 k-d Trees (k-dimensional trees)
70
5.2.5 Nearest neighbor trees (NN trees)
71
5.3 Building NN trees
74
5.4 Searching the NN tree
75
5.4.1 Searching the k nearest neighbors
76
5.4.2 Concluding a value from its neighbors
77
5.5 Practical considerations
79
5.5.1 Bucketing
79
5.5.2 Normalizing dimensions
79
5.5.3 Using random directions
79
5.6 NN trees used for clustering
80
6 Secondary structure prediction with linear programming (LP) 82
6.1 Introduction
82
6.1.1 An introductory example
83
6.2 The simplex algorithm
85
6.2.1 Slack variables
86
6.3 SSP formulated as linear programming
89
6.4 Inconsistent data
90
6.4.1 Using slack variables for each inconsistency
91
6.4.2 Alternative method — removing inconsistencies
91
6.5 Prediction
92
6.6 Comparison of linear programming with least squares
92
6.7 Example
93
Further reading
94
7 Stock market prediction 96
7.1 Introduction
96
7.1.1 Definitions
97
7.1.2 Examples of information available online
98
7.1.3 Rationale for stock market prediction
99
7.1.4 Organization of the modelling
100
7.2 The optimal transaction sequence (OTS)
101
7.2.1 Defining the problem
101
7.2.2 Calculating the OTS for the past
101
7.2.3 Interactive exercise "Optimal transaction sequence"
102
7.3 Approximating the OTS
103
7.3.1 Decision versus numerical approach
104
7.3.2 Computing good (optimal) parameters
109
7.3.3 From the OTS to reality
111
7.4 Simulation
111
7.4.1 Two simple models
112
7.4.2 Optimization of simulation parameters
114
7.5 Confidence analysis
124
7.5.1 The notion of "transfer of error"
125
7.5.2 Monte Carlo estimation
125
7.6 Dynamic programming
127
7.6.1 The idea of dynamic programming in more detail
128
7.7 Examples of dynamic programming
129
7.7.1 Optimal binary search trees
129
7.7.2 Optimal order of matrix multiplication
132
7.7.3 String matching
135
Further reading
138
8 Phylogenetic tree construction 139
8.1 Introduction
139
8.1.1 Applications
139
8.1.2 What it is about
142
8.1.3 Classification types
143
8.1.4 Phylogenies from protein sequences
144
8.2 Tree building
145
8.2.1 Rooted versus unrooted trees
146
8.2.2 The constructor—heuristic—evaluator (CHE) optimization strategy
148
8.3 Trees based on distance information
149
8.3.1 Measures of distance
149
8.3.2 PAM distance in detail
153
8.3.3 Calculating distances from sequence alignment
157
8.3.4 Dayhoff matrices
158
8.3.5 Sequence alignment using Dayhoff matrices and maximum likelihood
160
8.3.6 Estimating distances between sequences by maximum likelihood
162
8.3.7 Sequence alignment — global alignment
164
8.3.8 Sequence alignment — local alignment
164
8.3.9 Sequence alignment — cost-free end alignment
164
8.3.10 Distance and variance matrices
165
8.4 Creating an initial topology
166
8.4.1 The UPGMA algorithm
166
8.4.2 The WPGMA algorithm
168
8.4.3 Tree fitting by least squares
171
8.4.4 Improving a tree: swapping heuristics
173
8.5 Phylogenetic trees based on character information
175
8.5.1 Parsimony
176
8.5.2 General parsimony
177
8.6 Special cases allowing easy tree construction
181
8.6.1 Strict character compatibility
181
8.6.2 Ultrametric trees
184
8.6.3 Additive trees
185
8.7 Evolutionary algorithms or genetic algorithms
185
8.8 Randomness, parallelism and CHE procedures
190
8.8.1 Concluding additional information from different runs
190
8.8.2 Sensitivity analysis suitable for parallel CHE
191
8.8.3 Tradeoff between randomness and quality
194
Further reading
195
Appendix A Methods for function minimization 196
Appendix B Online resources 228
Index 231
Gaston H. Gonnet is a Professor in the Department of Informatik at ETH, Zurich. His research interests lie in symbolic and algebraic computation, system development, limit and series computation and heuristic algorithms, computational biochemistry algorithms, and the design and analysis of algorithms. Ralf Scholl teaches mathematics, physics, and science and technology at the Geschwister-Scholl-Gymnasium in Stuttgart. His interests lie in the didactics of teaching science and mathematics at high school to graduate level.
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