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Section
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Topic
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Page
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Chapter 5
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Data Fitting
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63
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5.1
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Problem: Fitting an Experimental Spectrum
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63
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5.2
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Theory: Curve Fitting
|
64
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|
5.3
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Method: Lagrange Interpolation
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65
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5.3.1
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Example
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66
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5.4
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Implementation: Lagrange Interpolation, lagrange.f
(.c)
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67
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5.5
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Assessment: Interpolating a Resonant Spectrum
|
67
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5.6
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Assessment: Exploration
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68
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|
5.7
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Method: Cubic Splines
|
69
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5.7.1
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Cubic Spline Boundary Conditions
|
70
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5.7.2
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Exploration: Cubic Spline Quadrature
|
71
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5.8
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Implementation: Packaged Spline Subprogram, spline.f
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71
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5.9
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Assessment: Spline Fit of Resonant Cross
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71
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5.10
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Problem: Fitting Exponential Decay
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72
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5.11
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Model: Exponential Decay
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72
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5.12
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Theory: Probability Theory
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73
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5.13
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Method: Least-Squares Fitting
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75
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5.14
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Theory: Goodness of Fit
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77
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5.15
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Implementation: Least-Squares Fits, fit.f
(.c)
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78
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5.16
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Assessment: Fitting Exponential Decay
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79
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5.17
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Assessment: Fitting Heat Flow
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80
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|
5.18
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Implementation: Linear Quadratic Fits
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81
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|
5.19
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Assessment: Quadratic fit
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82
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|
5.20
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Method: Nonlinear Least-Squares Fitting
|
82
|
|
5.21
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Assessment: Nonlinear Fitting
|
82
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|
Chapter 6
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Deterministic
Randomness
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83
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6.1
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Problem: Deterministic Randomness
|
83
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6.2
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Theory: Random
Sequences
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83
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6.3
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Method: Pseudo-Random-Number
Generators
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84
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6.4
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Assessment:
Random Sequences
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86
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6.5
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Implementation: Simple and Not [random.f
(.c);
call.f
(.c)]
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87
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|
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6.6
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Assessment: Randomness and Uniformity
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87
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6.7
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Assessment: Tests for Randomness and Uniformity
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88
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|
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6.8
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Problem: A
Random Walk
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89
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6.9
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Model: Random Walk Simulation
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89
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|
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6.10
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Method: Numerical
Random Walk
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90
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6.11
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Implementation: Random Walk, walk.f
(.c)
|
91
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6.12
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Assessment: Different Random Walkers
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91
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| - | Exploration:
2D Gas Simulation (SUHEP)
| - |
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Chapter 7
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Monte Carlo Applications
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93
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7.1
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Problem: Radioactive
Decay
|
93
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7.2
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Theory: Spontaneous Decay
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93
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7.3
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Model: Discrete Decay
|
94
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7.4
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Model: Continuous Decay
|
95
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7.5
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Method: Decay
Simulation
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95
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7.6
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Implementation:
Radioactive Decay, decay.f
(.c)
|
97
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7.7
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Assessment: Decay Visualization (sonification)
|
97
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7.8
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Problem: Measuring
by Stone Throwing
|
97
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7.9
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Theory: Integration
by Rejection
|
97
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7.10
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Implementation:
Stone Throwing, pond.f
(.c)
|
99
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7.11
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Problem: High-Dimensional Integration
|
99
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7.12
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Method: Integration by Mean Value
|
100
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7.12.1
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Multidimensional Monte Carlo
|
101
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7.13
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Assessment: Error in Multidimensional Integration
|
101
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7.14
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Implementation: Monte Carlo 10-D Integration, int_10d.f
(.c)
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102
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7.15
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Problem: MC Integration of Rapidly Varying Functions (O)
|
102
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7.16
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Method: Variance Reduction (O)
|
102
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7.17
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Method: Importance Sampling (O)
|
103
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7.18
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Implementation: Nonuniform Randomness (O)
|
103
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7.18.1
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Inverse Transform/Change of Variable Method
|
104
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7.18.2
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Uniform Weight Function
|
105
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7.18.3
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Exponential Weight
|
105
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7.18.4
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Gaussian (Normal) Distribution
|
106
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7.18.5
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Alternate Gaussian Distribution
|
107
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7.19
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Method: von Neumann Rejection (O)
|
107
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7.20
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Assessment (O)
|
108
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|
Chapter 8
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Differentiation
|
109
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8.1
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Problem 1: Numerical Limits
|
109
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8.2
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Method: Numeric
|
109
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8.2.1
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Method: Forward Difference
|
109
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8.2.2
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Method: Central Difference
|
111
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8.2.3
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Method: Extrapolated Difference
|
111
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8.3
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Assessment: Error Analysis
|
112
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8.4
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Implementation: Differentiation, diff.f
(.c)
|
114
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|
8.5
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Assessment: Error Analysis, Numerical
|
114
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|
8.6
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Problem 2: Second Derivatives
|
114
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8.7
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Theory: Newton II
|
114
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|
8.8
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Method: Numerical Second Derivatives
|
114
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|
8.9
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Assessment: Numerical Second Derivatives
|
115
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Chapter 9
|
Differential
Equations and Oscillations
|
117
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|
9.1
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Problem: A Forced Nonlinear Oscillator
|
117
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9.2
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Theory, Physics: Newton's Laws
|
117
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9.3
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Model: Nonlinear Oscillator
|
118
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9.4
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Theory, Mathematics: Types of Differential Equations
|
119
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9.4.1
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Order
|
119
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9.4.2
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Ordinary and Partial
|
120
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9.4.3
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Linear and Nonlinear
|
121
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9.4.4
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Initial and Boundary Conditions
|
121
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|
9.5
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Theory, Math, and Physics: The Dynamical Form for ODEs
|
122
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9.5.1
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Dynamical Form for Second-Order Equation
|
122
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9.6
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Implementation: Dynamical Form for Oscillator
|
123
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|
9.7
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Numerical Method: Differential Equation Algorithms
|
124
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|
9.8
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Method (Numerical): Euler's
Algorithm (euler.c),
(plots)
|
125
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|
9.9
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Method (Numerical): Second-Order
Runge-Kutta
|
126
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|
9.10
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Method (Numerical): Fourth-Order
Runge-Kutta
|
127
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|
9.11
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Implementation: ODE Solver, rk4.f
(.c)
|
128
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|
9.12
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Assessment: rk4
and Linear Oscillations
|
128
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|
9.13
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Assessment: rk4
and Nonlinear Oscillations
|
129
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|
9.14
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Exploration: Energy
Conservation
|
129
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|
Chapter 10
|
Quantum Eigenvalues; Zero-Finding and Matching
|
131
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|
10.1
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Problem: Binding A Quantum Particle
|
131
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|
10.2
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Theory: Quantum Waves
|
132
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|
10.3
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Model: Particle in a Box
|
133
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|
10.4
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Solution: Semianalytic
|
133
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|
10.5
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Method: Finding Zero via the Bi Algorithm
|
136
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|
10.6
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Method: Eigenvalues from an ODE Solver
|
136
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|
10.6.1
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Matching
|
139
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|
10.7
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Implementation: ODE Eigenvalues, numerov.c
|
140
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|
10.8
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Assessment: Explorations
Applet (SUHEP)
|
141
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|
10.9
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Extension: Newton's Rule for Finding Roots
|
142
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|
Chapter 11
|
Anharmonic Oscillations
|
143
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|
11.1
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Problem 1: Nonlinearly Perturbed Harmonic Oscillator
|
143
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|
11.2
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Theory: Newton II
|
144
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|
11.3
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Implementation: ODE Solver, rk4.f
(.c)
|
144
|
|
-
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Exploration: x^4 Perturbed Oscillator Applet
(SUHEP)
|
-
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|
11.4
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Assessment: Amplitude Dependence of Frequency
|
145
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|
11.5
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Problem 2: Realistic Pendulum
|
145
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11.6
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Theory: Newton II for Rotations
|
146
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|
11.7
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Method, Analytic: Elliptic Integrals
|
147
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|
11.8
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Implementation, rk4 for Pendulum
|
147
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|
11.9
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Exploration: Resonance and Beats
|
148
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|
11.10
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Exploration: Phase-Space Plot
|
149
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|
11.11
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Exploration: Damped Oscillator
|
150
|
|
Chapter 12
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Fourier Analysis of Nonlinear Oscillations
|
151
|
|
12.1
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Problem 1: The Harmonics of Nonlinear Oscillations
|
151
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12.2
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Theory: Fourier Analysis
|
152
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|
12.2.1
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Example 1: Sawtooth Function
|
154
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|
12.2.2
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Example 2: Half-Wave Function
|
154
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|
12.3
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Assessment: Summation of Fourier Series
|
155
|
|
12.4
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Theory: Fourier Transforms
|
156
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|
12.5
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Method: Discrete Fourier Transform
|
157
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|
12.6
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Method: DFT for Fourier Series
|
161
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|
12.7
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Implementation: DFT, fourier.f
(.c),
invfour.c
|
162
|
|
12.8
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Assessment: Simple Analytic Input
|
162
|
|
12.9
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Assessment: Highly Nonlinear Oscillator
|
163
|
|
12.10
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Assessment: Nonlinearly Perturbed Oscillator
|
163
|
|
12.11
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Exploration: DFT of Nonperiodic Functions
|
164
|
|
12.12
|
Exploration: Processing Noisy Signals
|
164
|
|
12.13
|
Model: Autocorrelation Function
|
164
|
|
12.14
|
Assessment: DFT and Autocorrelation Function
|
166
|
|
12.15
|
Problem 2: Model Dependence of Data Analysis (O)
|
167
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|
12.16
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Method: Model-Independent Data Analysis
|
167
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|
12.17
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Assessment
|
169
|
|
Chapter 13
|
Unusual Dynamics of Nonlinear Systems
|
171
|
|
13.1
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Problem: Variability of Bug Populations
|
171
|
|
13.2
|
Theory: Nonlinear Dynamics
|
171
|
|
13.3
|
Model: Nonlinear Growth, The Logistic Map
|
172
|
|
13.3.1
|
The Logistic Map
|
173
|
|
13.4
|
Theory: Properties of Nonlinear Maps
|
174
|
|
13.4.1
|
Fixed Points
|
174
|
|
13.4.2
|
Period Doubling, Attractors
|
175
|
|
13.5
|
Implementation and Assessment: Explicit Mapping
|
176
|
|
13.6
|
Assessment: Bifurcation Diagram (sonification)
|
177
|
|
13.7
|
Implementation: Bifurcation Diagram, bugs.f
(.c)
|
178
|
|
13.8
|
Exploration: Random Numbers from Logistic Map?
|
179
|
|
13.9
|
Exploration: Feigenbaum Constants
|
179
|
|
13.10
|
Exploration: Other Maps
|
180
|
|
Chapter 14
|
Differential Chaos in Phase Space
|
181
|
|
14.1
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Problem: A Pendulum Becomes Chaotic
|
181
|
|
14.2
|
Theory and Model: The Chaotic Pendulum (Java
animate)
|
182
|
|
14.3
|
Theory: Limit Cycles and Mode Locking (animation)
|
183
|
|
14.4
|
Implementation 1: Solve ODE, rk4.f
(.c)
|
184
|
|
14.5
|
Assessment and Visualization: Phase-Space Orbits
|
184
|
|
14.6
|
Implementation 2: Free Oscillations in Phase Space
|
187
|
|
14.7
|
Theory: Chaotic and Random Motion in Phase Space
|
188
|
|
14.8
|
Implementation 3: Chaotic Pendulum
|
188
|
|
14.9
|
Assessment: Chaotic Structure in Phase Space
|
191
|
|
14.10
|
Assessment: Fourier Analysis of Chaotic Pendulum
|
191
|
|
14.11
|
Exploration: Pendulum with Vibrating Pivot
|
192
|
|
14.11.1
|
Implementation: Bifurcation Diagram of Pendulum
|
193
|
|
14.12
|
Further Explorations
|
194
|
| Chapter 24 |
Fractals
|
323
|
|
24.1
|
Problem : Fractals
|
323
|
|
24.2
|
Theory: Fractional Dimension
|
324
|
|
24.3
|
Problem 1: The Sierpienski Gasket
|
324
|
|
24.4
|
Implementation: sierpin.c
|
325
|
|
24.5
|
Assessment: Determining a Fractal Dimension
|
326
|
|
24.6
|
Problem 2: How to Grow Beautiful Plants
|
327
|
|
24.7
|
Theory: Self-Affine Connections
|
328
|
|
24.8
|
Implementation: Barnsley's Fern, fern.c
|
329
|
|
24.9
|
Exploration: Self-Affinity in Trees
|
330
|
|
24.10
|
Implementation: Nice Trees, tree.c
|
330
|
|
24.11
|
Problem 3: Ballistic Deposition
|
330
|
|
24.12
|
Method
|
331
|
|
24.13
|
Implementation: Ballistic Deposition, film.c
|
333
|
|
24.14
|
Problem 4: Length Of The Coastline of Britain
|
333
|
|
24.15
|
Model: The Coast As A Fractal
|
333
|
|
24.16
|
Method: Box Counting
|
334
|
|
24.17
|
Problem 5: Correlated Growth in Forests and Films
|
335
|
|
24.18
|
Method: Correlated Ballistic Deposition
|
336
|
|
24.19
|
Implementation: Correlated Ballistic Deposition, column.c
|
337
|
|
24.20
|
Problem 6: A Globular Cluster
|
337
|
|
24.21
|
Model: Diffusion-Limited Aggregation
|
337
|
|
24.22
|
Method
|
338
|
|
24.23
|
Implementation: Diffusion-Limited Aggregation, dla.c
|
339
|
|
24.24
|
Assessment: Fractal Analysis Of The DLA Graph
|
339
|
|
24.25
|
Problem 7: Fractal Structures in Bifurcation Graph
|
340
|