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Numerical Analysis:
a compact compendium
Roberto De Leo
Contents
Index
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Contents
Prev
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Front Matter
Preface
1
Numbers
How much information is contained in a number?
Floating-point Systems
Double precision
Python recap
Exercises
References and Suggested Readings
2
What is Numerical Analysis
Case study 1: evaluating the value of a function
Case study 2: evaluating the derivative of a function
Error and conditioning
Python recap
Exercises
References and Suggested Readings
3
Root-finding methods
Roots and Conditioning
Bisection Method
Functions Iterations
Newton's method
Secant method
The
root_scalar
SciPy command
Python recap
Exercises
References and Suggested Readings
4
Linear systems
Linear spaces and linear maps
The LU Method
Pivoting
Solving a system
Error analysis
Iterative Methods
Example: A Boundary Value Problem
The SciPy linalg.solve algorithm
Python recap
Exercises
References and Suggested Readings
5
Eigenvalues
Eigenvalues and Eigenvectors
The Power method
The QR decomposition
The LR and QR methods
Python recap
Exercises
References and Suggested Readings
6
Optimization
What does it mean optimizing a function
Iterative methods
Gradient methods
Steepest Descent
Newton's Method
Conjugate Gradient
Scipy function minimize
Exercises
References and Suggested Readings
7
Interpolation
Motivation
Polynomial interpolation
How good can a polynomial approximation be?
Python recap
Exercises
References and Suggested Readings
8
Numerical Integration
Riemann sums
Newton-Cotes Quadrature
Midpoint rule (\(n=0\))
Trapezoidal rule (\(n=1\))
Simpson rule (\(n=2\))
Exercises
References and Suggested Readings
9
ODEs
Initial Value Problems
Numerical Methods
Explicit Euler Method
Heun Method
RK4 Method
Stiff ODEs
Implicit Euler method
Solving higher-order IVPs
Euler method in \(\mathbb R^n\)
Boundary Value Problems
Shooting Method
Finite Differences Method
Scipy's BVP method
Exercises
References and Suggested Readings
10
PDEs
Exercises
References and Suggested Readings
Back Matter
A
Can we trust the accuracy of functions evaluations on computers?
B
GNU Free Documentation License
References and Suggested Readings
Index
Colophon
Authored in PreTeXt
Section
9.13
Scipy's BVP method
Consider again the problem
\begin{equation*} \ddot x = -\dot x\cos t+x\sin t,\,x(0)=1,\,x(15\pi/2)=e. \end{equation*}
Now we will solve it with SciPy's own solve_bvp method.
