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Numerical Analysis:
a compact compendium
Roberto De Leo
Contents
Index
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Contents
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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
Chapter
4
Linear systems
In this chapter we present the main methods to solve numerically linear systems of equations. Before starting the discussion of the numerical algorithms, we recall in the first section the main facts we will use from the Linear Algebra field.
4.1
Linear spaces and linear maps
4.2
The LU Method
4.3
Pivoting
4.4
Solving a system
4.5
Error analysis
4.6
Iterative Methods
4.7
Example: A Boundary Value Problem
4.8
The SciPy linalg.solve algorithm
4.9
Python recap
4.10
Exercises
4.11
References and Suggested Readings
