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Introduction to Numerical Methods and Analysis with Julia (draft)
Preface
1. Introduction
2. Root-finding
2.1. Root Finding by Interval Halving (Bisection)
2.2. Solving Equations by Fixed Point Iteration (of Contraction Mappings)
2.3. Newton’s Method for Solving Equations
2.4. Taylor’s Theorem and the Accuracy of Linearization
2.5. Measures of Error and Order of Convergence
2.6. The Convergence Rate of Newton’s Method
2.7. Root-finding without Derivatives
3. Linear Algebra and Simultaneous Equations
3.1. Row Reduction/Gaussian Elimination
3.2. Machine Numbers, Rounding Error and Error Propagation
3.3. Partial Pivoting
3.4. Solving
\(Ax = b\)
with LU factorization
3.5. Solving
\(Ax = b\)
With Both Pivoting and LU Factorization
3.6. Error bounds for linear algebra, condition numbers, matrix norms, etc.
3.7. Iterative Methods for Simultaneous Linear Equations
3.8. Faster Methods for Solving
\(Ax = b\)
for Tridiagonal and Banded matrices, and Strict Diagonal Dominance
3.9. Computing Eigenvalues and Eigenvectors: the Power Method, and a bit beyond
3.10. Solving Nonlinear Systems of Equations by generalizations of Newton’s Method — a brief introduction
4. Polynomial Collocation and Approximation
4.1. Polynomial Collocation (Interpolation/Extrapolation) and Approximation
4.2. Error Formulas for Polynomial Collocation
4.3. Choosing the collocation points: the Chebyshev method
4.4. Piecewise Polynomial Approximating Functions: Splines and Hermite Cubics
4.5. Least-Squares Fitting to Data
4.6. Least-squares Fitting to Data: Appendix on The Geometrical Approach
5. Derivatives and Definite Integrals
5.1. Approximating Derivatives by the Method of Undetermined Coefficients
5.2. Richardson Extrapolation
5.3. Definite Integrals, Part 1: The Building Blocks
5.4. Definite Integrals, Part 2: The Composite Trapezoid and Midpoint Rules
5.5. Definite Integrals, Part 3: The (Composite) Simpson’s Rule and Richardson Extrapolation
5.6. Definite Integrals, Part 4: Romberg Integration
6. Minimization
6.1. Finding the Minimum of a Function of One Variable Without Using Derivatives – under construction
6.2. Finding the Minimum of a Function of Several Variables — Coming Soon
7. Initial Value Problems for Ordinary Differential Equations
7.1. Background and Some Examples
7.2. Euler’s Method
7.3. Runge-Kutta Methods
7.4. A Global Error Bound for One Step Methods
7.5. Systems of ODEs and Higher Order ODEs
7.6. Error Control and Variable Step Sizes
7.7. An Introduction to Multistep Methods
7.8. Adams-Bashforth Multistep Methods
7.9. Implicit Methods: Adams-Moulton
8. Bibliography
9. Appendices
9.1. Installing Julia and some useful add-ons
9.2. Notes on the Julia Language
9.3. Module
NumericalMethods
Index
