Numerical Analysis Lecture Notes
Peter J. Olver
7. Iterative Methods for Linear Systems
Linear iteration coincides with multiplication by successive powers of a matrix; convergence of the iterates depends on the magnitude of its eigenvalues. We discuss in
Numerical Analysis Lecture Notes
Peter J. Olver
4. Gaussian Elimination
In this part, our focus will be on the most basic method for solving linear algebraic
systems, known as Gaussian Elimination in honor of one of the all-time mathematical
greats the ea
Numerical Analysis Lecture Notes
Peter J. Olver
8. Numerical Computation of Eigenvalues
In this part, we discuss some practical methods for computing eigenvalues and eigenvectors of matrices. Needless to say, we completely avoid trying to solve (or even w
Numerical Analysis Lecture Notes
Peter J. Olver
13. Approximation and Interpolation
We will now apply our minimization results to the interpolation and least squares
tting of data and functions.
13.1. Least Squares.
Linear systems with more equations than
Numerical Analysis Lecture Notes
Peter J. Olver
5. Inner Products and Norms
The norm of a vector is a measure of its size. Besides the familiar Euclidean norm
based on the dot product, there are a number of other important norms that are used in
numerical
Numerical Analysis Lecture Notes
Peter J. Olver
3. Review of Matrix Algebra
Vectors and matrices are essential for modern analysis of systems of equations
algebrai, dierential, functional, etc. In this part, we will review the most basic facts of
matrix
Numerical Analysis Lecture Notes
Peter J. Olver
12. Minimization
In this part, we will introduce and solve the most basic mathematical optimization
problem: minimize a quadratic function depending on several variables. This will require
a short introducti
Numerical Analysis Lecture Notes
Peter J. Olver
11. Finite Dierence Methods for
Partial Dierential Equations
As you are well aware, most dierential equations are much too complicated to be
solved by an explicit analytic formula. Thus, the development of a
Numerical Analysis Lecture Notes
Peter J. Olver
14. Finite Elements
In this part, we introduce the powerful nite element method for nding numerical
approximations to the solutions to boundary value problems involving both ordinary and
partial dierential e
Numerical Analysis Lecture Notes
Peter J. Olver
1. Computer Arithmetic
The purpose of computing is insight, not numbers.
R.W. Hamming, [24]
The main goal of numerical analysis is to develop ecient algorithms for computing
precise numerical values of math
Numerical Analysis Lecture Notes
Peter J. Olver
6. Eigenvalues and Singular Values
In this section, we collect together the basic facts about eigenvalues and eigenvectors.
From a geometrical viewpoint, the eigenvectors indicate the directions of pure stre
Numerical Analysis Lecture Notes
Peter J. Olver
9. Numerical Solution of Algebraic Systems
In this part, we discuss basic iterative methods for solving systems of algebraic equations. By far the most common is a vector-valued version of Newtons Method, wh
Numerical Analysis Lecture Notes
Peter J. Olver
10. Numerical Solution of
Ordinary Dierential Equations
This part is concerned with the numerical solution of initial value problems for systems
of ordinary dierential equations. We will introduce the most b