NLA_2015
Numerical Linear Algebra
Senior Seminar in Scientific Computing
http://people.sc.fsu.edu/~jburkardt/classes/nla_2015/nla_2015.html
NLA_2015 is the home page for the class ISC4932,
"Numerical Linear Algebra", Senior Seminar in Scientific Computing,
an undergraduate seminar class offered by the Department of Scientific Computing
at Florida State University, Fall Session 2015.
This is a 1-credit seminar class, intended to introduce the students
to ideas, algorithms, and programs common to Scientific Computing.
The topic this semester will be numerical linear algebra.
You might think numerical linear algebra is simply the rules for
solving a system of linear equations, "A*x=b", in which case we
all know how to call Matlab to magically get an answer. But
numerical linear algebra will also tell you what happens when
your problem includes equations that contradict each other, or
where you have errors in your data, or too many or not enough
equations.
Numerical linear algebra will show you how you can end up needing
to solve systems of linear equations even though you started out
trying to predict the temperature of an iron beam that is being
heated by a torch, or if you are trying to estimate the future
population of California after someone tells you that, every year,
30 percent of the people in California move out of the state, and
10 percent of the people outside of California move into it.
Numerical linear algebra shows you how to set up and solve systems
of linear equations that are so large that it would seem your computer
doesn't even have enough space to set the problem up.
Numerical linear algebra allows you to study the process of iteration,
in which we follow the changes in a system over time as it is repeatedly
affected by some kind of physical process. This could be the aging of
a population, the vibration of a metal beam, or the attempts of a prisoner
to escape from a maze by taking one step at a time.
The class meets on ?days, from ?to?, in room ?.
A tentative schedule:
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What is linearity?
Linear and affine relationships between x and y;
Thinking of a linear relationship as a map or function;
Graphing a linear function.
Inverting a simple linear function.
Seeking a common solution to two linear functions.
The sensitivity of a solution.
Representing the problem in matrix/vector form.
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Linear relationships for x, y and z.
An (x,y,z) linear relationship is a plane.
Graphing an (x,y,z) linear relationship.
Will two (x,y,z) relationships have common solutions?
Will three (x,y,z) relationships have a common solution?
Representing the problem in matrix/vector form.
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Solving a system of 2 or 3 linear equations.
When the solving process breaks down.
When the solving process is easy (a triangular matrix).
How to automate the solving process.
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The matrix/vector form is A*x=b.
The matrix/vector form is a special kind of multiplication.
The rules of matrix-vector multiplication.
Special matrices: the identity, the zero matrix, the transpose.
A vector is a kind of matrix.
We can have a row vector or a column vector.
There is an inverse matrix that can solve our linear system.
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A matrix has an LU factorization.
The LU factorization can be discovered during the solving process.
We can compute the solution very easily if we already have the LU factors.
We can recover the matrix from the LU factors.
We can compute the inverse matrix from the LU factors.
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Mapping the Hand;
rotation, translation, dilation, reflection
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Linear maps, Affine maps, bilinear maps, inverse maps,
orthogonal maps, a linear space, a basis,
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Vectors, vector Norms
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Kinds of matrices: diagonal, symmetric, triangular, orthogonal
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Matrices, Matrix-vector multiplication, Matrix norms
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Matrix powers, iterative methods
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Discrete differentiation; discrete differentiation matrices;
the norm of differentiation matrices; heating up a metal beam.
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"Easy" systems of linear equations; Gauss elimination; LU factorization
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"Hard" systems of linear equations; over and underdetermined systems;
Badly conditioned systems
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Sparse systems of linear equations; direct and iterative methods
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The QR factorization
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The SVD factorization
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The eigenvalue problem
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Mathematical graphs; directed graphs; connectedness; counting paths;
escaping from a maze.
Last revised on 14 March 2015.