Linear Algebraic Equations

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This tutorial is made solely for the purpose of education and it is designed for students taking Applied Math 0340. It is primarily for students who have some experience using Mathematica. If you have never used Mathematica before and would like to learn more of the basics for this computer algebra system, it is strongly recommended looking at the APMA 0330 tutorial. As a friendly reminder, don't forget to clear variables in use and/or the kernel.

Finally, the commands in this tutorial are all written in bold black font, while Mathematica output is in normal font. This means that you can copy and paste all commands into Mathematica, change the parameters and run them. You, as the user, are free to use the scripts for your needs to learn the Mathematica program, and have the right to distribute this tutorial and refer to this tutorial as long as this tutorial is accredited appropriately.

This chapter is devoted to methods for solving linear algebraic equations. While they originated in the early nineteen century (by the work of Gauss), the field has seen an explosion of activity spurred by demand due to extraordinary technological advances in engineering and sciences. The reviews of the methods could be found in

http://www-users.cs.umn.edu/~saad/PDF/umsi-99-152.pdf

Vector Space

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We recall the definition of vector space. A vector space is a set \( V \) together with an inner operation called addition (a rule for adding two elements of \( V \) to obtain a third element of \( V \) ) and another outer operation called scalar multiplication (a rule for multiplying a real or complex number times an element of \( V \) to obtain a second element of \( V \) ) on which the following properties hold:

1. \( {\bf x} + {\bf y} = {\bf y} + {\bf x} \)
2. \( {\bf x} + ({\bf y} + {\bf z}) = ({\bf x} + {\bf y}) + {\bf z} \)
3. There is an element 0 of V so that \( {\bf 0} + {\bf y} = {\bf y} + {\bf 0} = {\bf y}\) for every \( {\bf y} \in V \)
4. There is an element \( -{\bf x} \) in V such that \( {\bf x} - {\bf x} = -{\bf x} + {\bf x} = {\bf 0}\)
5. \( a({\bf x} + {\bf y}) = a{\bf y} + a{\bf x} \)
6. \( (a+b)\,{\bf x} = a\,{\bf x} + b\,{\bf x} \)
7. \( (ab){\bf x} = a(b {\bf x} )\)
8. \( 1\,{\bf x} = {\bf x} \)

The elements of a vector space V are called vectors. It is a custom to denote vectors by lower case letter in bold font, while scalars/numbers by regular letters. Later on, when a system of coordinates (or basis) is introduced, vectors will be identified by ordered n-tuples written in either column form or row form. For the former, we will use the same notation as for vectors in general, but the later will be identified by an arrow above the vector ( \( \vec{x} \) ).

 

 

 

2.4.2. Euler's Algorithms

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Euler method for systems of differential equations

Using for simplicity the initial point to be the origin. we present another way to implement the Euler rule:

We can make several different types of plots. Here we plot on the phase plane using dots and then using line segments

Here we create the associated streamplot

 

We plot x-component versus t

We plot y-component versus t

Finally, we mash them all together

 

 

II. Basic Properties of Matrices

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2.4.3. LU Factorization

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The matrix A is a global variable and elements are changed when the LUfactor subroutine is executed. Hence, it is important to save a copy of A in the variable A0. Use one of the following two versions of the LUfactor subroutine and the SolveLU subroutine. The first version of LUfactor uses parallel programming and "row operations" instead of loops. This is the second version of LUfactor and it uses more loops and traditional programming. Use the subroutine SolveLU which is similar to the forward substitution and back substitution subroutines.

 

 

2.4.4. Runge-Kutta Methods

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2.4.5. Multistep Methods