1550078481-Ordinary_Differential_Equations__Roberts_

504 Ordinary Differential Equations

On the right-hand side of Figure A.5, the coefficients of the polynomial

and the roots of the polynomial appear in single precision, exponential form.

Since -7.414635E-17 is nearly 0 and small compared to 1, one root of the

polynomial is i. The second root is clearly 3 + 2i. And since -4.440892E-16

is nearly 0 and small compared to 4, the third root of the polynomial is 4.

Using EIGEN

The computer program EIGEN computes all eigenvalues and eigenvectors
of an n x n matrix with real entries where 2 :::; n :::; 6. The input for this
program is t he size, n, of the matrix and the n^2 entries of the matrix. The
output of the program is a set of n eigenvalues and associated eigenvectors.

When an eigenvalue has multiplicity m > 1, EIGEN generates m associated

vectors. When the eigenvalue h as m linearly independent eigenvectors associ-
ated with it, the vectors produced by the computer are linearly independent,

as they should be. When an eigenvalue of multiplicity m > 1 does not h ave m

associated linearly independent eigenvectors, the m vectors produced by the
computer will not be linearly independent. In general, if there are k < m lin-
early independent eigenvectors associated with a particular eigenvalue, then
k of the m vectors produced by the computer will be linearly independent
eigenvectors.

Let us use EIGEN to calculate the eigenvalues and eigenvectors of the
matrix

(1)

(

1.5 1)

A=

-2.5 .5

We double click the CSODE icon and when the selection screen (Figure A.l)
appears on the monitor, we single click on the EIGEN button. When the
screen shown in Figure A.6 appears, we enter 2 in the highlighted box after
"N = " and press the Enter key, because the size of the matrix A is 2.

ii EIGEN 1!!11!113

EIGEN finds all eigenvalues and eigenvectors of a real. square matrix A of order N where 2 <= N <= 6.

Enter the size of the N x N matriK A. N = f2" [After you have entered the vakle on N, press the Enter key.J

(Enter the non-zero entries of the matrix A below.)

Column 1 2

Row

1 11.5 r--;-

2 l-25E1 j5:D-01

VERIFY MA TRIX

ENTRIES AND

CALCUlATE

Figure A.6 Inputing the Size of the Matrix A and Its Entries