1550078481-Ordinary_Differential_Equations__Roberts_

CSODE User's Guide 507

• EIGEN r;)@~
  EISEN finds aD eigenvalues: and eigenvedots of a real. square inatrix A of order N whete 2 (: N < .. 6.
  EnterlhetizeoftheNxNmatriJ: A. N• [3 {Me1~haveentecedthe11ak.ieonN,preutheEnterkey.J
  {Enter the non-zero entries of the matrix A below.)
  Column l 2 3
  R0w

I er:

ice

3 0::

l.0000!+00 O.OOOOEtOO l.OOOOEtOO

-2.0000UOO 0.00001+00 1.00001+00 0.0000!+00 -1.00001+00 -1.00001+00

AH THI EIG!NVALUI ASSOCIATED IS IICINVECTOR 1.000001+00 IS + O.OOOOOltOO I

o.ooooouoo l.000001+00 0. 000001+00 O.OOOOOE+OO

0.000001+00 + 0.000001!+00 I

AH THI ilICINVALUI ASSOCIATED IS BICINVICTOR 0.000001+00 IS t LOOOOOltOO I

• S.000001S.000001-0l -0l +-S.000001-01 + S.000001-01 I I
  O.OOOOOBtOO t 1.00000ltOO I
  AN llIGl!WALUll IS 0.000001!+00 t -l.OOOOOUOO I
  THI ASSOCIATED iIGINVllC!Oll, S.OOOOOE-01 rs + S.OOOOOJ- 01 I
  -S.OOOOOE-01 +-S.OOOOOE- 01 I

CAl.C\JIATETHE

EIGENV.o!.UES ANO

EIGENVECTORS OFANE\11

MATRI<

Figure A .8 Screen Showing a 3 x 3 Matrix and Its Eigenvalues

and Associated Eigenvectors

Using GRAPH, DIRFIELD, SOLVEIVP, and SOLVESYS

In calculus and differential equations we study many functions and their
properties. Recall that a function is a set of ordered pairs such that to each
element in the domain there corresponds a unique element in the range. Often,
instead of defining a function by listing the ordered pairs of the function,
the function is defined by an equation from which the ordered pairs can be
generated. We will be interested in functions which can be expressed by an

equation of the form y = f (x) where x and y are real numbers. That is, we will

consider only real valued functions of a real variable. The variable, x, is called

the independent variable and the variable, y, is called the dependent

variable. In order to graph a function given by y = f(x) using the computer,

we first select a finite set of points Xi, i = 1, 2, ... , n ordered such that

Xi < Xi+i, then we compute f(xi), and finally, provided both Xi and Xi+l

are in the domain of the function f, we connect (xi, f(xi)) to (xi+1, f(xi+1))

by a straight line segment. If Xi and Xi+l are "sufficiently close" and if f is
defined on the interval [xi, xi+ 1 ], then the line segment joining (xi, f(xi)) and
(xi+ 1 , f(xi+ 1 )) will be "close" to (x, f(x)) for all x in the interval (xi, xi+1)
and the graph will closely approximate the graph of f(x).