View of /trunk/NationalProblemLibrary/Rochester/setLinearAlgebra11Eigenvalues/ur_la_11_16.pg

    1 ## DESCRIPTION
    2 ## Linear Algebra
    3 ## ENDDESCRIPTION
    4 
    5 ## KEYWORDS ('linear algebra','matrix','eigenvalue')
    6 ## Tagged by cmd6a 4/30/06
    7 
    8 ## DBsubject('Linear Algebra')
    9 ## DBchapter('Matrices')
   10 ## DBsection('Eigenvalues')
   11 ## Date('')
   12 ## Author('')
   13 ## Institution('Rochester')
   14 ## TitleText1('')
   15 ## EditionText1('')
   16 ## AuthorText1('')
   17 ## Section1('')
   18 ## Problem1('')
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGchoicemacros.pl",
   26 "PGanswermacros.pl",
   27 "PGgraphmacros.pl",
   28 "PGmatrixmacros.pl",
   29 "PGnumericalmacros.pl",
   30 "PGauxiliaryFunctions.pl",
   31 "PGmorematrixmacros.pl"
   32 );
   33 
   34 TEXT(beginproblem());
   35 $showPartialCorrectAnswers = 1;
   36 
   37 $a = non_zero_random(-2,2,1);
   38 $e = random(1,2,1) + 3*random(-1,1,1) - $a;
   39 $b = random(-1,1,2);
   40 $sqrD = random(1,2,1) + 3*random(0,2,1);
   41 $D = $sqrD**2;
   42 $d = ($D - ($a+$e)**2)*$b/3 + $a*$e*$b;
   43 
   44 # characteristic polynomial is - (lambda^3 - (a+e)lambda^2 + (ae-bd)lambda - b^2k)
   45 # now forget about the minus in front
   46 # we want 3 distinct real roots, so we want local max and min with pos and neg values respectively
   47 # find max and min:
   48 # derivative is 3lamda^2 - 2(a+e)lambda + (ae-bd)
   49 # need 2 disctinct real roots
   50 # discriminant/4 is (a+e)^2 - 3ae + 3bd = (a+e)^2 - 3ae + D - (a+e)^2 + 3ae = D
   51 # ok, the discriminant is positive
   52 # roots are
   53 
   54 $max_root = ($a + $e + $sqrD)/3;
   55 $min_root = ($a + $e - $sqrD)/3;
   56 
   57 # local max and min values of the cubic polynomial (without the minus in front) have to be pos and neg, so
   58 
   59 $ans_min = $max_root**3 - ($a+$e)*$max_root**2 + ($a*$e - $b*$d)*$max_root;
   60 $ans_max = $min_root**3 - ($a+$e)*$min_root**2 + ($a*$e - $b*$d)*$min_root;
   61 
   62 BEGIN_TEXT
   63 
   64 \{ mbox( 'The matrix \(A=\)', display_matrix([[$a, $b, 0], [$d, $e, $b], ['k', 0, 0]]) )\}
   65 $BR
   66 has three distinct real eigenvalues if and only if
   67 $BR
   68 \{ans_rule(20)\} \( < k < \) \{ans_rule(20)\}.
   69 
   70 END_TEXT
   71 
   72 ANS(num_cmp($ans_min));
   73 ANS(num_cmp($ans_max));
   74 
   75 ENDDOCUMENT();       # This should be the last executable line in the problem.
   76