[npl] / branches / UGA / 5.5.3.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

# View of /branches/UGA/5.5.3.pg

Sat Jul 24 17:09:50 2010 UTC (2 years, 10 months ago) by ted shifrin
File size: 2009 byte(s)
Log message

    1 ## DESCRIPTION
2 ## Linear Algebra
3 ## ENDDESCRIPTION
4
5 ## KEYWORDS ('linear algebra','linear system','least squares','inner product')
6 ## Tagged by cmd6a 5/3/06
7
8 ## DBsubject('Linear Algebra')
9 ## DBchapter('Systems of Linear Equations and Matrices')
10 ## DBsection('Least Squares')
11 ## Date('')
12 ## Author('modified by Shifrin')
13 ## Institution('UGA')
14 ## TitleText1('')
15 ## EditionText1('')
16 ## AuthorText1('')
17 ## Section1('')
18 ## Problem1('')
19
20 DOCUMENT();        # This should be the first executable line in the problem.
21
23 "PG.pl",
24 "PGbasicmacros.pl",
25 "PGchoicemacros.pl",
27 "PGgraphmacros.pl",
28 "PGmatrixmacros.pl",
29 "PGnumericalmacros.pl",
30 "PGauxiliaryFunctions.pl"
31 );
32
33 TEXT(beginproblem());
34 $showPartialCorrectAnswers = 1; 35 36 # we choose the matrix A, solution x, and a vector c orthogonal to the columns of A. 37 # we set b=Ax+c, then x is the least-squares solution of Ax=b. 38 39$a11 = random(1,2,1);
40 $a12 = random(-2,-1,1); 41$a21 = - $a11; 42$a22 = - $a12; 43$a31 = random(3,5,1);
44 $a32 = random(3,5,1); 45 46$x1 = non_zero_random(-5,5,1);
47 $x2 = random(-5,5,1); 48 49$c1 = non_zero_random(-6,6,1);
50 $c2 =$c1;
51 $c3 = 0; 52 53$b1 = $a11*$x1 + $a12*$x2 + $c1; 54$b2 = $a21*$x1 + $a22*$x2 + $c2; 55$b3 = $a31*$x1 + $a32*$x2 + $c3; 56 57 BEGIN_TEXT 58 59 Find the least-squares solution $$\overline{\mathbf x}$$ of the system 60$PAR
61 $$\{display_matrix_mm([[a11, a12], [a21, a22], [a31, a32]])\}\mathbf x= 62 \{display_matrix_mm([[b1], [b2], [b3]])\}$$ .
63 $BR 64 \{ mbox( '$$\overline{\mathbf x} =$$', answer_matrix(2,1,5), '.' ) \} 65 66$PAR
67 Use your answer to find the point on the plane spanned by
68 $$\{display_matrix_mm([[a11],[a21],[a31]])\}$$  and $$\{display_matrix_mm([[a12],[a22],[a32]])\}$$  that is closest to $$\{display_matrix_mm([[b1], [b2], [b3]])\}$$ .
69 $BR 70 \{ answer_matrix(3,1,8) \} 71$BR
72
73
74 END_TEXT
75
76 ANS(num_cmp($x1)); 77 ANS(num_cmp($x2));
78 ANS(num_cmp($a11*$x1 + $a12*$x2),num_cmp($a21*$x1 + $a22*$x2),num_cmp(\$b3));
79
80 ENDDOCUMENT();       # This should be the last executable line in the problem.
81


 aubreyja at gmail dot com ViewVC Help Powered by ViewVC 1.0.9