##DESCRIPTION
##  3x3 Eigenvalue/Eigenvector computation with 2-d eigenspace 
##ENDDESCRIPTION

## DBsubject('Linear Algebra')
## DBchapter('Eigenvalues and eigenvectors')
## DBsection('Computing eigenvalues and eigenvectors')
## Date('5/2/2018')
## Author('K. Andrew Parker')
## Institution('New York City College of Technology')

########################################################################

DOCUMENT();      

loadMacros(
   "PGstandard.pl",     # Standard macros for PG language
   "MathObjects.pl",
   "parserMultiAnswer.pl"
);

# Print problem number and point value (weight) for the problem
TEXT(beginproblem());

# Show which answers are correct and which ones are incorrect
$showPartialCorrectAnswers = 1;

##############################################################
#
#  Setup
#
#
Context("Vector");

# orthogonal basis for 2-d vector space associated to the repeated eigenvalue
do {
    $v = Vector(random(-2,2,1),random(-2,2,1),random(-2,2,1));
    $w = Vector(random(-2,2,1),random(-2,2,1),random(-2,2,1));
} until ( $v . $u == 0 );

# expand to a basis for R^3
# prevent the determinant from ballooning
do {
    $u = Vector(random(-2,2,1),random(-2,2,1),random(-2,2,1));
    $S = Matrix($u,$v,$w)->transpose;
    $detS = $S->det;
} until ($detS != 0 && abs($detS) <= 6);

# generate eigenvalues that are equivalent modulo determinant of S
# this will ensure an integer-valued matrix for the problem
# cap the repeated eigenvalue to ensure reasonable characteristic polynomial
$lambda2 = non_zero_random(-3,3,1);
do { $lambda1 = non_zero_random(-2*$detS+$lambda2,6,$detS); } until ($lambda1 != $lambda2 && abs($lambda1) <= 6);
$lambda3 = $lambda2;

# compute the matrix to be solved
$SInv = $S->inverse;
$M = $S * Matrix([$lambda1,0,0],[0,$lambda2,0],[0,0,$lambda3]) * $SInv;

# this should probably be re-written as a multiAns question
# since eigenvalues and eigenvectors ought to be paired
# and because the eigenspace is 2-d, a custom answer checker needs to be used
# check that M has the desired effect, and the two solutions provided
# by the student are orthogonal or normal depending on how strict you want it
$values = List($lambda1, $lambda2, $lambda3);
$vectors = List($u, $v, $w);

$multians = MultiAnswer($lambda1, $u, $lambda2, $v, $lambda3, $w)->with(
    singleResult => 1,
    checker => sub {
        my ( $correct, $student, $ans ) = @_;
        my ( $sVal1, $sVec1, $sVal2, $sVec2, $sVal3, $sVec3 ) = @{$student};
        my ( $cVal1, $cVec1, $cVal2, $cVec2, $cVal3, $cVec3 ) = @{$correct};
        return 0 if $ans->{isPreview};
        # First check for the correct list of eigenvalues
        Value::Error("At least one of your eigenvalues is incorrect") if ( List($sVal1, $sVal2, $sVal3) != List($cVal1, $cVal2, $cVal3) );
        # Then confirm eigenvector times M is the appropriate multiple.
        Value::Error("Your first eigenvector is incorrect.") if ( Vector($M*$sVec1) != Vector($sVal1*$sVec1) );
        Value::Error("Your second eigenvector is incorrect.") if ( Vector($M*$sVec2) != Vector($sVal2*$sVec2) );
        Value::Error("Your third eigenvector is incorrect.") if ( Vector($M*$sVec3) != Vector($sVal3*$sVec3) );
        # optionally, confirm that 2-d eigenspace has orthogonal basis
        # otherwise, use `!($sVec1->isParallel($sVec2))`
        Value::Error("You don't have an orthogonal basis for \(\lambda = $sVal1\)") if ($sVal1 == $sVal2 && $sVec1 . $sVec2 != 0);
        Value::Error("You don't have an orthogonal basis for \(\lambda = $sVal1\)") if ($sVal1 == $sVal3 && $sVec1 . $sVec3 != 0);
        Value::Error("You don't have an orthogonal basis for \(\lambda = $sVal2\)") if ($sVal2 == $sVal3 && $sVec2 . $sVec3 != 0);
        # optionally, confirm that all eigenvectors are unit vectors
        Value::Error("Your first eigenvector is not a unit vector.") unless ( norm($sVec1) == 1 );
        Value::Error("Your second eigenvector is not a unit vector.") unless ( norm($sVec2) == 1 );
        Value::Error("Your third eigenvector is not a unit vector.") unless ( norm($sVec3) == 1 );
        return 1;
    }
);

##############################################################
#
#  Text
#
#

Context()->texStrings;
BEGIN_TEXT

Find the eigenvalues and eigenvectors for \( $M \):
$BR
$PAR
First eigenvalue: \{$multians->ans_rule\} and its eigenvector: \{$multians->ans_rule\}
$BR
Second eigenvalue: \{$multians->ans_rule\} and its eigenvector: \{$multians->ans_rule\}
$BR
Third eigenvalue: \{$multians->ans_rule\} and its eigenvector: \{$multians->ans_rule\}
$BR



END_TEXT
Context()->normalStrings;

##############################################################
#
#  Answers
#
#

ANS($multians->cmp);
#ANS($values->cmp);
#ANS($vectors->cmp);

ENDDOCUMENT();