sub zero_check{ my $array = shift; my %options = @_; my$num = @$array; my$i; my $max = 0; my$mm; for ($i=0;$i< $num;$i++) { $mm =$array->[$i] ;$max = abs($mm) if abs($mm) > $max; } my$tol = $options{tol};$tol = 0.01*$options{reltol}*$average if defined($options{reltol});$tol = .000001 unless defined($tol); ($max <$tol) ? 1: 0; # 1 if the array is close to constant; } sub vec_dot{ my$vec1 = shift; my $vec2 = shift; warn "vectors must have the same length" unless @$vec1 == @$vec2; # the vectors must have the same length. my @vec1=@$vec1; my @vec2=@$vec2; my$sum = 0; while(@vec1) { $sum += shift(@vec1)*shift(@vec2); }$sum; } sub proj_vec { my $vec = shift; warn "First input must be a column matrix" unless ref($vec) eq 'Matrix' and ${$vec->dim()}[1] == 1; my $matrix = shift; # the matrix represents a set of vectors spanning the linear space # onto which we want to project the vector. warn "Second input must be a matrix" unless ref($matrix) eq 'Matrix' and ${$matrix->dim()}[1] == ${$vec->dim()}[0]; $matrix * transpose($matrix) * $vec; } sub vec_cmp{ #check to see that the submitted vector is a non-zero multiple of the correct vector my$correct_vector = shift; my %options = @_; $ans_eval = sub { my$in = shift @_; $ans_hash = new AnswerHash; my @in = split("\0",$in); my @correct_vector=@$correct_vector;$ans_hash->{student_ans} = "( " . join(", ", @in ) . " )"; $ans_hash->{correct_ans} = "( " . join(", ", @correct_vector ) . " )"; return($ans_hash) unless @$correct_vector == @in; # make sure the vectors are the same dimension my$correct_length = vec_dot($correct_vector,$correct_vector); my $in_length = vec_dot(\@in,\@in); return($ans_hash) if $in_length == 0; if (defined($correct_length) and $correct_length != 0) { my$constant = vec_dot($correct_vector,\@in)/$correct_length; my @difference = (); for(my $i=0;$i < @correct_vector; $i++ ) {$difference[$i]=$constant*$correct_vector[$i] - $in[$i]; } $ans_hash->{score} = zero_check(\@difference); } else {$ans_hash->{score} = 1 if vec_dot(\@in,\@in) == 0; } $ans_hash; };$ans_eval; } ############ =head4 display_matrix Useage $\{ display_matrix(A) \}$ $\{ display_matrix([ [ 1, 3], [4, 6] ]) \}$ Output is text which represents the matrix in TeX format used in math display mode. =cut sub display_matrix{ # will display a matrix in tex format. # the matrix can be either of type array or type 'Matrix' my $ra_matrix = shift; my$out=''; if (ref($ra_matrix) eq 'Matrix' ) { my ($rows, $cols) =$ra_matrix->dim(); $out = q!\\left(\\begin{array}{! . 'c'x$cols . q!}!; for( $i=1;$i<=$rows;$i++) { for ($j=1;$j<=$cols;$j++) { my $entry =$ra_matrix->element($i,$j); $entry = "#" unless defined($entry); $out.=$entry; $out .= ($j < $cols) ? ' & ' : "\\cr\n"; } }$out .= "\\end{array}\\right)"; } elsif( ref($ra_matrix) eq 'ARRAY') { my$rows = @$ra_matrix; my$cols = @{$ra_matrix->[0]};$out = q!\\left(\\begin{array}{! . 'c' x$cols . q!}!; for($i=0; $i<$rows; $i++) { my @row = @{$ra_matrix->[$i]}; while (@row) { my$entry = shift(@row); $entry = "#" unless defined($entry); $out.=$entry; if (@row) { $out .= "& "; } else { next; } }$out .= "\\cr\n"; } $out .= "\\end{array}\\right)"; } else { warn "The input" . ref($ra_matrix) . " doesn't make sense as input to display_matrix. "; } $out; } =head4 ra_flatten_matrix Useage: ra_flatten_matrix($A) where $A is a matrix object The output is a reference to an array. The matrix is placed in the array by iterating over columns on the inside loop, then over the rows. (e.g right to left and then down, as one reads text) =cut sub ra_flatten_matrix{ my$matrix = shift; warn "The argument must be a matrix object" unless ref($matrix) =~ /Matrix/; my @array = (); my ($rows, $cols ) =$matrix->dim(); foreach my $i (1..$rows) { foreach my $j (1..$cols) { push(@array, $matrix->element($i,$j) ); } } \@array; } sub apl_matrix_mult{$ra_a= shift; $ra_b= shift; %options = @_; my$rf_op_times= sub {$_[0] *$_[1]}; my $rf_op_plus = sub {my$sum = 0; my @in = @_; while(@in){ $sum =$sum + shift(@in) } $sum; };$rf_op_times = $options{'times'} if defined($options{'times'}) and ref($options{'times'}) eq 'CODE';$rf_op_plus = $options{'plus'} if defined($options{'plus'}) and ref($options{'plus'}) eq 'CODE'; my$rows = @$ra_a; my$cols = @{$ra_b->[0]}; my$k_size = @$ra_b; my$out ; my ($i,$j, $k); for($i=0;$i<$rows;$i++) { for($j=0;$j<$cols;$j++) { my @r = (); for($k=0;$k<$k_size;$k++) {$r[$k] = &$rf_op_times($ra_a->[$i]->[$k] ,$ra_b->[$k]->[$j]); } $out->[$i]->[$j] = &$rf_op_plus( @r ); } } $out; } sub matrix_mult { apl_matrix_mult($_[0], $_[1]); } sub make_matrix{ my$function = shift; my $rows = shift; my$cols = shift; my ($i,$j, $k); for($i=0;$i<$rows;$i++) { for($j=0;$j<$cols;$j++) {$ra_out->[$i]->[$j] = &$function($i,$j); } }$ra_out; } # sub format_answer{ # my $ra_eigenvalues = shift; # my$ra_eigenvectors = shift; # my $functionName = shift; # my @eigenvalues=@$ra_eigenvalues; # my $size= @eigenvalues; # my$ra_eigen = make_matrix( sub {my ($i,$j) = @_; ($i==$j) ? "e^{$eigenvalues[$j] t}": 0 }, $size,$size); # my $out = qq! #$functionName(t) =! . # displayMatrix(apl_matrix_mult($ra_eigenvectors,$ra_eigen, # 'times'=>sub{($_[0] and$_[1]) ? "$_[0]$_[1]" : ''}, # 'plus'=>sub{ my $out = join("",@_); ($out) ?$out : '0' } # ) ) ; #$out; # } # sub format_vector_answer{ # my $ra_eigenvalues = shift; # my$ra_eigenvectors = shift; # my $functionName = shift; # my @eigenvalues=@$ra_eigenvalues; # my $size= @eigenvalues; # my$ra_eigen = make_matrix( sub {my ($i,$j) = @_; ($i==$j) ? "e^{$eigenvalues[$j] t}": 0 }, $size,$size); # my $out = qq! #$functionName(t) =! . # displayMatrix($ra_eigenvectors)."e^{$eigenvalues[0] t}" ; # $out; # } # sub format_question{ # my$ra_matrix = shift; # my $out = qq! y'(t) = ! . displayMatrix($B). q! y(t)! # # } 1;