Algorithm::GoldenSection - Golden Section Search Algorithm for one-dimensional minimisation.

    use Algorithm::GoldenSection;

    # Create a Algorithm::GoldenSection object and pass it a CODE reference to the function to be minimised and initials values for x_low and x_int.
    $gs = Algorithm::GoldenSection->new( { function => sub { my $x = shift; my $b =  $x * sin($x) - 2 * cos($x); return $b },
                                        x_low    => 4,
                                        x_int    => 4.7,} ) ;

    # Call minimisation method to bracket and minimise.
    my ($x_min, $f_min, $iterations) = $gs->minimise;

    print qq{\nMinimisation results: x a minimum = $x_min, function value at minimum = $f_min. Calculation took $iterations iterations};

package Algorithm::GoldenSection;

use warnings;
use strict;
use Carp;
use Readonly;

use version; our $VERSION = qv('0.0.2');

# package-scoped lexicals
Readonly::Scalar my $ouro => 1.618034 ;
Readonly::Scalar my $glimite => 100.0 ;
Readonly::Scalar my $pequeninho => 1.0e-20 ;
Readonly::Scalar my $tolerancia => 3.0e-8;  # tolerance
Readonly::Scalar my $C => (3-sqrt(5))/2;
Readonly::Scalar my $R => 1-$C;

#/ I had leaving things for operator precedence. you won´t see A+B*(C-D) whe you mean: A+( B*(C-D) ) - i.e. * binds more tightly that +

sub new {
    my ( $class, $h_ref ) = @_;
    croak qq{\nArguments must be passed as HASH reference.} if ( ( $h_ref ) && ( ref $h_ref ne q{HASH} ) );
    my $self = {};
    bless $self, $class;
    $self->_check_options($h_ref);
    return $self;
}

sub _check_options {

    my ( $self, $h_ref ) = @_;

    croak qq{\nOption \x27function\x27 is obrigatory and accepts a CODE reference} 
      if ( ( !exists $h_ref->{function} ) || ( ref $h_ref->{function} ne q{CODE} ) );
    croak qq{\nOption \x27x_low\x27 requirements a numeric value} 
      if ( ( !exists $h_ref->{x_low} ) || ( $h_ref->{x_low} !~ /\A[+-]?\ *(\d+(\.\d*)?|\.\d+)([eE][+-]?\d+)?\z/xms ) );
    croak qq{\nOption \x27x_low\x27 requirements a numeric value} 
      if ( ( !exists $h_ref->{x_int} ) || ( $h_ref->{x_int} !~ /\A[+-]?\ *(\d+(\.\d*)?|\.\d+)([eE][+-]?\d+)?\z/xms ) );

    $self->{function} = $h_ref->{function};
    $self->{x_low} = $h_ref->{x_low};
    $self->{x_int} = $h_ref->{x_int};

}

sub _switch {
    # twat did you usual of forgetting @_ and then you didn´t even return from it!
    my ( $a, $b, $f_a, $f_b) = @_; 
    my $buf = $a;
    my $f_buf = $f_a;
    $a = $b;
    $f_a = $f_b;
    $b = $buf;
    $f_b = $f_buf;
    return ($a, $b, $f_a, $f_b); 
}

sub minimise {

    my $self = shift;
    
    #y bracket interval
    $self->_bracket;

    my $func = $self->{function};
    my $a = $self->{x_low};
    my $b = $self->{x_int};
    my $c = $self->{x_high};

    my $x1;
    my $x2;
    # this is not efficient code...
    my $x0 = $self->{x_low};
    my $x3 = $self->{x_high};

    if ( abs($c-$b) > abs($b-$a) ) {
        $x1 = $b;
        #y create new point to try
        $x2 = $b + ( $C * ($c-$b) );
    }
    else {
        $x2 = $b;
        #y create new point to try
        $x1 = $b - ( $C * ($b-$a) );
    }
    
    #y initial function evaluations
    my $f1 = $func->($x1);
    my $f2 = $func->($x2);

    my $counter = 0;

    #y start iterating...
    while ( abs($x3-$x0) > ( $tolerancia * ( abs($x1) + abs($x2) ) ) ) {

        #y lets increment here just to make it easier - hence start with 0
        $counter++;

        #y a possible outcome
        if ( $f2 < $f1 ) {

            #########################################
            #y choose one of the two - but why the fuck-up with $R multiplication?!?
            #########################################
            #y the following is identical to:
            $x0 = $x1;
            $x1 = $x2;
            $x2 = ($R*$x2) + ($C*$x3); #
            $f1 = $f2;
            $f2 = $func->($x2);
            #########################################
#            my $x_temp = ($R*$x2) + ($C*$x3);
#            &_shft3(\$x0,\$x1,\$x2,\$x_temp);
#            my $f_x_temp = $func->($x2);
#            &_shft2(\$f1,\$f2,\$f_x_temp);
            #########################################
        }
        #y other possibility
        else {

            #########################################
            $x3 = $x2;
            $x2 = $x1;
            $x1 = ($R*$x1) + ($C*$x0);
            $f2 = $f1;
            $f1 = $func->($x1);
            #########################################
#            my $x_temp = ($R*$x1) + ($C*$x0);
#            &_shft3(\$x3,\$x2,\$x1,\$x_temp);
#            my $f_x_temp = $func->($x1);
#            &_shft2(\$f2,\$f1,\$f_x_temp);
            #########################################
        }
    }

    my $xmin;
    my $fmin;
    
    #y set final values
    if ($f1 < $f2) { 
        $xmin = $x1;
        $fmin = $f1;
    }
    else {
        $xmin = $x2;
        $fmin = $f2;
    }

    return $xmin, $fmin, $counter;
}

sub _bracket {
    
    my $self = shift;
    
    my $function = $self->{function};
    $a = $self->{x_low};
    $b = $self->{x_int};
    
    my $f_u;
    my $f_a = $function->($a);
    my $f_b = $function->($b);

    #y that is downhill
    if ($f_b > $f_a ) { 
        #print qq{\n\n**** in this case fb is higher than fa - thus we are going uphill so we need to swap them****\n};
        print qq{\n\nswitch $a, $b, $f_a and $f_b};
        ( $a, $b, $f_a, $f_b) = _switch( $a, $b, $f_a, $f_b); 
        print qq{\n\nswitch $a, $b, $f_a and $f_b};
    }

    # has higher precedence that + thus: $c = $b+$ouro*($b-$a);  is the same as $c = $b+($gold*($b-$a)); - same in C/C++

    #y WE MAKE A GUESS AT A VALUE OF C
    my $c = $b + ( $ouro * ($b-$a) ); # c 26.18034 and f_c 21.6787847478271

    my $f_c = $function->($c);

    # (1) by SWAPPING we are sure that f(a) > f(b)! - (2) BUT we must also have f(b) < f(c) in order to have _bracketed our MINIMUM

    while ( $f_b > $f_c ) {
    
        #y compute u by parabolic extrapolation - tiny is there just to stop ilegal divisions by 0
        my $r = ($b-$a) * ($f_b-$f_c);
        my $q = ($b-$c) * ($f_b-$f_a);
        my $u = $b - ( ( $b - $c ) * $q - ( $b - $a ) * $r )  / ( 2.0 * &_sign ( &_max ( abs ($q-$r), $pequeninho ), $q-$r ) );
        my $ulim = $b + ( $glimite * ($c-$b) );

        #y test the possibilities!

        if ( ($b-$u)*($u-$c) > 0.0 ) {      #y parabolic u is between b and c 
            $f_u = $function->($u);
            
            #y have a minimium between b and c - i.e. is f(u) < f(c) - if so:
            if ( $f_u  < $f_c ) { 

                $a = $b;
                $b = $u;
                $f_a = $f_b;
                $f_b = $f_u;

                #/ we´re going to return early here so as we aren´t using any package-scoped vars we will need to feed the object here
                $self->{x_low} = $a;
                $self->{x_int} = $b;
                $self->{x_high} = $c;
                
                return
            }
            elsif ( $f_u > $f_b ) {
                $c = $u;
                $f_c = $f_u;
                
                #/ we´re going to return early here so as we aren´t using any package-scoped vars we will need to feed the object here
                $self->{x_low} = $a;
                $self->{x_int} = $b;
                $self->{x_high} = $c;
                
                return
            }

            #y parabolic fit was useless in this case - so we use a default magnification
            $u = $c + ( $ouro * ($c-$b) );
            $f_u = $function->($u);
        }

        #y parabolic fit is between c and is not allowed
        elsif  ( ($c-$u)*($u-$ulim) > 0 ) {

            $f_u = $function->($u);

            if ( $f_u < $f_c ) {

                my $u_other = $u + ( $ouro * ($u-$c) );
                #/ this should make b = c, c = u  and u = u_other
                &_shft3(\$b,\$c,\$u,$u_other); 
                #/ so as u is now u_other this shouldn´t be a prob
                my $f_u_other = $function->($u_other);
                &_shft3(\$f_b,\$f_c,\$f_u, \$f_u_other); 
            }
        }

        #y limit parabolic u to max allowed
        elsif ( ($u-$ulim)*($ulim-$c) >= 0.0 ) {
            $u = $ulim;
            $f_u = $function->($u);
        }

        #y reject parabolic u
        else { 
            $u = $c + ( $ouro * ($c-$b) );
            $f_u = $function->($u);
        }

        #y eliminate oldest points and will continue};

        &_shft3(\$a,\$b,\$c,\$u); 
        &_shft3(\$f_a,\$f_b,\$f_c,\$f_u); 
       
    }

    croak qq{\nThere is a problem - email dsth\@cantab.net.} if ( !$a || !$b || !$c );#|| ( $b > $a ) || ( $b > $c ) ); 
    $self->{x_low} = $a;
    $self->{x_int} = $b;
    $self->{x_high} = $c;
}

sub _sign {
    my ($a, $b) = @_;
    my $val = abs $a;
    my $sig = $b >= 0 ? q{+} : q{-};
    my $final = $sig.$val;
    # force numeric context - no real reason
    return 0+$final;
}

sub _max {
    my ($a, $b) = @_;
    my $ret = $a >= $b ? $a : $b;
    return $ret;
}

sub _shft3 {
    my ($a, $b, $c, $d) = @_;
    $$a = $$b;
    $$b = $$c;
    $$c = $$d;
    return;
}

sub _shft2 {
    my ($a, $b, $c) = @_;
    $$a = $$b;
    $$b = $$c;
    return;
}

1; # Magic true value required at end of module

__END__