/**
* Verify the input arguments are valid for correct use of the bisection
* method. If the tolerance is less than zero, an Exception will be thrown.
* If f($a) and f($b) have the same sign, we cannot use the intermediate
* value theorem to guarantee a root is between $a and $b. This exposes the
* risk of an endless loop, so we throw an Exception. If $a = $b, then clearly
* we cannot run our loop as $a and $b will themselves be the midpoint, so we
* throw an Exception.
*
* @param Callable $function f(x) callback function
* @param number $a The start of the interval which contains a root
* @param number $b The end of the interval which contains a root
* @param number $tol Tolerance; How close to the actual solution we would like.
*
* @throws Exception if $tol (the tolerance) is negative
* @throws Exception if $a = $b
* @throws Exception if f($a) and f($b) share the same sign
*/
private static function validate(callable $function, $a, $b, $tol)
{
Validation::tolerance($tol);
Validation::interval($a, $b);
$f⟮a⟯ = $function($a);
$f⟮b⟯ = $function($b);
if (Special::sgn($f⟮a⟯) === Special::sgn($f⟮b⟯)) {
throw new Exception\BadDataException('Input function has the same sign at the start and end of the interval. Choose start and end points such that the function evaluated at those points has a different sign (one positive, one negative).');
}
}
/**
* Verify the input arguments are valid for correct use of fixed point
* iteration. If the tolerance is less than zero, an Exception will be thrown.
* If $a = $b, then clearly we cannot run our loop as [$a, $b] will not be
* an interval, so we throw an Exception. If $a > $b, we simply reverse them
* as if the user input $b = $a and $a = $b so the new $a < $b.
*
* @param number $a The start of the interval which contains a root
* @param number $b The end of the interval which contains a root
* @param number $p The initial guess of our root
* @param number $tol Tolerance; How close to the actual solution we would like.
*
* @throws Exception if $tol (the tolerance) is negative
* @throws Exception if $a = $b
* @throws Exception if either $p > $a or $p < $b return false
*/
private static function validate($a, $b, $p, $tol)
{
Validation::tolerance($tol);
Validation::interval($a, $b);
if ($a > $b) {
list($a, $b) = [$b, $a];
}
if ($p < $a || $p > $b) {
throw new Exception\OutOfBoundsException("Initial guess {$p} must be in [{$a}, {$b}].");
}
}