/** * 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).'); } }