PHP Class MathPHP\NumericalAnalysis\RootFinding\FixedPointIteration
In numerical analysis, the fixed point method is a method for finding
successively better approximations to the roots (or zeroes) of a continuous,
real-valued function f(x). For fixed point iteration, we require that we can
rewrite f(x) = 0 as g(x) = x for some continuous, real-valued function g(x).
We then determine some interval [a, b] to which we will iterate over. To
guarantee a root is in [a, b], we should chose [a, b] such that g(x) is
continuous and g(x) is in [a, b], for all x in [a, b]. To guarantee our
iteration will converge to a single root, we should choose [a, b] such that
for some 0 < k < 1, the magnitude of the derivative |g'(x)| < k on all x
in [a, b].
https://en.wikipedia.org/wiki/Fixed-point_iteration
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Public Methods
| Method |
Description |
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solve ( callable $function, number $a, number $b, number $p, number $tol ) : number |
Use Fixed Point Iteration to find the x which produces f(x) = 0 by
rewriting f(x) = 0 as g(x) = x, where g(x) is our input function. |
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Private Methods
| Method |
Description |
|
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validate ( number $a, number $b, number $p, number $tol ) |
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. |
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Method Details
solve()
public static method
Use Fixed Point Iteration to find the x which produces f(x) = 0 by
rewriting f(x) = 0 as g(x) = x, where g(x) is our input function.
|
public static solve ( callable $function, number $a, number $b, number $p, number $tol ) : number |
|
$function |
callable |
g(x) callback function, obtained by rewriting
f(x) = 0 as g(x) = x |
|
$a |
number |
The start of the interval which contains a root |
|
$b |
number |
The end of the interval which contains a root |
|
$p |
number |
The initial guess of our root, in [$a, $b] |
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$tol |
number |
Tolerance; How close to the actual solution we would like. |
| return |
number |
|