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499 lines
17 KiB
Plaintext
499 lines
17 KiB
Plaintext
[section:roots_noderiv Root Finding Without Derivatives]
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[h4 Synopsis]
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``
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#include <boost/math/tools/roots.hpp>
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``
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namespace boost { namespace math {
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namespace tools { // Note namespace boost::math::tools.
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// Bisection
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template <class F, class T, class Tol>
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std::pair<T, T>
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bisect(
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F f,
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T min,
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T max,
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Tol tol,
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boost::uintmax_t& max_iter);
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template <class F, class T, class Tol>
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std::pair<T, T>
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bisect(
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F f,
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T min,
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T max,
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Tol tol);
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template <class F, class T, class Tol, class ``__Policy``>
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std::pair<T, T>
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bisect(
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F f,
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T min,
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T max,
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Tol tol,
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boost::uintmax_t& max_iter,
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const ``__Policy``&);
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// Bracket and Solve Root
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template <class F, class T, class Tol>
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std::pair<T, T>
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bracket_and_solve_root(
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F f,
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const T& guess,
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const T& factor,
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bool rising,
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Tol tol,
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boost::uintmax_t& max_iter);
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template <class F, class T, class Tol, class ``__Policy``>
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std::pair<T, T>
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bracket_and_solve_root(
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F f,
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const T& guess,
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const T& factor,
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bool rising,
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Tol tol,
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boost::uintmax_t& max_iter,
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const ``__Policy``&);
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// TOMS 748 algorithm
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template <class F, class T, class Tol>
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std::pair<T, T>
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toms748_solve(
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F f,
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const T& a,
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const T& b,
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Tol tol,
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boost::uintmax_t& max_iter);
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template <class F, class T, class Tol, class ``__Policy``>
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std::pair<T, T>
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toms748_solve(
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F f,
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const T& a,
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const T& b,
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Tol tol,
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boost::uintmax_t& max_iter,
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const ``__Policy``&);
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template <class F, class T, class Tol>
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std::pair<T, T>
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toms748_solve(
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F f,
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const T& a,
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const T& b,
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const T& fa,
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const T& fb,
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Tol tol,
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boost::uintmax_t& max_iter);
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template <class F, class T, class Tol, class ``__Policy``>
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std::pair<T, T>
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toms748_solve(
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F f,
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const T& a,
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const T& b,
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const T& fa,
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const T& fb,
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Tol tol,
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boost::uintmax_t& max_iter,
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const ``__Policy``&);
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// Termination conditions:
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template <class T>
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struct eps_tolerance;
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struct equal_floor;
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struct equal_ceil;
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struct equal_nearest_integer;
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}}} // boost::math::tools namespaces
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[h4 Description]
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These functions solve the root of some function ['f(x)] -
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['without the need for any derivatives of ['f(x)]].
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The `bracket_and_solve_root` functions use __root_finding_TOMS748
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by Alefeld, Potra and Shi that is asymptotically the most efficient known,
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and has been shown to be optimal for a certain classes of smooth functions.
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Variants with and without __policy_section are provided.
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Alternatively, __bisect is a simple __bisection_wikipedia routine which can be useful
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in its own right in some situations, or alternatively for narrowing
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down the range containing the root, prior to calling a more advanced
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algorithm.
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All the algorithms in this section reduce the diameter of the enclosing
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interval with the same asymptotic efficiency with which they locate the
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root. This is in contrast to the derivative based methods which may ['never]
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significantly reduce the enclosing interval, even though they rapidly approach
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the root. This is also in contrast to some other derivative-free methods
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(for example, Brent's method described at
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[@http://en.wikipedia.org/wiki/Brent%27s_method Brent-Dekker)]
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which only reduces the enclosing interval on the final step.
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Therefore these methods return a `std::pair` containing the enclosing interval found,
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and accept a function object specifying the termination condition.
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Three function objects are provided for ready-made termination conditions:
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* ['eps_tolerance] causes termination when the relative error in the enclosing
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interval is below a certain threshold.
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* ['equal_floor] and ['equal_ceil] are useful for certain statistical applications
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where the result is known to be an integer.
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* Other user-defined termination conditions are likely to be used
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only rarely, but may be useful in some specific circumstances.
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[section:bisect Bisection]
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template <class F, class T, class Tol>
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std::pair<T, T>
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bisect( // Unlimited iterations.
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F f,
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T min,
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T max,
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Tol tol);
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template <class F, class T, class Tol>
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std::pair<T, T>
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bisect( // Limited iterations.
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F f,
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T min,
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T max,
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Tol tol,
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boost::uintmax_t& max_iter);
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template <class F, class T, class Tol, class ``__Policy``>
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std::pair<T, T>
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bisect( // Specified policy.
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F f,
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T min,
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T max,
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Tol tol,
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boost::uintmax_t& max_iter,
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const ``__Policy``&);
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These functions locate the root using __bisection_wikipedia.
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`bisect` function arguments are:
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[variablelist
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[[f] [A unary functor which is the function ['f(x)] whose root is to be found.]]
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[[min] [The left bracket of the interval known to contain the root.]]
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[[max] [The right bracket of the interval known to contain the root.[br]
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It is a precondition that ['min < max] and ['f(min)*f(max) <= 0],
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the function raises an __evaluation_error if these preconditions are violated.
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The action taken on error is controlled by the __Policy template argument: the default behavior is to
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throw a ['boost::math::evaluation_error]. If the __Policy is changed to not throw
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then it returns ['std::pair<T>(min, min)].]]
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[[tol] [A binary functor that specifies the termination condition: the function
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will return the current brackets enclosing the root when ['tol(min, max)] becomes true.
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See also __root_termination.]]
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[[max_iter][The maximum number of invocations of ['f(x)] to make while searching for the root. On exit, this is updated to the actual number of invocations performed.]]
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]
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[optional_policy]
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[*Returns]: a pair of values ['r] that bracket the root so that:
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f(r.first) * f(r.second) <= 0
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and either
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tol(r.first, r.second) == true
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or
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max_iter >= m
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where ['m] is the initial value of ['max_iter] passed to the function.
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In other words, it's up to the caller to verify whether termination occurred
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as a result of exceeding ['max_iter] function invocations (easily done by
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checking the updated value of ['max_iter] when the function returns), rather than
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because the termination condition ['tol] was satisfied.
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[endsect]
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[section:bracket_solve Bracket and Solve Root]
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template <class F, class T, class Tol>
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std::pair<T, T>
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bracket_and_solve_root(
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F f,
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const T& guess,
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const T& factor,
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bool rising,
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Tol tol,
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boost::uintmax_t& max_iter);
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template <class F, class T, class Tol, class ``__Policy``>
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std::pair<T, T>
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bracket_and_solve_root(
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F f,
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const T& guess,
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const T& factor,
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bool rising,
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Tol tol,
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boost::uintmax_t& max_iter,
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const ``__Policy``&);
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`bracket_and_solve_root` is a convenience function that calls __root_finding_TOMS748 internally
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to find the root of ['f(x)]. It is generally much easier to use this function rather than __root_finding_TOMS748, since it
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does the hard work of bracketing the root for you. It's bracketing routines are quite robust and will
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usually be more foolproof than home-grown routines, unless the function can be analysed to yield tight
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brackets.
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Note that this routine can only be used when:
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* ['f(x)] is monotonic in the half of the real axis containing ['guess].
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* The value of the inital guess must have the same sign as the root: the function
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will ['never cross the origin] when searching for the root.
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* The location of the root should be known at least approximately,
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if the location of the root differs by many orders of magnitude
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from ['guess] then many iterations will be needed to bracket the root in spite of
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the special heuristics used to guard against this very situation. A typical example would be
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setting the initial guess to 0.1, when the root is at 1e-300.
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The `bracket_and_solve_root` parameters are:
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[variablelist
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[[f][A unary functor that is the function whose root is to be solved.
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['f(x)] must be uniformly increasing or decreasing on ['x].]]
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[[guess][An initial approximation to the root.]]
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[[factor][A scaling factor that is used to bracket the root: the value
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/guess/ is multiplied (or divided as appropriate) by /factor/
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until two values are found that bracket the root. A value
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such as 2 is a typical choice for ['factor].
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In addition ['factor] will be multiplied by 2 every 32 iterations:
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this is to guard against a really very bad initial guess, typically these occur
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when it's known the result is very large or small, but not the exact order
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of magnitude.]]
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[[rising][Set to ['true] if ['f(x)] is rising on /x/ and /false/ if ['f(x)]
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is falling on /x/. This value is used along with the result
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of /f(guess)/ to determine if /guess/ is
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above or below the root.]]
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[[tol] [A binary functor that determines the termination condition for the search
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for the root. /tol/ is passed the current brackets at each step,
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when it returns true then the current brackets are returned as the pair result.
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See also __root_termination.]]
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[[max_iter] [The maximum number of function invocations to perform in the search
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for the root. On exit is set to the actual number of invocations performed.]]
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]
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[optional_policy]
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[*Returns]: a pair of values ['r] that bracket the root so that:
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f(r.first) * f(r.second) <= 0
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and either
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tol(r.first, r.second) == true
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or
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max_iter >= m
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where ['m] is the initial value of ['max_iter] passed to the function.
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In other words, it's up to the caller to verify whether termination occurred
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as a result of exceeding ['max_iter] function invocations (easily done by
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checking the value of ['max_iter] when the function returns), rather than
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because the termination condition ['tol] was satisfied.
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[endsect]
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[section:TOMS748 Algorithm TOMS 748: Alefeld, Potra and Shi: Enclosing zeros of continuous functions]
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template <class F, class T, class Tol>
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std::pair<T, T>
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toms748_solve(
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F f,
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const T& a,
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const T& b,
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Tol tol,
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boost::uintmax_t& max_iter);
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template <class F, class T, class Tol, class ``__Policy``>
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std::pair<T, T>
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toms748_solve(
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F f,
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const T& a,
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const T& b,
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Tol tol,
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boost::uintmax_t& max_iter,
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const ``__Policy``&);
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template <class F, class T, class Tol>
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std::pair<T, T>
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toms748_solve(
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F f,
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const T& a,
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const T& b,
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const T& fa,
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const T& fb,
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Tol tol,
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boost::uintmax_t& max_iter);
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template <class F, class T, class Tol, class ``__Policy``>
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std::pair<T, T>
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toms748_solve(
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F f,
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const T& a,
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const T& b,
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const T& fa,
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const T& fb,
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Tol tol,
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boost::uintmax_t& max_iter,
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const ``__Policy``&);
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These functions implement TOMS Algorithm 748: it uses a mixture of
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cubic, quadratic and linear (secant) interpolation to locate the root of
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['f(x)]. The two pairs of functions differ only by whether values for ['f(a)] and
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['f(b)] are already available.
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Generally speaking it is easier (and often more efficient) to use __bracket_solve
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rather than trying to bracket the root yourself as this function requires.
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This function is provided rather than [@http://en.wikipedia.org/wiki/Brent%27s_method Brent's method] as it is known to be more
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effient in many cases (it is asymptotically the most efficient known,
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and has been shown to be optimal for a certain classes of smooth functions).
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It also has the useful property of decreasing the bracket size
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with each step, unlike Brent's method which only shrinks the enclosing interval in the
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final step. This makes it particularly useful when you need a result where the ends
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of the interval round to the same integer: as often happens in statistical applications
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for example. In this situation the function is able to exit after a much smaller
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number of iterations than would otherwise be possible.
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The __root_finding_TOMS748 parameters are:
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[variablelist
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[[f] [A unary functor that is the function whose root is to be solved.
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f(x) need not be uniformly increasing or decreasing on ['x] and
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may have multiple roots. However, the bounds given must bracket a single root.]]
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[[a] [The lower bound for the initial bracket of the root.]]
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[[b] [The upper bound for the initial bracket of the root.
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It is a precondition that ['a < b] and that ['a] and ['b]
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bracket the root to find so that ['f(a) * f(b) < 0].]]
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[[fa] [Optional: the value of ['f(a)].]]
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[[fb] [Optional: the value of ['f(b)].]]
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[[tol] [A binary functor that determines the termination condition for the search
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for the root. ['tol] is passed the current brackets at each step,
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when it returns true, then the current brackets are returned as the result.
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See also __root_termination.]]
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[[max_iter] [The maximum number of function invocations to perform in the search
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for the root. On exit, ['max_iter] is set to actual number of function
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invocations used.]]
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]
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[optional_policy]
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`toms748_solve` returns: a pair of values ['r] that bracket the root so that:
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f(r.first) * f(r.second) <= 0
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and either
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tol(r.first, r.second) == true
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or
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max_iter >= m
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where ['m] is the initial value of ['max_iter] passed to the function.
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In other words, it's up to the caller to verify whether termination occurred
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as a result of exceeding ['max_iter] function invocations (easily done by
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checking the updated value of ['max_iter]
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against its previous value passed as parameter),
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rather than because the termination condition ['tol] was satisfied.
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[endsect]
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[section:brent Brent-Decker Algorithm]
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The [@http://en.wikipedia.org/wiki/Brent%27s_method Brent-Dekker algorithm], although very well know,
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is not provided by this library as __root_finding_TOMS748 or
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its slightly easier to use variant __bracket_solve are superior and provide equivalent functionality.
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[endsect]
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[section:root_termination Termination Condition Functors]
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template <class T>
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struct eps_tolerance
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{
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eps_tolerance();
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eps_tolerance(int bits);
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bool operator()(const T& a, const T& b)const;
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};
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`eps_tolerance` is the usual termination condition used with these root finding functions.
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Its `operator()` will return true when the relative distance between ['a] and ['b]
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is less than four times the machine epsilon for T, or 2[super 1-bits], whichever is
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the larger. In other words, you set ['bits] to the number of bits of precision you
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want in the result. The minimal tolerance of ['four times the machine epsilon of type T] is
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required to ensure that we get back a bracketing interval, since this must clearly
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be at greater than one epsilon in size. While in theory a maximum distance of twice
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machine epsilon is possible to achieve, in practice this results in a great deal of "thrashing"
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given that the function whose root is being found can only ever be accurate to 1 epsilon at best.
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struct equal_floor
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{
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equal_floor();
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template <class T> bool operator()(const T& a, const T& b)const;
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};
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This termination condition is used when you want to find an integer result
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that is the ['floor] of the true root. It will terminate as soon as both ends
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of the interval have the same ['floor].
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struct equal_ceil
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{
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equal_ceil();
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template <class T> bool operator()(const T& a, const T& b)const;
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};
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This termination condition is used when you want to find an integer result
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that is the ['ceil] of the true root. It will terminate as soon as both ends
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of the interval have the same ['ceil].
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struct equal_nearest_integer
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{
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equal_nearest_integer();
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template <class T> bool operator()(const T& a, const T& b)const;
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};
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This termination condition is used when you want to find an integer result
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that is the /closest/ to the true root. It will terminate as soon as both ends
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of the interval round to the same nearest integer.
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[endsect]
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[section:implementation Implementation]
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The implementation of the bisection algorithm is extremely straightforward
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and not detailed here.
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__TOMS748 is described in detail in:
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['Algorithm 748: Enclosing Zeros of Continuous Functions,
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G. E. Alefeld, F. A. Potra and Yixun Shi,
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ACM Transactions on Mathematica1 Software, Vol. 21. No. 3. September 1995.
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Pages 327-344.]
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The implementation here is a faithful translation of this paper into C++.
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[endsect]
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[endsect] [/section:roots_noderiv Root Finding Without Derivatives]
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[/
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Copyright 2006, 2010, 2015 John Maddock and Paul A. Bristow.
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Distributed under the Boost Software License, Version 1.0.
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(See accompanying file LICENSE_1_0.txt or copy at
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http://www.boost.org/LICENSE_1_0.txt).
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]
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