426 lines
14 KiB
Plaintext
426 lines
14 KiB
Plaintext
[section:roots2 Root Finding Without Derivatives: Bisection, Bracket and TOMS748]
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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{
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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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}}} // namespaces
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[h4 Description]
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These functions solve the root of some function /f(x)/ ['without the
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need for any derivatives of /f(x)/].
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The `bracket_and_solve_root` functions use TOMS Algorithm 748 that is asymptotically the most efficient known,
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and have been shown to be optimal for a certain classes of smooth functions.
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Variants with and without __policies are provided.
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Alternatively, there is a simple bisection 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 the methods of [@http://en.wikipedia.org/wiki/Brent%27s_method Brent or Dekker)]
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which only reduce 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, while /equal_floor/ and /equal_ceil/ are
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useful for certain statistical applications where the result is known to be
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an integer. 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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[h6 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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These functions locate the root using bisection: function arguments are:
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[variablelist
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[[f] [A unary functor which is the function 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.
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It is a precondition that /min < max/ and /f(min)*f(max) <= 0/,
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the function signals evaluation error if these preconditions are violated.
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The action taken is controlled by the evaluation error policy.
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A best guess may be returned, perhaps significantly wrong.]]
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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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[[max_iter][The maximum number of invocations of /f(x)/ to make while searching for the root.]]
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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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[h6 Bracket and solve]
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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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This is a convenience function that calls /toms748_solve/ internally
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to find the root of /f(x)/. It's usable only when /f(x)/ is a monotonic
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function, and the location of the root is known approximately, and in
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particular it is known whether the root is occurs for positive or negative
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/x/. The 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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[[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 result.]]
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[[max_iter] [The maximum number of function invocations to perform in the search
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for the root.]]
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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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[h6 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 two 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 functions differ only by whether values for /f(a)/ and
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/f(b)/ are already available. The toms748_solve 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.]]
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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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[[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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template <class T>
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struct eps_tolerance
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{
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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 twice 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 twice the machine epsilon of 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 least 1 epsilon in size.
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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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[h4 Implementation]
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The implementation of the bisection algorithm is extremely straightforward
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and not detailed here. __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] [/section:roots2 Root Finding Without Derivatives]
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[/
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Copyright 2006, 2010 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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