findLocalMin

Find a real minimum of a real function f(x) via bracketing. Given a function f and a range (ax .. bx), returns the value of x in the range which is closest to a minimum of f(x). f is never evaluted at the endpoints of ax and bx. If f(x) has more than one minimum in the range, one will be chosen arbitrarily. If f(x) returns NaN or -Infinity, (x, f(x), NaN) will be returned; otherwise, this algorithm is guaranteed to succeed.

Tuple!(T, "x", Unqual!(ReturnType!DF), "y", T, "error")

findLocalMin

(

)

(

scope DF f

const T ax

const T bx

const T relTolerance = sqrt(T.epsilon)

const T absTolerance = sqrt(T.epsilon)

)

if (

isFloatingPoint!T &&

__traits(compiles, )

)

out (result) { assert (isFinite(result.x)); }

Parameters

f DF: Function to be analyzed
ax T: Left bound of initial range of f known to contain the minimum.
bx T: Right bound of initial range of f known to contain the minimum.
relTolerance T: Relative tolerance.
absTolerance T: Absolute tolerance.
Preconditions: ax and bx shall be finite reals.
relTolerance shall be normal positive real.
absTolerance shall be normal positive real no less then T.epsilon*2.

Return Value

Type: Tuple!(T, "x", Unqual!(ReturnType!DF), "y", T, "error")

A tuple consisting of x, y = f(x) and error = 3 * (absTolerance * fabs(x) + relTolerance).

The method used is a combination of golden section search and successive parabolic interpolation. Convergence is never much slower than that for a Fibonacci search.

References: "Algorithms for Minimization without Derivatives", Richard Brent, Prentice-Hall, Inc. (1973)

Examples

import std.math.operations : isClose;

auto ret = findLocalMin((double x) => (x-4)^^2, -1e7, 1e7);
assert(ret.x.isClose(4.0));
assert(ret.y.isClose(0.0, 0.0, 1e-10));

findLocalMin

Parameters

Return Value

Examples

See Also

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