ZerosFunction Class

ZerosFunction Class

Finds the real zeros of a real, continuous, univariate function, f(x).

Inheritance Hierarchy

SystemObject
Imsl.MathZerosFunction

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

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[SerializableAttribute]
public class ZerosFunction

<SerializableAttribute>
Public Class ZerosFunction

[SerializableAttribute]
public ref class ZerosFunction

[<SerializableAttribute>]
type ZerosFunction =  class end

The ZerosFunction type exposes the following members.

Constructors

	Name	Description
	ZerosFunction	Creates an instance of the solver.

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Methods

	Name	Description
	ComputeZeros	Returns the zeros of a univariate function.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	SetBounds	Sets the closed interval in which to search for the roots.
	SetGuess	Sets the initial guess for the zeros.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	AbsoluteError	The second convergence criterion.
	AllConverged	Returns true if the iterations for all of the roots have converged.
	Error	First convergence criterion.
	MaxEvaluations	The maximum number of function evaluations allowed.
	MinimumSeparation	Sets the minimum separation between accepted roots.
	MullerTolerance	Tolerance used during refinement to determine if Müllers method is started.
	NumberOfEvaluations	The actual number of function evaluations performed.
	NumberOfRoots	The requested number of roots to be found.
	NumberOfRootsFound	Returns the number of zeros found.
	XScale	Sets the the scaling in the x-coordinate.

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Remarks

ZerosFunction computes n real zeros of a real, continuous, univariate function f. The search for the zeros of the function can be limited to a specified interval, or extended over the entire real line. The algorithm is generally more efficient if an interval is specified. The user supplied function, f(x), must return valid results for all values in the specified interval. If no interval is given, the user-supplied function must return valid results for all real numbers.

The function has two convergence criteria. The first criterion accepts a root, x, if

$\left| f\left(x\right) \right|\le \tau$

where $\tau$ = Error, see property Error.

The second criterion accepts a root if it is known to be inside of an interval of length at most AbsoluteError, see property AbsoluteError.

A root is accepted if it satisfies either criteria and is not within MinimumSeparation of another accepted root, see property MinimumSeparation.

If initial guesses for the roots are given, Müller's method (Müller 1956) is used for each of these guesses. For each guess, the Müller iteration is stopped if the next step would be outside of the bound, if given. The iteration is also stopped if the algorithm cannot make further progress in finding a root.

If no guesses for the zeros were given, or if Müller's method with the guesses did not find the requested number of roots, a meta-algorithm, combining Müller's and Brent's methods, is used. Müller's method is used primarily to find the roots of functions, such as , where the function does not cross the y=0 line. Brent's method is used to find other types of roots.

The meta-algorithm successively refines the interval using a one-dimensional Faure low-discrepancy sequence. The Faure sequence may be scaled by setting the bound interval [a,b] using the SetBounds method. The Faure sequence will be scaled from (0,1) to (a,b).

If no bound on the function's domain is given, the entire real line must be searched for roots. In this case the Faure sequence is scaled from (0, 1) to $\left( -\infty, +\infty \right)$ using the mapping

$h(u) = \mbox{XScale} \cdot{\tan{(\pi( u - 1/2))}}$

where XScale is set by the XScale property.

At each step of the iteration, the next point in the Faure sequence is added to the list of breakpoints defining the subintervals. Call the points $x_0=a, x_1=b, x_2, x_3, \ldots$ . The new point, splits an existing subinterval, .

The function is evaluated at . If its value is small enough, specifically if

$\left|f\left(x_s\right)\right|\lt \hbox{MullerTolerance}$

then Müller's method is used with

and

as starting values. If a root is found, it is added to the list of roots. If more roots are required, the new Faure point is used.

If Müller's method did not find a root using the new point, the function value at the point is compared with the function values at the endpoints of the subinterval it divides. If $f\left(x_p\right)f\left(x_s\right) \lt 0$ and no root has previously been found in , then Brent's method is used to find a root in this interval. Similarly, if the function changes sign over the interval , and a root has not already been found in the subinterval, Brent's method is used.

Reference

Imsl.Math Namespace

Other Resources

Example