Function Delegates
Many calculations involve the repeated evaluation of one or more user-supplied functions eg Numerical integration. The EO MathLib provides delegate types for common function signatures and the FunctionFactory class can generate new delegates from existing ones.
RealFunction delegate - takes one Double parameter – can encapsulate most of the static methods of the System.Math class, as well as the classes in the Extreme.Mathematics.SpecialFunctions namespace:
var sin = new RealFunction(Math.Sin);
var result = sin(1);
BivariateRealFunction delegate - takes two Double parameters:
var atan2 = new BivariateRealFunction (Math.Atan2);
var result = atan2(1, 2);
TrivariateRealFunction delegate – represents a function takes three Double arguments
ParameterizedRealFunction delegate - represents a function taking one Integer and one Double argument that returns a real number. The Pow method implements such a function, but the arguments need order re-arrangement:
static double Power(int exponent, double x)
{
return ElementaryFunctions.Pow(x, exponent);
}
...
var power = new ParameterizedRealFunction(Power);
var result = power(6, 3.2);
A ComplexFunction delegate - represents a function that takes an Extreme.Mathematics.DoubleComplex argument and also returns a complex number.
MultivariateRealFunction delegate - represents a function that takes an Extreme.Mathematics.LinearAlgebra.Vector argument and returns a real number.
MultivariateVectorFunction delegate - represents a function that takes a Vector argument and returns a Vector.
FastMultivariateVectorFunction delegate - represents a function that takes an input Vector argument and an output Matrix argument – avoiding object construction
The FunctionFactory class
RealFromBivariateRealFunction and RealFromParameterizedRealFunction helper
methods - transform BivariateRealFunction or a ParameterizedRealFunction into a RealFunction delegate by fixing one of the arguments, and treating this as a new function of a single argument.
var tenthPower = FunctionFactory.RealFromParameterizedRealFunction(power, 10);
var result = tenthPower(x);
Note: There is no direct way to do this programmatically in C# - in F# you have partial value functions where you supply a subset of the arguments (as a travelling closure) that the function expects. When you omit arguments, F# generates a new function that holds onto/remembers the arguments you passed in and "waits" for the other parameters to be supplied.
let sumVals x y = x + y
let sumX = sumVals 10 // Note: no 2nd param supplied.
// sumX is a new function generated from partially applied sumVals.
// ie "sumX is a partial application of sumVals."
let sum = sumX 20
// Invokes sumX, passing in expected int (parameter y from original)
val sumVals : int -> int -> int
val sumX : (int -> int)
val sum : int = 30
RealFunctionsToVectorFunction and RealFunctionsToFastVectorFunction helper methods - combines an array of delegates returning a real number or a vector into vector or matrix functions.
The resulting vector function returns a vector whose components are the function values of the delegates in the array.
var funcVector = FunctionFactory.RealFunctionsToVectorFunction(
new MultivariateRealFunction(myFunc1),
new MultivariateRealFunction(myFunc2));
The IterativeAlgorithm<T> abstract base class
Iterative algorithms are common in numerical computing - a method is executed repeatedly until a certain condition is reached, approximating the result of a calculation with increasing accuracy until a certain threshold is reached. If the desired accuracy is achieved, the **algorithm** is said to converge.
This base class is derived by many classes in the Extreme.Mathematics.EquationSolvers and Extreme.Mathematics.Optimization namespaces, as well as the
ManagedIterativeAlgorithm
class which contains a driver method that manages the iteration process.
The ConvergenceTest abstract base class
This class is used to specify **algorithm** Termination , convergence and results - calculates an estimate for the error, and signals termination of the **algorithm** when the error is below a specified tolerance.
Termination Criteria - specify the success condition as the difference between some quantity and its actual value is within a certain tolerance – 2 ways:
absolute error - difference between the result and the actual value.
relative error is the difference between the result and the actual value relative to the size of the result.
Tolerance property - specify trade-off between accuracy and execution time. The lower the tolerance, the longer it will take for the **algorithm** to obtain a result within that tolerance. Most algorithms in the EO NumLib have a default value of MachineConstants.SqrtEpsilon - gives slightly less than 8 digits of accuracy.
ConvergenceCriterion property - specify under what condition the **algorithm** is assumed to converge. Using the ConvergenceCriterion enum: WithinAbsoluteTolerance / WithinRelativeTolerance / WithinAnyTolerance / NumberOfIterations
Active property - selectively ignore certain convergence tests
Error property - returns the estimated error after a run
MaxIterations / MaxEvaluations properties - Other Termination Criteria - If the **algorithm** cannot achieve the desired accuracy, the **algorithm** still has to end – according to an absolute boundary.
Status property - indicates how the **algorithm** terminated - the AlgorithmStatus enum values:NoResult / Busy / Converged (ended normally - The desired accuracy has been achieved) / IterationLimitExceeded / EvaluationLimitExceeded / RoundOffError / BadFunction / Divergent / ConvergedToFalseSolution. After the iteration terminates, the Status should be inspected to verify that the **algorithm** terminated normally. Alternatively, you can set the ThrowExceptionOnFailure to true.
Result property - returns the result of the **algorithm**. This property contains the best available estimate, even if the desired accuracy was not obtained.
IterationsNeeded / EvaluationsNeeded properties - returns the number of iterations required to obtain the result, number of function evaluations.
Concrete Types of Convergence Test classes
SimpleConvergenceTest class - test if a value is close to zero or very small compared to another value.
VectorConvergenceTest class - test convergence of vectors. This class has two additional properties. The Norm property specifies which norm is to be used when calculating the size of the vector - the VectorConvergenceNorm enum values: EuclidianNorm / Maximum / SumOfAbsoluteValues. The ErrorMeasure property specifies how the error is to be measured – VectorConvergenceErrorMeasure enum values: Norm / Componentwise
ConvergenceTestCollection class - represent a combination of tests. The Quantifier property is a ConvergenceTestQuantifier enum that specifies how the tests in the collection are to be combined: Any / All
The AlgorithmHelper Class
inherits from IterativeAlgorithm<T> and exposes two methods for convergence testing.
IsValueWithinTolerance<T> method - determines whether a value is close to another value to within an **algorithm**'s requested tolerance.
IsIntervalWithinTolerance<T> method - determines whether an interval is within an **algorithm**'s requested tolerance.