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.