de.jtem.riemann.theta

Class AbelianFunction

java.lang.Object

de.jtem.riemann.theta.AbelianFunction

public class AbelianFunction
extends Object

Example: An Implementation of an Abelian function.

Let a,b,c,d ∈ Cg with a+b=c+d mod 2πiZg, then

	θ(z+a|B)·θ(z+b|B)
f(z) =
	θ(z+c|B)·θ(z+d|B)

defines an Abelian function, which is represented by this class. To evaluate it you only need to envoke applyTo(ComplexVector) or #applyTo(ComplexVector,Field.Complex). Below the implementation of the last:

   public void applyTo( ComplexVector z, Field.Complex result ) {
 
        zPlusA.assignPlus( z, a );
        zPlusB.assignPlus( z, b );
        zPlusC.assignPlus( z, c );
        zPlusD.assignPlus( z, d );

        theta.theta( zPlusA, factorOfZPlusA, thetaSumOfZPlusA ); 
        theta.theta( zPlusB, factorOfZPlusB, thetaSumOfZPlusB ); 
        theta.theta( zPlusC, factorOfZPlusC, thetaSumOfZPlusC ); 
        theta.theta( zPlusD, factorOfZPlusD, thetaSumOfZPlusD ); 
                                        
        factor.assign(      factorOfZPlusA );                       
        factor.assignPlus(  factorOfZPlusB );                       
        factor.assignMinus( factorOfZPlusC );                       
        factor.assignMinus( factorOfZPlusD );                       
                                                            
        expOfFactor.assignExp( factor );

        result.assign(       expOfFactor      );                                 
        result.assignTimes(  thetaSumOfZPlusA );                    
        result.assignTimes(  thetaSumOfZPlusB );
        result.assignDivide( thetaSumOfZPlusC );
        result.assignDivide( thetaSumOfZPlusD );
   }
 

AbelianFunction is executable. Its main(String[]) performs a periodicity test by envoking testPeriodicity(ComplexVector,boolean) on a 100 different random arguments.

Author:

Markus Schmies

See Also:

Theta

Constructor Summary

Constructors

Constructor and Description
AbelianFunction(de.jtem.blas.ComplexMatrix periodMatrix, de.jtem.blas.ComplexVector a, de.jtem.blas.ComplexVector b, de.jtem.blas.ComplexVector c, de.jtem.blas.ComplexVector d, double tol) Creates Abelean function with prescribed periodMatrix, vectors a,b,c,d and, error tolerance tol.

Method Summary

Modifier and Type	Method and Description
de.jtem.mfc.field.Complex	applyTo(de.jtem.blas.ComplexVector z) Applies Abelian function to z.
void	applyTo(de.jtem.blas.ComplexVector z, de.jtem.mfc.field.Complex result) Applies Abelian function to z.
double	getErrorOfLastEvaluation() Returns error of last Evaluation.
static void	main(String[] argv) Performs a periodicity test for a 2-dimensional example by envoking testPeriodicity(ComplexVector,boolean) on a 100 different random arguments.
void	testPeriodicity(de.jtem.blas.ComplexVector Z, boolean verbose) Performs periodicity test.

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

AbelianFunction

public AbelianFunction(de.jtem.blas.ComplexMatrix periodMatrix,
                       de.jtem.blas.ComplexVector a,
                       de.jtem.blas.ComplexVector b,
                       de.jtem.blas.ComplexVector c,
                       de.jtem.blas.ComplexVector d,
                       double tol)

Creates Abelean function with prescribed periodMatrix, vectors a,b,c,d and, error tolerance tol. The vectors a,b,c,d e Cg have to fill:

a+b=c+d mod 2πiZg Unfortunately tol is only the error tolerance for the four individual theta sums. Use getErrorOfLastEvaluation() to recieve an error estimate for the Abelian function, which depends on the argument. See the description of Theta for a more detailed discussion.

Parameters:

periodMatrix - symmetric complex matrix with negative definite real part

a - vector compatible with periodMatrix

b - vector compatible with periodMatrix

c - vector compatible with periodMatrix

d - vector compatible with periodMatrix

tol - positive number

Method Detail

applyTo

public de.jtem.mfc.field.Complex applyTo(de.jtem.blas.ComplexVector z)

Applies Abelian function to z.

Parameters:

z - agrument vector

Returns:

value of Abelian function at z

applyTo

public void applyTo(de.jtem.blas.ComplexVector z,
                    de.jtem.mfc.field.Complex result)

Applies Abelian function to z.

Parameters:

z - agrument vector

result - value of Abelian function at z

getErrorOfLastEvaluation

public double getErrorOfLastEvaluation()

Returns error of last Evaluation.

main

public static void main(String[] argv)

Performs a periodicity test for a 2-dimensional example by envoking testPeriodicity(ComplexVector,boolean) on a 100 different random arguments.

Parameters:

argv - has no effect

testPeriodicity

public void testPeriodicity(de.jtem.blas.ComplexVector Z,
                            boolean verbose)

Performs periodicity test. The test is quite simple. It adds each of the 2g vectors of the basis of the lattice to the argument Z and checks if all these results coincide.

Parameters:

Z - argument vector

verbose - controles wheter the check is done verbose or silent.