de.jtem.riemann.theta

Class TransformPropertySupport

java.lang.Object

de.jtem.riemann.theta.TransformPropertySupport

All Implemented Interfaces:

Serializable, Cloneable

public final class TransformPropertySupport
extends Object
implements Serializable, Cloneable

Supports the use of the transform property of Riemann theta functions. From the definition of the Riemann theta function easily follows, that for all integer vectors M,N∈Zg holds:

θ(z|B) = exp( ½ MtBM + Mtz ) θ( z + 2πiN + BM | B) .

This property is know as the transform property of Riemann theta functions and it can be used to factor out the exponential growth of the Riemann theta function, by determing the integer vectors M and N such that the absolute value of the transformed argument z+2πiN+BM is as small as possible. This is done by the following vectors:

M(z|B)	= -[	re(B)-1 re(z) ]
N(z|B)	= -[	½ ( im(z) + im(BM(z|B) )/π ]

To use this class you have to provide a period matrix B (setPeriodMatrix(de.jtem.blas.ComplexMatrix)) and the argument vector z (setZ(de.jtem.blas.ComplexVector)). TransformPropertySupport than determines the above defined integer vectors M(z|B) and N(z|B). It further computes the transformed argument

T(z|B) = z + 2πiN(z|B) + BM(z|B)

and the exponential transformation factor

f(z|B) = ½ M(z|B)tBM(z|B) + M(z|B)tz

which can be accessed by calling getTransformedZ() and getFactor().

This class can also be used to check whether a vector z∈Cg is a grid point of

{2πiN+BM|M,N∈Zg}.

In that case T(z|B) will obviously be zero.

Author:

Markus Schmies

See Also:

Theta, Serialized Form

Constructor Summary

Constructors

Constructor and Description
TransformPropertySupport(de.jtem.blas.ComplexMatrix periodMatrix) Creates instance with presrcibed period matrix.

Method Summary

Modifier and Type	Method and Description
de.jtem.mfc.field.Complex	getFactor() Returns the exponential transformation factor.
de.jtem.blas.IntegerVector	getM() Returns the integer vector M(z|B).
de.jtem.blas.IntegerVector	getN() Returns the integer vector N(z|B).
de.jtem.blas.ComplexMatrix	getPeriodMatrix() Returns period matrix.
de.jtem.blas.ComplexVector	getTransformedZ() Returns the transformed argument vector.
de.jtem.blas.ComplexVector	getZ() Returns the argument vector which was transformed.
void	setPeriodMatrix(de.jtem.blas.ComplexMatrix periodMatrix) Sets the period matrix.
void	setZ(de.jtem.blas.ComplexVector z) Sets the argument vector which will be transformed.

Methods inherited from class java.lang.Object

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

Constructor Detail

TransformPropertySupport

public TransformPropertySupport(de.jtem.blas.ComplexMatrix periodMatrix)

Creates instance with presrcibed period matrix.

Parameters:

periodMatrix - symmetric complex matrix with negative definite real part

Method Detail

getFactor

public final de.jtem.mfc.field.Complex getFactor()

Returns the exponential transformation factor.

Returns:

f(z|B) = ½ M(z|B)tBM(z|B) + M(z|B)tz

getM

public final de.jtem.blas.IntegerVector getM()

Returns the integer vector M(z|B).

Returns:

M(z|B) = - [ re(B)^-1 re(z) ]

getN

public final de.jtem.blas.IntegerVector getN()

Returns the integer vector N(z|B).

Returns:

N(z|B) = - [ ½ ( im(z) + im(B)M(z|B )/π ]

getPeriodMatrix

public de.jtem.blas.ComplexMatrix getPeriodMatrix()

Returns period matrix.

getTransformedZ

public final de.jtem.blas.ComplexVector getTransformedZ()

Returns the transformed argument vector.

Returns:

T(z|B) = z + 2πiN(z|B) + BM(z|B)

getZ

public final de.jtem.blas.ComplexVector getZ()

Returns the argument vector which was transformed.

setPeriodMatrix

public final void setPeriodMatrix(de.jtem.blas.ComplexMatrix periodMatrix)

Sets the period matrix.

Parameters:

periodMatrix - symmetric complex matrix with negative definite real part

setZ

public final void setZ(de.jtem.blas.ComplexVector z)

Sets the argument vector which will be transformed.

Parameters:

z - argument vector