/*
* Redberry: symbolic tensor computations.
*
* Copyright (c) 2010-2015:
* Stanislav Poslavsky
* Bolotin Dmitriy
*
* This file is part of Redberry.
*
* Redberry is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* Redberry is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with Redberry. If not, see
* T^{ij..}_{kp..}*Tinv^{kp..}_{mn..} = d^{i}_{m}*d^{j}_{m}*.. + ... (combinations of kroneckers),
*
* where T is specified tensor, Tinv is unknown tensor
* and d - kronecker delta.
*
* The main goal of this class is to create the tensor Tinv of the
* most general form with unknown coefficients, and produce a system of
* linear equations on these coefficients. The resulting equations can
* then be solved and coefficients values substituted in generated
* Tinv tensor.
*
*
*
The following example demonstrates the usage of the {@code InverseTensor} to
* find out the photon propagator in Lorentz gauge:
*
* ...
* //expression specifies tensor, which need to inverse
* Expression toInverse = Tensors.parseExpression("D_mn = k_m*k_n-(1/a)*k_i*k^i*g_mn");
* //linear equation on the unknown tensor K
* Expression equation = Tensors.parseExpression("D_ab*K^ac=d_b^c");
* //samples from which inverse should be formed
* Tensor[] samples = {Tensors.parse("g_mn"), Tensors.parse("g^mn"), Tensors.parse("d_m^n"), Tensors.parse("k_m"), Tensors.parse("k^b")};
*
* InverseTensor inverseTensor = new InverseTensor(toInverse,equation,samples);
* System.out.println(inverseTensor.getGeneralInverseForm());
* System.out.println(Arrays.toString(inverseTensor.getEquations()));
*
*
The above code displays the inverse of specified tensor
*
* K^{ac} = a1*g^{ac}+a0*k^{a}*k^{c}
*
* and a system of equations on its coefficients
*
* [(-a**(-1)*a0+a0)*k^{i}*k_{i}+a1 = 0, -a**(-1)*a1*k_{i}*k^{i} = 1]
*
*
*
* @author Dmitry Bolotin
* @author Stanislav Poslavsky
* @see #findInverseWithMaple(cc.redberry.core.tensor.Expression, cc.redberry.core.tensor.Expression, cc.redberry.core.tensor.Tensor[], boolean, boolean, cc.redberry.core.transformations.Transformation[], String, String)
* @since 1.0
*/
public final class InverseTensor {
private final Expression[] equations;
private final SimpleTensor[] unknownCoefficients;
private final Expression generalInverse;
/**
* Creates the {@code InverseTensor} instance from the equation.
*
* @param toInverse expression specifies tensor, which need to inverse
* @param equation linear equation on the unknown tensor in the form
* T^{..}_{...}*Tinv^{...}_{...} = ...
* @param samples samples from which inverse should be formed
* into account when forming a system of linear equations
*/
public InverseTensor(Expression toInverse, Expression equation, Tensor[] samples) {
this(toInverse, equation, samples, false, new Transformation[0]);
}
/**
* Creates the {@code InverseTensor} instance from the equation.
*
* @param toInverse expression specifies tensor, which need to inverse
* @param equation linear equation on the unknown tensor in the form
* T^{..}_{...}*Tinv^{...}_{...} = ...
* @param samples samples from which inverse should be formed
* @param symmetricForm specifies whether inverse tensor should be symmetric
* @param transformations additional simplification rules, which can be taken
* into account when forming a system of linear equations
*/
public InverseTensor(Expression toInverse,
Expression equation,
Tensor[] samples,
boolean symmetricForm,
Transformation[] transformations) {
if (!(equation.get(0) instanceof Product))
throw new IllegalArgumentException("Equation l.h.s. is not a product of tensors.");
Product leftEq = (Product) equation.get(0);
//matching toInverse l.h.s in equation
SimpleTensor inverseLhs = null;
for (Tensor t : leftEq)
if (!IndexMappings.mappingExists(t, toInverse.get(0))) {
inverseLhs = (SimpleTensor) t;
break;
}
//creating tensor of the most general form from the specified samples
GeneratedTensor generatedTensor = TensorGenerator.generateStructure(inverseLhs.getIndices(), samples, symmetricForm, true, true);
unknownCoefficients = generatedTensor.coefficients;
//creating inverse tensor expression
generalInverse = Tensors.expression(inverseLhs, generatedTensor.generatedTensor);
//substituting toInverse and generalInverse into equation
Tensor temp = equation;
temp = toInverse.transform(temp);
temp = generalInverse.transform(temp);
//collecting all transformations in single array
transformations = ArraysUtils.addAll(new Transformation[]{EliminateMetricsTransformation.ELIMINATE_METRICS}, transformations);
//preparing equation
temp = ExpandTransformation.expand(temp, transformations);
for (Transformation transformation : transformations)
temp = transformation.transform(temp);
temp = CollectNonScalarsTransformation.collectNonScalars(temp);
equation = (Expression) temp;
//processing r.h.s. of the equation
List rightSplit = new ArrayList<>();
if (equation.get(1) instanceof Sum)
for (Tensor summand : equation.get(1))
rightSplit.add(Split.splitScalars(summand));
else
rightSplit.add(Split.splitScalars(equation.get(1)));
//forming system of linear equations
List equationsList = new ArrayList<>();
for (Tensor summand : equation.get(0)) {
Split current = Split.splitScalars(summand);
boolean one = false;
for (Split split : rightSplit)
if (TensorUtils.equals(current.factor, split.factor)) {
equationsList.add(Tensors.expression(current.summand, split.summand));
one = true;
break;
}
if (!one)
equationsList.add(Tensors.expression(current.summand, Complex.ZERO));
}
this.equations = equationsList.toArray(new Expression[equationsList.size()]);
}
private static String newCoefficientName(Tensor... tensors) {
Set simpleTensors = TensorUtils.getAllSymbols(tensors);
List forbidden = new ArrayList<>();
for (SimpleTensor tensor : simpleTensors) {
String name = CC.getNameDescriptor(tensor.getName()).getName(tensor.getIndices());
try {
Integer.parseInt(name.substring(1));
forbidden.add(name.charAt(0));
} catch (NumberFormatException e) {
}
}
Collections.sort(forbidden);
char c = 'a';
for (int i = 0; i < forbidden.size(); ++i) {
if (c != forbidden.get(i).charValue())
break;
else {
++c;
}
}
return String.valueOf(c);
}
/**
* Return the resulting equations on the unknown coefficients.
*
* @return the resulting equations on the unknown coefficients
*/
public Expression[] getEquations() {
return equations.clone();
}
/**
* Returns the inverse of the tensor with unknown coefficients.
*
* @return the inverse of the tensor with unknown coefficients
*/
public Expression getGeneralInverseForm() {
return generalInverse;
}
/**
* Returns the array of the unknown coefficients.
*
* @return the array of the unknown coefficients
*/
public SimpleTensor[] getUnknownCoefficients() {
return unknownCoefficients.clone();
}
public ReducedSystem toReducedSystem() {
return new ReducedSystem(equations, unknownCoefficients, new Expression[]{generalInverse});
}
/**
* This method calculates the tensor inverse to the specified tensor according
* to the specified equation using the Maple facilities to solve the system of
* linear equations. The Maple code will be placed in the specified temporary
* directory in {@code equations.maple} file. The solution of the linear system,
* produced by Maple, will be placed in the specified temporary directory in
* {@code equations.mapleOut} file.
*
*
The following example demonstrates the usage of this method to
* find out the photon propagator in Lorentz gauge:
*
* ...
* //expression specifies tensor, which need to inverse
* Expression toInverse = Tensors.parseExpression("D_mn = k_m*k_n-(1/a)*k_i*k^i*g_mn");
* //linear equation on the unknown tensor K
* Expression equation = Tensors.parseExpression("D_ab*K^ac=d_b^c");
* //samples from which inverse should be formed
* Tensor[] samples = {Tensors.parse("g_mn"), Tensors.parse("g^mn"), Tensors.parse("d_m^n"), Tensors.parse("k_m"), Tensors.parse("k^b")};
*
* Tensor inverse = InverseTensor.findInverseWithMaple(toInverse, equation, samples, false, new Transformation[0], mapleBinDir, temporaryDir);
* System.out.println(inverse);
*
*
The above code displays the inverse of specified tensor
*
* K^ac=-a*g^ac*(k_i*k^i)**(-1)+a**2/(a-1)*k^a*k^c*(k_i*k^i)**(-2)
*
*
*
* @param toInverse expression specifies tensor, which need to inverse
* @param equation linear equation on the unknown tensor in the form
* T^{..}_{...}*Tinv^{...}_{...} = ...
* @param samples samples from which inverse should be formed
* @param symmetricForm specifies whether inverse tensor should be symmetric
* @param transformations additional simplification rules, which can be taken
* into account when forming a system of linear equations
* @param mapleBinDir path to Maple bin directory (e.g. "/home/user/maple14/bin")
* @param path path to your temporary folder
* @return tensor inverse to the specified tensor according to the specified
* equation and null if inverse does not exist
* @throws IOException
* @throws InterruptedException when Maple fails to run
*/
public static Expression findInverseWithMaple(Expression toInverse,
Expression equation,
Tensor[] samples,
boolean symmetricForm,
Transformation[] transformations,
String mapleBinDir,
String path) throws IOException, InterruptedException {
return findInverseWithMaple(toInverse, equation, samples, symmetricForm, false, transformations, mapleBinDir, path);
}
/**
* This method calculates the tensor inverse to the specified tensor according
* to the specified equation using the Maple facilities to solve the system of
* linear equations. The Maple code will be placed in the specified temporary
* directory in {@code equations.maple} file. The solution of the linear system,
* produced by Maple, will be placed in the specified temporary directory in
* {@code equations.mapleOut} file.
*
*
*
The following example demonstrates the usage of this method to
* find out the photon propagator in Lorentz gauge:
*
* ...
* //expression specifies tensor, which need to inverse
* Expression toInverse = Tensors.parseExpression("D_mn = k_m*k_n-(1/a)*k_i*k^i*g_mn");
* //linear equation on the unknown tensor K
* Expression equation = Tensors.parseExpression("D_ab*K^ac=d_b^c");
* //samples from which inverse should be formed
* Tensor[] samples = {Tensors.parse("g_mn"), Tensors.parse("g^mn"), Tensors.parse("d_m^n"), Tensors.parse("k_m"), Tensors.parse("k^b")};
*
* Tensor inverse = InverseTensor.findInverseWithMaple(toInverse, equation, samples, false, new Transformation[0], false, mapleBinDir, temporaryDir);
* System.out.println(inverse);
*
*
The above code displays the inverse of specified tensor
*
* K^ac=-a*g^ac*(k_i*k^i)**(-1)+a**2/(a-1)*k^a*k^c*(k_i*k^i)**(-2)
*
*
* If the produced system of linear equations have infinitely many solutions, some
* of the coefficient cannot be determined exactly and remain as free parameters. The
* flag {@code keepFreeParameters} specifies whether this free parameters should be
* zeroed.
*
* @param toInverse expression specifies tensor, which need to inverse
* @param equation linear equation on the unknown tensor in the form
* T^{..}_{...}*Tinv^{...}_{...} = ...
* @param samples samples from which inverse should be formed
* @param symmetricForm specifies whether inverse tensor should be symmetric
* @param transformations additional simplification rules, which can be taken
* into account when forming a system of linear equations
* @param keepFreeParameters specifies whether the free parameters remaining from solution
* of linear system should be zeroed
* @param mapleBinDir path to Maple bin directory (e.g. "/home/user/maple14/bin")
* @param path path to your temporary folder
* @return tensor inverse to the specified tensor according to the specified
* equation and null if inverse does not exist
* @throws IOException
* @throws InterruptedException when Maple fails to run
*/
public static Expression findInverseWithMaple(Expression toInverse,
Expression equation,
Tensor[] samples,
boolean symmetricForm,
boolean keepFreeParameters,
Transformation[] transformations,
String mapleBinDir,
String path)
throws IOException, InterruptedException {
ReducedSystem reducedSystem = new InverseTensor(toInverse, equation, samples, symmetricForm, transformations).toReducedSystem();
return ExternalSolver.solveSystemWithExternalProgram(
ExternalSolver.MapleScriptCreator.INSTANCE,
reducedSystem, keepFreeParameters, mapleBinDir, path)[0][0];
}
/**
* This method calculates the tensor inverse to the specified tensor according
* to the specified equation using the Wolfram Mathematica facilities to solve the system of
* linear equations. The Wolfram Mathematica code will be placed in the specified temporary
* directory in {@code equations.mathematica} file. The solution of the linear system,
* produced by Wolfram Mathematica, will be placed in the specified temporary directory in
* {@code equations.mathematicaOut} file.
*
*
The following example demonstrates the usage of this method to
* find out the photon propagator in Lorentz gauge:
*
* ...
* //expression specifies tensor, which need to inverse
* Expression toInverse = Tensors.parseExpression("D_mn = k_m*k_n-(1/a)*k_i*k^i*g_mn");
* //linear equation on the unknown tensor K
* Expression equation = Tensors.parseExpression("D_ab*K^ac=d_b^c");
* //samples from which inverse should be formed
* Tensor[] samples = {Tensors.parse("g_mn"), Tensors.parse("g^mn"), Tensors.parse("d_m^n"), Tensors.parse("k_m"), Tensors.parse("k^b")};
*
* Tensor inverse = InverseTensor.findInverseWithmathematica(toInverse, equation, samples, false, new Transformation[0], mapleBinDir, temporaryDir);
* System.out.println(inverse);
*
*
The above code displays the inverse of specified tensor
*
* K^ac=-a*g^ac*(k_i*k^i)**(-1)+a**2/(a-1)*k^a*k^c*(k_i*k^i)**(-2)
*
*
*
* @param toInverse expression specifies tensor, which need to inverse
* @param equation linear equation on the unknown tensor in the form
* T^{..}_{...}*Tinv^{...}_{...} = ...
* @param samples samples from which inverse should be formed
* @param symmetricForm specifies whether inverse tensor should be symmetric
* @param transformations additional simplification rules, which can be taken
* into account when forming a system of linear equations
* @param mathematicaBinDir path to Mathematica bin directory (e.g. "/home/user/maple14/bin")
* @param path path to your temporary folder
* @return tensor inverse to the specified tensor according to the specified
* equation and null if inverse does not exist
* @throws IOException
* @throws InterruptedException when Mathematica fails to run
*/
public static Expression findInverseWithMathematica(Expression toInverse,
Expression equation,
Tensor[] samples,
boolean symmetricForm,
Transformation[] transformations,
String mathematicaBinDir,
String path) throws IOException, InterruptedException {
return findInverseWithMathematica(toInverse, equation, samples, symmetricForm, false, transformations, mathematicaBinDir, path);
}
/**
* This method calculates the tensor inverse to the specified tensor according
* to the specified equation using the Wolfram Mathematica facilities to solve the system of
* linear equations. The Wolfram Mathematica code will be placed in the specified temporary
* directory in {@code equations.mathematica} file. The solution of the linear system,
* produced by Wolfram Mathematica, will be placed in the specified temporary directory in
* {@code equations.mathematicaOut} file.
*
*
The following example demonstrates the usage of this method to
* find out the photon propagator in Lorentz gauge:
*
* ...
* //expression specifies tensor, which need to inverse
* Expression toInverse = Tensors.parseExpression("D_mn = k_m*k_n-(1/a)*k_i*k^i*g_mn");
* //linear equation on the unknown tensor K
* Expression equation = Tensors.parseExpression("D_ab*K^ac=d_b^c");
* //samples from which inverse should be formed
* Tensor[] samples = {Tensors.parse("g_mn"), Tensors.parse("g^mn"), Tensors.parse("d_m^n"), Tensors.parse("k_m"), Tensors.parse("k^b")};
*
* Tensor inverse = InverseTensor.findInverseWithmathematica(toInverse, equation, samples, false, new Transformation[0], mapleBinDir, temporaryDir);
* System.out.println(inverse);
*
*
The above code displays the inverse of specified tensor
*
* K^ac=-a*g^ac*(k_i*k^i)**(-1)+a**2/(a-1)*k^a*k^c*(k_i*k^i)**(-2)
*
*
*
* @param toInverse expression specifies tensor, which need to inverse
* @param equation linear equation on the unknown tensor in the form
* T^{..}_{...}*Tinv^{...}_{...} = ...
* @param samples samples from which inverse should be formed
* @param symmetricForm specifies whether inverse tensor should be symmetric
* @param keepFreeParameters specifies whether the free parameters remaining from solution
* of linear system should be zeroed
* @param transformations additional simplification rules, which can be taken
* into account when forming a system of linear equations
* @param mathematicaBinDir path to Mathematica bin directory (e.g. "/home/user/maple14/bin")
* @param path path to your temporary folder
* @return tensor inverse to the specified tensor according to the specified
* equation and null if inverse does not exist
* @throws IOException
* @throws InterruptedException when Mathematica fails to run
*/
public static Expression findInverseWithMathematica(Expression toInverse,
Expression equation,
Tensor[] samples,
boolean symmetricForm,
boolean keepFreeParameters,
Transformation[] transformations,
String mathematicaBinDir,
String path)
throws IOException, InterruptedException {
ReducedSystem reducedSystem = new InverseTensor(toInverse, equation, samples, symmetricForm, transformations).toReducedSystem();
return ExternalSolver.solveSystemWithExternalProgram(
ExternalSolver.MathematicaScriptCreator.INSTANCE,
reducedSystem, keepFreeParameters, mathematicaBinDir, path)[0][0];
}
}
Ads help maintain this website.