/// Verify that the matrix does not contain elements with abs(element) > 1 and
/// has an acceptable number of non-zero elements.
void SymmetryOperationSymbolParser::verifyMatrix(
const Kernel::IntMatrix &matrix) {
for (size_t i = 0; i < matrix.numRows(); ++i) {
if (!isValidMatrixRow(matrix[i], matrix.numCols())) {
std::ostringstream error;
error << "Matrix row " << i << " is invalid. Elements: [" << matrix[i][0]
<< ", " << matrix[i][1] << ", " << matrix[i][2] << "]";
throw Kernel::Exception::ParseError(error.str(), "", 0);
}
}
}
/**
* Returns the Jones faithful representation of a symmetry operation
*
* This method generates a Jones faithful string for the given matrix and
*vector.
* The string is generated bases on some rules:
*
* - No spaces:
* 'x + 1/2' -> 'x+1/2'
* - Matrix components occur before vector components:
* '1/2+x' -> 'x+1/2'
* - No leading '+' signs:
* '+x' -> 'x'
* - If more than one matrix element is present, they are ordered x, y, z:
* 'y-x' -> '-x+y'
*
* If the matrix is not 3x3, an std::runtime_error exception is thrown.
*
* @param matrix
* @param vector
* @return
*/
std::string SymmetryOperationSymbolParser::getNormalizedIdentifier(
const Kernel::IntMatrix &matrix, const V3R &vector) {
if (matrix.numCols() != 3 || matrix.numRows() != 3) {
throw std::runtime_error("Matrix is not a 3x3 matrix.");
}
std::vector<std::string> symbols{"x", "y", "z"};
std::vector<std::string> components;
for (size_t r = 0; r < 3; ++r) {
std::ostringstream currentComponent;
for (size_t c = 0; c < 3; ++c) {
if (matrix[r][c] != 0) {
if (matrix[r][c] < 0) {
currentComponent << "-";
} else {
if (!currentComponent.str().empty()) {
currentComponent << "+";
}
}
currentComponent << symbols[c];
}
}
if (vector[r] != 0) {
if (vector[r] > 0) {
currentComponent << "+";
}
if (vector[r].denominator() != 1) {
currentComponent << vector[r];
} else {
currentComponent << vector[r].numerator();
}
}
components.push_back(currentComponent.str());
}
return boost::join(components, ",");
}
/// Returns the order of the symmetry operation based on the matrix. From
/// "Introduction to Crystal Growth and Characterization, Benz and Neumann,
/// Wiley, 2014, p. 51."
size_t
SymmetryOperation::getOrderFromMatrix(const Kernel::IntMatrix &matrix) const {
int trace = matrix.Trace();
int determinant = matrix.determinant();
if (determinant == 1) {
switch (trace) {
case 3:
return 1;
case 2:
return 6;
case 1:
return 4;
case 0:
return 3;
case -1:
return 2;
default:
break;
}
} else if (determinant == -1) {
switch (trace) {
case -3:
return 2;
case -2:
return 6;
case -1:
return 4;
case 0:
return 6;
case 1:
return 2;
default:
break;
}
}
throw std::runtime_error("There is something wrong with supplied matrix.");
}
/**
* Returns the symmetry axis for the given matrix
*
* According to ITA, 11.2 the axis of a symmetry operation can be determined by
* solving the Eigenvalue problem \f$Wu = u\f$ for rotations or \f$Wu = -u\f$
* for rotoinversions. This is implemented using the general real non-symmetric
* eigen-problem solver provided by the GSL.
*
* @param matrix :: Matrix of a SymmetryOperation
* @return Axis of symmetry element.
*/
V3R SymmetryElementWithAxisGenerator::determineAxis(
const Kernel::IntMatrix &matrix) const {
gsl_matrix *eigenMatrix = getGSLMatrix(matrix);
gsl_matrix *identityMatrix =
getGSLIdentityMatrix(matrix.numRows(), matrix.numCols());
gsl_eigen_genv_workspace *eigenWs = gsl_eigen_genv_alloc(matrix.numRows());
gsl_vector_complex *alpha = gsl_vector_complex_alloc(3);
gsl_vector *beta = gsl_vector_alloc(3);
gsl_matrix_complex *eigenVectors = gsl_matrix_complex_alloc(3, 3);
gsl_eigen_genv(eigenMatrix, identityMatrix, alpha, beta, eigenVectors,
eigenWs);
gsl_eigen_genv_sort(alpha, beta, eigenVectors, GSL_EIGEN_SORT_ABS_DESC);
double determinant = matrix.determinant();
Kernel::V3D eigenVector;
for (size_t i = 0; i < matrix.numCols(); ++i) {
double eigenValue = GSL_REAL(gsl_complex_div_real(
gsl_vector_complex_get(alpha, i), gsl_vector_get(beta, i)));
if (fabs(eigenValue - determinant) < 1e-9) {
for (size_t j = 0; j < matrix.numRows(); ++j) {
double element = GSL_REAL(gsl_matrix_complex_get(eigenVectors, j, i));
eigenVector[j] = element;
}
}
}
eigenVector *= determinant;
double sumOfElements = eigenVector.X() + eigenVector.Y() + eigenVector.Z();
if (sumOfElements < 0) {
eigenVector *= -1.0;
}
gsl_matrix_free(eigenMatrix);
gsl_matrix_free(identityMatrix);
gsl_eigen_genv_free(eigenWs);
gsl_vector_complex_free(alpha);
gsl_vector_free(beta);
gsl_matrix_complex_free(eigenVectors);
double min = 1.0;
for (size_t i = 0; i < 3; ++i) {
double absoluteValue = fabs(eigenVector[i]);
if (absoluteValue != 0.0 &&
(eigenVector[i] < min && (absoluteValue - fabs(min)) < 1e-9)) {
min = eigenVector[i];
}
}
V3R axis;
for (size_t i = 0; i < 3; ++i) {
axis[i] = static_cast<int>(boost::math::round(eigenVector[i] / min));
}
return axis;
}
