NumericalMethods::Eigenanalysis Namespace Reference

Classes
struct	Eigenpair
struct	SolverResult
class	SquareMatrix

Typedefs
using	Real = double
	Real scalar type used by requests within this interface.
using	Complex = std::complex< double >
	Complex scalar type used by requests within this interface.

Enumerations
enum class	SolverOutcome { Invalid , Success , Failure , FailureInvalidInput }

Functions
SolverResult< Real, 3 >	Solver3x3RealSymmetric (const SquareMatrix< Real, 3 > &matrix)
SolverResult< Real, 3 >	NonIterativeSymmetricEigensolver3x3 (double a00, double a01, double a02, double a11, double a12, double a22)
VectorVariable	CrossProduct (const VectorVariable &lhs, const VectorVariable &rhs)
void	ComputeOrthogonalComplement (const VectorVariable &vecW, VectorVariable &vecU, VectorVariable &vecV)
VectorVariable	ComputeEigenvector0 (double a00, double a01, double a02, double a11, double a12, double a22, double val)
VectorVariable	ComputeEigenvector1 (double a00, double a01, double a02, double a11, double a12, double a22, double val, const VectorVariable &vec)
VectorVariable	ComputeEigenvector2 (const VectorVariable &vec0, const VectorVariable &vec1)

Detailed Description

Namespace for methods for finding eigenvalues and unit-length eigenvectors of matrices. The 3x3 symmetric eigensolver algorithm was adapted from A Robust Eigensolver for 3x3 Symmetric Matrices by Eberly.

Enumeration Type Documentation

SolverOutcome

enum class NumericalMethods::Eigenanalysis::SolverOutcome

strong

Used when returning the solver result to indicate if the solver was successful or to indicate failure reasons.

Enumerator
Invalid	Default value to which fields of this type should be initialized.
Success	The solver successfully found the eigenvalues and vectors.
Failure	The solver failed for unspecified reasons.
FailureInvalidInput	The solver failed because the input matrix was not valid.

Function Documentation

ComputeEigenvector0()

VectorVariable NumericalMethods::Eigenanalysis::ComputeEigenvector0	(	double	a00,
		double	a01,
		double	a02,
		double	a11,
		double	a12,
		double	a22,
		double	val
	)

Given elements of a symmetric 3x3 matrix and one of its eigenvalues, computes the corresponding eigenvector. For numerical stability, this function should only be used to find the eigenvector corresponding to eigenvalues that are unique and numerically not close to other eigenvalues.

Parameters

a<ij>	The element of the matrix in row i, column j.
val	One of the eigenvalues of the matrix.

Returns

The corresponding eigenvector.

ComputeEigenvector1()

VectorVariable NumericalMethods::Eigenanalysis::ComputeEigenvector1	(	double	a00,
		double	a01,
		double	a02,
		double	a11,
		double	a12,
		double	a22,
		double	val,
		const VectorVariable &	vec
	)

Given elements of a symmetric 3x3 matrix, one of its eigenvalues and an unrelated eigenvector, computes the eigenvector corresponding to the eigenvalue. This algorithm is numerically stable even if the eigenvalue is repeated.

Parameters

a<ij>	The element of the matrix in row i, column j.
val	The eigenvalue whose corresponding eigenvector is to be found.
vec	The unrelated eigenvector that is already known.

Returns

The eigenvector corresponding to the given eigenvalue.

ComputeEigenvector2()

VectorVariable NumericalMethods::Eigenanalysis::ComputeEigenvector2	(	const VectorVariable &	vec0,
		const VectorVariable &	vec1
	)

Parameters

vec0	The first of the already known eigenvectors.
vec1	The second of the already known eigenvectors.

Returns

The computed eigenvector.

ComputeOrthogonalComplement()

void NumericalMethods::Eigenanalysis::ComputeOrthogonalComplement	(	const VectorVariable &	vecW,
		VectorVariable &	vecU,
		VectorVariable &	vecV
	)

Robustly computes a right-handed orthonormal basis containing a given unit-length 3D input vector.

Parameters

vecW[in]	A 3D input vector which must be of unit length.
vecU[out]	The first of the computed unit-length orthogonal vectors.
vecV[out]	The second of the computed unit-length orthogonal vectors.

Returns

{vecU, vecV, vecW} will be a right-handed orthogonal set.

CrossProduct()

VectorVariable NumericalMethods::Eigenanalysis::CrossProduct	(	const VectorVariable &	lhs,
		const VectorVariable &	rhs
	)

Compute the cross product between two 3D vectors.

Parameters

lhs	The left-hand side vector.
rhs	The right-hand side vector.

Returns

The 3D vector equal to the cross product if both input vectors are 3-dimensional.

NonIterativeSymmetricEigensolver3x3()

SolverResult< Real, 3 > NumericalMethods::Eigenanalysis::NonIterativeSymmetricEigensolver3x3	(	double	a00,
		double	a01,
		double	a02,
		double	a11,
		double	a12,
		double	a22
	)

Finds the eigenvalues and vectors of the symmetric matrix whose unique elements are given (see Eberly).

Parameters

a<ij>

The element of the matrix in row i, column j.

Returns

Orthonormal eigenbasis of the matrix and the corresponding eigenvalues.

Solver3x3RealSymmetric()

SolverResult< Real, 3 > NumericalMethods::Eigenanalysis::Solver3x3RealSymmetric

(

const SquareMatrix< Real, 3 > &

matrix

)

Compute the eigenvalues and a corresponding eigenbasis for a real symmetric 3x3 matrix. The eigenvalues in this case are guaranteed to be real and the eigenbasis returned is guaranteed to be right-handed and orthonormal (within numerical precision).

Parameters

matrix

The input matrix which must be real and symmetric.

Returns

An instance of SolverResult. The SolverResult::m_outcome will be set to SolverOutcome::FailureInvalidInput if the given matrix is not real and symmetric. Otherwise, SolverResult::m_eigenpairs will contain an orthonormal basis.