Physsim::LinAlg Class Reference

Linear algebra routines. More...

#include <LinAlg.h>

List of all members.

Static Public Member Functions
static Real	cond (const MatrixN &x)
static MatrixN	to_matrix (const std::string &s)
	Constructs a matrix from a string.
static VectorN	to_vector (const std::string &s)
	Constructs a vector from a string.
static VectorN	solve (const MatrixNN &A, const VectorN &b)
	Solves the general system Ax = b.
static VectorN	solve_symmetric (const MatrixNN &A, const VectorN &b)
	Solves a symmetric, indefinite square matrix.
static VectorN	solve_PD (const MatrixNN &A, const VectorN &b)
	Solves a symmetric, positivie-definite square matrix.
static void	solve_iterative (const MatrixNN &A, const VectorN &b, VectorN &x, unsigned iter)
	Iteratively solves a system of equations using the Conjugate-Gradient method.
static void	svd (const MatrixN &A, MatrixN &U, VectorN &S, MatrixN &V)
	Performs singular value decomposition on A.
static MatrixNN	inverse (const MatrixNN &mat)
	Inverts the matrix A using LU decomposition.
static MatrixNN	inverse_symmetric (const MatrixNN &mat)
	Inverts the symmetric, indefinite matrix A.
static MatrixNN	inverse_PD (const MatrixNN &mat)
	Inverts the symmetric, positive-definite matrix A using Cholesky factorization.
static MatrixN	pseudo_inverse (const MatrixNN &A, Real tol=0)
	Computes the psuedo-inverse of a square matrix.
static MatrixN	pseudo_inverse (const MatrixN &A, Real tol=0)
	Computes the psuedo-inverse of a non-square matrix.
static MatrixN	right_pseudo_inverse (const MatrixN &A)
	Computes the right pseudo inverse of a matrix.
static MatrixN	left_pseudo_inverse (const MatrixN &A)
	Computes the left pseudo inverse of a matrix.
static unsigned	calc_rank (const MatrixN &x, Real tol=1e-10)
static Matrix3	rot_X (Real angle)
	Returns the rotation matrix of the specified angle about the X-axis.
static Matrix3	rot_Y (Real angle)
	Returns the rotation matrix of the specified angle about the Y-axis.
static Matrix3	rot_Z (Real angle)
	Returns the rotation matrix of the specified angle about the Z-axis.
static void	eig_symm (const MatrixNN &A, VectorN &evals)
	Computes the eigenvalues of the matrix A.
static bool	is_PSD (const MatrixNN &A, Real tolerance)
	Determines whether a symmetric matrix is positive semi-definite.
static bool	is_PD (const MatrixNN &A, Real tolerance)
	Determines whether a matrix is positive-definite.

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Detailed Description

Linear algebra routines.

LinAlg is a set of static routines that interface to LAPACK. I have included only very few routines here, however they should be some of the most utilized: SVD, (SVD-based) pseudo-inverse, linear equation solving, and matrix inverse.

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Member Function Documentation

MatrixN LinAlg::to_matrix

(

const std::string &

str

)

[static]

Constructs a matrix from a string.

Elements in a row are separated ustd::sing whitespace or commas. Rows are separated ustd::sing semicolons.

VectorN LinAlg::to_vector

(

const std::string &

str

)

[static]

Constructs a vector from a string.

Parameters:

str

a string representing the vector; elements are separated ustd::sing whitespace, comma, or semicolon

VectorN LinAlg::solve	(	const MatrixNN &	A,
		const VectorN &	b
	)			[static]

Solves the general system Ax = b.

Parameters:

	A	a square matrix
	b	a vector of length A.rows()

Returns:

the solution vector of length b.size()

void LinAlg::solve_iterative	(	const MatrixNN &	A,
		const VectorN &	b,
		VectorN &	x,
		unsigned	iter
	)			[static]

Iteratively solves a system of equations using the Conjugate-Gradient method.

Note:

A must have given form to ensure convergence (e.g., diagonally dominant, symmetric PSD, etc.)

void LinAlg::svd	(	const MatrixN &	A,
		MatrixN &	U,
		VectorN &	S,
		MatrixN &	V
	)			[static]

Performs singular value decomposition on A.

The singular value decomposition of A is U*S*V' (' is the transpose operator); to recompose A, it will be necessary to transpose V before multiplication (i.e., V is returned by the algorithm, not V'). Note: passed matrices and vectors U, S, and V are resized as necessary.

Parameters:

	A	the matrix on which the SVD will be performed
	U	on output, a A.rows() x A.rows() orthogonal matrix
	S	on output, a min(A.rows(), A.columns()) length vector of singular values
	V	on output, a A.columns() x A.columns() orthogonal matrix

MatrixNN LinAlg::inverse

(

const MatrixNN &

A

)

[static]

Inverts the matrix A using LU decomposition.

Parameters:

A

a square matrix

Returns:

the inverse of A

MatrixNN LinAlg::inverse_symmetric

(

const MatrixNN &

A

)

[static]

Inverts the symmetric, indefinite matrix A.

Parameters:

A

a square, symmetric indefinite matrix

Returns:

the inverse of A

MatrixNN LinAlg::inverse_PD

(

const MatrixNN &

A

)

[static]

Inverts the symmetric, positive-definite matrix A using Cholesky factorization.

Parameters:

A

a square, symmetric positive-definite matrix

Returns:

the inverse of A

void LinAlg::eig_symm	(	const MatrixNN &	A,
		VectorN &	evals
	)			[static]

Computes the eigenvalues of the matrix A.

Parameters:

	A	a matrix
	evals	on return, the eigenvalues will be stored here

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The documentation for this class was generated from the following files:

• /home/drumwrig/Physsim/include/Physsim/LinAlg.h
• /home/drumwrig/Physsim/src/LinAlg.cpp

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