iterative scaling algorithm to equilibrate rows and column norms in matrices More...

Classes
class	AlignedVector3
	A vectorization friendly 3D vector. More...

class	AutoDiffScalar
	A scalar type replacement with automatic differentation capability. More...

class	BDCSVD
	class Bidiagonal Divide and Conquer SVD More...

class	DGMRES
	A Restarted GMRES with deflation. This class implements a modification of the GMRES solver for sparse linear systems. The basis is built with modified Gram-Schmidt. At each restart, a few approximated eigenvectors corresponding to the smallest eigenvalues are used to build a preconditioner for the next cycle. This preconditioner for deflation can be combined with any other preconditioner, the IncompleteLUT for instance. The preconditioner is applied at right of the matrix and the combination is multiplicative. More...

class	DynamicSparseMatrix
	A sparse matrix class designed for matrix assembly purpose. More...

class	GMRES
	A GMRES solver for sparse square problems. More...

class	HybridNonLinearSolver
	Finds a zero of a system of n nonlinear functions in n variables by a modification of the Powell hybrid method ("dogleg"). More...

class	IncompleteCholesky
	Modified Incomplete Cholesky with dual threshold. More...

class	IterationController
	Controls the iterations of the iterative solvers. More...

class	JacobiSVD
	Two-sided Jacobi SVD decomposition of a rectangular matrix. More...

class	KdBVH
	A simple bounding volume hierarchy based on AlignedBox. More...

class	KroneckerProduct
	Kronecker tensor product helper class for dense matrices. More...

class	KroneckerProductSparse
	Kronecker tensor product helper class for sparse matrices. More...

class	LevenbergMarquardt
	Performs non linear optimization over a non-linear function, using a variant of the Levenberg Marquardt algorithm. More...

class	MatrixExponential
	Class for computing the matrix exponential. More...

struct	MatrixExponentialReturnValue
	Proxy for the matrix exponential of some matrix (expression). More...

class	MatrixFunction
	Class for computing matrix functions. More...

class	MatrixFunctionAtomic
	Helper class for computing matrix functions of atomic matrices. More...

class	MatrixFunctionReturnValue
	Proxy for the matrix function of some matrix (expression). More...

class	MatrixLogarithmAtomic
	Helper class for computing matrix logarithm of atomic matrices. More...

class	MatrixLogarithmReturnValue
	Proxy for the matrix logarithm of some matrix (expression). More...

class	MatrixMarketIterator
	Iterator to browse matrices from a specified folder. More...

class	MatrixPower
	Class for computing matrix powers. More...

class	MatrixPowerReturnValue
	Proxy for the matrix power of some matrix (expression). More...

class	MatrixSquareRoot
	Class for computing matrix square roots of general matrices. More...

class	MatrixSquareRootQuasiTriangular
	Class for computing matrix square roots of upper quasi-triangular matrices. More...

class	MatrixSquareRootReturnValue
	Proxy for the matrix square root of some matrix (expression). More...

class	MatrixSquareRootTriangular
	Class for computing matrix square roots of upper triangular matrices. More...

class	MINRES
	A minimal residual solver for sparse symmetric problems. More...

class	NumericalDiff

class	PolynomialSolver
	A polynomial solver. More...

class	PolynomialSolverBase
	Defined to be inherited by polynomial solvers: it provides convenient methods such as. More...

class	RandomSetter
	The RandomSetter is a wrapper object allowing to set/update a sparse matrix with random access. More...

class	SkylineInplaceLU
	Inplace LU decomposition of a skyline matrix and associated features. More...

class	SkylineMatrix
	The main skyline matrix class. More...

class	SkylineMatrixBase
	Base class of any skyline matrices or skyline expressions. More...

class	SkylineStorage

class	Spline
	A class representing multi-dimensional spline curves. More...

struct	SplineFitting
	Spline fitting methods. More...

struct	SplineTraits< Spline< _Scalar, _Dim, _Degree >, _DerivativeOrder >
	Compile-time attributes of the Spline class for fixed degree. More...

struct	SplineTraits< Spline< _Scalar, _Dim, _Degree >, Dynamic >
	Compile-time attributes of the Spline class for Dynamic degree. More...

struct	StdMapTraits

class	StdStemFunctions
	Stem functions corresponding to standard mathematical functions. More...

class	SVDBase
	Mother class of SVD classes algorithms. More...

Typedefs
typedef Spline< double, 2 >	Spline2d
	2D double B-spline with dynamic degree.

typedef Spline< float, 2 >	Spline2f
	2D float B-spline with dynamic degree.

typedef Spline< double, 3 >	Spline3d
	3D double B-spline with dynamic degree.

typedef Spline< float, 3 >	Spline3f
	3D float B-spline with dynamic degree.

Functions
template<typename BVH , typename Intersector >
void	BVIntersect (const BVH &tree, Intersector &intersector)

template<typename BVH1 , typename BVH2 , typename Intersector >
void	BVIntersect (const BVH1 &tree1, const BVH2 &tree2, Intersector &intersector)

template<typename BVH , typename Minimizer >
Minimizer::Scalar	BVMinimize (const BVH &tree, Minimizer &minimizer)

template<typename BVH1 , typename BVH2 , typename Minimizer >
Minimizer::Scalar	BVMinimize (const BVH1 &tree1, const BVH2 &tree2, Minimizer &minimizer)

template<typename Polynomial >
NumTraits< typename Polynomial::Scalar >::Real	cauchy_max_bound (const Polynomial &poly)

template<typename Polynomial >
NumTraits< typename Polynomial::Scalar >::Real	cauchy_min_bound (const Polynomial &poly)

template<typename PointArrayType , typename KnotVectorType >
void	ChordLengths (const PointArrayType &pts, KnotVectorType &chord_lengths)
	Computes chord length parameters which are required for spline interpolation. More...

template<typename KnotVectorType >
void	KnotAveraging (const KnotVectorType &parameters, DenseIndex degree, KnotVectorType &knots)
	Computes knot averages.The knots are computed as $\begin{align} u_0 & = \hdots = u_p = 0 \\ u_{m-p} & = \hdots = u_{m} = 1 \\ u_{j+p} & = \frac{1}{p}\sum_{i=j}^{j+p-1}\bar{u}_i \quad\quad j=1,\hdots,n-p \end{align}$ where is the degree and the number knots of the desired interpolating spline. More...

template<typename A , typename B >
KroneckerProduct< A, B >	kroneckerProduct (const MatrixBase< A > &a, const MatrixBase< B > &b)

template<typename A , typename B >
KroneckerProductSparse< A, B >	kroneckerProduct (const EigenBase< A > &a, const EigenBase< B > &b)

template<typename Polynomials , typename T >
T	poly_eval (const Polynomials &poly, const T &x)

template<typename Polynomials , typename T >
T	poly_eval_horner (const Polynomials &poly, const T &x)

template<typename RootVector , typename Polynomial >
void	roots_to_monicPolynomial (const RootVector &rv, Polynomial &poly)

Detailed Description

iterative scaling algorithm to equilibrate rows and column norms in matrices

This class can be used as a preprocessing tool to accelerate the convergence of iterative methods

This feature is useful to limit the pivoting amount during LU/ILU factorization The scaling strategy as presented here preserves the symmetry of the problem NOTE It is assumed that the matrix does not have empty row or column,

Example with key steps

* VectorXd x(n), b(n);
* SparseMatrix<double> A;
* // fill A and b;
* IterScaling<SparseMatrix<double> > scal; 
* // Compute the left and right scaling vectors. The matrix is equilibrated at output
* scal.computeRef(A); 
* // Scale the right hand side
* b = scal.LeftScaling().cwiseProduct(b); 
* // Now, solve the equilibrated linear system with any available solver
* 
* // Scale back the computed solution
* x = scal.RightScaling().cwiseProduct(x); 
* 

Template Parameters

_MatrixType the type of the matrix. It should be a real square sparsematrix

References : D. Ruiz and B. Ucar, A Symmetry Preserving Algorithm for Matrix Scaling, INRIA Research report RR-7552

See Also: IncompleteLUT

Function Documentation

void Eigen::BVIntersect	(	const BVH &	tree,
		Intersector &	intersector
	)

Given a BVH, runs the query encapsulated by intersector. The Intersector type must provide the following members:

bool intersectVolume(const BVH::Volume &volume) //returns true if volume intersects the query

bool intersectObject(const BVH::Object &object) //returns true if the search should terminate immediately

void Eigen::BVIntersect	(	const BVH1 &	tree1,
		const BVH2 &	tree2,
		Intersector &	intersector
	)

Given two BVH's, runs the query on their Cartesian product encapsulated by intersector. The Intersector type must provide the following members:

bool intersectVolumeVolume(const BVH1::Volume &v1, const BVH2::Volume &v2) //returns true if product of volumes intersects the query
bool intersectVolumeObject(const BVH1::Volume &v1, const BVH2::Object &o2) //returns true if the volume-object product intersects the query
bool intersectObjectVolume(const BVH1::Object &o1, const BVH2::Volume &v2) //returns true if the volume-object product intersects the query
bool intersectObjectObject(const BVH1::Object &o1, const BVH2::Object &o2) //returns true if the search should terminate immediately

Minimizer::Scalar Eigen::BVMinimize	(	const BVH &	tree,
		Minimizer &	minimizer
	)

Given a BVH, runs the query encapsulated by minimizer.

Returns: the minimum value. The Minimizer type must provide the following members:
typedef Scalar //the numeric type of what is being minimized--not necessarily the Scalar type of the BVH (if it has one)

Scalar minimumOnVolume(const BVH::Volume &volume)

Scalar minimumOnObject(const BVH::Object &object)

Minimizer::Scalar Eigen::BVMinimize	(	const BVH1 &	tree1,
		const BVH2 &	tree2,
		Minimizer &	minimizer
	)

Given two BVH's, runs the query on their cartesian product encapsulated by minimizer.

Returns: the minimum value. The Minimizer type must provide the following members:
typedef Scalar //the numeric type of what is being minimized--not necessarily the Scalar type of the BVH (if it has one)

Scalar minimumOnVolumeVolume(const BVH1::Volume &v1, const BVH2::Volume &v2)

Scalar minimumOnVolumeObject(const BVH1::Volume &v1, const BVH2::Object &o2)

Scalar minimumOnObjectVolume(const BVH1::Object &o1, const BVH2::Volume &v2)

Scalar minimumOnObjectObject(const BVH1::Object &o1, const BVH2::Object &o2)

Classes

Typedefs

Functions

Detailed Description

Function Documentation