Linear equations and similar problems

Functions
template<class Vector, class Matrix>
Vector	Kaskade::project (Vector const &y, Matrix const &A)
	Computes the projection of \( y \) onto the range of \( A \).
template<class MEntry, class Vector>
Vector	gesv (DynamicMatrix< MEntry > const &A, Vector const &b)
	Computes the solution of \( Ax = b \) for general invertible matrices.
template<class MEntry, class Vector>
Vector	posv (DynamicMatrix< MEntry > const &A, Vector const &b)
	Computes the solution of \( Ax = b \) if \( A \) is symmetric positive definite.
template<class MEntry, class Vector>
Vector	gelss (DynamicMatrix< MEntry > const &A, Vector const &b)
	Computes the solution of the standard least squares problem \( \|Ax-b\|=\min\).
template<class Scalar>
void	invert (DynamicMatrix< Scalar > &A)
	Computes the inverse \( A^{-1} \).
template<class Entry>
DynamicMatrix< Entry > &	invertSpd (DynamicMatrix< Entry > &A, bool symmetrize=true)
	In place computation of the inverse \( A^{-1} \).
template<class Entry>
DynamicMatrix< typename EntryTraits< Entry >::transpose_type >	pseudoinverse (DynamicMatrix< Entry > A)
	Computes the right pseudoinverse \( A^+ \) of a wide matrix such that \( A A^+ = I \).
template<class Entry>
std::tuple< std::vector< typename EntryTraits< Entry >::field_type >, DynamicMatrix< typename EntryTraits< Entry >::field_type >, std::vector< int > >	qr (DynamicMatrix< Entry > const &A)
	Computes a rank-revealing QR decomposition of \( A \) using column pivoting.

Detailed Description

Function Documentation

gelss()

template<class MEntry, class Vector>

Vector gelss ( DynamicMatrix< MEntry > const & A, Vector const & b )

related

Computes the solution of the standard least squares problem \( \|Ax-b\|=\min\).

This uses LAPACK *gelss for computing a norm-minimal solution to the least squares problem.

Parameters

A	a n x m matrix
b	a n-vector

Returns

a m-vector

gesv()

template<class MEntry, class Vector>

Vector gesv ( DynamicMatrix< MEntry > const & A, Vector const & b )

related

Computes the solution of \( Ax = b \) for general invertible matrices.

Template Parameters

Vector

a vector type, usually Dune::DynamicVector or Dune::BlockVector

The field types of MEntry and VEntry must be the same.

invert()

template<class Scalar>

void invert ( DynamicMatrix< Scalar > & A )

related

Computes the inverse \( A^{-1} \).

Parameters

A	the input matrix to be inverted, is overwritten with its inverse

invertSpd()

template<class Entry>

DynamicMatrix< Entry > & invertSpd ( DynamicMatrix< Entry > & A, bool symmetrize = true )

related

In place computation of the inverse \( A^{-1} \).

Template Parameters

Entry

the type of matrix entries, either a scalar or a FieldMatrix<scalar,n,n>

Parameters

A	the quadratic input matrix to be inverted (has to be spd), is overwritten with its inverse
symmetrize	if true, the complete inverse is computed, otherwise only the lower triangular part

Returns

a reference to the (inverted) input matrix

Can throw NonpositiveMatrixException or SingularMatrixException.

posv()

template<class MEntry, class Vector>

Vector posv ( DynamicMatrix< MEntry > const & A, Vector const & b )

related

Computes the solution of \( Ax = b \) if \( A \) is symmetric positive definite.

Template Parameters

Vector

a vector type, usually Dune::DynamicVector or Dune::BlockVector

Parameters

A	must be symmetric positive definite

The field types of MEntry and VEntry must be the same.

project()

template<class Vector, class Matrix>

Vector Kaskade::project	(	Vector const &	y,
		Matrix const &	A )

Computes the projection of \( y \) onto the range of \( A \).

Equivalently, this solves the problem \( \min_s \|y-As\| \) and returns \( As \).

Explicit instantiations for common vector and matrix types are provided.

pseudoinverse()

template<class Entry>

DynamicMatrix< typename EntryTraits< Entry >::transpose_type > pseudoinverse ( DynamicMatrix< Entry > A )

related

Computes the right pseudoinverse \( A^+ \) of a wide matrix such that \( A A^+ = I \).

Template Parameters

Entry

the type of matrix entries of \( A \). This needs to be scalar, i.e. either an arithmetic type or Dune::FieldMatrix<*,1,1>.

Parameters

A	the matrix \( A \in\mathbb{R}^{n\times m}\)

Returns

the matrix \( A^+ \in \mathbb{R}^{m\times n}\)

qr()

template<class Entry>

std::tuple< std::vector< typename EntryTraits< Entry >::field_type >, DynamicMatrix< typename EntryTraits< Entry >::field_type >, std::vector< int > > qr ( DynamicMatrix< Entry > const & A )

related

Computes a rank-revealing QR decomposition of \( A \) using column pivoting.

This computes a unitary matrix \( Q \) and an upper triangular matrix \( R \) such that \( A P = QR \) with a permutation matrix \( P \). The permutation matrix is given as an index permutation vector \( p \), such that \( P_{k,j} = \delta_{k,p_j} \), i.e. the \( j \)-th column of \( AP \) is the \( p_j\)-th column of \( A \).

Template Parameters

Entry

either a scalar arithmetic type or a square field matrix

Returns

\((\tau,QR,p)\) with a vector \( \tau \) of elementary Householder reflectors \( H_i = I - \tau_i v_i v_i^T \), a matrix \( QR \) containing \( R \) in the upper triangular part and the Householder vectors \( v_i \) in the lower triangular part, and the permutation vector \( p \). The entry type of the returned matrix is the scalar field type of \( A \).