Eigen::BjorckPereyra< Scalar_ > Class Template Reference

#include <contrib/Eigen/src/StructuredMatrices/Vandermonde.h>

Detailed Description

template<typename Scalar_>
class Eigen::BjorckPereyra< Scalar_ >

Björck-Pereyra O(n^2) solver for square Vandermonde systems.

Solves V*a = f – polynomial interpolation: find the coefficients of the polynomial taking values f at the nodes – in O(n^2) operations and O(n) storage, via divided differences in the Newton basis followed by the basis change to monomials (Björck & Pereyra, 1970; Golub & Van Loan, Alg. 4.6.2). The transposed (dual, or moment) system \( V^T w = b \) is solved by the companion dual recurrences through the standard SolverBase idiom:

BjorckPereyra<double> bp(V);            // or bp.compute(V);
VectorXd a = bp.solve(f);                // solve V   * a = f
VectorXd w = bp.transpose().solve(b);    // solve V^T * w = b
VectorXd u = bp.adjoint().solve(b);      // solve V^H * u = b

There is no factorization: compute() stores the nodes (and flags exactly repeated or non-finite nodes through info()), and each solve runs the O(n^2) recurrences directly. Genuinely complex node sets are put in a deterministic Leja order to control growth in the Newton representation; real nodes, including complex scalars with zero imaginary parts, retain their input order and its useful monotonicity properties.

Despite the exponential conditioning of real-node Vandermonde matrices, the computed solution is often far more accurate than the conditioning suggests: for monotonically ordered nodes and a right-hand side with alternating signs the forward error is governed by a small relative-perturbation bound independent of the condition number (Higham, ASNA ch. 22).

Template Parameters

Scalar_

a floating-point-like real or complex scalar supporting Eigen's scalar math hooks, including isfinite (and abs and log for complex node ordering). Integer types are rejected.

See also

class Vandermonde

Inheritance diagram for Eigen::BjorckPereyra< Scalar_ >:

Public Member Functions
	BjorckPereyra ()
template<int Rows_, int Cols_>
	BjorckPereyra (const Vandermonde< Scalar, Rows_, Cols_ > &V)
template<int Rows_, int Cols_>
BjorckPereyra &	compute (const Vandermonde< Scalar, Rows_, Cols_ > &V)
ComputationInfo	info () const
template<typename Rhs>
const Solve< BjorckPereyra, Rhs >	solve (const MatrixBase< Rhs > &f) const

Constructor & Destructor Documentation

BjorckPereyra() [1/2]

template<typename Scalar_>

Eigen::BjorckPereyra< Scalar_ >::BjorckPereyra ( )

inline

Default constructor; call compute before solve.

BjorckPereyra() [2/2]

template<typename Scalar_>

template<int Rows_, int Cols_>

Eigen::BjorckPereyra< Scalar_ >::BjorckPereyra ( const Vandermonde< Scalar, Rows_, Cols_ > & V )

inlineexplicit

Constructs the solver for the square Vandermonde matrix V.

Member Function Documentation

compute()

template<typename Scalar_>

template<int Rows_, int Cols_>

BjorckPereyra & Eigen::BjorckPereyra< Scalar_ >::compute ( const Vandermonde< Scalar, Rows_, Cols_ > & V )

inline

Stores the nodes of the square Vandermonde matrix V, checks that they are finite and distinct, and computes a Leja order for genuinely complex nodes.

info()

template<typename Scalar_>

ComputationInfo Eigen::BjorckPereyra< Scalar_ >::info ( ) const

inline

Returns

Success, NumericalIssue when the nodes contain an exact duplicate (the matrix is singular), or InvalidInput for a non-finite node.

solve()

template<typename Scalar_>

template<typename Rhs>

const Solve< BjorckPereyra, Rhs > Eigen::BjorckPereyra< Scalar_ >::solve ( const MatrixBase< Rhs > & f ) const

inline

Returns

the solution a of V*a = f, as a lazily evaluated expression. Supports multiple right-hand sides. The transposed and adjoint systems are solved through transpose().solve(b) and adjoint().solve(b).

Precondition

compute has been called.

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

• /builds/libeigen/eigen/contrib/Eigen/src/StructuredMatrices/Vandermonde.h