PolynomialFilterInterface Class Reference

An interface to define R(S), i.e. set R and S, and compute R(S)*v, where R is the residual polynomial, S is a sparse matrix, and v is a vector. More...

#include <polyfilt.h>

Static Public Member Functions
static Vector	setFilter (const SparseMatrix &S0, const Vector &frame, mkIndex polyDeg, mkIndex baseDeg, IntervalOptions &opts)
	This member function sets the sparse matrix S translated from S0, and also the base filter P(z). More...
static Vector	filteredSparseMatrixPolynomialVectorProduct (const Vector &v)
	This routine computes R(S)*v, where it is assumed that S and R(S) have been defined by setFilter(). More...

Static Public Attributes
static PolynomialFilterInfo	filterInfo
	The information of computing the polynomial filter. See class PolynomialFilterInfo in polyfilt.h for more information.

Static Private Attributes
static const SparseMatrix *	Sptr
	A pointer to the input sparse symmetric matrix via setFilter().
static SparseMatrix	S
	The sparse symmetric matrix S in computing R(S)*v; S is translated from the input sparse symmetric matrix S0 in setFilter().
static Vector	intervals
	The jth interval is [intervals(j),intervals(j+1)). More...
static Vector	intervalWeights
	intervalWeights(j) is the weight of jth interval [intervals(j),intervals(j+1)).
static mkIndex	polyDegree
	Maximum possible degree of s(z), with z*s(z) the polynomial filter.
static mkIndex	baseDegree
	Left-and-right degree of the base filter in each interval.
static Matrix	baseFilter
	A base filter, which is a piecewise polynomial typically from Hermite interpolation. More...
static Vector	zn
	A zero vector of length the number of rows/columns of S.

Detailed Description

An interface to define R(S), i.e. set R and S, and compute R(S)*v, where R is the residual polynomial, S is a sparse matrix, and v is a vector.

• S is a sparse symmetric matrix.
• R(z) is a polynomial which approximates a piecewise polynomial 1-P(z), with P(z) the base filter.
• v is the input vector.

In short, use setFilter() to define R(S) once, and use filteredSparseMatrixPolynomialVectorProduct() to compute R(S)*v for every v.

Member Function Documentation

filteredSparseMatrixPolynomialVectorProduct()

static Vector PolynomialFilterInterface::filteredSparseMatrixPolynomialVectorProduct ( const Vector &  v )

inlinestatic

This routine computes R(S)*v, where it is assumed that S and R(S) have been defined by setFilter().

Remarks

R(z) is in the form z*s(z) which minimizes ||(1-P(z))-z*s(z)||_w by a conjugate-residual-type algorithm, where P(z) is a piecewise polynomial as the base filter, and s(z) is a polynomial of degree up to PolynomialFilterInterface::polyDegree, and w-norm is defined by data members PolynomialFilterInterface::intervals and PolynomialFilterInterface::intervalWeights.

P(z) is defined by the data members PolynomialFilterInterface::baseFilter and PolynomialFilterInterface::intervals. It is expanded in the ‘translated’ (scale-and-shift) Chebyshev basis for each interval, and presented as a matrix of Chebyshev coefficients PolynomialFilterInterface::baseFilter.

See also

FilteredConjugateResidualMatrixPolynomialVectorProduct().

setFilter()

static Vector PolynomialFilterInterface::setFilter ( const SparseMatrix &  S0, const Vector &  frame, mkIndex  polyDeg, mkIndex  baseDeg, IntervalOptions &  opts  )

static

This member function sets the sparse matrix S translated from S0, and also the base filter P(z).

Parameters

S0	is the sparse symmetric matrix of concern.
frame	is a vector of 4 ordered elements. [frame(1),frame(4)] is the interval which (tightly) contains all eigenvalues of S0, and [frame(2),frame(3)] is the interval in which the eigenvalues of S0 are sought.
polyDeg	is the (maximum possible) degree of s(z), with z*s(z) the polynomial filter.
baseDeg	is the left-and-right degree of the base filter for each interval.
opts	is a collection of options to determine the intervals.

See also

GetIntervals().

Member Data Documentation

baseFilter

Matrix PolynomialFilterInterface::baseFilter

staticprivate

A base filter, which is a piecewise polynomial typically from Hermite interpolation.

For jth interval, the polynomial is expanded in the ‘translated’ (scale-and-shift) Chebyshev basis, with the Chebyshev coefficients stored in baseFilter(:,j).

intervals

Vector PolynomialFilterInterface::intervals

staticprivate

The jth interval is [intervals(j),intervals(j+1)).

The intervals are decided by GetIntervals() invoked by setFilter() which takes parameters frame, baseDeg, and polyDeg:

• frame is a vector consisting of 4 ordered elements: [frame(1),frame(4)] is the interval which (tightly) contains all eigenvalues of S, and [frame(2),frame(3)] is the interval in which the eigenvalues are sought.
• baseDeg is the left-and-right degree of the base filter for each interval.
• polyDeg is the (maximum possible) degree of s(z), with z*s(z) the polynomial filter.

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

• polyfilt.h