Base Filter

Classes
struct	IntervalOptions
	An instance of this class, taken by GetIntervals() and PolynomialFilterInterface::setFilter(), is a collection of options to determine the intervals which decides the base filter. More...
struct	PolynomialFilterInfo
	The routine GetIntervals() returns an instance of this class, which gives the information of a polynomial filter. More...

Functions
Matrix	HermiteBaseFilterInChebyshevBasis (const Vector &intv, const int *HiLowFlags, mkIndex baseDeg)
	Compute a base filter P(z) which is a continuous, piecewise polynomial P(z) expanded in a basis of ‘translated’ (i.e. scale-and-shift) Chebyshev polynomials in each interval. More...
PolynomialFilterInfo	GetIntervals (Vector &intv, const Vector &frame, mkIndex polyDeg, mkIndex baseDeg, IntervalOptions &opts)
	This routine determines the intervals (including the transition one(s)) by an interative process. More...
PolynomialFilterInfo	GetIntervals (Vector &intervals, const Vector &frame, mkIndex polyDeg, mkIndex baseDeg)
	Same as GetIntervals(), with default IntervalOptions.

Detailed Description

Function Documentation

GetIntervals()

PolynomialFilterInfo GetIntervals	(	Vector &	intv,
		const Vector &	frame,
		mkIndex	polyDeg,
		mkIndex	baseDeg,
		IntervalOptions &	opts
	)

This routine determines the intervals (including the transition one(s)) by an interative process.

Parameters

frame	is a vector consisting of 4 ordered elements: [frame(1),frame(4)] is the interval which (tightly) contains all eigenvalues. [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.
intv	is the output; the jth interval is [intv(j),intv(j+1)).
opts	is a collection of interval options.

The base filter P(z) is a piecewise polynomial from Hermite interpolation with degree baseDeg at each end point of intervals.

The polynomial filter Q(z) is in the form z*s(z), i.e. Q(0)==0, such that ||(1-P(z))-z*s(z)||_w is minimized with s(z) a polynomial of degree up to polyDeg.

The resulting polynomial filter Q(z) satisfies Q(x)>=Q(y) for x in [frame(2),frame(3)], and y in [frame(1),frame(4)] but not in [frame(2),frame(3)].

Returns

An instance of PolynomialFilterInfo which gives some information of the polynomial filter.

HermiteBaseFilterInChebyshevBasis()

Matrix HermiteBaseFilterInChebyshevBasis	(	const Vector &	intv,
		const int *	HiLowFlags,
		mkIndex	baseDeg
	)

Compute a base filter P(z) which is a continuous, piecewise polynomial P(z) expanded in a basis of ‘translated’ (i.e. scale-and-shift) Chebyshev polynomials in each interval.

The base filter P(z) equals P_j(z) for z in the jth interval [intv(j), intv(j+1)), where P_j(z) a Hermite interpolating polynomial.

Parameters

intv	is a vector which defines the intervals. The jth interval is [intv(j), intv(j+1)).
HiLowFlags	determines the shape of the base filter P(z). Consider the jth interval. If HighLowFlag[j-1]==1, then P(z)==1 for z in [intv(j), intv(j+1)]. If HighLowFlag[j-1]==1, then P(z)==1 for z in [intv(j), intv(j+1)]. If HighLowFlag[j-1]==-1, then [intv(j), intv(j+1)] is a transition interval; q(intv(j)) and q(intv(j+1)) are defined such that q(z) is continuous.
baseDeg	defines the smoothness of the piecewise polynomial P(z). To be precise, To be precise, the ith derivative of P(z) is zero, i.e. diP(z)/dzi==0, at all interval end points z=intv(j) for i=1,...,baseDeg.

The base filter P(z) expanded in a basis of ‘translated’ (scale-and-shift) Chebyshev polynomials, presented by a matrix of Chebyshev coefficients pp.

For z in the jth interval [intv(j), intv(j+1)), P(z) equals P_j(z) = pp(1,j)*S₀(z) + pp(2,j)*S₁(z) + ... + pp(n,j)*S_n-1(z).

Here S_i(z) is the ‘translated’ Chebyshev polynomial in the jth interval, with S_i((z-c)/h) = T_i(z), c = (intv(j) + intv(j+1))/2, h = (intv(j+1) - intv(j))/2,
where T_i(z) is the Chebyshev polynomial of the first kind, T₀(z) = 1, T₁(z) = z, and T_i(z) = 2*z*T_i-1(z) - T_i-2(Z) for i>=2.

Returns

The matrix pp of Chebyshev coefficients.

Remarks

The degree of P(z) in each interval is (at most) 2*baseDeg+1, with 2*baseDeg+2 coefficients.

Let n be the length of intv; then there are n-1 intervals. The return matrix pp is of size (2*baseDeg+2)-by-(n-1).