Chebyshev Polynomials

Functions
Vector	ExpandNewtonPolynomialInChebyshevBasis (Real aa, Real bb, const Vector &a, const Vector &x)
	Translate the coefficients of a Newton polynomial to the coefficients in a basis of the ‘translated’ (scale-and-shift) Chebyshev polynomials. More...
Real	PolynomialEvaluationInChebyshevBasis (const Vector &c, Real z0, Real aa=-1.0, Real bb=1.0)
	Evaluate P(z) at z=z0, where P(z) is a polynomial expanded in a basis of the ‘translated’ (i.e. scale-and-shift) Chebyshev polynomials. More...
Real	PiecewisePolynomialEvaluationInChebyshevBasis (const Matrix &pp, const Vector &intv, Real z0, bool basisTranslated=true)
	Evaluate P(z) at z=z0, where P(z) is a piecewise polynomial expanded in a basis of the (optionally translated, i.e. scale-and-shift) Chebyshev polynomials for each interval. More...
Real	PiecewisePolynomialInnerProductInChebyshevBasis (const Matrix &pp, const Matrix &qq, const Vector &intervalWeights)
	Compute the weighted inner product of two piecewise polynomials expanded in a basis of ‘translated’ (i.e. scale-and-shift) Chebyshev polynomials for each interval. More...
Matrix	PiecewisePolynomialInChebyshevBasisMultiplyX (const Matrix &pp, const Vector &intv)
	compute Q(z) = z*P(z), where P(z) and Q(z) are piecewise polynomials expanded in a basis of ‘translated’ (i.e. scale-and-shift) Chebyshev polynomials for each interval. More...

Detailed Description

Function Documentation

ExpandNewtonPolynomialInChebyshevBasis()

Vector ExpandNewtonPolynomialInChebyshevBasis	(	Real	aa,
		Real	bb,
		const Vector &	a,
		const Vector &	x
	)

Translate the coefficients of a Newton polynomial to the coefficients in a basis of the ‘translated’ (scale-and-shift) Chebyshev polynomials.

Parameters

a,x	define a Newton polynomial as follows: P(z) = a(1) + a(2)*(z-x(1)) + a(3)*(z-x(1))*(z-x(2)) + ... + a(n)*(z-x(1))*...*(z-x(n-1)).
[aa,bb]	is the interval which defines the ‘translated’ Chebyshev polynomials Si(z) = Ti((z-c)/h), where c=(aa+bb)/2 and h=(bb-aa)/2, and Ti(z) is the Chebyshev polynomial of the first kind T0(z)=1, T1(z)=z, and Ti(z)=2*z*Ti-1(z)+Ti-2(Z) for i>=2.

Returns

A vector q containing the Chebyshev coefficients, such that P(z) = q(1)*S₀(z) + q(2)*S₁(z) + ... + q(n)*S_n-1(z).

PiecewisePolynomialEvaluationInChebyshevBasis()

Real PiecewisePolynomialEvaluationInChebyshevBasis	(	const Matrix &	pp,
		const Vector &	intv,
		Real	z0,
		bool	basisTranslated = true
	)

Evaluate P(z) at z=z0, where P(z) is a piecewise polynomial expanded in a basis of the (optionally translated, i.e. scale-and-shift) Chebyshev polynomials for each interval.

Parameters

intv	is a vector which defines the intervals. The jth interval is [intv(j), intv(j+1)).
basisTranslated	tells whether the basis of Chebyshev polynomials are translated (basisTranslated==true) or not (basisTranslated==false).
pp	is a matrix of Chebyshev coefficients which defines a piecewise polynomial in a basis of the (optionally translated) Chebyshev polynomials in each interval. The polynomial Pj(z) in the jth interval, i.e. when z is in [intv(j), intv(j+1)), is defined by the jth column of pp. If basisTranslated==false, then Pj(z) = pp(1,j)*T0(z) + pp(2,j)*T1(z) + ... + pp(n,j)*Tn-1(z), where Ti(z) is the Chebyshev polynomial of the first kind T0(z)=1, T1(z)=z, and Ti(z)=2*z*Ti-1(z)+Ti-2(Z) for i>=2. If basisTranslated==true, then Pj(z) = pp(1,j)*S0(z) + pp(2,j)*S1(z) + ... + pp(n,j)*Sn-1(z), where Si(z) is the ‘translated’ Chebyshev polynomial Si((z-c)/h) = Ti(z), with c = (intv(j)+intv(j+1)) / 2 and h = (intv(j+1)-intv(j)) / 2.
z0	is a number at which the piecewise polynomial P(z) is evaluated.

Returns

The evaluated value of P(z) at z=z0.

Remarks

If z0 falls below the first interval, then the polynomial in the first interval will be used for evaluting P(z0).

If z0 flies over the last interval, then the polynomial in the last interval will be used for evaluting P(z0).

PiecewisePolynomialInChebyshevBasisMultiplyX()

Matrix PiecewisePolynomialInChebyshevBasisMultiplyX	(	const Matrix &	pp,
		const Vector &	intv
	)

compute Q(z) = z*P(z), where P(z) and Q(z) are piecewise polynomials expanded in a basis of ‘translated’ (i.e. scale-and-shift) Chebyshev polynomials for each interval.

P(z) and Q(z) are stored as matrices of Chebyshev coefficients pp and qq, respectively.

Parameters

intv	is a vector which defines the intervals. The jth interval is [intv(j), intv(j+1)).
pp	is a matrix of Chebyshev coefficients which defines the piecewise polynomial P(z). For z in the jth interval [intv(j), intv(j+1)), P(z) equals Pj(z) = pp(1,j)*S0(z) + pp(2,j)*S1(z) + ... + pp(n,j)*Sn-1(z).

Returns

A matrix qq of Chebyshev coefficients which defines the piecewise polynomial Q(z).
For z in the jth interval [intv(j), intv(j+1)), Q(z) equals Q_j(z) = qq(1,j)*S₀(z) + qq(2,j)*S₁(z) + ... + qq(n,j)*S_n-1(z).

Remarks

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 of coefficients qq which represents Q(z) = z*P(z).

PiecewisePolynomialInnerProductInChebyshevBasis()

Real PiecewisePolynomialInnerProductInChebyshevBasis	(	const Matrix &	pp,
		const Matrix &	qq,
		const Vector &	intervalWeights
	)

Compute the weighted inner product of two piecewise polynomials expanded in a basis of ‘translated’ (i.e. scale-and-shift) Chebyshev polynomials for each interval.

Parameters

intervalWeights	defines the interval weights; intervalWeights(j) is the weight of the jth interval.
pp	is a matrix of Chebyshev coefficients which defines the piecewise polynomial P(z). For z in the jth interval, P(z) equals Pj(z) = pp(1,j)*S0(z) + pp(2,j)*S1(z) + ... + pp(n,j)*Sn-1(z).
qq	is a matrix of Chebyshev coefficients which defines the piecewise polynomial Q(z). For z in the jth interval, Q(z) equals Qj(z) = qq(1,j)*S0(z) + qq(2,j)*S1(z) + ... + qq(n,j)*Sn-1(z).

Remarks

Here S_i(z) is the ‘translated’ Chebyshev polynomial in that interval [aa,bb], S_i((z-c)/h) = T_i(z), c = (aa+bb))/2, h = (bb-aa)/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.

The (scaled) jth interval inner product is defined by <P_j,Q_j> = ( /2) * [ pp(1,j)*qq(1,j) + sum_k pp(k,j)*qq(k,j) ],
which comes from the property <T₀,T₀>= , <T_i,T_i>= /2 for i>=1, and <T_i,T_j>=0 for i j.

Returns

The weighted inner product is <P,Q> = sum_j [ intervalWeights(j) * <P_j,Q_j> ].

Remarks

For unit weights, pass an empty vector of intervalWeights (i.e. of length 0).

PolynomialEvaluationInChebyshevBasis()

Real PolynomialEvaluationInChebyshevBasis	(	const Vector &	c,
		Real	z0,
		Real	aa = -1.0,
		Real	bb = 1.0
	)

Evaluate P(z) at z=z0, where P(z) is a polynomial expanded in a basis of the ‘translated’ (i.e. scale-and-shift) Chebyshev polynomials.

Parameters

c	is a vector of Chebyshev coefficients which defines the polynomial P(z) = c(1)*S0(z) + c(2)*S1(z) + ... + c(n)*Sn-1(z), where Si(z) is the ‘translated’ Chebyshev polynomial Si((z-c)/h) = Ti(z), with c = (aa+bb) / 2 and h = (bb-aa) / 2.
[aa,bb]	is the interval which defines the ‘translated’ Chebyshev polynomials Si(z) = Ti((z-c)/h), where c=(aa+bb)/2 and h=(bb-aa)/2, and Ti(z) is the Chebyshev polynomial of the first kind T0(z)=1, T1(z)=z, and Ti(z)=2*z*Ti-1(z)+Ti-2(Z) for i>=2.
z0	is a number at which the piecewise polynomial P(z) is evaluated.

Returns

The evaluated value of P(z) at z=z0.