gridripper::multipole Namespace Reference

Multipole expansion classes and methods. More...

Classes

class Y

Class for spherical harmonic functions. More...

class Yre

Class for real spherical harmonic functions. More...

class Coeff< GReal_t >

Class for real spherical harmonic (Yre) expansion coefficients. More...

class Coeff< GComplex_t >

Class for complex spherical harmonic (Y) expansion coefficients. More...

class ScalarFieldMP< GReal_t >

Class for real scalar fields on sphere. More...

class ScalarFieldMP< GComplex_t >

Class for complex scalar fields on sphere. More...

Functions

Coeff< GReal_t > operator* (const GReal_t, const Yre &)

Tensor product of a number and an Yre.

Coeff< GReal_t > operator* (const Yre &, const GReal_t)

Coeff< GReal_t > operator/ (const Yre &, const GReal_t)

Tensor quotient of a number and an Yre.

Coeff< GReal_t > operator* (const GReal_t, const Coeff< GReal_t > &)

Multiply a coefficient by a number.

Coeff< GReal_t > operator* (const Coeff< GReal_t > &, const GReal_t)

Coeff< GReal_t > operator/ (const Coeff< GReal_t > &, const GReal_t)

Divide a coefficient by a number.

Coeff< GReal_t > operator- (const Yre &)

Negative of a coefficient.

Coeff< GReal_t > operator- (const Coeff< GReal_t > &)

Coeff< GComplex_t > operator* (const GReal_t, const Y &)

Tensor product of a number and an Y.

Coeff< GComplex_t > operator* (const Y &, const GReal_t)

Coeff< GComplex_t > operator* (const GComplex_t &, const Y &)

Coeff< GComplex_t > operator* (const Y &, const GComplex_t &)

Coeff< GComplex_t > operator/ (const Y &, const GReal_t)

Tensor quotient of a number and an Y.

Coeff< GComplex_t > operator/ (const Y &, const GComplex_t &)

Coeff< GComplex_t > operator* (const GReal_t, const Coeff< GComplex_t > &)

Multiply a coefficient by a number.

Coeff< GComplex_t > operator* (const Coeff< GComplex_t > &, const GReal_t)

Coeff< GComplex_t > operator* (const GComplex_t &, const Coeff< GComplex_t > &)

Coeff< GComplex_t > operator* (const Coeff< GComplex_t > &, const GComplex_t &)

Coeff< GComplex_t > operator/ (const Coeff< GComplex_t > &, const GReal_t)

Divide a coefficient by a number.

Coeff< GComplex_t > operator/ (const Coeff< GComplex_t > &, const GComplex_t &)

Coeff< GComplex_t > operator- (const Y &)

Negative of a coefficient.

Coeff< GComplex_t > operator- (const Coeff< GComplex_t > &)

Coeff< GComplex_t > conj (const Y &)

Complex conjugate a coefficient.

Coeff< GComplex_t > conj (const Coeff< GComplex_t > &)

GReal_t YYY_coupling (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3)

The $\int_{0}^{\pi}\int_{0}^{2\pi} Y_{l_1}^{m_1}(\vartheta, \varphi) Y_{l_2}^{m_2}(\vartheta, \varphi) Y_{l_3}^{m_3}(\vartheta, \varphi) \;\sin(\vartheta)\mathrm{d}\vartheta\,\mathrm{d}\varphi$ spherical harmonic function Gaunt coefficient calculator routine.

unsigned int YYY_coupling_size ()

Show table size currently used in YYY_coupling.

GReal_t YreYreYre_coupling (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3)

The same coupling for Yre basis.

GComplex_t YYreY_coupling (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3)

Similar coupling for Y and Yre basis.

GReal_t intYl1ExpMIPhi (const int l)

Calculation of $\int_{0}^{2\pi}\int_{0}^{\pi} Y_{l}^{1}(\vartheta, \varphi) \exp(-\mathrm{i}\varphi) \sin(\vartheta)\;\mathrm{d}\vartheta\,\mathrm{d}\varphi$ .

GReal_t ExpPmIPhiYY_coupling (const int l1, const int m1, const int l2, const int sign)

The overlap integrals $\int_{0}^{\pi}\int_{0}^{2\pi} \mathrm{e}^{\pm\mathrm{i}\varphi} Y_{l_1}^{m_1}(\vartheta, \varphi) Y_{l_2}^{-(m_1\pm1)}(\vartheta, \varphi) \;\sin(\vartheta)\mathrm{d}\vartheta\,\mathrm{d}\varphi$ .

unsigned int ExpPmIPhiYY_coupling_size ()

Show table size currently used in ExpPmIPhiYY_coupling.

GReal_t intYl0 (const int l, const GReal_t theta)

Calculation of $\int_{[0,\theta]\times[0,2\pi]} Y_{l}^{0} \;\mathrm{d}\lambda_{S^2}$ .

GReal_t intBarYl1mYl2m (const int l1, const int l2, const int m, const GReal_t theta, tvector< tvector< tvector< GReal_t > > > *coeffstorage)

Calculation of $\int_{[0,\theta]\times[0,2\pi]} \bar{Y}_{l_1}^{m} Y_{l_2}^{m} \;\mathrm{d}\lambda_{S^2}$ .

GReal_t SobolevConstantS2 (const int m)

The minimal Sobolev constant in the inequality over the sphere: $\Vert\cdot\Vert_{C^{0}} \leq C \Vert\cdot\Vert_{H^{2}_{\mu}}$ .

GReal_t tailSumS2 (const int mup, const int mu, const int L)

The sum $\sum_{l=L}^{\infty} \sum_{m=-l}^{l} \frac{(l(l+1))^{\mu+1}-1}{(l(l+1))^{\mu^{'}+1}-1} \frac{1}{(2l+1)^{2}}$ .

ScalarFieldMP< GReal_t > operator+ (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &)

ScalarFieldMP<GReal_t> algebraic operators.

ScalarFieldMP< GReal_t > operator- (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &)

ScalarFieldMP< GReal_t > operator- (const ScalarFieldMP< GReal_t > &)

ScalarFieldMP< GReal_t > operator* (const ScalarFieldMP< GReal_t > &, const GReal_t)

ScalarFieldMP< GReal_t > operator* (const GReal_t, const ScalarFieldMP< GReal_t > &)

ScalarFieldMP< GReal_t > operator* (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &)

ScalarFieldMP< GReal_t > operator/ (const ScalarFieldMP< GReal_t > &, const GReal_t)

GReal_t L2S2 (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &)

L^2 scalarproduct of two fields on sphere.

GReal_t H2S2 (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &, const int mu)

$H_{\mu}^2$ Sobolev scalarproduct of two fields on sphere.

ScalarFieldMP< GReal_t > reflect (const ScalarFieldMP< GReal_t > &)

Reflect with respect to origin (parity operator).

ScalarFieldMP< GReal_t > mulCosTheta (const ScalarFieldMP< GReal_t > &)

Multiplication operator by $\cos(\vartheta)$ .

ScalarFieldMP< GReal_t > mulCosThetaSqr (const ScalarFieldMP< GReal_t > &)

Multiplication operator by $\cos(\vartheta)^2$ .

ScalarFieldMP< GReal_t > mulSinThetaSqr (const ScalarFieldMP< GReal_t > &)

Multiplication operator by $\sin(\vartheta)^2$ .

ScalarFieldMP< GReal_t > partial_phi (const ScalarFieldMP< GReal_t > &)

$\varphi$ derivative on sphere for real scalar field.

ScalarFieldMP< GReal_t > Sqr_partial_phi (const ScalarFieldMP< GReal_t > &)

Second $\varphi$ derivative on sphere for real scalar field.

ScalarFieldMP< GReal_t > sinTheta_partial_theta (const ScalarFieldMP< GReal_t > &)

$\sin(\vartheta)\partial_{\vartheta}$ derivative on sphere for real scalar field.

ScalarFieldMP< GReal_t > partial_x (const ScalarFieldMP< GReal_t > &)

( $\partial_{1}(\cdot)\circ(\arccos,\mathrm{id}))\circ(\cos,\mathrm{id})$ derivative on sphere for real scalar field.

ScalarFieldMP< GReal_t > LaplaceS2 (const ScalarFieldMP< GReal_t > &)

Laplace operator on unit sphere for real scalar field.

GReal_t normBoundNeumann (const GReal_t a00, const GReal_t H2S2aa)

Upper estimate of the $C^{0}$ norm of $1 -$ a real field for Neumann series expansion.

GReal_t optimalScalingNeumann (const GReal_t a00, const GReal_t H2S2aa)

Scaling factor $a$ , with which $\Vert1-a^{-1}A\Vert^2_{H_2^2}$ is minimized for a real field $A$ on sphere.

ScalarFieldMP< GReal_t > divideByNeumann (const ScalarFieldMP< GReal_t > &a, const ScalarFieldMP< GReal_t > &f, const GReal_t norm, const GReal_t errortolerance)

Division of a real field by a real field, using Neumann series.

GReal_t tailErrorBoundS2 (const ScalarFieldMP< GReal_t > &, const int mup, const int mu, int *lmax, int *mmax)

An upper bound for the tail sum in $H_{\mu}^{2}$ norm, assuming that the field is in the space $H_{\mu'}^2$ .

ScalarFieldMP< GComplex_t > operator+ (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &)

ScalarFieldMP<GComplex_t> algebraic operators.

ScalarFieldMP< GComplex_t > operator+ (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &)

ScalarFieldMP< GComplex_t > operator+ (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &)

ScalarFieldMP< GComplex_t > operator- (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &)

ScalarFieldMP< GComplex_t > operator- (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &)

ScalarFieldMP< GComplex_t > operator- (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &)

ScalarFieldMP< GComplex_t > operator- (const ScalarFieldMP< GComplex_t > &)

ScalarFieldMP< GComplex_t > operator* (const ScalarFieldMP< GComplex_t > &, const GReal_t)

ScalarFieldMP< GComplex_t > operator* (const GReal_t, const ScalarFieldMP< GComplex_t > &)

ScalarFieldMP< GComplex_t > operator* (const ScalarFieldMP< GComplex_t > &, const GComplex_t &)

ScalarFieldMP< GComplex_t > operator* (const GComplex_t &, const ScalarFieldMP< GComplex_t > &)

ScalarFieldMP< GComplex_t > operator* (const ScalarFieldMP< GReal_t > &, const GComplex_t &)

ScalarFieldMP< GComplex_t > operator* (const GComplex_t &, const ScalarFieldMP< GReal_t > &)

ScalarFieldMP< GComplex_t > operator* (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &)

ScalarFieldMP< GComplex_t > operator* (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &)

ScalarFieldMP< GComplex_t > operator* (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &)

ScalarFieldMP< GComplex_t > operator/ (const ScalarFieldMP< GComplex_t > &, const GReal_t)

ScalarFieldMP< GComplex_t > operator/ (const ScalarFieldMP< GComplex_t > &, const GComplex_t &)

ScalarFieldMP< GComplex_t > operator/ (const ScalarFieldMP< GReal_t > &, const GComplex_t &)

ScalarFieldMP< GComplex_t > conj (const ScalarFieldMP< GComplex_t > &)

Complex conjugate a scalarfield on sphere.

ScalarFieldMP< GReal_t > real (const ScalarFieldMP< GComplex_t > &)

Real part of a scalarfield on sphere.

ScalarFieldMP< GReal_t > imag (const ScalarFieldMP< GComplex_t > &)

Imaginary part of a scalarfield on sphere.

GComplex_t L2S2 (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &)

L^2 scalarproduct of fields.

GComplex_t L2S2 (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &)

GComplex_t L2S2 (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &)

GComplex_t H2S2 (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &, const int mu)

$H_{\mu}^2$ Sobolev scalarproduct of two fields on sphere.

GComplex_t H2S2 (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &, const int mu)

GComplex_t H2S2 (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &, const int mu)

GComplex_t L2S2limited (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &, const GReal_t theta, tvector< tvector< tvector< GReal_t > > > *coeffstorage)

$L^2$ scalarproduct of fields, with limited integration domain to $[0,\theta]\times[0,2\pi]$ .

ScalarFieldMP< GComplex_t > reflect (const ScalarFieldMP< GComplex_t > &)

Reflect with respect to origin (parity operator).

ScalarFieldMP< GComplex_t > stepping (const ScalarFieldMP< GComplex_t > &, const int upOrDown)

Stepping up and down operators.

ScalarFieldMP< GComplex_t > mulCosTheta (const ScalarFieldMP< GComplex_t > &)

Multiplication operator by $\cos(\vartheta)$ .

ScalarFieldMP< GComplex_t > mulCosThetaSqr (const ScalarFieldMP< GComplex_t > &)

Multiplication operator by $\cos(\vartheta)^2$ .

ScalarFieldMP< GComplex_t > mulSinThetaSqr (const ScalarFieldMP< GComplex_t > &)

Multiplication operator by $\sin(\vartheta)^2$ .

ScalarFieldMP< GComplex_t > mulExpPmIPhi (const ScalarFieldMP< GComplex_t > &, const int sign)

Multiplication operator by $\mathrm{e}^{\pm\mathrm{i}\varphi}$ .

ScalarFieldMP< GComplex_t > partial_phi (const ScalarFieldMP< GComplex_t > &)

$\varphi$ derivative on sphere for complex scalar field.

ScalarFieldMP< GComplex_t > Sqr_partial_phi (const ScalarFieldMP< GComplex_t > &)

Second $\varphi$ derivative on sphere for complex scalar field.

ScalarFieldMP< GComplex_t > sinTheta_partial_theta (const ScalarFieldMP< GComplex_t > &)

$\sin(\vartheta)\partial_{\vartheta}$ derivative on sphere for complex scalar field.

ScalarFieldMP< GComplex_t > partial_theta (const ScalarFieldMP< GComplex_t > &)

$\partial_{\vartheta}$ derivative on sphere for complex field.

ScalarFieldMP< GComplex_t > partial_x (const ScalarFieldMP< GComplex_t > &)

$(\partial_{1}(\cdot)\circ(\arccos,\mathrm{id}))\circ(\cos,\mathrm{id})$ derivative on sphere for complex scalar field.

ScalarFieldMP< GComplex_t > LaplaceS2 (const ScalarFieldMP< GComplex_t > &)

Laplace operator on unit sphere for complex scalar field.

GReal_t normBoundNeumann (const GComplex_t &a00, const GComplex_t &H2S2aa)

Upper estimate of the $C^{0}$ norm of $1 -$ a complex field for Neumann series.

GComplex_t optimalScalingNeumann (const GComplex_t &a00, const GComplex_t &H2S2aa)

Scaling factor $a$ , with which $\Vert1-a^{-1}A\Vert^2_{H_2^2}$ is minimized for a complex field $A$ on sphere.

ScalarFieldMP< GComplex_t > divideByNeumann (const ScalarFieldMP< GComplex_t > &a, const ScalarFieldMP< GReal_t > &f, const GReal_t norm, const GReal_t errortolerance)

Division of a complex field by a complex field, using Neumann series.

ScalarFieldMP< GComplex_t > divideByNeumann (const ScalarFieldMP< GReal_t > &a, const ScalarFieldMP< GComplex_t > &f, const GReal_t norm, const GReal_t errortolerance)

ScalarFieldMP< GComplex_t > divideByNeumann (const ScalarFieldMP< GComplex_t > &a, const ScalarFieldMP< GComplex_t > &f, const GReal_t norm, const GReal_t errortolerance)

GReal_t tailErrorBoundS2 (const ScalarFieldMP< GComplex_t > &a, const int mup, const int mu, int *lmax, int *mmax)

An upper bound for the tail sum in the $H_{\mu}^{2}$ norm, assuming that the field is in the space $H_{\mu'}^{2}$ .

void maxEval (const unsigned int)

Integration error tolerance parameters for functional field expansion by AGM routine.

unsigned int maxEval ()

void reqAbsError (const GReal_t)

GReal_t reqAbsError ()

void reqRelError (const GReal_t)

GReal_t reqRelError ()

Variables

const int SobolevOrderS2 = 1000000

Maximal series expansion order for the numerical calculation of Sobolev related constants over the sphere.

Detailed Description

Multipole expansion classes and methods.

Function Documentation

ScalarFieldMP<GComplex_t> gridripper::multipole::conj ( const ScalarFieldMP< GComplex_t > & )

Complex conjugate a scalarfield on sphere.

Based on: $\bar{Y}_{l}^{m} = (-1)^{m} Y_{l}^{-m}$ .

Coeff<GComplex_t> gridripper::multipole::conj ( const Y & )

Complex conjugate a coefficient.

ScalarFieldMP<GComplex_t> gridripper::multipole::divideByNeumann	(	const ScalarFieldMP< GComplex_t > &	a,
		const ScalarFieldMP< GReal_t > &	f,
		const GReal_t	norm,
		const GReal_t	errortolerance
	)

Division of a complex field by a complex field, using Neumann series.

Based on: $\sum_{n=0}^{N} (I-A)^{n} A = I - (I-A)^{N+1}$ , where $I$ is the identity operator and $A$ is such an operator that $Ran(A)\in Dom(A)$ . Specially, if $A$ is continuous, then it is absolute convergent if and only if $\Vert I-A\Vert<1$ , furthermore $\Vert I - \sum_{n=0}^{N} (I-A)^{n} A\Vert \leq \Vert I-A\Vert^{N+1}$ . If $A$ is a multiplication by a measurable function $a$ , then equality holds instead of inequality, and $\Vert I-A\Vert^{N+1} = (\mathrm{ess-sup}|1-a|)^{N+1}$ .

Parameters:

	a	: the field, which is in the nominator
	f	: the field to be divided
	norm	: an estimate for the essential supremum of
	tolerance	: an error tolarance

Returns:: f/a : up to the specified error tolerance

ScalarFieldMP<GReal_t> gridripper::multipole::divideByNeumann	(	const ScalarFieldMP< GReal_t > &	a,
		const ScalarFieldMP< GReal_t > &	f,
		const GReal_t	norm,
		const GReal_t	errortolerance
	)

Division of a real field by a real field, using Neumann series.

Based on: same rule as for complex.

GReal_t gridripper::multipole::ExpPmIPhiYY_coupling	(	const int	l1,
		const int	m1,
		const int	l2,
		const int	sign
	)

The overlap integrals $\int_{0}^{\pi}\int_{0}^{2\pi} \mathrm{e}^{\pm\mathrm{i}\varphi} Y_{l_1}^{m_1}(\vartheta, \varphi) Y_{l_2}^{-(m_1\pm1)}(\vartheta, \varphi) \;\sin(\vartheta)\mathrm{d}\vartheta\,\mathrm{d}\varphi$ .

Based on: idea from J. Phys. A31 (1998) 1101, and J. Phys. A32 (1999) 2601, and J. Phys. A35 (2002) 4187. Namely, on expanding of products of Y-s by Gaunt coefficients $G$ . $\int_{0}^{\pi}\int_{0}^{2\pi} \mathrm{e}^{\pm\mathrm{i}\varphi} Y_{l_1}^{m_1}(\vartheta, \varphi) Y_{l_2}^{-(m_1\pm1)}(\vartheta, \varphi) \;\sin(\vartheta)\mathrm{d}\vartheta\,\mathrm{d}\varphi = \sum_{k=0}^{\min(l_1,l_2)} \pm G^{m_1,-(m_1\pm1),\pm1}_{l_1,l_2,|l_1-l_2|+2k} \int_{0}^{\pi}\int_{0}^{2\pi} Y^{1}_{|l_1-l_2|+2k}(\vartheta, \varphi) \exp(-\mathrm{i}\varphi) \;\sin{\vartheta}\mathrm{d}\vartheta\,\mathrm{d}\varphi$ . This quantity is zero for: 1) $l_1<0$ or $l_2<0$ , 2) $|m_1|>l_1$ or $|m_1\pm1|>l_2$ , 3) $l_1+l_2$ even.

Numerically verified using a Maple code.

Version:: 0.5, 04/01/2008

Since:: GridRipper 0.5

Author:: Andras Laszlo

unsigned int gridripper::multipole::ExpPmIPhiYY_coupling_size ( )

Show table size currently used in ExpPmIPhiYY_coupling.

ScalarFieldMP<GReal_t> gridripper::multipole::imag ( const ScalarFieldMP< GComplex_t > & )

Imaginary part of a scalarfield on sphere.

GReal_t gridripper::multipole::intBarYl1mYl2m	(	const int	l1,
		const int	l2,
		const int	m,
		const GReal_t	theta,
		tvector< tvector< tvector< GReal_t > > > *	coeffstorage
	)

Calculation of $\int_{[0,\theta]\times[0,2\pi]} \bar{Y}_{l_1}^{m} Y_{l_2}^{m} \;\mathrm{d}\lambda_{S^2}$ .

Based on: expansion of $\bar{Y}_{l_1}^{m}Y_{l_2}^{m}$ product by using Gaunt coefficients. Result: $\int_{[0,\theta]\times[0,2\pi]} \bar{Y}_{l_1}^{m} Y_{l_2}^{m} \;\mathrm{d}\lambda_{S^2} = (-1)^{m} \sum_{k=0}^{\min(l_1, l_2)} G_{l_1,l_2,|l_1-l_2|+|l_1-l_2|\%2+2k}^{-m,m,0} \int_{[0,\theta]\times[0,2\pi]} Y_{|l_1-l_2|+|l_1-l_2|\%2+2k}^{0} \;\mathrm{d}\lambda_{S^2}$ , $G$ being the Gaunt coefficients.

To avoid expensive evaluation, a table is stored (appended) to 'coeffstorage'. Initially should be empty, and should be different for every ' $\theta$ '. If the pointer to 'coeffstorage' is NULL, the coefficients are not stored, but calculated each time.

Numerically verified using Maple.

Version:: 0.5, 22/07/2008

Since:: GridRipper 0.5

Author:: Andras Laszlo

GReal_t gridripper::multipole::intYl0	(	const int	l,
		const GReal_t	theta
	)

Calculation of $\int_{[0,\theta]\times[0,2\pi]} Y_{l}^{0} \;\mathrm{d}\lambda_{S^2}$ .

Based on: $P_{l+1}^{0'}-P_{l-1}^{0'} = (2l+1) P_{l}^{0}$ (from Journal of Modern Optics 36 (1989) 685, but may be easily verified from Rodriguez formula). Result is: $\int_{[0,\theta]\times[0,2\pi]} Y_{l}^{0} \;\mathrm{d}\lambda_{S^2} = 2\pi ( \frac{1}{\sqrt{4\pi}}\delta_{l,0} - \frac{1}{\sqrt{(2l+1)(2l+3)}}Y_{l+1}^{0}(\theta, 0) + \frac{1}{\sqrt{(2l-1)(2l+1)}}Y_{l-1}^{0}(\theta, 0) )$ .

Numerically verified using Maple.

Version:: 0.5, 22/07/2008

Since:: GridRipper 0.5

Author:: Andras Laszlo

GReal_t gridripper::multipole::intYl1ExpMIPhi ( const int l )

Calculation of $\int_{0}^{2\pi}\int_{0}^{\pi} Y_{l}^{1}(\vartheta, \varphi) \exp(-\mathrm{i}\varphi) \sin(\vartheta)\;\mathrm{d}\vartheta\,\mathrm{d}\varphi$ .

Based on: formula in J. Phys. A32 (1999) 2601. Namely: $\int_{-1}^{1} P^{1}_{l}(x) \;\mathrm{d}x = \frac{(-1)^{l}-1}{2} \frac{l}{l+1} \left(\frac{\Gamma(l/2)}{((l-1)/2)!}\right)^{2} = \frac{(-1)^{l}-1}{2} \pi \frac{l}{l+1} \left(\prod_{k=1}^{(l-1)/2} \frac{l/2-k}{(l+1)/2-k}\right)^{2} = \frac{(-1)^{l}-1}{2} \pi \frac{l}{l+1} \prod_{n=1}^{(l-1)/2} \left(\frac{n-1/2}{n}\right)^{2}$ delivers final result (which is zero for even $l$ ). Thus: $\int_{0}^{2\pi}\int_{0}^{\pi} Y_{l}^{1}(\vartheta, \varphi) \exp(-\mathrm{i}\varphi) \sin(\vartheta)\;\mathrm{d}\vartheta\,\mathrm{d}\varphi = \frac{(-1)^{l}-1}{2} \sqrt{\pi^{3} \frac{l(2l+1)}{(l+1)^{3}}} \prod_{n=1}^{(l-1)/2} \left(\frac{n-1/2}{n}\right)^{2}$ delivers final result (which is zero for even $l$ ).

Numerically verified using a Maple and Mathematica code.

Version:: 0.5, 04/01/2008

Since:: GridRipper 0.5

Author:: Andras Laszlo

GComplex_t gridripper::multipole::L2S2	(	const ScalarFieldMP< GComplex_t > &	,
		const ScalarFieldMP< GReal_t > &
	)

L^2 scalarproduct of fields.

Based on: $\int_{0}^{\pi}\int_{0}^{2\pi} \bar{Y_{l_1}^{m_1}(\vartheta,\varphi)} Y_{l_2}^{m_2}(\vartheta,\varphi) \;\sin(\vartheta)\mathrm{d}\vartheta\,\mathrm{d}\varphi = \delta_{l_1, l_2}\delta_{m_1, m_2}$ .

GReal_t gridripper::multipole::L2S2	(	const ScalarFieldMP< GReal_t > &	,
		const ScalarFieldMP< GReal_t > &
	)

L^2 scalarproduct of two fields on sphere.

Based on: $\int_{0}^{\pi}\int_{0}^{2\pi} Yre_{l_1}^{m_1}(\vartheta,\varphi) Yre_{l_2}^{m_2}(\vartheta,\varphi) \;\sin(\vartheta)\mathrm{d}\vartheta\,\mathrm{d}\varphi = \delta_{l_1, l_2}\delta_{m_1, m_2}$ .

GComplex_t gridripper::multipole::L2S2limited	(	const ScalarFieldMP< GComplex_t > &	,
		const ScalarFieldMP< GComplex_t > &	,
		const GReal_t	theta,
		tvector< tvector< tvector< GReal_t > > > *	coeffstorage
	)

$L^2$ scalarproduct of fields, with limited integration domain to $[0,\theta]\times[0,2\pi]$ .

Based on the Gaunt expansion of spherical harmonic products. (See Gaunt.h and Gaunt.cxx.) The necessary coefficient matrix is stored (appended) to 'coeffstorage', for a given ' $\vartheta$ '. Should be empty initially for a given ' $\vartheta$ '. If the pointer to 'coeffstorage' is NULL, the coefficient matrix is not stored in table, but calculated every time.

ScalarFieldMP<GComplex_t> gridripper::multipole::LaplaceS2 ( const ScalarFieldMP< GComplex_t > & )

Laplace operator on unit sphere for complex scalar field.

Based on: $(\frac{1}{\sin(\vartheta)}\partial_{\vartheta}\sin(\vartheta)\partial_{\vartheta} + \frac{1}{\sin^{2}(\vartheta)}\partial_{\varphi}\partial_{\varphi}) Y_{l}^{m} = -l(l+1) Y_{l}^{m}$ .

ScalarFieldMP<GReal_t> gridripper::multipole::LaplaceS2 ( const ScalarFieldMP< GReal_t > & )

Laplace operator on unit sphere for real scalar field.

Based on: $(\frac{1}{\sin(\vartheta)}\partial_{\vartheta}\sin(\vartheta)\partial_{\vartheta} + \frac{1}{\sin^{2}(\vartheta)}\partial_{\varphi}\partial_{\varphi}) Yre_{l}^{m} = -l(l+1) Yre_{l}^{m}$ .

ScalarFieldMP<GComplex_t> gridripper::multipole::mulCosTheta ( const ScalarFieldMP< GComplex_t > & )

Multiplication operator by $\cos(\vartheta)$ .

Based on: $\cos(\vartheta) Y_{l}^{m} = \sqrt{\frac{(l-m)(l+m)}{(2l-1)(2l+1)}} Y_{l-1}^{m} + + \sqrt{\frac{(l+1-m)(l+1+m)}{(2l+1)(2l+3)}} Y_{l+1}^{m}$ . Numerically verified by a C++ code.

ScalarFieldMP<GReal_t> gridripper::multipole::mulCosTheta ( const ScalarFieldMP< GReal_t > & )

Multiplication operator by $\cos(\vartheta)$ .

Based on the rule for Y basis. Numerically verified using a C++ code.

ScalarFieldMP<GComplex_t> gridripper::multipole::mulCosThetaSqr ( const ScalarFieldMP< GComplex_t > & )

Multiplication operator by $\cos(\vartheta)^2$ .

Based on: $\cos(\vartheta)^{2} Y_{l}^{m} = \sqrt{\frac{(l-1-m)(l-m)(l-1+m)(l+m)} {(2l-3)(2l-1)(2l-1)(2l+1)}} Y_{l-2}^{m} + \left(\frac{(l-m)(l+m)}{(2l-1)(2l+1)}+\frac{(l+1-m)(l+1+m)}{(2l+1)(2l+3)}\right) Y_{l}^{m} + \sqrt{\frac{(l+1-m)(l+2-m)(l+1+m)(l+2+m)} {(2l+1)(2l+3)(2l+3)(2l+5)}} Y_{l+2}^{m}$ . Numerically verified by a C++ code.

ScalarFieldMP<GReal_t> gridripper::multipole::mulCosThetaSqr ( const ScalarFieldMP< GReal_t > & )

Multiplication operator by $\cos(\vartheta)^2$ .

Based on the rule for Y basis. Numerically verified using a C++ code.

ScalarFieldMP<GComplex_t> gridripper::multipole::mulExpPmIPhi	(	const ScalarFieldMP< GComplex_t > &	,
		const int	sign
	)

Multiplication operator by $\mathrm{e}^{\pm\mathrm{i}\varphi}$ .

Based on: matrix elements is Gaunt.h, Gaunt.cxx. Numerically verified by a Maple code.

ScalarFieldMP<GComplex_t> gridripper::multipole::mulSinThetaSqr ( const ScalarFieldMP< GComplex_t > & )

Multiplication operator by $\sin(\vartheta)^2$ .

Based on: $\cos(\vartheta)^2$ . Numerically verified by a C++ code.

ScalarFieldMP<GReal_t> gridripper::multipole::mulSinThetaSqr ( const ScalarFieldMP< GReal_t > & )

Multiplication operator by $\sin(\vartheta)^2$ .

Based on: $\cos(\vartheta)^2$ . Numerically verified using a C++ code.

GReal_t gridripper::multipole::normBoundNeumann	(	const GComplex_t &	a00,
		const GComplex_t &	H2S2aa
	)

Upper estimate of the $C^{0}$ norm of $1 -$ a complex field for Neumann series.

Based on: the Sobolev inequality, namely $\Vert\cdot\Vert_{C^{0}} \leq C\Vert\cdot\Vert_{H_2^2}$ .

Parameters:

	a00	: the (0, 0) multipole component of the field
	H2S2aa	: the $\left<a, a\right>_{H_2^2}$ scalar product

GReal_t gridripper::multipole::normBoundNeumann	(	const GReal_t	a00,
		const GReal_t	H2S2aa
	)

Upper estimate of the $C^{0}$ norm of $1 -$ a real field for Neumann series expansion.

Based on: the $H_2^2$ Sobolev inequality, namely $\Vert\cdot\Vert_{C^{0}} \leq C\Vert\cdot\Vert_{H_2^2}$ .

Parameters:

	a00	: the (0, 0) multipole component of the field
	H2S2aa	: the $\left<a, a\right>_{H_2^2}$ scalar product

Coeff<GComplex_t> gridripper::multipole::operator*	(	const	GReal_t,
		const Coeff< GComplex_t > &
	)

Multiply a coefficient by a number.

Coeff<GComplex_t> gridripper::multipole::operator*	(	const	GReal_t,
		const Y &
	)

Tensor product of a number and an Y.

Coeff<GReal_t> gridripper::multipole::operator*	(	const	GReal_t,
		const Coeff< GReal_t > &
	)

Multiply a coefficient by a number.

Coeff<GReal_t> gridripper::multipole::operator*	(	const	GReal_t,
		const Yre &
	)

Tensor product of a number and an Yre.

ScalarFieldMP<GComplex_t> gridripper::multipole::operator+	(	const ScalarFieldMP< GComplex_t > &	,
		const ScalarFieldMP< GComplex_t > &
	)

ScalarFieldMP<GComplex_t> algebraic operators.

ScalarFieldMP<GReal_t> gridripper::multipole::operator+	(	const ScalarFieldMP< GReal_t > &	,
		const ScalarFieldMP< GReal_t > &
	)

ScalarFieldMP<GReal_t> algebraic operators.

Coeff<GComplex_t> gridripper::multipole::operator- ( const Y & )

Negative of a coefficient.

Coeff<GReal_t> gridripper::multipole::operator- ( const Yre & )

Negative of a coefficient.

Coeff<GComplex_t> gridripper::multipole::operator/	(	const Coeff< GComplex_t > &	,
		const	GReal_t
	)

Divide a coefficient by a number.

Coeff<GComplex_t> gridripper::multipole::operator/	(	const Y &	,
		const	GReal_t
	)

Tensor quotient of a number and an Y.

Coeff<GReal_t> gridripper::multipole::operator/	(	const Coeff< GReal_t > &	,
		const	GReal_t
	)

Divide a coefficient by a number.

Coeff<GReal_t> gridripper::multipole::operator/	(	const Yre &	,
		const	GReal_t
	)

Tensor quotient of a number and an Yre.

GComplex_t gridripper::multipole::optimalScalingNeumann	(	const GComplex_t &	a00,
		const GComplex_t &	H2S2aa
	)

Scaling factor $a$ , with which $\Vert1-a^{-1}A\Vert^2_{H_2^2}$ is minimized for a complex field $A$ on sphere.

It is: $a=\frac{\Vert A\Vert^{2}_{H_2^2}}{\sqrt{4\pi}|A_{0}^{0}|^{2}}A_{0}^{0}$ .

Parameters:

	a00	: the (0, 0) multipole component of the field
	H2S2aa	: the $\left<a, a\right>_{H_2^2}$ scalar product

GReal_t gridripper::multipole::optimalScalingNeumann	(	const GReal_t	a00,
		const GReal_t	H2S2aa
	)

Scaling factor $a$ , with which $\Vert1-a^{-1}A\Vert^2_{H_2^2}$ is minimized for a real field $A$ on sphere.

It is: $a=\frac{\Vert A\Vert^{2}_{H_2^2}}{\sqrt{4\pi}|A_{0}^{0}|^{2}}A_{0}^{0}$ .

Parameters:

	a00	: the (0, 0) multipole component of the field
	H2S2aa	: the $\left<a, a\right>_{H_2^2}$ scalar product

ScalarFieldMP<GComplex_t> gridripper::multipole::partial_phi ( const ScalarFieldMP< GComplex_t > & )

$\varphi$ derivative on sphere for complex scalar field.

Based on: $\partial_{\varphi} Y_{l}^{m} = \mathrm{i}m Y_{l}^{m}$ .

ScalarFieldMP<GReal_t> gridripper::multipole::partial_phi ( const ScalarFieldMP< GReal_t > & )

$\varphi$ derivative on sphere for real scalar field.

Based on: $\partial_{\varphi} Yre_{l}^{m} = -m Yre_{l}^{-m}$ .

ScalarFieldMP<GComplex_t> gridripper::multipole::partial_theta ( const ScalarFieldMP< GComplex_t > & )

$\partial_{\vartheta}$ derivative on sphere for complex field.

Based on the stepping operators and on the matrix elements of the multiplication operator by $\mathrm{e}^{\pm\mathrm{i}\varphi}$ (Gaunt.h and Gaunt.cxx). Numerically verified by a Maple code.

ScalarFieldMP<GComplex_t> gridripper::multipole::partial_x ( const ScalarFieldMP< GComplex_t > & )

$(\partial_{1}(\cdot)\circ(\arccos,\mathrm{id}))\circ(\cos,\mathrm{id})$ derivative on sphere for complex scalar field.

(' $x$ '-derivative, $x=\cos(\vartheta)$ .) From Journal of Modern Optics 36 (1989) 685: $P_{l+1}^{0'}-P_{l-1}^{0'} = (2l+1) P_{l}^{0}$ (easily verified from Rodriguez formula). Thus: $P_{l}^{0'} = \sum_{k=0}^{(l-l\%2)/2-(1-l\%2)} (2(2k+(1-l\%2))+1) P_{2k+(1-l\%2)}^{0}$ . Thus: $(\partial_{1} Y_{l}^{0}\circ(\arccos,\mathrm{id}))\circ(\cos,\mathrm{id}) = (\sqrt{2l+1} \sum_{k=0}^{(l-l\%2)/2-(1-l\%2)} \sqrt{2(2k+(1-l\%2))+1} Y_{2k+(1-l\%2)}^{0})$ . Not implemented for $m\neq0$ . Not even all $Y_{l}^{m}$ is in its domain in the $L^2$ sense. Numerically verified using a C++ code.

ScalarFieldMP<GReal_t> gridripper::multipole::partial_x ( const ScalarFieldMP< GReal_t > & )

( $\partial_{1}(\cdot)\circ(\arccos,\mathrm{id}))\circ(\cos,\mathrm{id})$ derivative on sphere for real scalar field.

(' $x$ '-derivative, $x=\cos(\vartheta)$ .) Based on the rule for Y basis. Not implemented for $m\neq0$ . Not even all $Yre_{l}^{m}$ is in its domain in the L^2 sense if $m\neq0$ . Numerically verified using a C++ code.

ScalarFieldMP<GReal_t> gridripper::multipole::real ( const ScalarFieldMP< GComplex_t > & )

Real part of a scalarfield on sphere.

ScalarFieldMP<GComplex_t> gridripper::multipole::reflect ( const ScalarFieldMP< GComplex_t > & )

Reflect with respect to origin (parity operator).

Based on: $P Y_{l}^{m} = (-1)^{l} Y_{l}^{m}$ .

ScalarFieldMP<GReal_t> gridripper::multipole::reflect ( const ScalarFieldMP< GReal_t > & )

Reflect with respect to origin (parity operator).

Based on: $P Yre_{l}^{m} = (-1)^{l} Yre_{l}^{m}$ .

ScalarFieldMP<GComplex_t> gridripper::multipole::sinTheta_partial_theta ( const ScalarFieldMP< GComplex_t > & )

$\sin(\vartheta)\partial_{\vartheta}$ derivative on sphere for complex scalar field.

Based on: $-\sin(\vartheta)\partial_{\vartheta} Y_{l}^{m} = (l+1)\sqrt{\frac{(l-m)(l+m)}{(2l-1)(2l+1)}} Y_{l-1}^{m} - l\sqrt{\frac{(l+1-m)(l+1+m)}{(2l+1)(2l+3)}} Y_{l+1}^{m}$ . Numerically verified using a C++ code.

ScalarFieldMP<GReal_t> gridripper::multipole::sinTheta_partial_theta ( const ScalarFieldMP< GReal_t > & )

$\sin(\vartheta)\partial_{\vartheta}$ derivative on sphere for real scalar field.

Based on the rule for Y basis. Numerically verified using a C++ code.

GReal_t gridripper::multipole::SobolevConstantS2 ( const int m )

The minimal Sobolev constant in the inequality over the sphere: $\Vert\cdot\Vert_{C^{0}} \leq C \Vert\cdot\Vert_{H^{2}_{\mu}}$ .

It is the sum $\sqrt{\frac{1}{4\pi}\sum_{l=0}^{\infty} (2l+1)\frac{l(l+1)-1}{(l(l+1))^{\mu+1}-1}}$ .

ScalarFieldMP<GComplex_t> gridripper::multipole::Sqr_partial_phi ( const ScalarFieldMP< GComplex_t > & )

Second $\varphi$ derivative on sphere for complex scalar field.

Based on: $\partial_{\varphi}\partial_{\varphi} Y_{l}^{m} = -m^{2} Y_{l}^{m}$ .

ScalarFieldMP<GReal_t> gridripper::multipole::Sqr_partial_phi ( const ScalarFieldMP< GReal_t > & )

Second $\varphi$ derivative on sphere for real scalar field.

Based on: $\partial_{\varphi}\partial_{\varphi} Yre_{l}^{m} = -m^{2} Yre_{l}^{m}$ .

ScalarFieldMP<GComplex_t> gridripper::multipole::stepping	(	const ScalarFieldMP< GComplex_t > &	,
		const int	upOrDown
	)

Stepping up and down operators.

Based on: $\mathrm{e}^{\mathrm{i}\varphi} (\partial_{\vartheta} + \cot(\vartheta)\mathrm{i}\partial_{\phi}) Y_{l}^{m} = \sqrt{l(l+1)-m(m+1)} Y_{l}^{m+1}$ (stepping up, $L_{+}$ operator). $\mathrm{e}^{-\mathrm{i}\varphi} (-\partial_{\vartheta} + \cot(\vartheta)\mathrm{i}\partial_{\phi}) Y_{l}^{m} = \sqrt{l(l+1)-m(m-1)} Y_{l}^{m-1}$ (stepping down, $L_{-}$ operator).

Numerically verified using a C++ code.

GReal_t gridripper::multipole::tailErrorBoundS2	(	const ScalarFieldMP< GComplex_t > &	a,
		const int	mup,
		const int	mu,
		int *	lmax,
		int *	mmax
	)

An upper bound for the tail sum in the $H_{\mu}^{2}$ norm, assuming that the field is in the space $H_{\mu'}^{2}$ .

In this case, its multipole coefficients may be bound by the sequence $l \mapsto K \cdot \sqrt{\frac{l(l+1)-1}{(l(l+1))^{\mu'+1}-1}}\frac{1}{2l+1}$ . By adjusting $K$ to the stored coefficients, and by summing up the tail of this bound in the $H_{\mu}^2$ norm, one gets an upper estimate for the $H_{\mu}^2$ norm of the tail error.

GReal_t gridripper::multipole::tailErrorBoundS2	(	const ScalarFieldMP< GReal_t > &	,
		const int	mup,
		const int	mu,
		int *	lmax,
		int *	mmax
	)

An upper bound for the tail sum in $H_{\mu}^{2}$ norm, assuming that the field is in the space $H_{\mu'}^2$ .

Based on: same rule for complex.

GReal_t gridripper::multipole::YreYreYre_coupling	(	const int	l1,
		const int	l2,
		const int	l3,
		const int	m1,
		const int	m2,
		const int	m3
	)

The same coupling for Yre basis.

They are real valued as Yre -s are real valued. Zero if not: 1) $l_1\geq 0$ and $l_2\geq 0$ and $l_3\geq 0$ , and 2) $|m_1|\leq l_1$ and $|m2|\leq l_2$ and $|m_3|\leq l_3$ , and 3) $|l_1-l_2|\leq l_3$ and $l_3\leq l_1+l_2$ , and 4) only zero or two of the $\{m_1, m_2, m_3\}$ are negative, and 5) $(l_1+l_2+l_3)\%2=0$ . See Bases.h for definition of Yre basis.

Not tested numerically. Not used so far in multiplication operator.

Version:: 0.5, 01/31/2007

Since:: GridRipper 0.5

Author:: Andras Laszlo

GComplex_t gridripper::multipole::YYreY_coupling	(	const int	l1,
		const int	l2,
		const int	l3,
		const int	m1,
		const int	m2,
		const int	m3
	)

Similar coupling for Y and Yre basis.

They are not necessarily real valued. Zero if not: 1) $l_1\geq 0$ and $l_2\geq 0$ and $l_3\geq 0$ , and 2) $|m_1|\leq l_1$ and $|m_2|\leq l_2$ and $|m_3|\leq l_3$ , and 3) $|l_1-l_2|\leq l_3$ and $l_3\leq l_1+l_2$ , and 4) $m_1+m_2+m_3=0$ or $m_1-m_2+m_3=0$ , and 5) $(l_1+l_2+l_3)\%2=0$ .

Not tested numerically. Not used so far in multiplication operator.

Version:: 0.5, 01/31/2007

Since:: GridRipper 0.5

Author:: Andras Laszlo

GReal_t gridripper::multipole::YYY_coupling	(	const int	l1,
		const int	l2,
		const int	l3,
		const int	m1,
		const int	m2,
		const int	m3
	)

The $\int_{0}^{\pi}\int_{0}^{2\pi} Y_{l_1}^{m_1}(\vartheta, \varphi) Y_{l_2}^{m_2}(\vartheta, \varphi) Y_{l_3}^{m_3}(\vartheta, \varphi) \;\sin(\vartheta)\mathrm{d}\vartheta\,\mathrm{d}\varphi$ spherical harmonic function Gaunt coefficient calculator routine.

Zero if not: 1) $l_1\geq 0$ and $l_2\geq 0$ and $l_3\geq 0$ , and 2) $|m_1|\leq l_1$ and $|m_2|\leq l_2$ and $|m_3|\leq l_3$ , and 3) $|l_1-l_2|\leq l_3$ and $l_3\leq l_1+l_2$ , and 4) $m_1+m_2+m_3=0$ , and 5) $(l_1+l_2+l_3)\%2=0$ . Thus, by 4), they are real valued, which is not obvious otherwise. See Bases.h for definition of Y basis.

Numerically verified using Maple.

Version:: 0.5, 01/31/2007

Since:: GridRipper 0.5

Author:: Andras Laszlo

unsigned int gridripper::multipole::YYY_coupling_size ( )

Show table size currently used in YYY_coupling.


Classes
class	Y
	Class for spherical harmonic functions. More...
class	Yre
	Class for real spherical harmonic functions. More...
class	Coeff< GReal_t >
	Class for real spherical harmonic (Yre) expansion coefficients. More...
class	Coeff< GComplex_t >
	Class for complex spherical harmonic (Y) expansion coefficients. More...
class	ScalarFieldMP< GReal_t >
	Class for real scalar fields on sphere. More...
class	ScalarFieldMP< GComplex_t >
	Class for complex scalar fields on sphere. More...
Functions
Coeff< GReal_t >	operator* (const GReal_t, const Yre &)
	Tensor product of a number and an Yre.
Coeff< GReal_t >	operator* (const Yre &, const GReal_t)
Coeff< GReal_t >	operator/ (const Yre &, const GReal_t)
	Tensor quotient of a number and an Yre.
Coeff< GReal_t >	operator* (const GReal_t, const Coeff< GReal_t > &)
	Multiply a coefficient by a number.
Coeff< GReal_t >	operator* (const Coeff< GReal_t > &, const GReal_t)
Coeff< GReal_t >	operator/ (const Coeff< GReal_t > &, const GReal_t)
	Divide a coefficient by a number.
Coeff< GReal_t >	operator- (const Yre &)
	Negative of a coefficient.
Coeff< GReal_t >	operator- (const Coeff< GReal_t > &)
Coeff< GComplex_t >	operator* (const GReal_t, const Y &)
	Tensor product of a number and an Y.
Coeff< GComplex_t >	operator* (const Y &, const GReal_t)
Coeff< GComplex_t >	operator* (const GComplex_t &, const Y &)
Coeff< GComplex_t >	operator* (const Y &, const GComplex_t &)
Coeff< GComplex_t >	operator/ (const Y &, const GReal_t)
	Tensor quotient of a number and an Y.
Coeff< GComplex_t >	operator/ (const Y &, const GComplex_t &)
Coeff< GComplex_t >	operator* (const GReal_t, const Coeff< GComplex_t > &)
	Multiply a coefficient by a number.
Coeff< GComplex_t >	operator* (const Coeff< GComplex_t > &, const GReal_t)
Coeff< GComplex_t >	operator* (const GComplex_t &, const Coeff< GComplex_t > &)
Coeff< GComplex_t >	operator* (const Coeff< GComplex_t > &, const GComplex_t &)
Coeff< GComplex_t >	operator/ (const Coeff< GComplex_t > &, const GReal_t)
	Divide a coefficient by a number.
Coeff< GComplex_t >	operator/ (const Coeff< GComplex_t > &, const GComplex_t &)
Coeff< GComplex_t >	operator- (const Y &)
	Negative of a coefficient.
Coeff< GComplex_t >	operator- (const Coeff< GComplex_t > &)
Coeff< GComplex_t >	conj (const Y &)
	Complex conjugate a coefficient.
Coeff< GComplex_t >	conj (const Coeff< GComplex_t > &)
GReal_t	YYY_coupling (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3)
	The $\int_{0}^{\pi}\int_{0}^{2\pi} Y_{l_1}^{m_1}(\vartheta, \varphi) Y_{l_2}^{m_2}(\vartheta, \varphi) Y_{l_3}^{m_3}(\vartheta, \varphi) \;\sin(\vartheta)\mathrm{d}\vartheta\,\mathrm{d}\varphi$ spherical harmonic function Gaunt coefficient calculator routine.
unsigned int	YYY_coupling_size ()
	Show table size currently used in YYY_coupling.
GReal_t	YreYreYre_coupling (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3)
	The same coupling for Yre basis.
GComplex_t	YYreY_coupling (const int l1, const int l2, const int l3, const int m1, const int m2, const int m3)
	Similar coupling for Y and Yre basis.
GReal_t	intYl1ExpMIPhi (const int l)
	Calculation of $\int_{0}^{2\pi}\int_{0}^{\pi} Y_{l}^{1}(\vartheta, \varphi) \exp(-\mathrm{i}\varphi) \sin(\vartheta)\;\mathrm{d}\vartheta\,\mathrm{d}\varphi$ .
GReal_t	ExpPmIPhiYY_coupling (const int l1, const int m1, const int l2, const int sign)
	The overlap integrals $\int_{0}^{\pi}\int_{0}^{2\pi} \mathrm{e}^{\pm\mathrm{i}\varphi} Y_{l_1}^{m_1}(\vartheta, \varphi) Y_{l_2}^{-(m_1\pm1)}(\vartheta, \varphi) \;\sin(\vartheta)\mathrm{d}\vartheta\,\mathrm{d}\varphi$ .
unsigned int	ExpPmIPhiYY_coupling_size ()
	Show table size currently used in ExpPmIPhiYY_coupling.
GReal_t	intYl0 (const int l, const GReal_t theta)
	Calculation of $\int_{[0,\theta]\times[0,2\pi]} Y_{l}^{0} \;\mathrm{d}\lambda_{S^2}$ .
GReal_t	intBarYl1mYl2m (const int l1, const int l2, const int m, const GReal_t theta, tvector< tvector< tvector< GReal_t > > > *coeffstorage)
	Calculation of $\int_{[0,\theta]\times[0,2\pi]} \bar{Y}_{l_1}^{m} Y_{l_2}^{m} \;\mathrm{d}\lambda_{S^2}$ .
GReal_t	SobolevConstantS2 (const int m)
	The minimal Sobolev constant in the inequality over the sphere: $\Vert\cdot\Vert_{C^{0}} \leq C \Vert\cdot\Vert_{H^{2}_{\mu}}$ .
GReal_t	tailSumS2 (const int mup, const int mu, const int L)
	The sum $\sum_{l=L}^{\infty} \sum_{m=-l}^{l} \frac{(l(l+1))^{\mu+1}-1}{(l(l+1))^{\mu^{'}+1}-1} \frac{1}{(2l+1)^{2}}$ .
ScalarFieldMP< GReal_t >	operator+ (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &)
	ScalarFieldMP<GReal_t> algebraic operators.
ScalarFieldMP< GReal_t >	operator- (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &)
ScalarFieldMP< GReal_t >	operator- (const ScalarFieldMP< GReal_t > &)
ScalarFieldMP< GReal_t >	operator* (const ScalarFieldMP< GReal_t > &, const GReal_t)
ScalarFieldMP< GReal_t >	operator* (const GReal_t, const ScalarFieldMP< GReal_t > &)
ScalarFieldMP< GReal_t >	operator* (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &)
ScalarFieldMP< GReal_t >	operator/ (const ScalarFieldMP< GReal_t > &, const GReal_t)
GReal_t	L2S2 (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &)
	L^2 scalarproduct of two fields on sphere.
GReal_t	H2S2 (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GReal_t > &, const int mu)
	$H_{\mu}^2$ Sobolev scalarproduct of two fields on sphere.
ScalarFieldMP< GReal_t >	reflect (const ScalarFieldMP< GReal_t > &)
	Reflect with respect to origin (parity operator).
ScalarFieldMP< GReal_t >	mulCosTheta (const ScalarFieldMP< GReal_t > &)
	Multiplication operator by $\cos(\vartheta)$ .
ScalarFieldMP< GReal_t >	mulCosThetaSqr (const ScalarFieldMP< GReal_t > &)
	Multiplication operator by $\cos(\vartheta)^2$ .
ScalarFieldMP< GReal_t >	mulSinThetaSqr (const ScalarFieldMP< GReal_t > &)
	Multiplication operator by $\sin(\vartheta)^2$ .
ScalarFieldMP< GReal_t >	partial_phi (const ScalarFieldMP< GReal_t > &)
	$\varphi$ derivative on sphere for real scalar field.
ScalarFieldMP< GReal_t >	Sqr_partial_phi (const ScalarFieldMP< GReal_t > &)
	Second $\varphi$ derivative on sphere for real scalar field.
ScalarFieldMP< GReal_t >	sinTheta_partial_theta (const ScalarFieldMP< GReal_t > &)
	$\sin(\vartheta)\partial_{\vartheta}$ derivative on sphere for real scalar field.
ScalarFieldMP< GReal_t >	partial_x (const ScalarFieldMP< GReal_t > &)
	( $\partial_{1}(\cdot)\circ(\arccos,\mathrm{id}))\circ(\cos,\mathrm{id})$ derivative on sphere for real scalar field.
ScalarFieldMP< GReal_t >	LaplaceS2 (const ScalarFieldMP< GReal_t > &)
	Laplace operator on unit sphere for real scalar field.
GReal_t	normBoundNeumann (const GReal_t a00, const GReal_t H2S2aa)
	Upper estimate of the $C^{0}$ norm of a real field for Neumann series expansion.
GReal_t	optimalScalingNeumann (const GReal_t a00, const GReal_t H2S2aa)
	Scaling factor , with which $\Vert1-a^{-1}A\Vert^2_{H_2^2}$ is minimized for a real field on sphere.
ScalarFieldMP< GReal_t >	divideByNeumann (const ScalarFieldMP< GReal_t > &a, const ScalarFieldMP< GReal_t > &f, const GReal_t norm, const GReal_t errortolerance)
	Division of a real field by a real field, using Neumann series.
GReal_t	tailErrorBoundS2 (const ScalarFieldMP< GReal_t > &, const int mup, const int mu, int lmax, int mmax)
	An upper bound for the tail sum in $H_{\mu}^{2}$ norm, assuming that the field is in the space $H_{\mu'}^2$ .
ScalarFieldMP< GComplex_t >	operator+ (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &)
	ScalarFieldMP<GComplex_t> algebraic operators.
ScalarFieldMP< GComplex_t >	operator+ (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &)
ScalarFieldMP< GComplex_t >	operator+ (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &)
ScalarFieldMP< GComplex_t >	operator- (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &)
ScalarFieldMP< GComplex_t >	operator- (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &)
ScalarFieldMP< GComplex_t >	operator- (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &)
ScalarFieldMP< GComplex_t >	operator- (const ScalarFieldMP< GComplex_t > &)
ScalarFieldMP< GComplex_t >	operator* (const ScalarFieldMP< GComplex_t > &, const GReal_t)
ScalarFieldMP< GComplex_t >	operator* (const GReal_t, const ScalarFieldMP< GComplex_t > &)
ScalarFieldMP< GComplex_t >	operator* (const ScalarFieldMP< GComplex_t > &, const GComplex_t &)
ScalarFieldMP< GComplex_t >	operator* (const GComplex_t &, const ScalarFieldMP< GComplex_t > &)
ScalarFieldMP< GComplex_t >	operator* (const ScalarFieldMP< GReal_t > &, const GComplex_t &)
ScalarFieldMP< GComplex_t >	operator* (const GComplex_t &, const ScalarFieldMP< GReal_t > &)
ScalarFieldMP< GComplex_t >	operator* (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &)
ScalarFieldMP< GComplex_t >	operator* (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &)
ScalarFieldMP< GComplex_t >	operator* (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &)
ScalarFieldMP< GComplex_t >	operator/ (const ScalarFieldMP< GComplex_t > &, const GReal_t)
ScalarFieldMP< GComplex_t >	operator/ (const ScalarFieldMP< GComplex_t > &, const GComplex_t &)
ScalarFieldMP< GComplex_t >	operator/ (const ScalarFieldMP< GReal_t > &, const GComplex_t &)
ScalarFieldMP< GComplex_t >	conj (const ScalarFieldMP< GComplex_t > &)
	Complex conjugate a scalarfield on sphere.
ScalarFieldMP< GReal_t >	real (const ScalarFieldMP< GComplex_t > &)
	Real part of a scalarfield on sphere.
ScalarFieldMP< GReal_t >	imag (const ScalarFieldMP< GComplex_t > &)
	Imaginary part of a scalarfield on sphere.
GComplex_t	L2S2 (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &)
	L^2 scalarproduct of fields.
GComplex_t	L2S2 (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &)
GComplex_t	L2S2 (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &)
GComplex_t	H2S2 (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GReal_t > &, const int mu)
	$H_{\mu}^2$ Sobolev scalarproduct of two fields on sphere.
GComplex_t	H2S2 (const ScalarFieldMP< GReal_t > &, const ScalarFieldMP< GComplex_t > &, const int mu)
GComplex_t	H2S2 (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &, const int mu)
GComplex_t	L2S2limited (const ScalarFieldMP< GComplex_t > &, const ScalarFieldMP< GComplex_t > &, const GReal_t theta, tvector< tvector< tvector< GReal_t > > > *coeffstorage)
	scalarproduct of fields, with limited integration domain to $[0,\theta]\times[0,2\pi]$ .
ScalarFieldMP< GComplex_t >	reflect (const ScalarFieldMP< GComplex_t > &)
	Reflect with respect to origin (parity operator).
ScalarFieldMP< GComplex_t >	stepping (const ScalarFieldMP< GComplex_t > &, const int upOrDown)
	Stepping up and down operators.
ScalarFieldMP< GComplex_t >	mulCosTheta (const ScalarFieldMP< GComplex_t > &)
	Multiplication operator by $\cos(\vartheta)$ .
ScalarFieldMP< GComplex_t >	mulCosThetaSqr (const ScalarFieldMP< GComplex_t > &)
	Multiplication operator by $\cos(\vartheta)^2$ .
ScalarFieldMP< GComplex_t >	mulSinThetaSqr (const ScalarFieldMP< GComplex_t > &)
	Multiplication operator by $\sin(\vartheta)^2$ .
ScalarFieldMP< GComplex_t >	mulExpPmIPhi (const ScalarFieldMP< GComplex_t > &, const int sign)
	Multiplication operator by $\mathrm{e}^{\pm\mathrm{i}\varphi}$ .
ScalarFieldMP< GComplex_t >	partial_phi (const ScalarFieldMP< GComplex_t > &)
	$\varphi$ derivative on sphere for complex scalar field.
ScalarFieldMP< GComplex_t >	Sqr_partial_phi (const ScalarFieldMP< GComplex_t > &)
	Second $\varphi$ derivative on sphere for complex scalar field.
ScalarFieldMP< GComplex_t >	sinTheta_partial_theta (const ScalarFieldMP< GComplex_t > &)
	$\sin(\vartheta)\partial_{\vartheta}$ derivative on sphere for complex scalar field.
ScalarFieldMP< GComplex_t >	partial_theta (const ScalarFieldMP< GComplex_t > &)
	$\partial_{\vartheta}$ derivative on sphere for complex field.
ScalarFieldMP< GComplex_t >	partial_x (const ScalarFieldMP< GComplex_t > &)
	$(\partial_{1}(\cdot)\circ(\arccos,\mathrm{id}))\circ(\cos,\mathrm{id})$ derivative on sphere for complex scalar field.
ScalarFieldMP< GComplex_t >	LaplaceS2 (const ScalarFieldMP< GComplex_t > &)
	Laplace operator on unit sphere for complex scalar field.
GReal_t	normBoundNeumann (const GComplex_t &a00, const GComplex_t &H2S2aa)
	Upper estimate of the $C^{0}$ norm of a complex field for Neumann series.
GComplex_t	optimalScalingNeumann (const GComplex_t &a00, const GComplex_t &H2S2aa)
	Scaling factor , with which $\Vert1-a^{-1}A\Vert^2_{H_2^2}$ is minimized for a complex field $A$ on sphere.
ScalarFieldMP< GComplex_t >	divideByNeumann (const ScalarFieldMP< GComplex_t > &a, const ScalarFieldMP< GReal_t > &f, const GReal_t norm, const GReal_t errortolerance)
	Division of a complex field by a complex field, using Neumann series.
ScalarFieldMP< GComplex_t >	divideByNeumann (const ScalarFieldMP< GReal_t > &a, const ScalarFieldMP< GComplex_t > &f, const GReal_t norm, const GReal_t errortolerance)
ScalarFieldMP< GComplex_t >	divideByNeumann (const ScalarFieldMP< GComplex_t > &a, const ScalarFieldMP< GComplex_t > &f, const GReal_t norm, const GReal_t errortolerance)
GReal_t	tailErrorBoundS2 (const ScalarFieldMP< GComplex_t > &a, const int mup, const int mu, int lmax, int mmax)
	An upper bound for the tail sum in the $H_{\mu}^{2}$ norm, assuming that the field is in the space $H_{\mu'}^{2}$ .
void	maxEval (const unsigned int)
	Integration error tolerance parameters for functional field expansion by AGM routine.
unsigned int	maxEval ()
void	reqAbsError (const GReal_t)
GReal_t	reqAbsError ()
void	reqRelError (const GReal_t)
GReal_t	reqRelError ()
Variables
const int	SobolevOrderS2 = 1000000
	Maximal series expansion order for the numerical calculation of Sobolev related constants over the sphere.