Spherical Harmonic transform functions.

All these function perform a global spherical harmonic transform. More...

Macros
#define	SH_to_grad_spat(shtns, S, Gt, Gp)
	Compute the spatial representation of the gradient of a scalar SH field.
#define	SH_to_grad_spat_l(shtns, S, Gt, Gp, ltr)
	Compute the spatial representation of the gradient of a scalar SH field.
#define	SH_to_grad_spat_ml(shtns, im, S, Gt, Gp, ltr)
	Compute the spatial representation of the gradient of a scalar SH field.

Scalar transforms
void	spat_to_SH (shtns_cfg shtns, double *Vr, cplx *Qlm)
	transform the scalar field Vr into its spherical harmonic representation Qlm.
void	SH_to_spat (shtns_cfg shtns, cplx *Qlm, double *Vr)
	transform the spherical harmonic coefficients Qlm into its spatial representation Vr.
void	SH_to_spat_cplx (shtns_cfg shtns, cplx *alm, cplx *z)
	complex scalar synthesis.
void	spat_cplx_to_SH (shtns_cfg shtns, cplx *z, cplx *alm)
	complex scalar analysis.

2D vector transforms
void	spat_to_SHsphtor (shtns_cfg, double *Vt, double *Vp, cplx *Slm, cplx *Tlm)
	transform the theta and phi components (Vt,Vp) of a vector into its spheroidal-toroidal spherical harmonic representation (Slm,Tlm).
void	SHsphtor_to_spat (shtns_cfg, cplx *Slm, cplx *Tlm, double *Vt, double *Vp)
	transform spheroidal-toroidal spherical harmonic coefficients (Slm,Tlm) to the spatial theta and phi components (Vt,Vp).
void	SHsph_to_spat (shtns_cfg, cplx *Slm, double *Vt, double *Vp)
	transform spheroidal spherical harmonic coefficients Slm to the spatial theta and phi components (Vt,Vp), effectively computing the gradient of S.
void	SHtor_to_spat (shtns_cfg, cplx *Tlm, double *Vt, double *Vp)
	transform toroidal spherical harmonic coefficients Tlm to the spatial theta and phi components (Vt,Vp).
void	spat_cplx_to_SHsphtor (shtns_cfg, cplx *Vt, cplx *Vp, cplx *Slm, cplx *Tlm)
	transform the theta and phi components (Vt,Vp) of a complex valued vector into its spheroidal-toroidal spherical harmonic representation (Slm,Tlm).
void	SHsphtor_to_spat_cplx (shtns_cfg, cplx *Slm, cplx *Tlm, cplx *Vt, cplx *Vp)
	transform spheroidal-toroidal spherical harmonic coefficients (Slm,Tlm) to the spatial theta and phi complex-valued components (Vt,Vp).

3D transforms (combine scalar and vector)
void	spat_to_SHqst (shtns_cfg, double *Vr, double *Vt, double *Vp, cplx *Qlm, cplx *Slm, cplx *Tlm)
	3D vector transform from spherical coordinates to radial-spheroidal-toroidal spectral components (see Radial - Spheroidal - Toroidal decomposition).
void	SHqst_to_spat (shtns_cfg, cplx *Qlm, cplx *Slm, cplx *Tlm, double *Vr, double *Vt, double *Vp)
	3D vector transform from radial-spheroidal-toroidal spectral components (see Radial - Spheroidal - Toroidal decomposition) to spherical coordinates.
void	spat_cplx_to_SHqst (shtns_cfg, cplx *Vr, cplx *Vt, cplx *Vp, cplx *Qlm, cplx *Slm, cplx *Tlm)
	complex vector transform (3D).
void	SHqst_to_spat_cplx (shtns_cfg, cplx *Qlm, cplx *Slm, cplx *Tlm, cplx *Vr, cplx *Vt, cplx *Vp)
	complex vector transform (3D).

Truncated transforms at given degree l
with ltr <= lmax used for setup.
void	spat_to_SH_l (shtns_cfg, double *Vr, cplx *Qlm, int ltr)
void	SH_to_spat_l (shtns_cfg, cplx *Qlm, double *Vr, int ltr)
void	SHsphtor_to_spat_l (shtns_cfg, cplx *Slm, cplx *Tlm, double *Vt, double *Vp, int ltr)
void	SHsph_to_spat_l (shtns_cfg, cplx *Slm, double *Vt, double *Vp, int ltr)
void	SHtor_to_spat_l (shtns_cfg, cplx *Tlm, double *Vt, double *Vp, int ltr)
void	spat_to_SHsphtor_l (shtns_cfg, double *Vt, double *Vp, cplx *Slm, cplx *Tlm, int ltr)
void	spat_to_SHqst_l (shtns_cfg, double *Vr, double *Vt, double *Vp, cplx *Qlm, cplx *Slm, cplx *Tlm, int ltr)
void	SHqst_to_spat_l (shtns_cfg, cplx *Qlm, cplx *Slm, cplx *Tlm, double *Vr, double *Vt, double *Vp, int ltr)

Legendre transform at given m (no fft) and truncated at given degree l <= lmax
The input and output arrays contain only the specified m=im*mres, that is spatial size is nlat and spectral size is lmax+1-m, containing only the coefficients of the given m.
void	spat_to_SH_ml (shtns_cfg, int im, cplx *Vr, cplx *Ql, int ltr)
void	SH_to_spat_ml (shtns_cfg, int im, cplx *Ql, cplx *Vr, int ltr)
void	spat_to_SHsphtor_ml (shtns_cfg, int im, cplx *Vt, cplx *Vp, cplx *Sl, cplx *Tl, int ltr)
void	SHsphtor_to_spat_ml (shtns_cfg, int im, cplx *Sl, cplx *Tl, cplx *Vt, cplx *Vp, int ltr)
void	SHsph_to_spat_ml (shtns_cfg, int im, cplx *Sl, cplx *Vt, cplx *Vp, int ltr)
void	SHtor_to_spat_ml (shtns_cfg, int im, cplx *Tl, cplx *Vt, cplx *Vp, int ltr)
void	spat_to_SHqst_ml (shtns_cfg, int im, cplx *Vr, cplx *Vt, cplx *Vp, cplx *Ql, cplx *Sl, cplx *Tl, int ltr)
void	SHqst_to_spat_ml (shtns_cfg, int im, cplx *Ql, cplx *Sl, cplx *Tl, cplx *Vr, cplx *Vt, cplx *Vp, int ltr)

Detailed Description

All these function perform a global spherical harmonic transform.

Their first argument is a shtns_cfg variable (which is a pointer to a shtns_info struct) obtained by a previous call to shtns_create or shtns_init.

See also

Spatial data layouts and grids used by SHTns

Spherical Harmonics storage and normalization

Vector Spherical Harmonics as implemented in SHTns

Macro Definition Documentation

SH_to_grad_spat

#define SH_to_grad_spat	(		shtns,
			S,
			Gt,
			Gp )

Value:

SHsph_to_spat(shtns, S, Gt, Gp)

Compute the spatial representation of the gradient of a scalar SH field.

Alias for SHsph_to_spat

SH_to_grad_spat_l

#define SH_to_grad_spat_l	(		shtns,
			S,
			Gt,
			Gp,
			ltr )

Value:

SHsph_to_spat_l(shtns, S, Gt, Gp, ltr)

Compute the spatial representation of the gradient of a scalar SH field.

Alias for SHsph_to_spat_l

SH_to_grad_spat_ml

#define SH_to_grad_spat_ml	(		shtns,
			im,
			S,
			Gt,
			Gp,
			ltr )

Value:

SHsph_to_spat_ml(shtns, im, S, Gt, Gp, ltr)

Compute the spatial representation of the gradient of a scalar SH field.

Alias for SHsph_to_spat_l

Function Documentation

SH_to_spat()

void SH_to_spat	(	shtns_cfg	shtns,
		cplx *	Qlm,
		double *	Vr )

transform the spherical harmonic coefficients Qlm into its spatial representation Vr.

Parameters

[in]	shtns	= a configuration created by shtns_create with a grid set by shtns_set_grid or shtns_set_grid_auto
[in]	Qlm	= spherical harmonics coefficients : cplx array of size shtns->nlm.
[out]	Vr	= spatial scalar field : double array of size shtns->nspat.

Examples

SHT_example.c.

SH_to_spat_cplx()

void SH_to_spat_cplx	(	shtns_cfg	shtns,
		cplx *	alm,
		cplx *	z )

complex scalar synthesis.

Parameters

[in]	shtns	= a configuration created by shtns_create with a grid set by shtns_set_grid or shtns_set_grid_auto
[in]	alm[l*(l+1)+m]	is the SH coefficient of order l and degree m (with -l <= m <= l) [total of (LMAX+1)^2 coefficients]
[out]	z	= complex spatial field

complex scalar synthesis.

in: alm[LM_cplx(shtns,l,m)] is the SH coefficients of order l and degree m (with -l <= m <= l) for a total of nlm_cplx_calc(lmax,mmax,mres) coefficients. out: complex spatial field z.

SHqst_to_spat()

void SHqst_to_spat	(	shtns_cfg	shtns,
		cplx *	Qlm,
		cplx *	Slm,
		cplx *	Tlm,
		double *	Vr,
		double *	Vt,
		double *	Vp )

3D vector transform from radial-spheroidal-toroidal spectral components (see Radial - Spheroidal - Toroidal decomposition) to spherical coordinates.

They should be prefered over separate calls to scalar and 2D vector transforms as they can be significantly faster.

SHqst_to_spat_cplx()

void SHqst_to_spat_cplx	(	shtns_cfg	shtns,
		cplx *	qlm,
		cplx *	slm,
		cplx *	tlm,
		cplx *	zr,
		cplx *	zt,
		cplx *	zp )

complex vector transform (3D).

in: {qlm,slm,tlm}[LM_cplx(l,m)] are the SH coefficients of order l and degree m (with -l <= m <= l) out: zr,zt,zp: r,theta,phi components of the complex spatial vector field.

SHsph_to_spat()

void SHsph_to_spat	(	shtns_cfg	shtns,
		cplx *	Slm,
		double *	Vt,
		double *	Vp )

transform spheroidal spherical harmonic coefficients Slm to the spatial theta and phi components (Vt,Vp), effectively computing the gradient of S.

See also

Vector Spherical Harmonics as implemented in SHTns

SHsphtor_to_spat()

void SHsphtor_to_spat	(	shtns_cfg	shtns,
		cplx *	Slm,
		cplx *	Tlm,
		double *	Vt,
		double *	Vp )

transform spheroidal-toroidal spherical harmonic coefficients (Slm,Tlm) to the spatial theta and phi components (Vt,Vp).

See also

Vector Spherical Harmonics as implemented in SHTns

Examples

SHT_example.c.

SHsphtor_to_spat_cplx()

void SHsphtor_to_spat_cplx	(	shtns_cfg	shtns,
		cplx *	slm,
		cplx *	tlm,
		cplx *	zt,
		cplx *	zp )

transform spheroidal-toroidal spherical harmonic coefficients (Slm,Tlm) to the spatial theta and phi complex-valued components (Vt,Vp).

See also

Vector Spherical Harmonics as implemented in SHTns

transform spheroidal-toroidal spherical harmonic coefficients (Slm,Tlm) to the spatial theta and phi complex-valued components (Vt,Vp).

in: slm, tlm are the spheroidal/toroidal SH coefficients of order l and degree m (with -l <= m <= l) out: zt, zp are respectively the theta and phi components of the complex spatial vector field.

SHtor_to_spat()

void SHtor_to_spat	(	shtns_cfg	shtns,
		cplx *	Tlm,
		double *	Vt,
		double *	Vp )

transform toroidal spherical harmonic coefficients Tlm to the spatial theta and phi components (Vt,Vp).

See also

Vector Spherical Harmonics as implemented in SHTns

Examples

SHT_example.c.

spat_cplx_to_SH()

void spat_cplx_to_SH	(	shtns_cfg	shtns,
		cplx *	z,
		cplx *	alm )

complex scalar analysis.

Parameters

[in]	shtns	= a configuration created by shtns_create with a grid set by shtns_set_grid or shtns_set_grid_auto
[in]	z	= complex spatial field
[out]	alm[l*(l+1)+m]	is the SH coefficient of order l and degree m (with -l <= m <= l) [total of (LMAX+1)^2 coefficients]

complex scalar analysis.

in: complex spatial field z. out: alm[LM(shtns,l,m)] is the SH coefficients of order l and degree m (with -l <= m <= l) for a total of nlm_cplx_calc(lmax,mmax,mres) coefficients.

spat_cplx_to_SHqst()

void spat_cplx_to_SHqst	(	shtns_cfg	shtns,
		cplx *	zr,
		cplx *	zt,
		cplx *	zp,
		cplx *	qlm,
		cplx *	slm,
		cplx *	tlm )

complex vector transform (3D).

in: zr,zt,zp are the r,theta,phi components of the complex spatial vector field. out: {qlm,slm,tlm}[LM_cplx(l,m)] are the SH coefficients of order l and degree m (with -l <= m <= l)

spat_cplx_to_SHsphtor()

void spat_cplx_to_SHsphtor	(	shtns_cfg	shtns,
		cplx *	zt,
		cplx *	zp,
		cplx *	slm,
		cplx *	tlm )

transform the theta and phi components (Vt,Vp) of a complex valued vector into its spheroidal-toroidal spherical harmonic representation (Slm,Tlm).

See also

Vector Spherical Harmonics as implemented in SHTns

transform the theta and phi components (Vt,Vp) of a complex valued vector into its spheroidal-toroidal spherical harmonic representation (Slm,Tlm).

zt,zp: theta,phi components of the complex spatial vector field. out: slm[LM_cplx(l,m)] and tlm[LM_cplx(l,m)] are the SH coefficients of order l and degree m (with -l <= m <= l) for a total of shtns->nlm_cplx = nlm_cplx_calc(lmax, mmax, mres) coefficients.

spat_to_SH()

void spat_to_SH	(	shtns_cfg	shtns,
		double *	Vr,
		cplx *	Qlm )

transform the scalar field Vr into its spherical harmonic representation Qlm.

Parameters

[in]	shtns	= a configuration created by shtns_create with a grid set by shtns_set_grid or shtns_set_grid_auto
[in]	Vr	= spatial scalar field : double array of size shtns->nspat; NOT GUARANTEED TO BE PRESERVED.
[out]	Qlm	= spherical harmonics coefficients : cplx array of size shtns->nlm.

Examples

SHT_example.c.

spat_to_SHqst()

void spat_to_SHqst	(	shtns_cfg	shtns,
		double *	Vr,
		double *	Vt,
		double *	Vp,
		cplx *	Qlm,
		cplx *	Slm,
		cplx *	Tlm )

3D vector transform from spherical coordinates to radial-spheroidal-toroidal spectral components (see Radial - Spheroidal - Toroidal decomposition).

They should be prefered over separate calls to scalar and 2D vector transforms as they can be significantly faster.

spat_to_SHsphtor()

void spat_to_SHsphtor	(	shtns_cfg	shtns,
		double *	Vt,
		double *	Vp,
		cplx *	Slm,
		cplx *	Tlm )

transform the theta and phi components (Vt,Vp) of a vector into its spheroidal-toroidal spherical harmonic representation (Slm,Tlm).

See also

Vector Spherical Harmonics as implemented in SHTns