#include "math_funcs.h"
#include <cmath>
#include <iostream>
#include <stdexcept>
#include "array.h"
#include "arts_constants.h"
#include "arts_conversions.h"
#include "logic.h"
#include "mystring.h"

Functions
Numeric	fac (const Index n)
	fac

Index	integer_div (const Index &x, const Index &y)
	integer_div

Numeric	LagrangeInterpol4 (ConstVectorView x, ConstVectorView y, const Numeric a)
	Lagrange Interpolation (internal function).

Numeric	last (ConstVectorView x)
	last

Index	last (const ArrayOfIndex &x)
	last

void	linspace (Vector &x, const Numeric start, const Numeric stop, const Numeric step)
	linspace

void	nlinspace (Vector &x, const Numeric start, const Numeric stop, const Index n)
	nlinspace

void	nlinspace (VectorView x, const Numeric start, const Numeric stop, const Index n)

void	nlogspace (Vector &x, const Numeric start, const Numeric stop, const Index n)
	nlogspace

Numeric	trapz (ConstVectorView x, ConstVectorView y)
	trapz

void	cumsum (VectorView csum, ConstVectorView x)
	cumsum

Numeric	AngIntegrate_trapezoid (ConstMatrixView Integrand, ConstVectorView za_grid, ConstVectorView aa_grid)
	AngIntegrate_trapezoid.

Numeric	AngIntegrate_trapezoid_opti (ConstMatrixView Integrand, ConstVectorView za_grid, ConstVectorView aa_grid, ConstVectorView grid_stepsize)
	AngIntegrate_trapezoid_opti.

Numeric	AngIntegrate_trapezoid (ConstVectorView Integrand, ConstVectorView za_grid)
	AngIntegrate_trapezoid.

Numeric	sign (const Numeric &x)
	sign

void	mgd (VectorView psd, const Vector &x, const Numeric &n0, const Numeric &mu, const Numeric &la, const Numeric &ga)

void	mgd_with_derivatives (VectorView psd, MatrixView jac_data, const Vector &x, const Numeric &n0, const Numeric &mu, const Numeric &la, const Numeric &ga, const bool &do_n0_jac, const bool &do_mu_jac, const bool &do_la_jac, const bool &do_ga_jac)

void	delanoe_shape_with_derivative (VectorView psd, MatrixView jac_data, const Vector &x, const Numeric &alpha, const Numeric &beta)
	! Shape functions for normalized PSD.

Numeric	mod_gamma_dist (Numeric x, Numeric N0, Numeric Lambda, Numeric mu, Numeric gamma)
	Generalized Modified Gamma Distribution.

void	unitl (Vector &x)
	unitl

void	flat (VectorView x, ConstMatrixView X)
	flat

void	flat (VectorView x, ConstTensor3View X)
	flat

void	reshape (Tensor3View X, ConstVectorView x)
	reshape

void	reshape (MatrixView X, ConstVectorView x)
	reshape

void	calculate_weights_linear (Vector &x, Vector &w, const Index nph)
	calculate_weights_linear

void	calculate_int_weights_arbitrary_grid (Vector &w, const Vector &x)
	Calculates trapezoidal integration weights for arbitray grid.

Variables
constexpr Numeric	DEG2RAD =Conversion::deg2rad(1)

constexpr Numeric	PI =Constant::pi

Detailed Description

Author: Patrick Eriksson Patri.nosp@m.ck.E.nosp@m.rikss.nosp@m.on@c.nosp@m.halme.nosp@m.rs.s.nosp@m.e

Date: 2000-09-18

Contains basic mathematical functions.

Definition in file math_funcs.cc.

Function Documentation

◆ AngIntegrate_trapezoid() [1/2]

Numeric AngIntegrate_trapezoid	(	ConstMatrixView	Integrand,
		ConstVectorView	za_grid,
		ConstVectorView	aa_grid
	)

AngIntegrate_trapezoid.

Performs an integration of a matrix over all directions defined in angular grids using the trapezoidal integration method.

Parameters

Integrand	The Matrix to be integrated
za_grid	The zenith angle grid
aa_grid	The azimuth angle grid

Returns: The resulting integral

Definition at line 318 of file math_funcs.cc.

References ARTS_ASSERT, and DEG2RAD.

Referenced by AngIntegrate_trapezoid_opti(), doit_scat_fieldCalc(), doit_scat_fieldCalcLimb(), doit_scat_fieldNormalize(), get_stepwise_scattersky_source(), scat_dataCheck(), and scat_dataReduceT().

◆ AngIntegrate_trapezoid() [2/2]

Numeric AngIntegrate_trapezoid	(	ConstVectorView	Integrand,
		ConstVectorView	za_grid
	)

AngIntegrate_trapezoid.

Performs an integration of a matrix over all directions defined in angular grids using the trapezoidal integration method. The integrand is independant of the azimuth angle. The integration over the azimuth angle gives a 2*PI

Parameters

Integrand	Input : The vector to be integrated
za_grid	Input : The zenith angle grid

Author: Claas Teichmann

Date: 2003-05-13

Returns: The resulting integral

Definition at line 415 of file math_funcs.cc.

References ARTS_ASSERT, DEG2RAD, and PI.

◆ AngIntegrate_trapezoid_opti()

Numeric AngIntegrate_trapezoid_opti	(	ConstMatrixView	Integrand,
		ConstVectorView	za_grid,
		ConstVectorView	aa_grid,
		ConstVectorView	grid_stepsize
	)

AngIntegrate_trapezoid_opti.

Performs an integration of a matrix over all directions defined in angular grids using the trapezoidal integration method.

In addition to the "old fashined" integration method, it checks whether the stepsize is constant. If it is, it uses a faster method, if not, it uses the old one.

Parameters

Integrand	Input : The Matrix to be integrated
za_grid	Input : The zenith angle grid
aa_grid	Input : The azimuth angle grid
grid_stepsize	Input : stepsize of the grid

Returns: The resulting integral

Author: Claas Teichmann claas.nosp@m.@sat.nosp@m..phys.nosp@m.ik.u.nosp@m.ni-br.nosp@m.emen.nosp@m..de

Date: 2003/05/28

Definition at line 362 of file math_funcs.cc.

References AngIntegrate_trapezoid(), ARTS_ASSERT, and DEG2RAD.

Referenced by doit_scat_fieldCalc(), and doit_scat_fieldCalcLimb().

◆ calculate_int_weights_arbitrary_grid()

void calculate_int_weights_arbitrary_grid	(	Vector &	w,
		const Vector &	x
	)

Calculates trapezoidal integration weights for arbitray grid.

Parameters

w	Vector integration weights
x	Vector evaluation points

Definition at line 833 of file math_funcs.cc.

References ARTS_USER_ERROR_IF, and w.

Referenced by iySurfaceLambertian().

◆ calculate_weights_linear()

void calculate_weights_linear	(	Vector &	x,
		Vector &	w,
		const Index	nph
	)

calculate_weights_linear

Function to set the evaluation points and the corresponding weights for numerical integration on the domain from [-1,1]

Parameters:

Parameters

[out]	x	evaluation points \paran[out] w integration weights
[in]	nph	number of evaluation points per hemisphere

Definition at line 815 of file math_funcs.cc.

References nlinspace(), and w.

Referenced by AngularGridsSetFluxCalc().

◆ cumsum()

void cumsum	(	VectorView	csum,
		ConstVectorView	x
	)

cumsum

Cumulative sum of a vector

csum[0] becomes x[0] and last value in csum equals the full sum of x

Parameters

csum	Vector with cumulative values
x	Input vector

Author: Patrick Eriksson

Date: 2022-12-22

Definition at line 297 of file math_funcs.cc.

References ARTS_ASSERT.

Referenced by dlosGauss().

◆ delanoe_shape_with_derivative()

void delanoe_shape_with_derivative	(	VectorView	psd,
		MatrixView	jac_data,
		const Vector &	x,
		const Numeric &	alpha,
		const Numeric &	beta
	)

! Shape functions for normalized PSD.

This function implements the shape function F(X, alpha, beta) from as proposed by Delanoe et al. in "Normalized particle size distribution for remote sensing application".

Parameters

[OUT]	psd On return contains the values of F corresponding to the values in x.
[OUT]	jac_data On return contains the first derivative of F w.r.t x evaluated at the values in x.
[IN]	x The values at which to evaluate the shape functions and and derivatives.
[IN]	alpha The alpha parameter of the shape function.
[IN]	beta The beta parameter of the shape function.

Definition at line 623 of file math_funcs.cc.

Referenced by psdDelanoeEtAl14().

◆ fac()

Numeric fac ( const Index n )

fac

Calculates the factorial.

The function asserts that n must be >= 0

Returns: The factorial

Parameters

n Nominator

Author: Oliver Lemke

Date: 2003-08-15

Definition at line 46 of file math_funcs.cc.

Referenced by LineShape::RosenkranzQuadratic::dNdf(), MatrixGaussian(), MCRadar(), Absorption::LineMixing::SpeciesErrorCorrectedSuddenData::Omega(), LineShape::RosenkranzQuadratic::operator()(), specular_losCalc(), VectorGaussian(), and ze_cfac().

◆ flat() [1/2]

void flat	(	VectorView	x,
		ConstMatrixView	X
	)

flat

Flattens a matrix to a vector

The matrix is read from front, i.e. rows are looped first. In Matlab this equals x=X(:).

Parameters

[out]	x	The vector. Should already be sized
[in]	X	The matrix.

Author: Patrick Eriksson

Date: 2015-09-09

Definition at line 709 of file math_funcs.cc.

References ARTS_ASSERT, and c.

◆ flat() [2/2]

void flat	(	VectorView	x,
		ConstTensor3View	X
	)

flat

Converts Tensor3 to a vector

The matrix is read from front, i.e. pages are looped first, followed by rows. In Matlab this equals x=X(:).

Parameters

[out]	x	The vector. Should already be sized
[in]	X	The tensor.

Author: Patrick Eriksson

Date: 2015-09-09

Definition at line 735 of file math_funcs.cc.

References ARTS_ASSERT, and c.

◆ integer_div()

Index integer_div	(	const Index &	x,
		const Index &	y
	)

integer_div

Performs an integer division.

The function asserts that the reminder of the division x/y is 0.

Returns: The quotient

Parameters

x	Nominator
y	Denominator

Author: Patrick Eriksson

Date: 2002-08-11

Definition at line 70 of file math_funcs.cc.

References ARTS_ASSERT.

◆ LagrangeInterpol4()

Numeric LagrangeInterpol4	(	ConstVectorView	x,
		ConstVectorView	y,
		const Numeric	a
	)

Lagrange Interpolation (internal function).

This function calculates the Lagrange interpolation of four interpolation points as described in Lagrange Interpolating Polynomial.
The input are the four x-axis values [x0,x1,x2,x3] and their associated y-axis values [y0,y1,y2,y3]. Furthermore the x-axis point "a" at which the interpolation should be calculated must be given as input. NOTE that the relation x2 =< x < x3 MUST hold!

Parameters

x	x-vector with four elements [x0,x1,x2,x3]
y	y-vector with four elements: yj = y(xj), j=0,1,2,3
a	interpolation point on the x-axis with x1 =< a < x2

Returns: FIXME

Author: Thomas Kuhn

Date: 2003-11-25

Definition at line 96 of file math_funcs.cc.

References a, ARTS_USER_ERROR_IF, and b.

◆ last() [1/2]

Index last ( const ArrayOfIndex & x )

last

Returns the last value of an index array.

Returns: The last value of x.

Parameters

x	An index array.

Author: Patrick Eriksson

Date: 2000-06-27

Definition at line 157 of file math_funcs.cc.

References ARTS_ASSERT, and Array< base >::nelem().

◆ last() [2/2]

Numeric last ( ConstVectorView x )

last

Returns the last value of a vector.

Returns: The last value of x.

Parameters

x A vector.

Author: Patrick Eriksson

Date: 2000-06-27

Definition at line 142 of file math_funcs.cc.

References ARTS_ASSERT.

Referenced by chk_scat_data(), ext_abs_pfun_from_tro(), find_effective_channel_boundaries(), Raw::Reduce::find_first_and_last_1(), mixer_matrix(), particle_bulkpropRadarOnionPeeling(), pos2refell_r(), refr_index_airFreeElectrons(), scat_data_singleTmatrix(), ScatSpeciesExtendTemperature(), sensor_responseMixer(), summation_by_vecmult(), VectorNLinSpaceVector(), ybatchDoublingMeanFocus(), yDoublingMeanFocus(), and ySimpleSpectrometer().

◆ linspace()

void linspace	(	Vector &	x,
		const Numeric	start,
		const Numeric	stop,
		const Numeric	step
	)

linspace

Linearly spaced vector with specified spacing.

The first element of x is always start. The next value is start+step etc. Note that the last value can deviate from stop. The step can be both positive and negative. (in Matlab notation: start:step:stop)

Size of result is adjusted within this function!

Parameters

x	Output: linearly spaced vector
start	first value in x
stop	last value of x <= stop
step	distance between values in x

Author: Patrick Eriksson

Date: 2000-06-27

Definition at line 181 of file math_funcs.cc.

◆ mgd()

void mgd	(	VectorView	psd,
		const Vector &	x,
		const Numeric &	n0,
		const Numeric &	mu,
		const Numeric &	la,
		const Numeric &	ga
	)

Modified gamma distribution

Uses all four free parameters (n0, mu, la, ga) to calculate psd(D) = n0 * D^mu * exp( -la * x^ga )

Reference: Eq 1 of Petty & Huang, JAS, (2011).

Parameters

psd	Particle number density per x-interval. Sizing of vector should be done before calling the function.
x	Mass or size.
n0	See above.
mu	See above.
la	See above.
ga	See above.

Author: Jana Mendrok, Patrick Eriksson

Date: 2017-06-07

Definition at line 473 of file math_funcs.cc.

References ARTS_ASSERT, and ARTS_USER_ERROR_IF.

◆ mgd_with_derivatives()

void mgd_with_derivatives	(	VectorView	psd,
		MatrixView	jac_data,
		const Vector &	x,
		const Numeric &	n0,
		const Numeric &	mu,
		const Numeric &	la,
		const Numeric &	ga,
		const bool &	do_n0_jac,
		const bool &	do_mu_jac,
		const bool &	do_la_jac,
		const bool &	do_ga_jac
	)

Modified gamma distribution, and derivatives

As mgd, but this version can also return the derivate of psd with respect to the four parameters.

Parameters

psd	Particle number density per x-interval. Sizing of vector should be done before calling the function.
jac_data	Container for returning jacobian data. Shall be a matrix with four rows, where the rows match n0, mu, la and ga. Number of columns same as length of psd.
x	Mass or size.
n0	See above.
mu	See above.
la	See above.
ga	See above.
do_n0_jac	Flag to actually calculate d_psd/d_n0
do_mu_jac	Flag to actually calculate d_psd/d_mu
do_la_jac	Flag to actually calculate d_psd/d_la
do_ga_jac	Flag to actually calculate d_psd/d_ga

Author: Patrick Eriksson

Date: 2017-06-07

Definition at line 543 of file math_funcs.cc.

References ARTS_ASSERT, and ARTS_USER_ERROR_IF.

Referenced by psd_mgd_mass_and_something(), psd_mgd_smm_common(), psdModifiedGamma(), and psdModifiedGammaMass().

◆ mod_gamma_dist()

Numeric mod_gamma_dist	(	Numeric	x,
		Numeric	N0,
		Numeric	Lambda,
		Numeric	mu,
		Numeric	gamma
	)

Generalized Modified Gamma Distribution.

Returns number density per unit of 'x' as function of 'x'.

Returns: dN Number density as function of x.

Parameters

x	Numeric
N0	Numeric, Scaling parameter
Lambda	Numeric, Shape parameter
mu	Numeric, Shape parameter
gamma	Numeric, Shape parameter

Author: Manfred Brath

Date: 2015-01-19

Definition at line 657 of file math_funcs.cc.

References ARTS_USER_ERROR.

Referenced by psd_MY05(), and psd_SB06().

◆ nlinspace() [1/2]

void nlinspace	(	Vector &	x,
		const Numeric	start,
		const Numeric	stop,
		const Index	n
	)

nlinspace

Linearly spaced vector with specified length.

Returns a vector equally and linearly spaced between start and stop of length n. (equals the Matlab function linspace)

The length must be > 1.

Parameters

x	Output: linearly spaced vector
start	first value in x
stop	last value of x <= stop
n	length of x

Author: Patrick Eriksson

Date: 2000-06-27

Definition at line 208 of file math_funcs.cc.

References ARTS_ASSERT.

Referenced by AngularGridsSetFluxCalc(), calc_ssp_fixed_test(), calc_ssp_random_test(), calculate_weights_linear(), dlosGauss(), DOAngularGridsSet(), doit_scat_fieldCalcLimb(), DoitScatteringDataPrepare(), get_angs(), HydrotableCalc(), iyEmissionHybrid(), line_irradianceCalcForSingleSpeciesNonOverlappingLinesPseudo2D(), pha_mat_sptFromDataDOITOpt(), pha_mat_sptFromMonoData(), pnd_fieldZero(), run_cdisort(), run_cdisort_flux(), sensor_responseFillFgrid(), VectorNLinSpace(), and WriteBuiltinPartitionFunctionsXML().

◆ nlinspace() [2/2]

void nlinspace	(	VectorView	x,
		const Numeric	start,
		const Numeric	stop,
		const Index	n
	)

Definition at line 218 of file math_funcs.cc.

◆ nlogspace()

void nlogspace	(	Vector &	x,
		const Numeric	start,
		const Numeric	stop,
		const Index	n
	)

nlogspace

Logarithmically spaced vector with specified length.

Returns a vector logarithmically spaced vector between start and stop of length n (equals the Matlab function logspace)

The length must be > 1.

Parameters

x	Output: logarithmically spaced vector
start	first value in x
stop	last value of x <= stop
n	length of x

Author: Patrick Eriksson

Date: 2000-06-27

Definition at line 244 of file math_funcs.cc.

References a, and ARTS_ASSERT.

◆ reshape() [1/2]

void reshape	(	MatrixView	X,
		ConstVectorView	x
	)

reshape

Converts vector to Matrix

The matrix is filled from front, i.e. rows are looped first, followed by cols. In Matlab this equals X = reshape( x, [ X.nrows(), X.ncols() ]

Parameters

[out]	X	The matrix. Should already be sized
[in]	x	The vector.

Author: Patrick Eriksson

Date: 2015-09-10

Definition at line 791 of file math_funcs.cc.

References ARTS_ASSERT, and c.

◆ reshape() [2/2]

void reshape	(	Tensor3View	X,
		ConstVectorView	x
	)

reshape

Converts vector to Tensor3

The tensor is filled from front, i.e. pages are looped first, followed by rows. In Matlab this equals X = reshape( x, [ X.npages(), X.nrows(), X.ncols() ]

Parameters

[out]	X	The tensor. Should already be sized
[in]	x	The vector.

Author: Patrick Eriksson

Date: 2015-09-10

Definition at line 763 of file math_funcs.cc.

References ARTS_ASSERT, and c.

Referenced by AtmFieldPerturb(), and x2artsAtmAndSurf().

◆ sign()

Numeric sign ( const Numeric & x )

sign

Returns the sign of a numeric value.

The function returns 1 if the value is greater than zero, 0 if it equals zero and -1 if it is less than zero.

Returns: The sign of x (see above).

Parameters

x	A Numeric.

Author: Patrick Eriksson

Date: 2000-06-27

Definition at line 445 of file math_funcs.cc.

Referenced by error_if_limb_ppath(), geodeticposlos2cart(), geompath_from_r1_to_r2(), poslos2cart(), r_crossing_2d(), raytrace_2d_linear_basic(), specular_losCalc(), and specular_losCalcNoTopography().

◆ trapz()

Numeric trapz	(	ConstVectorView	x,
		ConstVectorView	y
	)

trapz

Integration by the basic trapezoidal rule

Parameters

x	Grid values
y	Integrand

Returns: The resulting integral

Author: Patrick Eriksson

Date: 2022-03-06

Definition at line 274 of file math_funcs.cc.

References ARTS_ASSERT.

Referenced by asymmetry_parameter().

◆ unitl()

void unitl ( Vector & x )

unitl

Normalises a vector to have unit length.

The standard Euclidean norm is used (2-norm).

param x In/Out: A vector.

Author: Patrick Eriksson

Date: 2012-02-12

Definition at line 689 of file math_funcs.cc.

References ARTS_ASSERT.

Variable Documentation

◆ DEG2RAD

constexpr Numeric DEG2RAD =Conversion::deg2rad(1)

inlineconstexpr

Definition at line 27 of file math_funcs.cc.

Referenced by AngIntegrate_trapezoid(), and AngIntegrate_trapezoid_opti().

◆ PI

constexpr Numeric PI =Constant::pi

inlineconstexpr

Definition at line 28 of file math_funcs.cc.

Referenced by AngIntegrate_trapezoid().

Functions

Variables

Detailed Description

Function Documentation

◆ AngIntegrate_trapezoid() [1/2]

◆ AngIntegrate_trapezoid() [2/2]

◆ AngIntegrate_trapezoid_opti()

◆ calculate_int_weights_arbitrary_grid()

◆ calculate_weights_linear()

◆ cumsum()

◆ delanoe_shape_with_derivative()

◆ fac()

◆ flat() [1/2]

◆ flat() [2/2]

◆ integer_div()

◆ LagrangeInterpol4()

◆ last() [1/2]

◆ last() [2/2]

◆ linspace()

◆ mgd()

◆ mgd_with_derivatives()

◆ mod_gamma_dist()

◆ nlinspace() [1/2]

◆ nlinspace() [2/2]

◆ nlogspace()

◆ reshape() [1/2]

◆ reshape() [2/2]

◆ sign()

◆ trapz()

◆ unitl()

Variable Documentation

◆ DEG2RAD

◆ PI