Module inter_module with interpolation routines. More...

Functions/Subroutines
subroutine, private	lininter (n, ref_array, ref_value, array, res, flin)
	Linear interpolation in lin-log space. More...

subroutine, public	interp1d (n, xp, xb, yp, res, flin, itype)
	Interface for 1D interpolation routines. More...

subroutine, private	cubinter (n, xp, xb, yp, res, flin)
	Cubic interpolation in log-log space. Here, it is assumed that. More...

subroutine, private	pchipinter (n, xp, xb, yp, res, flin)
	PCHIP interpolation in lin-log space. More...

subroutine, private	akimainter (n, xp, xb, yp, res, flin)
	Interpolation using prescription of Akima. More...

subroutine, private	makimainter (n, xp, xb, yp, res, flin)
	Interpolation using modified prescription of Akima. More...

subroutine, public	get_indice_2d (xval, yval, x, y, x_dim, y_dim, indices)
	Get indices of the grid for given x- and y- values. More...

subroutine, public	calc_derivative_2d (f, x, y, dfx, dfy, dfxy, x_dim, y_dim)
	Calculate derivatives at all points in a given grid. More...

real(r_kind) function, public	lininter_2d (xin, yin, x, y, f, x_dim, y_dim, indice, extrapolation)
	Interpolate linearly on 2 Dimensional grid. More...

real(r_kind) function, public	cubinter_2d (xin, yin, x, y, f, dfx, dfy, dfxy, x_dim, y_dim, indice, extrapolation)
	Cubic interpolate on 2-Dimensional grid. More...

subroutine, public	inverse_interp1d (n, xp, xb, yp, res, flin, itype, reverse_type)
	The inverse of the 1D interpolation function. More...

subroutine, private	inverse_nr (n, xp, xb, yp, res, flin, itype)
	The inverse of the 1D interpolation function. More...

subroutine, private	inverse_brent (n, xp, xb, yp, res, flin, itype)
	The inverse of the 1D interpolation function. More...

Detailed Description

Module inter_module with interpolation routines.

This routines are used to interpolate the hydrodynamic quantaties (e.g., temperature, density,..)

Function/Subroutine Documentation

◆ akimainter()

subroutine, private inter_module::akimainter	(	integer, intent(in)	n,
		real(r_kind), dimension(n), intent(in)	xp,
		real(r_kind), intent(in)	xb,
		real(r_kind), dimension(n), intent(in)	yp,
		real(r_kind), intent(out)	res,
		logical, intent(in), optional	flin
	)

private

Interpolation using prescription of Akima.

The interpolation is a spline interpolation that tries to avoid overshoots.

Here, it is assumed that the x and y arrays are given in the form of {x_i, log(y_i)}, but before interpolating the x coordinate, it is transformed to a log space: {log(x_i), log(y_i)}. An optional argument flin allows to control the return value:

if flin is absent: return exp(log(y)) = y;
if flin is present: return log(y).

See also: Akima, Journal of the ACM, 17: 589–602, 1970, Wikipedia: Akima Spline

Author: M. Reichert

Date: 30.04.2023

Parameters

[in]	n	array sizes
[in]	xp	array of x-values
[in]	xb	value of x
[in]	yp	array of log(y)-values
[out]	res	interpolated value of y
[in]	flin	if present, return log(y) instead of y

Definition at line 364 of file inter_module.f90.

◆ calc_derivative_2d()

subroutine, public inter_module::calc_derivative_2d	(	real(r_kind), dimension(x_dim, y_dim), intent(in)	f,
		real(r_kind), dimension(x_dim), intent(in)	x,
		real(r_kind), dimension(y_dim), intent(in)	y,
		real(r_kind), dimension(x_dim,y_dim), intent(inout)	dfx,
		real(r_kind), dimension(x_dim,y_dim), intent(inout)	dfy,
		real(r_kind), dimension(x_dim,y_dim), intent(inout)	dfxy,
		integer, intent(in)	x_dim,
		integer, intent(in)	y_dim
	)

Calculate derivatives at all points in a given grid.

These derivatives are necessary for the cubic interpolation. The subroutine returns the derivative df/dx, df/dy, and ddf/dxdy. They are second order accurate, i.e.,

\[ df/dx = \frac{h_1^2 f(x+h_2) + (h_2^2-h_1^2)f(x)-h_2^2 f(x-h_1) }{h_1 h_2 (h_1+h_2)} \]

Author: : M. Reichert

Date: 06.07.22

Parameters

[in]	y_dim	Dimension of x- and y-
[in]	f	Function values
[in]	x	x values
[in]	y	y values
[in,out]	dfx	Derivative with respect to x
[in,out]	dfy	Derivative with respect to y
[in,out]	dfxy	Derivative with respect to x and y

Definition at line 623 of file inter_module.f90.

◆ cubinter()

subroutine, private inter_module::cubinter	(	integer, intent(in)	n,
		real(r_kind), dimension(n), intent(in)	xp,
		real(r_kind), intent(in)	xb,
		real(r_kind), dimension(n), intent(in)	yp,
		real(r_kind), intent(out)	res,
		logical, intent(in), optional	flin
	)

private

Cubic interpolation in log-log space. Here, it is assumed that.

the x and y arrays are given in the form of {x_i, log(y_i)}, but before interpolating the x coordinate, it is transformed to a log space: {log(x_i), log(y_i)}. An optional argument flin allows to control the return value:

if flin is absent: return exp(log(y)) = y;
if flin is present: return log(y).

Edited:

14.01.14

Parameters

[in]	n	array sizes
[in]	xp	array of x-values
[in]	xb	value of x
[in]	yp	array of log(y)-values
[out]	res	interpolated value of y
[in]	flin	if present, return log(y) instead of y

Definition at line 161 of file inter_module.f90.

◆ cubinter_2d()

real(r_kind) function, public inter_module::cubinter_2d	(	real(r_kind), intent(in)	xin,
		real(r_kind), intent(in)	yin,
		real(r_kind), dimension(x_dim), intent(in)	x,
		real(r_kind), dimension(y_dim), intent(in)	y,
		real(r_kind), dimension(x_dim,y_dim), intent(in)	f,
		real(r_kind), dimension(x_dim,y_dim), intent(in)	dfx,
		real(r_kind), dimension(x_dim,y_dim), intent(in)	dfy,
		real(r_kind), dimension(x_dim,y_dim), intent(in)	dfxy,
		integer, intent(in)	x_dim,
		integer, intent(in)	y_dim,
		integer, dimension(2,2), intent(in), optional	indice,
		integer, intent(in), optional	extrapolation
	)

Cubic interpolate on 2-Dimensional grid.

This interpolation is a bicubic interpolation. For values outside the grid, either a constant extrapolation is performed or an error is raised.

See also: weak_inter_bilinear, weak_inter, readweak_logft

Author: : Moritz Reichert

Date: : 14.03.22

See also: Bicubic interpolation

Parameters

[in]	y_dim	Dimension of x- and y-
[in]	yin	Function input values
[in]	x	X values
[in]	y	Y values
[in]	dfxy	Function and derivatives
[in]	extrapolation	1: None; 2: constant
[in]	indice	Index where to interpolate

Returns: Interpolated value (output)

Definition at line 795 of file inter_module.f90.

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◆ get_indice_2d()

subroutine, public inter_module::get_indice_2d	(	real(r_kind), intent(in)	xval,
		real(r_kind), intent(in)	yval,
		real(r_kind), dimension(x_dim), intent(in)	x,
		real(r_kind), dimension(y_dim), intent(in)	y,
		integer, intent(in)	x_dim,
		integer, intent(in)	y_dim,
		integer, dimension(2,2), intent(out)	indices
	)

Get indices of the grid for given x- and y- values.

Author: : M. Reichert

Date: : 06.07.22

Parameters

[in]	y_dim	Dimension of x- and y-
[in]	yval	Function values
[in]	x	X-grid values
[in]	y	Y-grid values
[out]	indices	Index in the grid

Definition at line 587 of file inter_module.f90.

◆ interp1d()

subroutine, public inter_module::interp1d	(	integer, intent(in)	n,
		real(r_kind), dimension(n), intent(in)	xp,
		real(r_kind), intent(in)	xb,
		real(r_kind), dimension(n), intent(in)	yp,
		real(r_kind), intent(out)	res,
		logical, intent(in), optional	flin,
		integer, intent(in), optional	itype
	)

Interface for 1D interpolation routines.

This routine is a wrapper for the 1D interpolation routines lininter, cubinter, akimainter, makimainter, and pchipinter. The interpolation type is controlled by the optional argument itype. If itype is absent, interp_mode is used. The optional argument flin allows to control the return value:

if flin is absent: return exp(log(y)) = y;
if flin is present: return log(y).

Author: M. Reichert

Date: 01.05.2023

Parameters

[in]	n	array sizes
[in]	xp	array of x-values
[in]	xb	value of x
[in]	yp	array of log(y)-values
[out]	res	interpolated value of y
[in]	flin	if present, return log(y) instead of y
[in]	itype	Type of interpolation (1: linear, 2: cubic)

Definition at line 99 of file inter_module.f90.

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◆ inverse_brent()

subroutine, private inter_module::inverse_brent	(	integer, intent(in)	n,
		real(r_kind), dimension(n), intent(in)	xp,
		real(r_kind), intent(in)	xb,
		real(r_kind), dimension(n), intent(in)	yp,
		real(r_kind), intent(inout)	res,
		logical, intent(in)	flin,
		integer, intent(in)	itype
	)

private

The inverse of the 1D interpolation function.

Finds the inverse of interp1d and returns the value of x using Brent's method, a hybrid root-finding algorithm that combines the robustness of the bisection method with the speed of inverse quadratic interpolation and the secant method. Brent's method ensures that the root is bracketed at every iteration, providing a reliable and efficient way to converge on the solution. For more details, see the wikipedia article.

Note: The result depends on xb, because x_i does not necessarily have to be monotonic. The result should be the closest value to xb.

Author: M. Reichert

Date: 16.10.2024

Parameters

[in]	n	size of the array
[in]	xp	array of x-values
[in]	yp	array of y-values
[in]	xb	target x-value for root finding
[in,out]	res	interpolated root result
[in]	flin	flag for linear interpolation
[in]	itype	interpolation type

Definition at line 1085 of file inter_module.f90.

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◆ inverse_interp1d()

subroutine, public inter_module::inverse_interp1d	(	integer, intent(in)	n,
		real(r_kind), dimension(n), intent(in)	xp,
		real(r_kind), intent(in)	xb,
		real(r_kind), dimension(n), intent(in)	yp,
		real(r_kind), intent(inout)	res,
		logical, intent(in), optional	flin,
		integer, intent(in), optional	itype,
		integer, intent(in), optional	reverse_type
	)

The inverse of the 1D interpolation function.

Interface to the inverse interpolation functions using either Newton-Raphson or Brent's method.

Author: M. Reichert

Date: 16.10.2024

See also: timestep_module::restrict_timestep

Parameters

[in]	n	array sizes
[in]	xp	array of log(x)-values
[in]	xb	value of x
[in]	yp	array of y-values
[in,out]	res	interpolated value of y
[in]	flin	flag for linear interpolation
[in]	itype	interpolation type
[in]	reverse_type	flag for reverse interpolation (1: NR, 2: Brent)

Definition at line 925 of file inter_module.f90.

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◆ inverse_nr()

subroutine, private inter_module::inverse_nr	(	integer, intent(in)	n,
		real(r_kind), dimension(n), intent(in)	xp,
		real(r_kind), intent(in)	xb,
		real(r_kind), dimension(n), intent(in)	yp,
		real(r_kind), intent(inout)	res,
		logical, intent(in)	flin,
		integer, intent(in)	itype
	)

private

The inverse of the 1D interpolation function.

Finds the inverse of interp1d and returns the value of x. This is done by using a Newton-Raphson method.

Note: The result depends on xb, because x_i does not necessarily have to be monotonic. The result should be the closest value to xb. This is however not always the case in which brents method might be more suitable.

Author: M. Reichert

Edited:

30.01.23

Parameters

[in]	n	array sizes
[in]	xp	array of log(x)-values
[in]	xb	value of x
[in]	yp	array of y-values
[in,out]	res	interpolated value of y
[in]	flin	flag for linear interpolation
[in]	itype	interpolation type

Definition at line 997 of file inter_module.f90.

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◆ lininter()

subroutine, private inter_module::lininter	(	integer, intent(in)	n,
		real(r_kind), dimension(n), intent(in)	ref_array,
		real(r_kind), intent(in)	ref_value,
		real(r_kind), dimension(n), intent(in)	array,
		real(r_kind), intent(out)	res,
		logical, intent(in), optional	flin
	)

private

Linear interpolation in lin-log space.

This routine assumes that the x and y arrays are given in the form of {x_i, log(y_i)}. An optional argument flin allows to control the return value:

if flin is absent: return exp(log(y)) = y;
if flin is present: return log(y).

Edited:

11.01.14

Parameters

[in]	n	array sizes
[in]	ref_array	array of x-values {x_i}
[in]	array	array of log(y)-values {log(y_i)}
[in]	ref_value	value of log(x)
[out]	res	interpolated value of y
[in]	flin	if present, return log(y) instead of y

Definition at line 40 of file inter_module.f90.

◆ lininter_2d()

real(r_kind) function, public inter_module::lininter_2d	(	real(r_kind), intent(in)	xin,
		real(r_kind), intent(in)	yin,
		real(r_kind), dimension(x_dim), intent(in)	x,
		real(r_kind), dimension(y_dim), intent(in)	y,
		real(r_kind), dimension(x_dim,y_dim), intent(in)	f,
		integer, intent(in)	x_dim,
		integer, intent(in)	y_dim,
		integer, dimension(2,2), intent(in), optional	indice,
		integer, intent(in), optional	extrapolation
	)

Interpolate linearly on 2 Dimensional grid.

Bilinear interpolation

See also: weak_inter

Author: : M. Reichert

Date: : 06.07.22

Parameters

[in]	y_dim	Dimension of x- and y-
[in]	yin	Function input values
[in]	x	X values
[in]	y	Y values
[in]	f	Function and derivatives
[in]	extrapolation	1: None; 2: constant
[in]	indice	Index where to interpolate

Returns: Interpolated value (output)

Definition at line 699 of file inter_module.f90.

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◆ makimainter()

subroutine, private inter_module::makimainter	(	integer, intent(in)	n,
		real(r_kind), dimension(n), intent(in)	xp,
		real(r_kind), intent(in)	xb,
		real(r_kind), dimension(n), intent(in)	yp,
		real(r_kind), intent(out)	res,
		logical, intent(in), optional	flin
	)

private

Interpolation using modified prescription of Akima.

The interpolation is a spline interpolation that tries to avoid overshoots. It is doing this slightly more radically than the Akima interpolation, but not as strong as the pchip interpolation.

Here, it is assumed that the x and y arrays are given in the form of {x_i, log(y_i)}, but before interpolating the x coordinate, it is transformed to a log space: {log(x_i), log(y_i)}. An optional argument flin allows to control the return value:

if flin is absent: return exp(log(y)) = y;
if flin is present: return log(y).

See also: https://blogs.mathworks.com/cleve/2019/04/29/makima-piecewise-cubic-interpolation/

Author: M. Reichert

Date: 30.04.2023

Parameters

[in]	n	array sizes
[in]	xp	array of x-values
[in]	xb	value of x
[in]	yp	array of log(y)-values
[out]	res	interpolated value of y
[in]	flin	if present, return log(y) instead of y

Definition at line 481 of file inter_module.f90.

◆ pchipinter()

subroutine, private inter_module::pchipinter	(	integer, intent(in)	n,
		real(r_kind), dimension(n), intent(in)	xp,
		real(r_kind), intent(in)	xb,
		real(r_kind), dimension(n), intent(in)	yp,
		real(r_kind), intent(out)	res,
		logical, intent(in), optional	flin
	)

private

PCHIP interpolation in lin-log space.

The Piecewise Cubic Hermite Interpolating Polynomial (PCHIP) interpolation is a method for interpolating data points where every section between points is monoton. The interpolation avoids overshoots and undershoots and is close to the linear interpolation, but having the first derivative continuous.

See also: F. N. Fritsch and R. E. Carlson, Monotone Piecewise Cubic Interpolation, SIAM Journal on Numerical Analysis, 17(2), 1980, pp. 238-246. Steffen 1990

Author: : M. Reichert

Date: : 01.05.2023

Parameters

[in]	n	array sizes
[in]	xp	array of x-values
[in]	xb	value of x
[in]	yp	array of log(y)-values
[out]	res	interpolated value of y
[in]	flin	if present, return log(y) instead of y

Definition at line 243 of file inter_module.f90.

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Functions/Subroutines

Detailed Description

Function/Subroutine Documentation

◆ akimainter()

◆ calc_derivative_2d()

◆ cubinter()

◆ cubinter_2d()

◆ get_indice_2d()

◆ interp1d()

◆ inverse_brent()

◆ inverse_interp1d()

◆ inverse_nr()

◆ lininter()

◆ lininter_2d()

◆ makimainter()

◆ pchipinter()