Univariate spline curve fitting module. More...

Data Types
interface	fitpack_curve
	Univariate spline curve \( y = s(x) \). More...

type	fitpack_periodic_curve
	Derived type describing a periodic curve. No changes are made to the storage, but the appropriate package functions will be called depending on the type. More...

Functions/Subroutines
type(fitpack_curve) function	new_from_points (x, y, w, ierr)
	Construct a fitpack_curve from data points and perform a default fit.

integer(fp_flag) function	new_fit (this, x, y, w, smoothing, order)
	Load new data points and perform a fresh fit.

elemental subroutine	destroy (this)
	Destroy a curve object and release all allocated memory.

subroutine	new_points (this, x, y, w)
	Load new data points into the curve, resetting all internal state.

real(fp_real) function	curve_eval_one (this, x, ierr)
	Evaluate the spline at a single point (with error flag).

pure real(fp_real) function	curve_eval_one_noerr (this, x)
	Evaluate the spline at a single point (pure, returns NaN on error).

real(fp_real) function, dimension(size(x))	curve_eval_many (this, x, ierr)
	Evaluate the spline at multiple points (with error flag).

pure real(fp_real) function, dimension(size(x))	curve_eval_many_pure (this, x)
	Evaluate the spline at multiple points (pure, returns NaN on error).

integer(fp_flag) function	interpolating_curve (this, order, reset_knots)
	Fit an interpolating spline ( \( s = 0 \)) through the data points.

integer(fp_flag) function	curve_fit_least_squares (this, smoothing, reset_knots)
	Fit a least-squares spline with fixed knots.

integer(fp_flag) function	curve_fit_automatic_knots (this, smoothing, order, keep_knots)
	Fit a smoothing spline with automatic knot placement (curfit / percur).

real(fp_real) function, dimension(size(x))	curve_derivatives (this, x, order, ierr)
	Evaluate the k-th derivative at multiple points (with error flag).

real(fp_real) function, dimension(this%order+1)	curve_all_derivatives (this, x, ierr)
	Evaluate all derivatives \( s^{(j)}(x) \) for \( j = 0, \ldots, k \) (with error flag).

real(fp_real) function, dimension(this%order+1)	curve_all_derivatives_pure (this, x)
	Evaluate all derivatives \( s^{(j)}(x) \) for \( j = 0, \ldots, k \) (pure, NaN on error).

real(fp_real) function	curve_derivative (this, x, order, ierr)
	Evaluate the k-th derivative at a single point (with optional error flag).

real(fp_real) function	integral (this, from, to)
	Compute the definite integral \( \int_a^b s(x)\,dx \) via splint.

subroutine	fourier_coefficients (this, alpha, a, b, ierr)
	Compute Fourier sine and cosine coefficients of the spline.

real(fp_real) function, dimension(:), allocatable	zeros (this, ierr)
	Find all real zeros of a cubic spline \( s(x) = 0 \).

subroutine	grow_knot_storage (this)
	Grow the knot and coefficient arrays when storage is insufficient.

subroutine	curve_insert_knot_one (this, x, ierr)
	Insert a single knot into the B-spline representation.

subroutine	curve_insert_knot_many (this, x, ierr)
	Insert multiple knots into the B-spline representation.

elemental integer(fp_size) function	curve_comm_size (this)
	Return the communication buffer size (number of FP_COMM elements).

pure subroutine	curve_comm_pack (this, buffer)
	Pack curve data into a communication buffer for parallel transfer.

pure subroutine	curve_comm_expand (this, buffer)
	Expand curve data from a communication buffer.

Variables
integer, parameter	max_k = 5

Detailed Description

Univariate spline curve fitting module.

Provides fitpack_curve for fitting and evaluating a spline \( y = s(x) \) to data, and fitpack_periodic_curve for periodic splines. Wraps curfit / percur for fitting, splev for evaluation, splder / spalde for derivatives, splint for integration, sproot for zero-finding, and fourco for Fourier coefficients.

See also: Dierckx, Ch. 4–5 (smoothing curves), Ch. 6 §6.1–6.2 (periodic curves)

Function/Subroutine Documentation

◆ curve_all_derivatives()

real(fp_real) function, dimension(this%order+1) fitpack_curves::curve_all_derivatives	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), intent(in)	x,
		integer(fp_flag), intent(out)	ierr )

private

Evaluate all derivatives \( s^{(j)}(x) \) for \( j = 0, \ldots, k \) (with error flag).

Parameters

[in,out]	this	The fitted curve object.
[in]	x	Scalar evaluation point.
[out]	ierr	Error flag.

Returns: Array of size \( k+1 \) containing \( s(x), s'(x), \ldots, s^{(k)}(x) \).

See also: spalde

◆ curve_all_derivatives_pure()

real(fp_real) function, dimension(this%order+1) fitpack_curves::curve_all_derivatives_pure	(	class(fitpack_curve), intent(in)	this,
		real(fp_real), intent(in)	x )

private

Evaluate all derivatives \( s^{(j)}(x) \) for \( j = 0, \ldots, k \) (pure, NaN on error).

Parameters

[in]	this	The fitted curve object.
[in]	x	Scalar evaluation point.

Returns: Array of size \( k+1 \) containing \( s(x), s'(x), \ldots, s^{(k)}(x) \).

See also: spalde

◆ curve_comm_expand()

pure subroutine fitpack_curves::curve_comm_expand	(	class(fitpack_curve), intent(inout)	this,
		real(fp_comm), dimension(:), intent(in)	buffer )

private

Expand curve data from a communication buffer.

◆ curve_comm_pack()

pure subroutine fitpack_curves::curve_comm_pack	(	class(fitpack_curve), intent(in)	this,
		real(fp_comm), dimension(:), intent(out)	buffer )

private

Pack curve data into a communication buffer for parallel transfer.

◆ curve_comm_size()

elemental integer(fp_size) function fitpack_curves::curve_comm_size ( class(fitpack_curve), intent(in) this )

private

Return the communication buffer size (number of FP_COMM elements).

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◆ curve_derivative()

real(fp_real) function fitpack_curves::curve_derivative	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), intent(in)	x,
		integer, intent(in)	order,
		integer(fp_flag), intent(out), optional	ierr )

private

Evaluate the k-th derivative at a single point (with optional error flag).

Parameters

[in,out]	this	The fitted curve object.
[in]	x	Scalar evaluation point.
[in]	order	Derivative order ( \( 0 \leq \text{order} \leq k \)).
[out]	ierr	Optional error flag.

Returns: Scalar derivative value \( s^{(\text{order})}(x) \).

See also: splder

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◆ curve_derivatives()

real(fp_real) function, dimension(size(x)) fitpack_curves::curve_derivatives	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), dimension(:), intent(in)	x,
		integer, intent(in)	order,
		integer(fp_flag), intent(out), optional	ierr )

private

Evaluate the k-th derivative at multiple points (with error flag).

Parameters

[in,out]	this	The fitted curve object.
[in]	x	Array of evaluation points.
[in]	order	Derivative order ( \( 0 \leq \text{order} \leq k \)).
[out]	ierr	Optional error flag.

Returns: Array of derivative values \( s^{(\text{order})}(x_i) \).

See also: splder

◆ curve_eval_many()

real(fp_real) function, dimension(size(x)) fitpack_curves::curve_eval_many	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), dimension(:), intent(in)	x,
		integer(fp_flag), intent(out)	ierr )

private

Evaluate the spline at multiple points (with error flag).

◆ curve_eval_many_pure()

pure real(fp_real) function, dimension(size(x)) fitpack_curves::curve_eval_many_pure	(	class(fitpack_curve), intent(in)	this,
		real(fp_real), dimension(:), intent(in)	x )

private

Evaluate the spline at multiple points (pure, returns NaN on error).

◆ curve_eval_one()

real(fp_real) function fitpack_curves::curve_eval_one	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), intent(in)	x,
		integer(fp_flag), intent(out)	ierr )

private

Evaluate the spline at a single point (with error flag).

Parameters

[in,out]	this	The fitted curve object.
[in]	x	Evaluation point.
[out]	ierr	Error flag.

Returns: Spline value \( s(x) \).

See also: splev

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◆ curve_eval_one_noerr()

pure real(fp_real) function fitpack_curves::curve_eval_one_noerr	(	class(fitpack_curve), intent(in)	this,
		real(fp_real), intent(in)	x )

private

Evaluate the spline at a single point (pure, returns NaN on error).

Parameters

[in]	this	The fitted curve object.
[in]	x	Evaluation point.

Returns: Spline value \( s(x) \), or NaN on error.

See also: splev

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◆ curve_fit_automatic_knots()

integer(fp_flag) function fitpack_curves::curve_fit_automatic_knots	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), intent(in), optional	smoothing,
		integer(fp_size), intent(in), optional	order,
		logical, intent(in), optional	keep_knots )

private

Fit a smoothing spline with automatic knot placement (curfit / percur).

Get smoothing trajectory

Ensure we start with new knots (unless caller wants to keep them)

Set/update order

◆ curve_fit_least_squares()

integer(fp_flag) function fitpack_curves::curve_fit_least_squares	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), intent(in), optional	smoothing,
		logical, intent(in), optional	reset_knots )

private

Fit a least-squares spline with fixed knots.

◆ curve_insert_knot_many()

subroutine fitpack_curves::curve_insert_knot_many	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), dimension(:), intent(in)	x,
		integer(fp_flag), intent(out), optional	ierr )

private

Insert multiple knots into the B-spline representation.

Parameters

[in,out]	this	The fitted curve.
[in]	x	Array of knot positions to insert.
[out]	ierr	Optional error flag.

See also: insert_inplace

◆ curve_insert_knot_one()

subroutine fitpack_curves::curve_insert_knot_one	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), intent(in)	x,
		integer(fp_flag), intent(out), optional	ierr )

private

Insert a single knot into the B-spline representation.

The spline value is unchanged; only the knot vector and coefficients are updated. Workspace state is reset after insertion.

Parameters

[in,out]	this	The fitted curve.
[in]	x	Knot position to insert.
[out]	ierr	Optional error flag.

See also: insert_inplace

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◆ destroy()

elemental subroutine fitpack_curves::destroy ( class(fitpack_curve), intent(inout) this )

private

Destroy a curve object and release all allocated memory.

◆ fourier_coefficients()

subroutine fitpack_curves::fourier_coefficients	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), dimension(:), intent(in)	alpha,
		real(fp_real), dimension(size(alpha)), intent(out)	a,
		real(fp_real), dimension(size(alpha)), intent(out)	b,
		integer(fp_flag), intent(out), optional	ierr )

private

Compute Fourier sine and cosine coefficients of the spline.

Computes integrals over the interior knot range \( [t_4, t_{n-3}] \):

\[ A_i = \int_{t_4}^{t_{n-3}} s(x) \sin(\alpha_i x) \, dx, \quad B_i = \int_{t_4}^{t_{n-3}} s(x) \cos(\alpha_i x) \, dx \]

Requires at least 10 knots.

Parameters

[in,out]	this	The fitted curve (must have \( \geq 10 \) knots).
[in]	alpha	Array of frequency values \( \alpha_i \).
[out]	A	Sine coefficients, same size as `alpha`.
[out]	B	Cosine coefficients, same size as `alpha`.
[out]	ierr	Optional error flag.

See also: fourco

◆ grow_knot_storage()

subroutine fitpack_curves::grow_knot_storage ( class(fitpack_curve), intent(inout) this )

private

Grow the knot and coefficient arrays when storage is insufficient.

◆ integral()

real(fp_real) function fitpack_curves::integral	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), intent(in)	from,
		real(fp_real), intent(in)	to )

private

Compute the definite integral \( \int_a^b s(x)\,dx \) via splint.

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◆ interpolating_curve()

integer(fp_flag) function fitpack_curves::interpolating_curve	(	class(fitpack_curve), intent(inout)	this,
		integer(fp_size), intent(in), optional	order,
		logical, intent(in), optional	reset_knots )

private

Fit an interpolating spline ( \( s = 0 \)) through the data points.

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◆ new_fit()

integer(fp_flag) function fitpack_curves::new_fit	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), dimension(:), intent(in)	x,
		real(fp_real), dimension(size(x)), intent(in)	y,
		real(fp_real), dimension(size(x)), intent(in), optional	w,
		real(fp_real), intent(in), optional	smoothing,
		integer(fp_size), intent(in), optional	order )

private

Load new data points and perform a fresh fit.

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◆ new_from_points()

type(fitpack_curve) function fitpack_curves::new_from_points	(	real(fp_real), dimension(:), intent(in)	x,
		real(fp_real), dimension(size(x)), intent(in)	y,
		real(fp_real), dimension(size(x)), intent(in), optional	w,
		integer(fp_flag), intent(out), optional	ierr )

private

Construct a fitpack_curve from data points and perform a default fit.

Parameters

[in]	x	Independent variable values (must be sortable).
[in]	y	Dependent variable values, same size as `x`.
[in]	w	Optional weights (positive), same size as `x`.
[out]	ierr	Optional error flag.

Returns: Fitted curve object.

◆ new_points()

subroutine fitpack_curves::new_points	(	class(fitpack_curve), intent(inout)	this,
		real(fp_real), dimension(:), intent(in)	x,
		real(fp_real), dimension(size(x)), intent(in)	y,
		real(fp_real), dimension(size(x)), intent(in), optional	w )

private

Load new data points into the curve, resetting all internal state.

Sorts data by x, allocates knot/coefficient storage, and prepares workspaces. A subsequent call to fit, interpolate, or least_squares is required to compute the spline representation.

Parameters

[in,out]	this	The curve object (destroyed and reinitialized).
[in]	x	Independent variable values (unsorted OK; will be sorted internally).
[in]	y	Dependent variable values, same size as `x`.
[in]	w	Optional positive weights, same size as `x`. Default: uniform.

See also: curfit, percur

◆ zeros()

real(fp_real) function, dimension(:), allocatable fitpack_curves::zeros	(	class(fitpack_curve), intent(in)	this,
		integer(fp_flag), intent(out), optional	ierr )

private

Find all real zeros of a cubic spline \( s(x) = 0 \).

Only valid for cubic splines ( \( k = 3 \)). Iteratively grows storage until all zeros are found.

Parameters

[in]	this	The fitted cubic curve.
[out]	ierr	Optional error flag.

Returns: Allocatable array of zero locations (empty if none found).

See also: sproot

Always return an allocated array

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Variable Documentation

◆ max_k

integer, parameter fitpack_curves::max_k = 5

private

Data Types

Functions/Subroutines

Variables

Detailed Description

Function/Subroutine Documentation

◆ curve_all_derivatives()

◆ curve_all_derivatives_pure()

◆ curve_comm_expand()

◆ curve_comm_pack()

◆ curve_comm_size()

◆ curve_derivative()

◆ curve_derivatives()

◆ curve_eval_many()

◆ curve_eval_many_pure()

◆ curve_eval_one()

◆ curve_eval_one_noerr()

◆ curve_fit_automatic_knots()

◆ curve_fit_least_squares()

◆ curve_insert_knot_many()

◆ curve_insert_knot_one()

◆ destroy()

◆ fourier_coefficients()

◆ grow_knot_storage()

◆ integral()

◆ interpolating_curve()

◆ new_fit()

◆ new_from_points()

◆ new_points()

◆ zeros()

Variable Documentation

◆ max_k