The B-spline basis module. More...

Data Types
type	bsplinefun
	The B-spline function type. More...

interface	bsplinespace
	The B-spline space type. More...

interface	evaluate
	Evaluate a B-spline (function) More...

interface	evaluate_support

interface	get_index_ranges
	Get the index ranges for integrating a B-spline or the product of two B-splines. More...

interface	get_petsc
	Convert a B-spline function to a PETSc vector. More...

interface	quadrature
	Compute quadrature of a B-spline. More...

interface	quadrature_product
	Compute quadrature of the product of two B-splines. More...

interface	size
	Get the size of the B-spline space (i.e., the number of linearly independent B-splines) More...

Functions/Subroutines
pure type(bsplinespace) function, public	derivative (bspline, allow_periodic_degree0)
	Returns the B-spline space containing the derivatives of the given B-spline space.

pure type(bsplinespace) function, public	anti_derivative (bspline)
	Returns the B-spline space containing the anti-derivatives of the given B-spline space.

pure real(wp) function, public	evaluate_single (this, x, j, derivative)
	Evaluate the j-th B-spline of the B-spline space 'this' at x.

real(wp) function, public	find_root_newton (this, x0, rhs, max_iter, tol, success)
	Find a root of the B-spline function 'this' minus rhs using Newton's method.

real(wp) function, public	find_root_brent (this, x0, x1, rhs, max_iter, tol, success, res0, res1)
	Find a root of the B-spline function 'this' minus rhs using Brent's method.

pure integer function, public	get_j_minus_i_quad (this, j, i)
	Determine j_actual - i, where the input j may exceed the number of B-splines.

pure subroutine, public	get_j_internal (this, j, j_internal, bspline_mirrored)
	Get the internal B-spline index and whether the B-spline is mirrored.

pure subroutine, public	get_internal_and_actual_inds (bspline, i, j_minus_i, j, j_internal, bspline_mirrored)
	Get the internal B-spline index and the actual B-spline index.

pure integer function, public	get_interval_index (bspline, x)
	Get the interval index for a point in the B-spline space.

Detailed Description

The B-spline basis module.

This module provides

BSplineSpace: the space of B-spline basis functions
BSplineFun: the B-spline function which is represented by a linear combination of the basis functions from a space

The space is defined by the number of uniform intervals, the B-spline degree, as well as the boundary type:

The B-spline space can be clamped meaning that only the first B-spline is nonzero at the left boundary and only the last B-spline is nonzero at the right boundary.
Or the B-spline space can be periodic meaning that the first 'degree' B-splines are combined with the last 'degree' B-splines, thereby enforcing periodicity.

Function/Subroutine Documentation

◆ anti_derivative()

pure type(bsplinespace) function, public m_bspline_basis::anti_derivative ( type(bsplinespace), intent(in) bspline )

Returns the B-spline space containing the anti-derivatives of the given B-spline space.

Parameters

[in] bspline The B-spline space

Returns: The B-spline space containing the anti-derivatives of the given B-spline space

Note: This requires the B-spline space to be scaled

◆ derivative()

pure type(bsplinespace) function, public m_bspline_basis::derivative	(	type(bsplinespace), intent(in)	bspline,
		logical, intent(in), optional	allow_periodic_degree0 )

Returns the B-spline space containing the derivatives of the given B-spline space.

Parameters

[in]	bspline	The B-spline space
[in]	allow_periodic_degree0	Whether to allow periodic degree 0 B-splines in the 0-form space (default: false)

Returns: The B-spline space containing the derivatives of the given B-spline space

Note: This requires the B-spline space to be unscaled

◆ evaluate_single()

pure real(wp) function, public m_bspline_basis::evaluate_single	(	type(bsplinespace), intent(in)	this,
		real(wp), intent(in)	x,
		integer, intent(in)	j,
		logical, intent(in), optional	derivative )

Evaluate the j-th B-spline of the B-spline space 'this' at x.

Parameters

[in]	this	The B-spline space
[in]	x	The evaluation point
[in]	j	The B-spline index
[in]	derivative	_(optional)_ If true, the derivative of the B-spline is evaluated

Returns: The value of the B-spline at x

◆ find_root_brent()

real(wp) function, public m_bspline_basis::find_root_brent	(	type(bsplinefun), intent(in)	this,
		real(wp), intent(in)	x0,
		real(wp), intent(in)	x1,
		real(wp), intent(in), optional	rhs,
		integer, intent(in), optional	max_iter,
		real(wp), intent(in), optional	tol,
		logical, intent(out), optional	success,
		real(wp), intent(in), optional	res0,
		real(wp), intent(in), optional	res1 )

Find a root of the B-spline function 'this' minus rhs using Brent's method.

Parameters

[in]	this	The B-spline function
[in]	x0	The left-hand side of the bracketed interval for the root
[in]	x1	The right-hand side of the bracketed interval for the root
[in]	rhs	_(optional)_ The right-hand side value to find the root of (default: 0)
[in]	max_iter	_(optional)_ The maximum number of iterations for the root-finding method (default: 20)
[in]	tol	_(optional)_ The tolerance for convergence of the root-finding method (default: 1e-6)
[out]	success	_(optional)_ A logical flag indicating whether the root-finding method converged successfully
[in]	res0	_(optional)_ The value of the function minus rhs at x0 (if cheaply available)
[in]	res1	_(optional)_ The value of the function minus rhs at x1 (if cheaply available)

◆ find_root_newton()

real(wp) function, public m_bspline_basis::find_root_newton	(	type(bsplinefun), intent(in)	this,
		real(wp), intent(in), optional	x0,
		real(wp), intent(in), optional	rhs,
		integer, intent(in), optional	max_iter,
		real(wp), intent(in), optional	tol,
		logical, intent(out), optional	success )

Find a root of the B-spline function 'this' minus rhs using Newton's method.

Parameters

[in]	this	The B-spline function
[in]	x0	_(optional)_ The initial guess for the root (default: 0.5)
[in]	rhs	_(optional)_ The right-hand side value to find the root of (default: 0)
[in]	max_iter	_(optional)_ The maximum number of iterations for the root-finding method (default: 20)
[in]	tol	_(optional)_ The tolerance for convergence of the root-finding method (default: 1e-8)
[out]	success	_(optional)_ A logical flag indicating whether the root-finding method converged successfully

Returns: The root of the B-spline function 'this' minus rhs

◆ get_internal_and_actual_inds()

pure subroutine, public m_bspline_basis::get_internal_and_actual_inds	(	type(bsplinespace), intent(in)	bspline,
		integer, intent(in)	i,
		integer, intent(in)	j_minus_i,
		integer, intent(out)	j,
		integer, intent(out), optional	j_internal,
		logical, intent(out), optional	bspline_mirrored )

Get the internal B-spline index and the actual B-spline index.

Parameters

[in]	bspline	The B-spline space
[in]	i	The interval index
[in]	j_minus_i	The difference between the B-spline index and the interval index
[out]	j	The actual B-spline index (0:nr_bsplines-1)
[out]	j_internal	_(optional)_ The internal B-spline index (0:degree)
[out]	bspline_mirrored	_(optional)_ Whether the B-spline is mirrored (i.e., the B-spline is on the right boundary of the domain)

◆ get_interval_index()

pure integer function, public m_bspline_basis::get_interval_index	(	type(bsplinespace), intent(in)	bspline,
		real(wp), intent(in)	x )

Get the interval index for a point in the B-spline space.

Parameters

[in]	bspline	The B-spline space
[in]	x	The point

Returns: The interval index for the point in the B-spline space (0:nr_intervals-1)

◆ get_j_internal()

pure subroutine, public m_bspline_basis::get_j_internal	(	type(bsplinespace), intent(in)	this,
		integer, intent(in)	j,
		integer, intent(out)	j_internal,
		logical, intent(out)	bspline_mirrored )

Get the internal B-spline index and whether the B-spline is mirrored.

Parameters

[in]	this	The B-spline space
[in]	j	The B-spline index
[out]	j_internal	The internal B-spline index (0:degree)
[out]	bspline_mirrored	_(optional)_ Whether the B-spline is mirrored (i.e., the B-spline is on the right boundary of the domain)

◆ get_j_minus_i_quad()

pure integer function, public m_bspline_basis::get_j_minus_i_quad	(	type(bsplinespace), intent(in)	this,
		integer, intent(in)	j,
		integer, intent(in)	i )

Determine j_actual - i, where the input j may exceed the number of B-splines.

Parameters

[in]	this	The B-spline basis
[in]	j	The B-spline index
[in]	i	The interval index

Returns: The j_actual - i

Data Types

Functions/Subroutines

Detailed Description

Function/Subroutine Documentation

◆ anti_derivative()

◆ derivative()

◆ evaluate_single()

◆ find_root_brent()

◆ find_root_newton()

◆ get_internal_and_actual_inds()

◆ get_interval_index()

◆ get_j_internal()

◆ get_j_minus_i_quad()