csplines Module Reference

csplines implements interpolation using cubic splines Description: Submodule interpolate More...

Data Types
interface	csplev
	csplev Performs a spline interpolation in a point or in a table More...
interface	csplevder
	csplev Performs a spline interpolation in a point or in a table More...
interface	cubicspline
	Type used to keep all information on splines. More...

Functions/Subroutines
subroutine, public	csplrep (x, y, s1, sn, csp)
	cubic spline interpolation between tabulated data After calling this function the result may be used to evaluate the function as: ynew= csplev(xnew, csp)
type(cubicspline) function, public	csplder (csp, m)
	csplder Computes the derivative of the cubic spline
type(cubicspline) function, public	csplantider (csp, m)
	Computes the antiderivative of the CubicSpline approximation.
subroutine, public	cspl_clean (r)
	Clean-up a spline representation.
subroutine, public	cspleps (x, y, err)
	cspleps Estimates the error produced by using a spline approximation
real(dp) function, public	csplint (xl, xu, csp, extrapolate)
	Definite integral of a cubic spline function.
real(dp) function, public	csplint_square (xl, xu, csp)
	Integral of the square of a function expressed as a cubic spline.
real(dp) function, dimension(:), allocatable, public	csplroots (csp)
	csplroots Computes the roots of the Spline approximation

Detailed Description

csplines implements interpolation using cubic splines Description: Submodule interpolate

Function/Subroutine Documentation

cspl_clean()

subroutine, public cspl_clean

(

type(cubicspline), intent(inout)

r

)

Clean-up a spline representation.

Parameters

[in,out]

r

CubicSpline

References basic::print_msg().

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csplantider()

type(cubicspline) function, public csplantider	(	type(cubicspline), intent(in)	csp,
		integer, intent(in)	m )

Computes the antiderivative of the CubicSpline approximation.

The result is a CubicSpline (not exactly, it is a polynomial of order m+4)

Parameters

[in]	csp	CubicSpline object holding the spline
[in]	m	order of integration

Returns

spline holding the antiderivative

References polynomial::polyint().

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csplder()

type(cubicspline) function, public csplder	(	type(cubicspline), intent(in)	csp,
		integer, intent(in)	m )

csplder Computes the derivative of the cubic spline

Parameters

[in]	csp	Interpolating object
[in]	m	order of derivation (must be 1 or 2)

Returns

Spline representation of derivative

References polynomial::polyder(), and basic::print_msg().

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cspleps()

subroutine, public cspleps	(	real(dp), dimension(:), intent(in)	x,
		real(dp), dimension(size(x)), intent(in)	y,
		real(dp), dimension(size(x)), intent(out)	err )

cspleps Estimates the error produced by using a spline approximation

This subroutine estimates the error introduced by natural cubic spline interpolation in a table x(i),y(i) (i=1,...,n). the interpolation error in the vicinity of x(k) is approximated by the difference between y(k) and the value obtained from the spline that interpolates the table with the k-th point removed. err is the largest relative error along the table.

Note

Some tests seems to show that the error is about one order of magnitude better than estimated by this routine

Modified from Salvat et al, Comp. Phys. Comm. (1995)

Parameters

[in]	x	grid points
[in]	y	value of function at grid points
[out]	err	Vector with error estimates

csplint()

real(dp) function, public csplint	(	real(dp), intent(in)	xl,
		real(dp), intent(in)	xu,
		class(cubicspline), intent(in)	csp,
		logical, intent(in), optional	extrapolate )

Definite integral of a cubic spline function.

Todo

Testing when extrapolate is True

Parameters

[in]	xl	Lower limit in the integral.
[in]	xu	Upper limit in the integral.
[in]	csp	Interpolating object
[in]	extrapolate	Flag signaling if we extrapolate outside interval

Returns

Value of integral

csplint_square()

real(dp) function, public csplint_square	(	real(dp), intent(in)	xl,
		real(dp), intent(in)	xu,
		type(cubicspline), intent(in)	csp )

Integral of the square of a function expressed as a cubic spline.

Todo

Testing

Parameters

[in]	xl	Lower limit in the integral.
[in]	xu	Upper limit in the integral.
[in]	csp	Interpolating object

Returns

Value of integral

csplrep()

subroutine, public csplrep	(	real(dp), dimension(:), intent(in)	x,
		real(dp), dimension(:), intent(in)	y,
		real(dp), intent(in)	s1,
		real(dp), intent(in)	sn,
		type(cubicspline), intent(out)	csp )

cubic spline interpolation between tabulated data After calling this function the result may be used to evaluate the function as:
ynew= csplev(xnew, csp)

REF.: M.J. Maron, 'Numerical Analysis: A Practical Approach', Macmillan Publ. Co., New York 1982.

Parameters

[in]	x	independent grid points
[in]	y	corresponding function values
[in]	s1	second derivative at x(1)
[in]	sn	second derivative at x(n) (Natural spline: s1=sn=0)
[out]	csp	Coefficients stored in

References basic::print_msg().

Referenced by cubicspline::init().

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csplroots()

real(dp) function, dimension(:), allocatable, public csplroots

(

class(cubicspline), intent(in)

csp

)

csplroots Computes the roots of the Spline approximation

Parameters

[in]

csp

Spline approximation to consider

Returns

Roots (zeros) of the spline function

Examples:

real(dp), dimension(:), allocatable :: zeros
zeros = csp%roots()