FortranConsoleSpline

Utility module for cubic spline interpolation, with example program.

Usage

Enable the module in your code with

use mod_splines

and then define a type(spline) object in your variable declaration section. For example:

type(spline) :: sp

Initialize the spline with two arrays, one for x values and one for y values. Both arrays must have the same number of elements

! Create a natural spline
sp = spline(xi, yi)

If you know the start and end slope of the spline, you can add them as optional parameters

! Create a spline with flat ends
sp = spline(xi, yi, 0.0d0, 0.0d0)

To extract a value for the spline use the elemental type bound function value(x)

y = sp%value(x)

Here x and y are scalar values. But since it is an elemental function, it can work with arrays directly. For example

xi= [(x0+i*h, i=0, n-1)]
yi = sp%value(xi)

where xi and yi are arrays

Additionaly, you can interpolate the slope and the second derivative using the slope(x) and slope2(x) functions. For example

y = sp%value(x)
yp = sp%slope(x)
ypp = sp%slope2(x)

As a note, if the evaluation point x is outside the defined range for the spline, then a linear iterpolation is done from the ends of the spline.

Finally, if a local extrema point is needed (local minimum or maximum) a bisection method is used to find the x value where the slope is zero using the extrema(x_low,x_high) function

x_max = sp%extrema()
y_max = sp%value(x_max)

The high and low limits of the extrema function are optional and the start and end of the spline are used by default.

Example

The included example program below defines an spline using 9 points between 0 and 2 with an interval of 0.25.

The interplation output has an interval of 0.20.

In the end the local maximum is found.

Program

! Variables
real(wp), allocatable :: xi(:), yi(:), h, x, y, yp, x0
type(spline) :: sp
integer :: i, n
! compile with /fpconstant
xi = [0.0,0.25,0.5,0.75,1.0,1.25,1.5,1.75,2.0]
yi = [18.0,18.4921875,18.9375,19.2890625,19.5,19.5234375,19.3125,18.8203125,18.0]

print *, 'Cubic Spline Interpolation Demo'
    
n = 11
h = (xi(size(xi))-xi(1))/(n-1)
x0 = xi(1)
sp = spline(xi, yi)
print *, ""
print '(1x,a6,1x,a18,1x,a18,1x,a18)', "Index", "x", "y", "yp"
do i=0,n-1
    x = x0 + i*h
    y = sp%value(x)
    yp = sp%slope(x)
    print '(1x,i6,1x,g18.11,1x,g18.6,1x,g18.6)', i, x, y, yp
end do
print *, ""
    
x = sp%extrema()
i = sp%indexof(x)
y = sp%value(x)
yp = sp%slope(x)
    
print *, "Local Extrema"
print '(1x,a6,1x,a18,1x,a18,1x,a18)', "Index", "x", "y", "yp"
print '(1x,i6,1x,g18.11,1x,g18.6,1x,g18.6)', i, x, y, yp

Output

The output of the example is as follows:

 Cubic Spline Interpolation Demo

  Index                  x                  y                 yp
      0   0.0000000000            18.0000            2.06799
      1  0.20000000000            18.4009            1.87745
      2  0.40000000000            18.7637            1.73300
      3  0.60000000000            19.0861            1.47687
      4  0.80000000000            19.3398           0.943939
      5   1.0000000000            19.5000           0.209478
      6   1.2000000000            19.5304          -0.461936E-01
      7   1.4000000000            19.4106          -0.938651
      8   1.6000000000            19.1328           -1.85224
      9   1.8000000000            18.6726           -3.07239
     10   2.0000000000            18.0000           -3.50827

 Local Extrema
  Index                  x                  y                 yp
      5   1.1857554913            19.5308           0.738816E-07

License

MIT © John Alexiou