Interpolation

This notebook uses the Interpolations package. As an alternative, consider Dierckx.

Load Packages



In [1]:

    
using Dates, Interpolations

include("printmat.jl")









    Out[1]:





printyellow (generic function with 1 method)



In [2]:

    
using Plots

#pyplot(size=(600,400))    
gr(size=(480,320))
default(fmt = :svg)

Interpolation of y = f(x)

Interpolations are particularly useful when we repeatedly want to evaluate a function $f(x)$, or when we only know $f(x)$ for a grid of $x$ values but we have strong beliefs that the function is pretty smooth. We then do something like this:

Calculate $f(x)$ values for a grid of $x$. This creates a "look-up" table.
Replace the expensive calculations of $f(x_i)$ by interpolating $f(x_i)$ from the "look-up" table.

Some Values to Be Interpolated

As a simple illustration, we interpolate the sine function. (In practice, the interpolation technique is typically applied to more complicated functions.)



In [3]:

    
xGrid = range(-pi,stop=pi,length=101)  #uniformly spaced grid
yGrid = sin.(xGrid)                    #y values at xGrid 

plot( xGrid,yGrid,
      linecolor = :red,
      linewidth = 2,
      legend = nothing,
      title = "the sine function",
      xlabel = "x",
      ylabel = "y" )









    Out[3]:

Interpolate

The next cell calls on CubicSplineInterpolation() to create the "look-up" table (more precisely, create an interpolation object).

To use a cubic spline it is required that the $x_i$ grid is uniformly spaced (for instance, 0.1,0.2,...). The case of a non-uniformly spaced $x$ grid is discussed later.

The option extrapolation_bc=... determines how extrapolation beyond the range of the $x_i$ grid is done.

The second cell interpolates and extrapolates $y$ at some specific $x$ values.



In [4]:

    
itp = CubicSplineInterpolation(xGrid,yGrid,extrapolation_bc=Flat());



In [5]:

    
x  = [0.25,0.75]                     #to interpolate the y values at

y_interpolated = itp(x) 
                         
println("x, interpolated y values and true y values")
printmat([x y_interpolated sin.(x)])


x2             = [1.25,pi+0.1,pi+0.5]            #to extrapolate the y values at
y_extrapolated = itp(x2)                         #"extrapolation" can be done
                                                 #inside and outside the range xGrid
println("x2 and extrapolated values")
printmat([x2 y_extrapolated])









    



x, interpolated y values and true y values
     0.250     0.247     0.247
     0.750     0.682     0.682

x2 and extrapolated values
     1.250     0.949
     3.242     0.000
     3.642     0.000

Plotting the Results



In [6]:

    
plot( xGrid,yGrid,
      linecolor = :red,
      linewidth = 2,
      label = "sine function",
      title = "the sine function",
      xlabel = "x",
      ylabel = "y" )    
    
scatter!(x,y_interpolated,markercolor=:magenta,markersize=5,marker=:square,label="interpolated")
scatter!(x2,y_extrapolated,markercolor=:blue,markersize=8,label="extrapolated")









    Out[6]:

Interpolation of y = f(x) for General Vectors

That is, when we cannot guarantee that the look-up table of $y_i=f(x_i)$ is from uniformly spaced $x_i$ values. This is useful, for instance, when we have empirical data on $(x_i,y_i)$.

The approach works similar to before, except that the CubicSplineInterpolation must be replaced by LinearInterpolation.



In [7]:

    
xGrid2 = deleteat!(collect(xGrid),55:60)         #non-uniformly spaced grid     
yGrid2 = sin.(xGrid2)

scatter( xGrid2,yGrid2,
         linecolor = :red,
         legend = false,
         title = "y",
         xlabel = "x" )









    Out[7]:



In [8]:

    
itp2 = LinearInterpolation(xGrid2,yGrid2,extrapolation_bc=Flat()) 
                                           
y_interpolated = itp2(x) 
                         
println("x, interpolated y values and true y values")
printmat([x y_interpolated sin.(x)])

y_extrapolated = itp2(x2)   
                            
println("x2 and extrapolated values")
printmat([x2 y_extrapolated])









    



x, interpolated y values and true y values
     0.250     0.243     0.247
     0.750     0.682     0.682

x2 and extrapolated values
     1.250     0.949
     3.242     0.000
     3.642     0.000



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