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Lagrange Interpolation with Maple
Lagrange Interpolation with Maple
# This Maple session shows how to implement
# Lagrange interpolation for the function
# exp(-10x^2) (a Gaussian distribution)
# at 5 nodes (4 intervals) on the interval [-1,1]. To import
# this into a Maple session, first save
# it as a text file (click on File, then Save As,
# and type in a file name with a .txt suffix).
# Then open a Maple session and
# click on File, and then click on Import Text.
# A dialogue window will appear and you can
# click on the file name in your home directory
# that you gave earlier.
# As an interesting exercise, try decreasing the step size
# to 1/4 and increasing the number of nodes to 9 (8 intervals).
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> f:=x->exp(-10*x^2);
2
f := x -> exp(- 10 x )
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> for i from 0 to 4 do
> x[i]:=-1+i*.5
> od;
x[0] := -1.
x[1] := -.5
x[2] := 0
x[3] := .5
x[4] := 1.0
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> for i from 0 to 4 do
> L[i]:=product((x-x[j])/(x[i]-x[j]),j=0..i-1)*product((x-x[k])/(x[i]-x[k]),k=i+1..4)
> od;
L[0] := - 1. ( - 2.000000000 x - 1.000000000) x
( - .6666666667 x + .3333333334) ( - .5000000000 x + .5000000000)
L[1] := - 2.000000000 (2.000000000 x + 2.000000000) x
( - 1.000000000 x + .5000000000) ( - .6666666667 x + .6666666667)
L[2] := (1. x + 1.) (2.000000000 x + 1.000000000)
( - 2.000000000 x + 1.000000000) ( - 1.000000000 x + 1.000000000)
L[3] := 2.000000000 (.6666666667 x + .6666666667) (1.000000000 x + .5000000000)
x ( - 2.000000000 x + 2.000000000)
L[4] := 1.000000000 (.5000000000 x + .5000000000) (.6666666667 x + .3333333334)
x (2.000000000 x - 1.000000000)
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> lagr:=sum(f(x[p])*L[p],p=0..4);
lagr := - .00004539992976 ( - 2.000000000 x - 1.000000000) x
( - .6666666667 x + .3333333334) ( - .5000000000 x + .5000000000)
- .1641699972 (2.000000000 x + 2.000000000) x
( - 1.000000000 x + .5000000000) ( - .6666666667 x + .6666666667) +
(1. x + 1.) (2.000000000 x + 1.000000000) ( - 2.000000000 x + 1.000000000)
( - 1.000000000 x + 1.000000000) + .1641699972 (.6666666667 x + .6666666667)
(1.000000000 x + .5000000000) x ( - 2.000000000 x + 2.000000000)
+ .00004539992976 (.5000000000 x + .5000000000)
(.6666666667 x + .3333333334) x (2.000000000 x - 1.000000000)
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> plot({f(x),lagr},x=-1..1,y=-2..2);
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> i:='i';
i := i
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