| Re: replace integration by summation [message #84957 is a reply to message #84953] |
Wed, 19 June 2013 14:04   |
Phillip Bitzer
Messages: 223
Registered: June 2006
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Senior Member |
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OK, let's go with a simple example.
Let:
x=FINDGEN(4)
y=x^2
Your loop over INT_TABULATED will give the area under the curve for x=0->x[i]; this means
tab = FLTARR(4)
FOR i=1, 3 DO tab[i] = INT_TABULATED(x[0:i], y[0:i])
yields
print, tab
0.00000 0.500000 2.75556 9.10000
You can check these aren't quite the correct answers, *for the integration*, given the (known) underlying function, but are fairly close. The discrepancy is caused by the rather coarse grid of dx=1.
For example, the integral of x^2 between 0 and 2 (i=2) is analytically 8/3=2.67. This method is off by about 5%.
The method of using the cumulative total is *not* the area under the curve, i.e., it's not integration. In this example,
tot = total(y, /cum)*(x[1]-x[0])
print, tot
0.00000 1.00000 5.00000 14.0000
Clearly, this is not doing the same thing as integration. In this case, for i=2 you are finding the area of a rectangle 5 tics tall and 1 tics wide. This is not the same as the integration of x^2 between 0 and 2.
The underlying answer to your question is these two methods should not give the same answer - they are different operations.
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