Hi all,
I have never been particularly fond of numerical integration and
generally do it pretty infrequently these days. Nevertheless, I am
trying to do a 'quick-and-dirty' atmospheric refraction/ray-trace
calculation and I'm stumped on the integration. The integral reads:
s = \int_{r1}^{r2} ( n(r) * r * dr ) / sqrt( n(r)^2 - c^2 )
where c is a constant. The trouble I'm having is that n is a function
of r. Thus, I have a set of discrete points for r:
0.00000 1.00000 2.00000 3.00000 4.00000
5.00000 6.00000
7.00000 8.00000 9.00000 10.0000 11.0000
12.0000 13.0000
14.0000 15.0000 16.0000 17.0000 18.0000
19.0000 20.0000
21.0000 22.0000 23.0000 24.0000 25.0000
26.0000 27.0000
28.0000 29.0000 30.0000 31.0000 32.0000
33.0000 34.0000
35.0000 36.0000 37.0000 38.0000 39.0000
40.0000 41.0000
42.0000 43.0000 44.0000 45.0000 46.0000
47.0000 48.0000
49.0000 50.0000 51.0000 52.0000 53.0000
54.0000 55.0000
56.0000 57.0000 58.0000 59.0000 60.0000
61.0000 62.0000
63.0000 64.0000 65.0000 66.0000 67.0000
68.0000 69.0000
70.0000 71.0000 72.0000 73.0000 74.0000
75.0000 76.0000
77.0000 78.0000 79.0000 80.0000 81.0000
82.0000 83.0000
84.0000 85.0000 86.0000 87.0000 88.0000
89.0000 90.0000
91.0000 92.0000 93.0000 94.0000 95.0000
96.0000 97.0000
98.0000 99.0000 100.000 101.000 102.000
103.000 104.000
105.000 106.000 107.000 108.000 109.000
110.000 111.000
112.000 113.000 114.000 115.000 116.000
117.000 118.000
119.000 120.000
and a set of corresponding points for n(r):
3210.0997 1915.7633 1031.6773 492.45952
253.87699 125.97528
60.358510 28.798265 14.395186 7.6860971
3.7610915 1.4102413
1.0227767 1.0044470 1.0012146 1.0003848
1.0001596 1.0001319
1.0001193 1.0001084 1.0001147 1.0001217
1.0001248 1.0001266
1.0001274 1.0001257 1.0001244 1.0001259
1.0001277 1.0001302
1.0001338 1.0001371 1.0001398 1.0001418
1.0001443 1.0001467
1.0001496 1.0001525 1.0001557 1.0001591
1.0001626 1.0001662
1.0001701 1.0001742 1.0001787 1.0001837
1.0001883 1.0001915
1.0001952 1.0001991 1.0002034 1.0002080
1.0002133 1.0002177
1.0002231 1.0002281 1.0002309 1.0002338
1.0002362 1.0002342
1.0002335 1.0002329 1.0002331 1.0002304
1.0002273 1.0002229
1.0002154 1.0002079 1.0002032 1.0001983
1.0001917 1.0001836
1.0001748 1.0001659 1.0001570 1.0001472
1.0001364 1.0001261
1.0001162 1.0001071 1.0000989 1.0000926
1.0000869 1.0000819
1.0000777 1.0000739 1.0000705 1.0000671
1.0000638 1.0000605
1.0000570 1.0000535 1.0000501 1.0000466
1.0000432 1.0000401
1.0000372 1.0000344 1.0000319 1.0000295
1.0000272 1.0000251
1.0000232 1.0000213 1.0000195 1.0000177
1.0000159 1.0000143
1.0000127 1.0000112 1.0000097 1.0000083
1.0000072 1.0000060
1.0000052 1.0000044 1.0000039 1.0000034
1.0000030 1.0000026
1.0000024
I have three similar integrals to evaluate but I'm pretty lost at the
moment. Sadly, I suspect this isn't as difficult as I am making it.
Any ideas?
Cheers,
Dave
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 Confused.Scientist (28)
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12/4/2006 10:31:47 PM |
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While it probably doesn't matter too much, the integral should have
read:
s = \int_{r1}^{r2} ( n(r) * r * dr ) / sqrt( n(r)^2 * r^2 - c^2 )
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Confused.Scientist (28)
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12/4/2006 10:39:52 PM
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In article <1165271507.802034.200410@j72g2000cwa.googlegroups.com>, "Dave" <Confused.Scientist@gmail.com> writes:
>Hi all,
>
>I have never been particularly fond of numerical integration and
>generally do it pretty infrequently these days. Nevertheless, I am
>trying to do a 'quick-and-dirty' atmospheric refraction/ray-trace
>calculation and I'm stumped on the integration. The integral reads:
>
> s = \int_{r1}^{r2} ( n(r) * r * dr ) / sqrt( n(r)^2 - c^2 )
>
>where c is a constant. The trouble I'm having is that n is a function
>of r. Thus, I have a set of discrete points for r:
>
> 0.00000 1.00000 2.00000 3.00000 4.00000
>5.00000 6.00000
> 7.00000 8.00000 9.00000 10.0000 11.0000
>12.0000 13.0000
> 14.0000 15.0000 16.0000 17.0000 18.0000
>19.0000 20.0000
> 21.0000 22.0000 23.0000 24.0000 25.0000
>26.0000 27.0000
> 28.0000 29.0000 30.0000 31.0000 32.0000
>33.0000 34.0000
> 35.0000 36.0000 37.0000 38.0000 39.0000
>40.0000 41.0000
> 42.0000 43.0000 44.0000 45.0000 46.0000
>47.0000 48.0000
> 49.0000 50.0000 51.0000 52.0000 53.0000
>54.0000 55.0000
> 56.0000 57.0000 58.0000 59.0000 60.0000
>61.0000 62.0000
> 63.0000 64.0000 65.0000 66.0000 67.0000
>68.0000 69.0000
> 70.0000 71.0000 72.0000 73.0000 74.0000
>75.0000 76.0000
> 77.0000 78.0000 79.0000 80.0000 81.0000
>82.0000 83.0000
> 84.0000 85.0000 86.0000 87.0000 88.0000
>89.0000 90.0000
> 91.0000 92.0000 93.0000 94.0000 95.0000
>96.0000 97.0000
> 98.0000 99.0000 100.000 101.000 102.000
>103.000 104.000
> 105.000 106.000 107.000 108.000 109.000
>110.000 111.000
> 112.000 113.000 114.000 115.000 116.000
>117.000 118.000
> 119.000 120.000
>
>and a set of corresponding points for n(r):
>
> 3210.0997 1915.7633 1031.6773 492.45952
>253.87699 125.97528
> 60.358510 28.798265 14.395186 7.6860971
>3.7610915 1.4102413
> 1.0227767 1.0044470 1.0012146 1.0003848
>1.0001596 1.0001319
> 1.0001193 1.0001084 1.0001147 1.0001217
>1.0001248 1.0001266
> 1.0001274 1.0001257 1.0001244 1.0001259
>1.0001277 1.0001302
> 1.0001338 1.0001371 1.0001398 1.0001418
>1.0001443 1.0001467
> 1.0001496 1.0001525 1.0001557 1.0001591
>1.0001626 1.0001662
> 1.0001701 1.0001742 1.0001787 1.0001837
>1.0001883 1.0001915
> 1.0001952 1.0001991 1.0002034 1.0002080
>1.0002133 1.0002177
> 1.0002231 1.0002281 1.0002309 1.0002338
>1.0002362 1.0002342
> 1.0002335 1.0002329 1.0002331 1.0002304
>1.0002273 1.0002229
> 1.0002154 1.0002079 1.0002032 1.0001983
>1.0001917 1.0001836
> 1.0001748 1.0001659 1.0001570 1.0001472
>1.0001364 1.0001261
> 1.0001162 1.0001071 1.0000989 1.0000926
>1.0000869 1.0000819
> 1.0000777 1.0000739 1.0000705 1.0000671
>1.0000638 1.0000605
> 1.0000570 1.0000535 1.0000501 1.0000466
>1.0000432 1.0000401
> 1.0000372 1.0000344 1.0000319 1.0000295
>1.0000272 1.0000251
> 1.0000232 1.0000213 1.0000195 1.0000177
>1.0000159 1.0000143
> 1.0000127 1.0000112 1.0000097 1.0000083
>1.0000072 1.0000060
> 1.0000052 1.0000044 1.0000039 1.0000034
>1.0000030 1.0000026
> 1.0000024
>
>I have three similar integrals to evaluate but I'm pretty lost at the
>moment. Sadly, I suspect this isn't as difficult as I am making it.
>
>Any ideas?
>
Equally spaced data, any variant on Simpson will do well here. I'm
using my own, INTEG, you can find it in Midl_lib on the users
contributions page of RSI. but, as I said, any Simpson will do.
Mati Meron | "When you argue with a fool,
meron@cars.uchicago.edu | chances are he is doing just the same"
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 mmeron (89)
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12/4/2006 11:13:42 PM
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On 4 Dec 2006 14:31:47 -0800, "Dave" <Confused.Scientist@gmail.com>
wrote:
>Hi all,
>
>I have never been particularly fond of numerical integration and
>generally do it pretty infrequently these days. Nevertheless, I am
>trying to do a 'quick-and-dirty' atmospheric refraction/ray-trace
>calculation and I'm stumped on the integration. The integral reads:
>
> s = \int_{r1}^{r2} ( n(r) * r * dr ) / sqrt( n(r)^2 - c^2 )
>
>where c is a constant. The trouble I'm having is that n is a function
>of r. Thus, I have a set of discrete points for r:
I'm not sure why "n is a function of r" would be a problem for
numerical integration. As long as you can evaluate it, as you clearly
can, there is no problem. Or am I missing something?
You can for example use IDL's INT_TABULATED where
X=r and F = ( n(r) * r) / sqrt( n(r)^2 * r^2 - c^2 )
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 nomail37 (248)
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12/5/2006 9:28:32 AM
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On Tue, 05 Dec 2006 10:28:32 +0100, Wox <nomail@hotmail.com> wrote:
>You can for example use IDL's INT_TABULATED where
>X=r and F = ( n(r) * r) / sqrt( n(r)^2 * r^2 - c^2 )
Maybe QROMB or QSIMP are better, I don't know. Check the manual and
try them out :-).
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nomail37 (248)
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12/5/2006 9:41:14 AM
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4 Replies
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