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Re: A Problem To Solve :-)

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On 2/6/2024 3:00 PM, Farley Flud wrote:
> On Mon, 5 Feb 2024 19:39:57 -0600, Physfitfreak wrote:
> 
>>
>> If the drone calculated coordinates of the two points on that line of
>> sight as (x_1, y_1) and (x_2, y_2), with x being the distance and y the
>> height, then the distance of the target to launch area would have any of
>> the two forms below:
>>
>>       x  . y  - x  . y
>>        1    2    2    1
>> x = -----------------
>>            y  - y
>>             2    1
>>
>> or:
>>
>>
>>            /x  - x \ . y
>>            \ 2    1/    1
>> x = x  - --------------
>>        1       y  - y
>>                 2    1
>>
>>
>> And the Americans in charge being cro-magnons (and Nazi), seeing that
>> one of the above formulas can be derived from the other one, were
>> assuming it didn't matter which one to use for calculating x-intercept
>> for their missiles to hit Houthis targets. But Houthis, being Modern
>> Humans, always chose one of the above, and never the other.
>>
>> So Physfit found out why about half of American missiles narrowly miss
>> their targets, while Houthis missiles always hit the targets.
>>
>> 1- Verify that the two formulas are correct, and algebraically related.
>>
>> 2- Assuming the drone calculated (x_1, y_1) as (1.31, 4.76) kilometers,
>> and (x_2, y_2) as (1.93, 3.24) kilometers, and assuming that for
>> shortest computation time, numbers are rounded to three digits in the
>> codes imbedded in drones and missiles, then calculate the x-intercept in
>> both ways.
>>
>> 3- Explain which of the two methods is being used by the Houthis, and why.
>>
> 
> 1)
> 
> Multiply first term of second expression by 1 = (y2 - y1)/(y2 - y1):
> 
> x1 ⋅ (y2 - y1)   (x2 - x1) ⋅ y1
> ────────────── - ──────────────
>      y2 - y1          y2 - y1
> 
> 
> Combine fractions:
> 
> x1 ⋅ y2 - x1 ⋅ y1 - x2 ⋅ y1 + x1 ⋅ y1
> ─────────────────────────────────────
>                 y2 - y1
> 
> Cancel terms:
> 
> x1 ⋅ y2 - x2 ⋅ y1
> ─────────────────
>       y2 - y1
> 
> 
> They are equivalent.
> 
> 
> 2)
> 
> Using exact rational arithmetic based on the values given
> (e.g. 1.31 = 131/100), the x-intercept for both expressions
> is:
> 
> x = 3089/950 = 3.25
> 
> For an image of this, see this link:
> 
> https://postimg.cc/2LS0KRHr
> 
> 
> 3)
> 
> Because of potential catastrophic cancellation in distance coordinates,
> the first expression is preferable due to the absence of subtraction
> (i.e. x2 - x1) in the numerator.  For the record:
> 
> x1 ⋅ y2 - x2 ⋅ y1
> ─────────────────
>       y2 - y1
> 
> 


Your reasoning is correct of course, but if you had numerically 
calculated the relative errors and compared them to each other, it would 
be more convincing.

The second expression, as you said, has a potential problem. If the 
drone when recording its own coordinates two times, is very far from the 
launch site, then x_1 and x_2 will be represented in 3-digit decimal 
machine numbers by two numbers very close to each other, and their 
difference when rounded produces a number with lower than 3 significant 
digits. If we calculate the relative error compared to case of using 
exact numbers (no rounding off), we see that it is much larger than same 
relative errors for closer distances to launch site.

So, subtracting two large numbers that are very close to each other is 
like asking for failure if you have it done by a machine.

But using the first expression, the closeness of the numbers x_1 and x_2 
gets to a good degree ineffective because each is multiplied by the 
other point's associated height _before_ subtraction is performed. So, 
the difference between first and second terms in the numerator gets much 
larger, causing it to lose fewer significant digits when rounded.

That's why Houthis kicked ass :-)

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Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-01-27 17:22 -0600
  Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-01-28 12:31 +0000
    Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-01-28 16:44 -0600
      Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-01-29 08:28 +0000
        Re: A Problem To Solve :-) Athel Cornish-Bowden <me@yahoo.com> - 2024-01-29 11:53 +0100
          Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-02 23:56 -0600
        Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-01-30 01:25 -0600
        Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-01-31 18:18 -0600
      Re: A Problem To Solve :-) Stéphane CARPENTIER <sc@fiat-linux.fr> - 2024-02-02 21:49 +0000
  Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-03 04:00 -0600
    Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-02-03 16:19 +0000
      Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-03 16:17 -0600
      Re: A Problem To Solve :-) Stéphane CARPENTIER <sc@fiat-linux.fr> - 2024-02-04 14:24 +0000
    Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-04 19:53 -0600
    Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-05 01:51 -0600
      Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-05 19:39 -0600
        Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-02-06 21:00 +0000
          Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-06 15:43 -0600
            Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-06 18:19 -0600
              Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-09 10:58 -0600
          Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-06 21:55 -0600
            Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-06 22:06 -0600
        Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-12 01:39 -0600
          Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-14 12:50 -0600
            Re: A Problem To Solve :-) Lester Thorpe <lt@gnu.rocks> - 2024-02-14 20:33 +0000
              Re: A Problem To Solve :-) Athel Cornish-Bowden <me@yahoo.com> - 2024-02-15 11:11 +0100
              Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-15 13:22 -0600
            Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-18 01:51 -0600
              Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-21 00:38 -0600
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-02-27 16:13 -0600
                Re: A Problem To Solve :-) Kuang Tze Kui <ugdiun@kgknudu.cn> - 2024-02-27 22:46 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-01 20:50 -0600
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-02 12:10 +0000
                Re: A Problem To Solve :-) Physfitfreak <Physfitfreak@gmail.com> - 2024-03-02 12:06 -0600
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-02 19:44 +0000
                Re: A Problem To Solve :-) Farley Flud <ff@linux.rocks> - 2024-03-02 14:09 +0000
                Re: A Problem To Solve :-) Farley Flud <fflud@gnu.rocks> - 2024-03-02 19:18 +0000
  Re: A Problem To Solve :-) Arindam Banerjee <banerjeeadda1234@gmail.com> - 2024-02-09 09:47 -0800
    Re: A Problem To Solve :-) Arindam Banerjee <banerjeeadda1234@gmail.com> - 2024-02-12 04:41 -0800

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