# Numeric root-finding in Python

Dave Angel d at davea.name
Mon Feb 13 00:52:47 CET 2012

```On 02/12/2012 06:05 PM, Steven D'Aprano wrote:
> On Sun, 12 Feb 2012 12:18:15 -0800, Mark Dickinson wrote:
>
>> On Feb 12, 6:41 am, Steven D'Aprano<steve
>> +comp.lang.pyt... at pearwood.info>  wrote:
>>
>>>              err = -a/b  # Estimate of the error in the current w.
>>>              if abs(err)<= 1e-16:
>>>                  break
>> If the result you're expecting is around -1.005, this exit condition is
>> rather optimistic:  the difference between the two Python floats either
>> side of this value is already 2.22e-16, so you're asking for less than
>> half a ulp of error!
> I was gradually coming to the conclusion on my own that I was being
> overly optimistic with my error condition, although I couldn't put it
> into words *why*. Thanks for this Mark, this is exactly the sort of thing
> I need to learn -- as is obvious, I'm no expert on numeric programming.
>
>
>> As to the rest;  your error estimate simply doesn't have enough
>> precision.  The main problem is in the computation of a, where you're
>> subtracting two almost identical values.<SNIP>
> <SNIP>
>
Two pieces of my history that come to mind.  40+ years ago I got a
letter from a user of our computer stating that our math seemed to be
imprecise in certain places.   He was very polte about it, and admitted
to maybe needing a different algorithm.  The letter was so polite that I
(as author of the math microcode) worked on his problem, and found the
difficulty, as well as a solution.

The problem was figuring out the difference in a machining table between
being level every place on its surface (in which case it would be
slightly curved to match the earth), or being perfectly flat (in which
case some parts of the table would be further from the earth's center
than others)  The table was 200 feet long, and we were talking
millionths of an inch.  He solved it three ways, and got three different
answers.  The first two differed in the 3rd place, which he thought far
too big an error, and the third answer was just about exactly half the
others.

Well the 2:1 discrepancy just happens when you change your assumption of
what part of the flat table is level.  If the center is level, then the
edges are only 100 feet out, while if the edge is level, the other edge
is 200 feet out.

But the other solution was very interesting.  Turns out he sketched a
right triangle, with narrow angle at the center of the earth, side
opposite being 200 feet.  He then calculated the difference between the
other two sides.  one 8000 miles, and the other 8000 miles plus a few
microinches.  He got that distance by subtracting the sine from the
tangent, or something similar to that.  I had microcoded both those
functions, and was proud of their accuracy.  But if you subtract two 13
digit numbers that only differ in the last 3, you only get 3 digits
worth of accuracy, best case.

Solution was to apply some similar triangles, and some trivial
approximations, and the problem turned out not to need trig at all, and
accurate to at least 12 places.  if I recall, it was something like
8000mi is to 200 feet, as 200 feet is to X.  Cross multiply and it's
just arithmetic.

The other problem was even earlier.  It was high school physics, and the
challenge was to experimentally determine the index of refraction of air
to 5 places.  Problem is our measurements can't be that accurate.  So
this is the same thing in reverse.  Find a way to measure the difference
of the index of refraction of air and vacuum, to one or two places, and

taken together with lots of other experience, i try to avoid commiting
an algorithm to code before thinking about errors, convergence, and
exceptional conditions.  I've no experience with Lambert, but I suspect
it can be attacked similarly.

--

DaveA

```