Numerical Methods

Sometimes, while I am working, I will run across sentences like this:
...which can be computed according to the following formula given by SoAndSo, SomeDate:  {SomeFormula}.

I will stop reading, close the paper, and work out the relationship for myself.

There are several reasons.

The first is about me.  I doubt solutions until I can demonstrate them to myself.  I do not value memorization at the expense of comprehension.
We may rearrange a problem however we wish using the rules of algebra and geometry.  If mathematics is a language, it has only a few words.  The rest is concision and clarity.  I think the student who can manipulate 12th century Persian Algebra and weild a protractor, compass and straightedge, and know what she is up to, may gain ground as quickly as she like on University coursework in computer science and programming.

I seek to describe and solve problems in acoustics, wave propagation, and signal processing.  The state of sound design tools is poor, and their advancement is hindered by errors of logic and observation, accepted as common knowledge and understanding.  To identify, describe, and solve these problems, it is sufficient to construct mathematical descriptions of sound, and account for the physical properties in a consistent way.  The errors are basic.  They can be demonstrated with a walk down the street; walking around a streetlight; listening to a train.  To make the properties of waves more intuitive and accessible --and to insure that they are correctly accounted for-- I have constructed a series of analogies to light: construction of an image, its properties, and the mechanics of wave propagation.  My solutions will follow directly from known mathematical results, and observation of the physical world.  I work out each problem from scratch. I teach myself the necessary math, mostly out of old Dover press paperbacks.

Patents and Mathematics
The second reason is about mathematics.  All mathematical results follow inescapably from the definitions and rules of manipulation.  (Which is not to say easy.)  A valid result is one which can [and will] be obtained independently by anyone [and everyone].  The proof is demonstration.  The bracketed words are the heart of the matter.

Not everyone is clear about this.  That is a problem, and each person has to decide how to handle it.   I choose not to be slowed down.  I do not have time to pretend that a person can own math.  I sequester myself; I will not copy another person's mind. I have always worked this way, and I love it.  The history of mathematics is free to use, and any results I derive from it are equally free.  So therein I will work.

I am not overly fond of Modern Abstract Mathematics.  Proponents often confuse themselves with the tools they advocate, and say what anyone can understand  in a way that only a few people can.  The proofs usually run backward, never identifying the structure of the problem.  So I practice going backward in time, to obtain the answer.  Each time I go back, what I find is worth the trip.

There is more at stake than money, time, or the enjoyment of mathematics.  Math belongs to no one.  We are ethically responsible to make it accessible.  We also have a rational responsibility.  Mathematics is a logically elaborated structure; within it appeal to authority is baseless.  Any declaration of ownership can be shown to be invalid.
The last two sentences are exact.  There is no other ground to take.

It's alright.  I won't use or examine methods which claim to ownership.  If you Own algorithms, I won't read your paper; it can sit quiet on the shelf. I do not want to know anything about what you are up to.  If we stumble into each other, there will be no question.   All I will need is a chalkboard.

Onward.

Example, With Train

 Recently, I was reading a paper on efficient computation of caustics.  Here is where I stopped:

"Let's forget about reflection for a moment and see how transmitted photons are refracted according to Snell's Law, which states that:  η₁ sin θ₁ = η₂ sin θ₂,   otherwise written as:
IOR = η₁/η ₂ = sin θ₂/sinθ₁
In the preceding equations, η₁ and η₂ are the indices of refraction for the respective materials, and θ₁ and θ₂ are the incident and refraction angles, as shown in Figure 2-2. The index of refraction, IOR, can then simply be written as the ratio of the sines of the angles of the incident and refracted rays.
Snell's Law is not easy to code with this formulation, because it only imposes one restriction, making the computation of the refracted ray nontrivial. Assuming that the incident, transmitted, and surface normal rays are co-planar, a variety of coder-friendly formulas can be used, such as the one in Foley et al. 1996: [....]"

[from, http://http.developer.nvidia.com/GPUGems/gpugems_ch02.html . Italics mine.  Diagram omited.]

I have left out the next line.  I glanced at it, but it was unfamiliar so I stopped, closed the paper, and began working from scratch.  The citation was just the icing.

   This is from a book called GPUGems which is available for free on nVidia's website. This is a great idea, and I hope more companies catch on.  I work alone on a limited budget and the GPUGems series, and nVidia in general, distinguish themselves for free numerical methods, heaped with examples and explanations.  When I am employed, one of my first treats to myself will be to purchase the whole series.
It makes me think selling numerical methods is easy, as long as you don't patent them.  You print them, and then you can sell them.  People will buy them so they can use them.  There's a word for this kind of cooperative-competitive market model, but I just can't remember what it is....


In this case, I do not think there is a problem.  Citation is essential to research and the propagation of knowledge.  I can investigate further if I want, and the author is a good professional  neighbor to boot.  But why don't we solve the problem anyway?  How do people resist?

Problem:  Assuming that the incident, transmitted, and surface normal rays are co-planar, derive a [varitey of] coder-friendly formula[s] for the direction of the transmitted ray.

___________

Update:  I am deeply indebted to Professor Herbert Gross for his clear and concise language.

  Inescapable at last supplies what is wanting in the troublesome expression, self-evident.

Professor Gross' video lectures at MIT's Open Course Ware can be found here.