Reduction of Order (Mark II)

[topsquark]

(No, not that reduction of order method! I couldn't think of a better name, though. Sorry for any confusion.)

I've been taking a break from my usual study programme to do some work on solving recursion relations with non-constant coefficients. I'm sure there is a theory of how to approach these but as I haven't really found anything about them on the net I thought I'd share my findings. So there are a few reasons why I wrote this::
1) It's been a very long time since I've written one and I thought it would be good to exercise some muscles
2) I wanted to try out my new LaTeX based word processor
3) I wanted to get some feedback on an application to Differential Equations that cropped up.

I'm looking for someone to do a critical review of the process and layout of the paper.

It's original work but I figure anything I might have come up with has been known for some two to three hundred years, so if someone wants to steal the idea and work on it themselves, go for it!

As to the last part, the paper is on solving recursion equations but I found that the method works equally well for solving Differential Equations. But I've never come across the method in any of my research so I wanted to see what someone has to say about that as well.

It's basically a Senior college level paper on Linear Algebra, but it's written at the College Algebra Mathematics level so it should be quite readable for the layman who is interested.

Please let me know what you think.

-Dan

 

[Cubist]

Thanks for sharing your paper, it looks like a lot of work has gone into it!

I think that my first goal is to fully understand section 2. Appendix A.1 will probably be very helpful with this goal in mind.

For now, I'll just post my initial thoughts rather than blasting you with one big post. Hope that these comments are useful...

• no page numbers are shown on the pages after page 1
• doesn't use "pdf indexing" (not sure of the proper name for this) which would appear as a separate tab/ window in a decent pdf viewer (in addition to the expected list of contents that appears on page 2)
• The page 2 contents doesn't have clickable links (a minor point, but would this be an expected feature nowadays?)
• I, personally, would choose to include a (virtually empty) bibliography section. And in there you could just use the text that currently appears in the foreward to explain why the bibliography is otherwise empty.

Content-wise, in section 2.1 I think that this sentence might need to be reworded, "This paradox does not occur for any other order and as there are many ways to solve a first order recursion reduction of order will be used simply to show how the technique is to work." I'm not naturally great at language and tend to prefer smaller sentences! How about, "This paradox does not occur for higher orders. Since there are many simpler ways to solve a first order recursion reduction, the work in this section should only be viewed as an introduction to this technique. The real payoff comes when solving higher order systems." But possibly I have misunderstood your intent, due to me not fully understanding the topic as a whole yet!

I think there's no need to post an updated pdf on the back of these small suggestions... that's if you decide to take any of them on board. I'll try to read more of it over the next days.

 

[topsquark]

Thanks for sharing your paper, it looks like a lot of work has gone into it!

I think that my first goal is to fully understand section 2. Appendix A.1 will probably be very helpful with this goal in mind.

For now, I'll just post my initial thoughts rather than blasting you with one big post. Hope that these comments are useful...

• no page numbers are shown on the pages after page 1
• doesn't use "pdf indexing" (not sure of the proper name for this) which would appear as a separate tab/ window in a decent pdf viewer (in addition to the expected list of contents that appears on page 2)
• The page 2 contents doesn't have clickable links (a minor point, but would this be an expected feature nowadays?)
• I, personally, would choose to include a (virtually empty) bibliography section. And in there you could just use the text that currently appears in the foreward to explain why the bibliography is otherwise empty.

Content-wise, in section 2.1 I think that this sentence might need to be reworded, "This paradox does not occur for any other order and as there are many ways to solve a first order recursion reduction of order will be used simply to show how the technique is to work." I'm not naturally great at language and tend to prefer smaller sentences! How about, "This paradox does not occur for higher orders. Since there are many simpler ways to solve a first order recursion reduction, the work in this section should only be viewed as an introduction to this technique. The real payoff comes when solving higher order systems." But possibly I have misunderstood your intent, due to me not fully understanding the topic as a whole yet!

I think there's no need to post an updated pdf on the back of these small suggestions... that's if you decide to take any of them on board. I'll try to read more of it over the next days.

Your response is appreciated. I don't know that I can do the clickable links. I'll look. My word processor can do just about anything, but I haven't seen a section on that.

The page numbers are there. It's just, in my last draft, I added the word "solution" to the title and it's now overwriting the page numbers. I have also found a couple of typos in the Appendix. Grrrrr.... If I find any more I'm likely to post an update.

I see what you mean about the sentence. I'll have to transfer some of the text to Word (or something) and do a grammar check on it. I should have thought of that before.

Thanks again! Keep the comments coming.

-Dan

 

[topsquark]

Thanks for sharing your paper, it looks like a lot of work has gone into it!

I think that my first goal is to fully understand section 2. Appendix A.1 will probably be very helpful with this goal in mind.

For now, I'll just post my initial thoughts rather than blasting you with one big post. Hope that these comments are useful...

• no page numbers are shown on the pages after page 1
• doesn't use "pdf indexing" (not sure of the proper name for this) which would appear as a separate tab/ window in a decent pdf viewer (in addition to the expected list of contents that appears on page 2)
• The page 2 contents doesn't have clickable links (a minor point, but would this be an expected feature nowadays?)

Here's an update with a few small corrections.

I don't think I can do a TOC with links. That appears to be an Adobe feature and I haven't bought that. Sorry!

-Dan

 

[topsquark]

I lied about the TOC.

-Dan

 

[Cubist]

I can see the page numbers very clearly now! You're right, they existed before, but they were hiding behind your header text

I've been reading appendix A.1 I've thrown some numbers into the initial and final equations and I've verified the result

I would have said...
Let g(n) = h_n
...because shouldn't the first variable be the new one that we're defining and assigning? But maybe I'm suggesting this because much of my background is in computing and BASIC language has a keyword "LET <variable>=<expression>" which assigns a value to a variable.

I think "A" can just take any real number value, right? Does this variable need to be introduced, or perhaps A is a standard variable that requires no introduction in this field.

I was a bit confused because I thought A must be 1, since it was introduced like this
g(n) = <something>
h_n = g(n) = A * <the same something as above> which implies A=1
...but the final equation seems to work when plugged into the initial equation even if A is a different value (and non-integers work also)

Sorry if my feedback is too basic, and perhaps too picky. Please tell me if these kind of comments are useful or not (I won't be offended!)


FYI: Regarding LaTeX, I've occasionally used "LyX" to create a document. And recently, while at Uni, my son has used a web-based LaTeX editor called Overleaf.- I think it's free but a paid for version offers extra features.

 

[Cubist]

Also this sentence needs a correction...

The recursion is linear (as all will be all examples) so we may put...

Suggested alternative, "All examples will be linear recursions. Therefore, we may put..." but please feel free to use your own wording or just simply remove the extra "all"!

My daughter recommends Grammarly. I think the basic version is free to use. Perhaps the plain text of your document can be copied out of the pdf into this utility, or perhaps into word or another similar checking software, just to perform an initial sweep for things like this which are SO easy to miss when you're the author. (I kind of scoffed when she suggested Grammarly because I've seen it advertised SO much on youtube. This level of advertising can put me off a product. But, apparently it's worth a shot, who knew ?‍. I haven't tried it myself, yet )

 

[topsquark]

I can see the page numbers very clearly now! You're right, they existed before, but they were hiding behind your header text

I've been reading appendix A.1 I've thrown some numbers into the initial and final equations and I've verified the result

I would have said...
Let g(n) = h_n
...because shouldn't the first variable be the new one that we're defining and assigning? But maybe I'm suggesting this because much of my background is in computing and BASIC language has a keyword "LET <variable>=<expression>" which assigns a value to a variable.

I think "A" can just take any real number value, right? Does this variable need to be introduced, or perhaps A is a standard variable that requires no introduction in this field.

I was a bit confused because I thought A must be 1, since it was introduced like this
g(n) = <something>
h_n = g(n) = A * <the same something as above> which implies A=1
...but the final equation seems to work when plugged into the initial equation even if A is a different value (and non-integers work also)

Sorry if my feedback is too basic, and perhaps too picky. Please tell me if these kind of comments are useful or not (I won't be offended!)


FYI: Regarding LaTeX, I've occasionally used "LyX" to create a document. And recently, while at Uni, my son has used a web-based LaTeX editor called Overleaf.- I think it's free but a paid for version offers extra features.

A is an arbitrary constant. (Actually, it's g(0). But we have the freedom to define that to be anything we like, as an initial condition.)

I'll consider the rest. Thanks for the Grammarly tip. I'll look it up.

And, FYI: I want you to be picky!

-Dan