Comments on: Finite differences http://www.johndcook.com/blog/2009/02/01/finite-differences/ The blog of John D. Cook Wed, 10 Mar 2010 01:04:08 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: John http://www.johndcook.com/blog/2009/02/01/finite-differences/comment-page-1/#comment-29203 John Sat, 19 Dec 2009 00:28:51 +0000 http://www.johndcook.com/blog/?p=1399#comment-29203 Jack, it depends on which analogy you want to draw. Log base 2 is analogous to the natural logarithm in that it is the inverse of 2<sup>x</sup>. But the harmonic numbers are analogous to the natural logarithm in that Δ H<sub>n</sub> = 1/n, just as the derivative of ln(x) is 1/x. Jack, it depends on which analogy you want to draw. Log base 2 is analogous to the natural logarithm in that it is the inverse of 2x. But the harmonic numbers are analogous to the natural logarithm in that Δ Hn = 1/n, just as the derivative of ln(x) is 1/x.

]]> By: Jack http://www.johndcook.com/blog/2009/02/01/finite-differences/comment-page-1/#comment-29200 Jack Sat, 19 Dec 2009 00:19:44 +0000 http://www.johndcook.com/blog/?p=1399#comment-29200 So which function should be considered the discrete analogue (no pun) of the logarithm function? Log base 2 , or H(n), the nth harmonic number? So which function should be considered the discrete analogue (no pun) of the logarithm function? Log base 2 , or H(n), the nth harmonic number?

]]> By: Rod Carvalho http://www.johndcook.com/blog/2009/02/01/finite-differences/comment-page-1/#comment-13307 Rod Carvalho Sat, 14 Feb 2009 07:00:05 +0000 http://www.johndcook.com/blog/?p=1399#comment-13307 John, "Discrete Calculus" is indeed very interesting. I only discovered its existence last year, and I was immediately intrigued by it. You mentioned the "summation by parts" technique. Another cool technique of discrete calculus that bears resemblance with continuous calculus is the "discrete l'Hôpital's rule", i.e., the <a href="http://topologicalmusings.wordpress.com/2008/05/08/stolz-cesaro-theorem/" rel="nofollow">Stolz-Cesàro theorem</a>. John,

“Discrete Calculus” is indeed very interesting. I only discovered its existence last year, and I was immediately intrigued by it.

You mentioned the “summation by parts” technique. Another cool technique of discrete calculus that bears resemblance with continuous calculus is the “discrete l’Hôpital’s rule”, i.e., the Stolz-Cesàro theorem.

]]> By: The 49th Carnival of Mathematics! « 360 http://www.johndcook.com/blog/2009/02/01/finite-differences/comment-page-1/#comment-13292 The 49th Carnival of Mathematics! « 360 Sat, 14 Feb 2009 01:53:21 +0000 http://www.johndcook.com/blog/?p=1399#comment-13292 [...] to ordinary calculus, but yields a few surprises itself, as illustrated by John Cook in  Finite differences at The Endeavour.     The number 49 is (4+3)2, a familiar fact, but it’s also the 4th [...] [...] to ordinary calculus, but yields a few surprises itself, as illustrated by John Cook in  Finite differences at The Endeavour.     The number 49 is (4+3)2, a familiar fact, but it’s also the 4th [...]

]]> By: RP http://www.johndcook.com/blog/2009/02/01/finite-differences/comment-page-1/#comment-12824 RP Thu, 05 Feb 2009 00:13:28 +0000 http://www.johndcook.com/blog/?p=1399#comment-12824 I've listened some older teachers telling about this but I never had the chance to study with details. I've been using foward and backward operators but didn't know there were calculus-like rules. Thanks for the post. I’ve listened some older teachers telling about this but I never had the chance to study with details. I’ve been using foward and backward operators but didn’t know there were calculus-like rules. Thanks for the post.

]]> By: John http://www.johndcook.com/blog/2009/02/01/finite-differences/comment-page-1/#comment-12707 John Mon, 02 Feb 2009 10:24:16 +0000 http://www.johndcook.com/blog/?p=1399#comment-12707 Sam, I got a PhD in math without ever seeing this. I worked with finite differences as approximations, but I never saw this elaborate calculus for working with finite differences until sometime after I was out of school. Notice there's no approximation going on here, only exact results. These tools can be used to solve discrete problems with no reference to ordinary calculus. Sam, I got a PhD in math without ever seeing this. I worked with finite differences as approximations, but I never saw this elaborate calculus for working with finite differences until sometime after I was out of school.

Notice there’s no approximation going on here, only exact results. These tools can be used to solve discrete problems with no reference to ordinary calculus.

]]> By: Samuel Jack http://www.johndcook.com/blog/2009/02/01/finite-differences/comment-page-1/#comment-12706 Samuel Jack Mon, 02 Feb 2009 09:57:44 +0000 http://www.johndcook.com/blog/?p=1399#comment-12706 John, Thanks for posting this: I would be interested to see some applications. Would you believe that in my four year degree course on Mathematics we never covered this subject! Sam John,
Thanks for posting this: I would be interested to see some applications. Would you believe that in my four year degree course on Mathematics we never covered this subject!

Sam

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