http://www.mapleprimes.com/posts/43762-Two-Illustrations-Of-Is-G-The-Same-As-F en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Wed, 10 Jun 2026 21:34:21 GMT Wed, 10 Jun 2026 21:34:21 GMT The latest comments added to the Post, Two Illustrations of "Is g the same as f?" http://www.mapleprimes.com/images/mapleprimeswhite.jpg http://www.mapleprimes.com/posts/43762-Two-Illustrations-Of-Is-G-The-Same-As-F http://www.mapleprimes.com/posts/43762-Two-Illustrations-Of-Is-G-The-Same-As-F?ref=Feed:MaplePrimes:Two Illustrations of "Is g the same as f?":Comments#comment80758 Hello, I just made use of the ability to post <a href="http://beta.mapleprimes.com/book/posting_maple_html_export">Maple HTML export</a> and modified your post to contain the contents of your worksheet. I then posted it to the <a href="http://beta.mapleprimes.com">Front page</a>. Thanks for your excellent posting. ____ William Spaetzel Applications Developer, Maplesoft The latest comments added to the Post, Two Illustrations of "Is g the same as f?" 80758 Tue, 09 Aug 2005 00:07:33 Z Will Will http://www.mapleprimes.com/posts/43762-Two-Illustrations-Of-Is-G-The-Same-As-F?ref=Feed:MaplePrimes:Two Illustrations of "Is g the same as f?":Comments#comment80757 Whether the question is "Is Pi and 3.14 the same?" or "To what extent is the residual of the approximate solution of the PDE zero?" I think the mathematical issues are the same. Unless one poses a definition of "the same" (or equivalently, of "zero") the questions are not well defined. It's like asking if two vectors are the same without a definition of the norm being used to measure the difference between them. Jim's "easy" question yields to an evalf, which shows the extent to which 3.14 approximates Pi. Jim's PDE example is more difficult because it requires finding the maximum of a function of two variables. The residual for the approximate solution is a function of two variables, and any viable techniques for finding the maximum of its absolute value would suffice. For the function in this example, the greatest difficulty seems to be the division by r. However, each such divisor sits under BesselJ(1, lambda*r), and as r goes to zero, so does BesselJ(1,alpha*r). The issue is whether r=0 is a removable singularity or not. It turns out that it is, and there is a finite upper bound to the maximum of the residual. But if we change the norm (say, rms norm) then the maximum of the residual would most likely change, so the question as to whether the residual of the approximate solution is sufficiently close to zero to warrant approval must be linked to a definition of the norm and to the measure of tolerance that will be accepted as sufficient. Using the infinity norm I determined the max was about 1.27 * 10(-9), and occurred in the limit as both r and t approached zero. RJL Maplesoft The latest comments added to the Post, Two Illustrations of "Is g the same as f?" 80757 Tue, 09 Aug 2005 01:44:26 Z rlopez rlopez http://www.mapleprimes.com/posts/43762-Two-Illustrations-Of-Is-G-The-Same-As-F?ref=Feed:MaplePrimes:Two Illustrations of "Is g the same as f?":Comments#comment86922 Hello, Robert. Thanks for stepping into this discussion. I think that, especially for young engineering students, it is important to distinguish equality ("the same as") and approximation. In the second example, one must be aware that the zeros for the Bessel functions which I listed in that worksheet are approximations of those zeros and that, as a consequence, the "solution" is going to be an approximation for the solution. By doing the 3.14 thing, I intended to emphasize that. All kind of good discussions arise from this second example. For example, it leads to discussions about where the maximum value of the solution for such a PDE can occur, and about how to measure how far an approximation misses the real solution ... especially in case the real solution is not computable for some reason. These are important ideas for engineering students. It's a whole hour lecture. Right? During my afternoon walk, I made examples in my head of bad approximations for which different norms hide how bad they were. Essentially, I ended the walk saying, again, "A graph is worth a thousand equations." You will note that I posted this in the Math Education Section of Maple Primes. Jim The latest comments added to the Post, Two Illustrations of "Is g the same as f?" 86922 Tue, 09 Aug 2005 03:58:06 Z jherod jherod http://www.mapleprimes.com/posts/43762-Two-Illustrations-Of-Is-G-The-Same-As-F?ref=Feed:MaplePrimes:Two Illustrations of "Is g the same as f?":Comments#comment85781 This looked like a great post, until I realized that all the inline Maple was gone... What happened? The latest comments added to the Post, Two Illustrations of "Is g the same as f?" 85781 Tue, 20 Mar 2007 07:12:36 Z JacquesC JacquesC