## Maple/Symbolic Math Package/Commercial

From:Len Stys (aa399@cleveland.Freenet.Edu)

Date:07/25/91-10:58:24 AM Z

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From: aa399@cleveland.Freenet.Edu (Len Stys) Subject: Maple/Symbolic Math Package/Commercial Date: Thu Jul 25 10:58:24 1991 Contributed by: sytang@lamar.ColoState.EDU (Shoou-yu tang) This review was obtained by using FTP at: atari.archive.umich.edu (reviews) Maple and Me Leroy J. (Lee) Dickey 1989 12 07 My first encounter with Maple came after someone put a mathematical question to me that question involved an equation with all sorts of wonderful mathematical objects: the square root function, the sine function, the tangent function, the number pi, multiplication, addition, division, and the whole numbers 2, 3, 4, and 11. The person who asked had already done some calculations on their microcomputer and discovered that two expressions, when evaluated, gave more or less the same value. The question was whether this was because the two expressions represented real numbers that were equal, or whether the apparent equality was it just a lucky coincidence. That is, "Was it an identity or was it an accident?" Here is the mathematical statement: tan (3*theta) + 4 * sin ( 2*theta) = sqrt (11) What had been done was the calculation of the left side tan (3*theta) + 4 * sin ( 2*theta) and the calculation of right side sqrt (11) separately. When the value of these two expressions were printed, the same number was shown on the screen. It looked pretty nice, but there was still some doubt, because only 10 decimal places were showing. Maybe the sides agree up to 10 places, and yet they differ somewhere further down the line, maybe in the 100th place, or in the 1000th place. We know that whether using BASIC or some other programming language like C, FORTRAN, or APL, there is always a round-off error when calculating a number like the square root of 11. This is because numbers like the square root of 11 have infinite decimal representations and when these languages store any number, (including the square root of 11), they are restricted to some pre-determined maximum number of digits, depending on how big the number is and how that particular computer language stores its numbers. (Not all computers store numbers the same way. But that is another story.) What happens in these cases is that the number gets ``truncated'', or chopped off, and what we see is an approximation to our number. Well, for many practical purposes, this may be OK, and in fact, a lot of useful computing has been done in those languages and others of their types. Much good scientific work, many practical business problems, and a lot of fun games. But in this case, our question has not been answered! We had an approximation, and this was not good enough. If the two numbers were close, but not equal, maybe we could prove that they were not equal by working with the difference or something. We wanted some more evidence before we wasted too much trying to prove the result. Any computing language that gives us an approximation to the right answer instead of the exact answer is giving us just that, an approximation. We would like to control the number of digits. And this brings us to Maple. Maple is an example of a "Symbolic Algebraic Manipulator''. When Maple works with the square root of 11, it works with a symbol standing for the square root of 11, and it has the property that the square of that symbol is 11, not 10.99999 or 11.000001. It is 11, exactly. Does that mean that I can see thousands of digits for sqrt(11)? Yes, if I choose to. The number of digits is controlled by a variable that shows printing precision. In any case, the number sqrt(11) is the same symbol internally; When I ask to have it displayed, Maple cranks out as many decimal places as I ask for. What was my plan? My idea was to repeat the calculations with some large number of digits. If I found left side and right side agreed, I would be willing to spend a little more time trying to prove the result. So, I decided to try for 50 digits of printing precision. I wanted to have a variable for the angle that was Pi/11, because the expressions (3*Pi)/11 and (4*Pi)/11 appeared in the left side. So here, exactly, is the Maple program that I wrote: theta := Pi / 11; rs := sqrt (11); ls := tan (3*theta) + 4 * sin ( 2*theta) ; lv := evalf (ls,50); rv := evalf (rs,50); lv-rv; What is "evalf"? The "ls" stands for the exact value of the left side of my equation. The "lv" gives the value of the expression "ls" to 50 digits. What did I see as a result? Here is a listing of my output: theta := Pi / 11; theta := 1/11 Pi rs := sqrt (11); 1/2 rs := 11 ls := tan (3*theta) + 4 * sin ( 2*theta) ; ls := tan(3/11 Pi) + 4 sin(2/11 Pi) lv := evalf (ls,50); lv := 3.3166247903553998491149327366706866839270885455893 rv := evalf (rs,50); rv := 3.3166247903553998491149327366706866839270885455894 lv-rv; -48 -.1*10 lv := evalf (ls,250); lv := 3.3166247903553998491149327366706866839270885455893535970586821461164846426090\ 438467088433991282906509070125578495274565922754397848575474797793249330447288\ 473028739748286556825773944446120980444771931123571441329715210988326604957100\ 37248520738106820 rv := evalf (rs,250); rv := 3.3166247903553998491149327366706866839270885455893535970586821461164846426090\ 438467088433991282906509070125578495274565922754397848575474797793249330447288\ 473028739748286556825773944446120980444771931123571441329715210988326604957100\ 37248520738106821 lv-rv; -248 -.1*10 quit; Wow, what accuracy! That was enough to convince me that I should look for a proof. I could imagine numbers being equal, say up to the 12 or 13 place and then differing after that. But accuracy to 249 decimal places convinced me that we had an identity here, and that I sould look for a proof. A proof is possible, and it is known, and Maple can help, but that is another story. What else can Maple do, besides producing screens full of numbers? The answer is just about every sort of mathematical computation. As an example, one exciting application is differentiation and integration of elementary functions: it is not to hard to imagine how differentiation can be done, but the exciting news is that integration can be done to. Here is a very simple example of a maple statement and its result: integrate (x^2,x); 3 1/3 x This is the famous "anti-derivative" operator at work. You put in the function "x squared" and you get back a function whose derivative is "x squared". Of course, there is much more that Maple does. My most recent experiments have been with three dimensional Euclidean Geometry, calculating formulae for the angle between two circles on the sphere. In coming to my final result, I passed through some incredible output that has to be measured, not just by the number of digits or the number of terms, but by the number of screens required to show the result. What is the minimum you need to run Maple on your Atari ST? You should have a 1 Meg machine, such as the 1040ST, and you should have a hard disk to hold the tons of software in the 6 Megabytes or so of files in the Maple library. To find out more about the latest release of Maple, contact: Waterloo Maple Software 160 Columbia St. West Waterloo, Ontario CANADA N2L 3L3 Tel: (519) 747-2373 wmsi@daisy.waterloo.edu Fax: (519) 747-5284 wmsi@daisy.uwaterloo.ca --

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