MaplePrimes - comments on Blog Entry, Yet More Gems from the Little Red Book of Maple Magic

MaplePrimes - comments on Blog Entry, Yet More Gems from the Little Red Book of Maple Magic http://www.mapleprimes.com/maplesoftblog/103464-Yet-More-Gems-From-The-Little-Red-Book en-us 2026 Maplesoft, A Division of Waterloo Maple Inc. Maplesoft Document System Sat, 13 Jun 2026 21:17:07 GMT Sat, 13 Jun 2026 21:17:07 GMT The latest comments added to the Blog Entry, Yet More Gems from the Little Red Book of Maple Magic http://www.mapleprimes.com/images/mapleprimeswhite.jpg MaplePrimes - comments on Blog Entry, Yet More Gems from the Little Red Book of Maple Magic http://www.mapleprimes.com/maplesoftblog/103464-Yet-More-Gems-From-The-Little-Red-Book Trig Identitities http://www.mapleprimes.com/maplesoftblog/103464-Yet-More-Gems-From-The-Little-Red-Book?ref=Feed:MaplePrimes:Yet More Gems from the Little Red Book of Maple Magic:Comments#comment103513 I do not agree, it may be the user who struggles here, not Maple... In principle, trigsubs was intended to handle trigonometric identities. And indeed, it works for this particular expression: <pre>trigsubs(sin(x)+sin(y)); [2 sin(x/2 + y/2) cos(x/2 - y/2)] </pre> But there is not much joy, as  trigsubs is fragile and fails under a slight change: <pre>trigsubs(-sin(x)-sin(y)); Error, (in trigsubs) expecting a sum or difference of two functions but got -sin(x)-sin(y) </pre> I say that it is fragile because automatic function normalization "pulls out" the minus sign: <pre>trigsubs(sin(-x)+sin(-y)); Error, (in trigsubs) expecting a sum or difference of two functions but got -sin(x)-sin(y) trigsubs(sin(y-x)+sin(z-w)); Error, (in trigsubs) expecting a sum or difference of two functions but got -sin(-y+x)-sin(-z+w) </pre> Likely, this error message arises because the normalization changes the type of the terms from function to `*`: <pre>whattype(sin(-x)); * </pre> And the primitive pattern matcher of trigsubs seems unable to cope this with case. I observe an error message arising already in Maple V Release 2 (the oldest version that I have installed): <pre>readlib(trigsubs): trigsubs(-sin(a)-sin(b)); Error, (in trigsubs) sum or difference of two functions expected not , -sin (a)-sin(b) </pre> And I see that the code in Maple 14 and Maple V Release 2 is the same, besides cosmetic changes. Now, Maple V Release 2 was released in Nov. 1992 and trigsubs is copyrighted 1990. So, I find quite likely that the its code has not changed, basically, from its release, over 20 years ago. [Michael Komma should be credited for recently reminding elsewhere about this longstanding bug] Trigonometric identities are a very basic subject. So, that the routine devoted to handling them, has not been really improved, and its bugs corrected, in about 20 years (2/3 of the lifetime of Maple) is a shame, in my opinion. Maple has a serious deficit in the critical area of pattern matching. But yet, better faciliies became available in the late 20 years. Indeed, for about 15 years, rules like these ones have been lurking around my Maple initialization files: <pre>trig_sumtoprod:=[ sin(a::algebraic)+sin(b::algebraic)=2*sin((a+b)/2)*cos((a-b)/2), sin(a::algebraic)-sin(b::algebraic)=2*cos((a+b)/2)*sin((a-b)/2), cos(a::algebraic)+cos(b::algebraic)=2*cos((a+b)/2)*cos((a-b)/2), cos(a::algebraic)-cos(b::algebraic)=2*sin((a+b)/2)*sin((b-a)/2) ]: </pre> So that <pre>applyrule(trig_sumtoprod,sin(x)+sin(y)); 2 sin(x/2 + y/2) cos(x/2 - y/2) </pre> The first two rules may be combined and extended like this: <pre>sin_sumtoprod:=conditional(A::algebraic*sin(a::algebraic)+B::algebraic*sin(b::algebraic)= 2*A*sin((a+signum(0,B/A,0)*b)/2)*cos((a-signum(0,B/A,0)*b)/2),A<>0 and _verify(B^2-A^2,0,equal)): applyrule(sin_sumtoprod,exp(x)*sin(x)+exp(x)*sin(y)); 2 exp(x) sin(x/2 + y/2) cos(x/2 - y/2) applyrule(sin_sumtoprod,4*sin(y-x)+4*sin(z-w)); -8 sin(- y/2 + x/2 - z/2 + w/2) cos(y/2 - x/2 - z/2 + w/2) </pre>   Certainly, some of the items that you have raised in this blog series are very useful in spotting at serious defficiences of Maple in the area of symbolic computations. Note, however, that I do not post comments to them for the purpose of my recipes beeing included in your red book (though it is fine for me if you do so with proper attribution). Rather, I post here for awakening the Maplesoft management about their priority and planning problems. Indeed, "magic", the key word in the title, is very significative in describing the current status: if performing elementary symbolic operations with Maple require "magic recipes", and even a seasoned user as you needs collecting and keeping such recipes in a handwritten book, then something is very wrong with the design of the whole system. And from a more practical point of view, frequently I tend to avoid answering to some of you items, even though they may deserve a comment, because I do edit and preview a lot. And your blogs, so loaded with gifs, makes that each preview cycle take 1-2 mins. Only exceptionally I can afford the extra time required by such an overhead. The latest comments added to the Blog Entry, Yet More Gems from the Little Red Book of Maple Magic 103513 Sat, 02 Apr 2011 01:18:19 Z Alejandro Jakubi Alejandro Jakubi Longing for more... http://www.mapleprimes.com/maplesoftblog/103464-Yet-More-Gems-From-The-Little-Red-Book?ref=Feed:MaplePrimes:Yet More Gems from the Little Red Book of Maple Magic:Comments#comment104087 Longing for more magic gems. They are of educational. Very magic. Best regards! The latest comments added to the Blog Entry, Yet More Gems from the Little Red Book of Maple Magic 104087 Mon, 11 Apr 2011 12:28:22 Z Wang Gaoteng Wang Gaoteng Gem 11 http://www.mapleprimes.com/maplesoftblog/103464-Yet-More-Gems-From-The-Little-Red-Book?ref=Feed:MaplePrimes:Yet More Gems from the Little Red Book of Maple Magic:Comments#comment120729 It can be done without guessing noticing that in this example <pre> x / | phi[k + 1] - phi[k] = | phi[k] - phi[k - 1] dx | / 0</pre> so every next difference is the integral of the previous one, and φk is the sum of the differences, so the limit of φk is <pre>sum(diff(x^2,x$(-n)),n=1..infinity); 2 2 exp(x) - 2 - 2 x - x </pre> (Knowing the Taylor series for exp(x) and the integral of x^m, this sum can be calculated mentally, without Maple.) _______________ Alec Mihailovs, PhD The latest comments added to the Blog Entry, Yet More Gems from the Little Red Book of Maple Magic 120729 Mon, 30 May 2011 23:26:01 Z Alec Mihailovs Alec Mihailovs Gem 12 http://www.mapleprimes.com/maplesoftblog/103464-Yet-More-Gems-From-The-Little-Red-Book?ref=Feed:MaplePrimes:Yet More Gems from the Little Red Book of Maple Magic:Comments#comment120733 Not only for real x. Also, evalc would give the same result in the original post even without assumptions (evalc automatically assumes that all variables in it are real). To do this properly, one has to look at the branch cuts of sqrt(z^2+1), sqrt(z+I), and sqrt(z-I). <pre>z:=x+y*I: solve({Re(z^2+1)<=0,Im(z^2+1)=0}) assuming real; {x = 0, y <= -1}, {x = 0, 1 <= y} solve({Re(z+I)<=0,Im(z+I)=0}) assuming real; {y = -1, x <= 0} solve({Re(z-I)<=0,Im(z-I)=0}) assuming real; {y = 1, x <= 0} </pre> They divide complex plane in 3 domains, in one of which (containing the real axis) the identity holds, and in 2 others - doesn't. That can be visualized with densityplot, <pre>plots:-densityplot(abs(sqrt(z^2+1)-sqrt(z+I)*sqrt(z-I)), x=-4..4,y=-4..4,colorstyle=HUE,style=patchnogrid,axes=box);</pre> <a href="/view.aspx?sf=120733/384195/dplot1.png"><img style="display: block; margin-left: auto; margin-right: auto;" src="/view.aspx?sf=120733/384195/dplot1.png" alt=""></a> _______________ Alec Mihailovs, PhD The latest comments added to the Blog Entry, Yet More Gems from the Little Red Book of Maple Magic 120733 Tue, 31 May 2011 01:13:19 Z Alec Mihailovs Alec Mihailovs Gem 11 http://www.mapleprimes.com/maplesoftblog/103464-Yet-More-Gems-From-The-Little-Red-Book?ref=Feed:MaplePrimes:Yet More Gems from the Little Red Book of Maple Magic:Comments#comment120736 Of course, what the Picard iteration is doing in this case is grinding out, one term at a time, the Maclaurin series of the solution.  That could be obtained more simply with dsolve(..., type=series).  As for guessgf, how does it come up with an answer?  Let's use infolevel[gfun] to get information on what it did. <pre>> infolevel[gfun]:= 2; guessgf(L, x); guessgf: Trying to find a rational generating function guessgf: Trying to find an hypergeometric generating function guessgf: Trying to find an algebraic generating function guessgf: Trying to find a linear differential equation listtodiffeq: The ogf seems to satisfy x^2+y(x)-diff(y(x),x) guessgf: Trying to solve it 2 [-2 - 2 x - x + 2 exp(x), ogf] Surprise! It did it by solving the same differential equation we started with.</pre> The latest comments added to the Blog Entry, Yet More Gems from the Little Red Book of Maple Magic 120736 Tue, 31 May 2011 02:53:00 Z Robert Israel Robert Israel Gem 14 http://www.mapleprimes.com/maplesoftblog/103464-Yet-More-Gems-From-The-Little-Red-Book?ref=Feed:MaplePrimes:Yet More Gems from the Little Red Book of Maple Magic:Comments#comment120941 That can be written in 2 lines, <pre>f:=sin(u+v)+sin(u-v): eval(f=expand(f),solve({u+v=x,u-v=y},{u,v})); sin(x) + sin(y) = 2 sin(x/2 + y/2) cos(x/2 - y/2) </pre> Alec The latest comments added to the Blog Entry, Yet More Gems from the Little Red Book of Maple Magic 120941 Thu, 02 Jun 2011 08:43:32 Z Alec Mihailovs Alec Mihailovs