f:=sin(t)*x*(1-x);
g:=cos(t)*x*(1-x);
plots[animate](plot,[[f,g],x=0..1],t=0..2*Pi);

This blog entry was created essentially for a possibility to post pictures in comments (where attachments are not allowed.)

OK, in posting a new blog, there is an "Attachment" section... So I could add the three GIFs. They don't show up in the preview, but they did come out when I actually posted...

>	sum((-2)^k, k=1..infinity);

 

 

>	_EnvFormal := true;

 

 

As moderator of the How Do I?? (Newbies) forum, I thought I would post some information for my viewers. As a somewhat "Newbie" to Maple myself, I thought I'd boast about how amazing I think Maple 10 is (and no... this is not just because I work for Maplesoft ;). My first experience using Maple was a year ago, when I started with Maple 9.5. As expected, like any scientific software, there was a learning curve that I had to conquer.

Wen-Xiu Ma recently asked in the maple-assist group,

How to color the following region?
a:=plot([t^2,t^3-1,t=-3..3):
b:=plot([t^3+1,t^2-3,t=-3..3):
c:=plot([t^2+2,t^5-4,t=-3..3):
display([a,b,c],view=[0..5,-3.5..11]);

Early this season, after the Maplesoft team came out on the wrong side of a 13-6 loss, we were frustrated by the team's inability to score more runs. The previous year we averaged 14 runs a game. This started me wondering, just how many runs can our team expect with a given lineup? Suppose you assume that it takes three hits in an inning to start scoring runs. Now, let's assume you have five .500 hitters coming up to bat. What is the probability that you'll get 3 hits among those five batters, thus scoring one run?

Joe Riel and Thomas Richard posted in their blogs solutions to the latest IBM Ponder This challenge and some related questions. From the name of the attached files, one can deduct that Thomas Richard is user No. 50 and Joe Riel is user No. 84. Being curious about my user number, I also decided to attach a worksheet.

A question was asked in the forums about series tests. I saw that this would also make an excellent weblog entry as well. In answer to the questions: How can I get Maple to determine if a series converges or diverges? and How can I obtain the general representation of a formal power series for a function? I offer the following advice:

RSS feeds allow people to easily keep track of updates to their favorite websites. Say that you have 50 websites that you check every day for updates. What if instead of going to each of those individual websites, you can simply load your RSS reader and see which websites have been updated at a glance.

To use RSS feeds you will need an RSS reader. Here are two pages that list many different readers: http://blogspace.com/rss/readers and http://allrss.com/rssreaders.html. My favorite reader is Bloglines which is a web based reader that allows me to view my RSS feeds at any computer with a web connection.

Here's a quick and dirty solution to the following task:
Arrange digits 1,...,9 so that the first two form a number divisible by 2, the first three form a number divisible by 3, etc..

See Joe Riel's blog for more information.

The standard Maple worksheet uses color to distinguish elements. This is convenient when viewed on the screen, however, when printing to a monochromatic printer, colored text frequently is too light. I prefer to have it printed as dark black. This tip describes one way to do that. The technique is to apply a previously created style file that specifies all the fonts as being black. The easiest way to do this is to first create a user style file that has has your default settings, then manually edit this file to change the foreground color of all fonts to black. You could manually change the colors using the Maple Format -> Styles submenu; however, doing so is more work than editing the file in an external editor.

Previously I described how to change the default zoom setting for the Maple gui by modifying the appropriate initialization file. Another useful setting to change is the default background color of the help browser. This is done by modifying, in the initialization file, the line HelpBGColor=. I set it to HelpBGColor=240 240 240, that gives a light gray background that is less harsh on my eyes. The three fields should be integers from 0 to 255; they correspond to the red, green, and blue components of the color.

A poster on comp.soft-sys.math.maple recently asked how to set the default zoom for worksheets to a non-standard value; the jump from 100% to 150% being rather large. Standard GUI This can be readily accomplished by modifying the proper Maple initialization file.

Linux

the file is ~/.maple10rc

Mac OS X

the file is Maple 10 Preferences in Library/Preferences under your user directory (thanks to Tim Lahey).

Windows

the file is Maple10.ini (not maple.ini). The maple help pages do not mention this file and I do not have Windows, so cannot check. I have received reports that it is located in the users subdirectory under the Maple installation directory, but also reports that it is in c:\Documents and Settings\JoeUser, where JoeUser is your user name. My advice, until this is cleared up, is to search for all instances of this file and use the newest one. Maple writes to this file whenever the gui exits, so the newest one should be the right one. You might send me a note on what you've found (along with which version of Maple you are using); I'll update this accordingly.

When I upgraded my Mac G5 from OS 10.3 to OS 10.4, the OS X versions of Maple (Maple 9.5 and Maple 10) continued to work. But the classic versions of Maple (Maple 7 and Maple Vr5) quit working. It turns out that the classic versions will run in OS 10.4 if the following files (perhaps from an old "System Folder") are moved into the "Extensions" folder of the active classic OS 9 "System Folder":

I've been completely unsatisfied with the Maple LaTeX output. So, with the help of Joe Riel on comp.soft-sys.math.maple, I've created a procedure to use instead of a call to the latex command called simpleLaTeX. I've attached to this entry a Maple 9.5 worksheet that contains the procedure and a sample of its use. I'd appreciate any feedback people might have or further suggestions. Please send me any modifications you might make to the procedure. It seems to work quite well for all the test expressions I've sent to it.

A while ago, I was trying to determine the fastest way, using Maple, to compute a list of a particular sequence of integers defined by a recurrence relation. This isn't a new idea of course, but I think the exercise is a useful introduction into the various ways of coding algorithms in Maple, and why some may be better than others.

My original example was a bit esoteric, so for simplicity I've redone it using the standard example of a recurrence relation, the Fibonacci sequence. We'll fix N, the number of Fibonacci numbers to compute, at 40000.

> N := 40000:


Below, we'll try to find the fastest way, in Maple, to compute a list of these numbers from scratch. Note that the speed comparisons might be specific to this machine, OS, or Maple version, and even though the runtime of most of these implementations is about the same, the fastest method at N=40000 might not be the fastest one for larger N.

The first idea is the completely naive approach. We'll store the Fibonacci numbers in a Maple list L, step through a loop counting up the sequence, and in each loop iteration add to the list with L := [op(L), new_element].

f1 := proc(N)
    local i, L;
    L := [1, 1];
    for i from 2 to N do
        L := [op(L), L[-2] + L[-1]];
    end do;
    L
end proc:

The problem with this approach is that Maple lists are not intended to be grown in this way, and each redefinition of the list L costs O(N) time, making the whole procedure O(N).

> time(f1(N));
31.440

From now on we'll use only implementations that take O(N) time, and see how they compare.

The next implementation uses a Maple table stored as a local variable of a module. We compute the sequence and assign it to table entries, and at the end generate a Maple list from these table entries. The use of tables is advantageous because tables can be dynamically grown more efficiently than lists can, but this approach has the disdvantage that the table must be completely re-traversed in the last step to generate the list.

f2 := module()
    local ModuleApply, T;

    T := table([0=1, 1=1]);

    ModuleApply := proc(N)
        local i;
        for i from 2 to N do
            T[i] := T[i-2] + T[i-1];
        end do:
        [seq(T[i], i=0..N)]
    end proc:
end module:


You'll see that since this code avoids list-appending, it's quite a bit faster:

> time(f2(N));
1.110

The next idea is to do essentially what we did above, but using remember tables rather than built-in Maple tables. We'll pre-populate the remember table for the utility procedure p with entries for the base cases, F₀ and F₁, and then build our list through recursive calls.

f3 := module()
    local p, ModuleApply;

    p := proc(n) option remember;
        procname(n-2) + procname(n-1)
    end proc:

    p(0) := 1;
    p(1) := 1;
 
    ModuleApply := proc(N)
        [1, 1, seq(p(i), i=2..N)]
    end proc:
end module:


As we might expect, the computed result is pretty close (but a bit higher than) our last number.

> time(f3(N));
1.229

One disadvantage of the last two approaches is that they both rely on creating an intermediate data structure of size O(N) (the table or remember table), which is promptly discarded, since what we're interested in is a list. Instead we can use a 'cache', which is a new feature of Maple 9.5, and is essentially a remember table of bounded size.

f4 := module()
    local p, ModuleApply;

    p := proc(n) option cache;
        procname(n-2) + procname(n-1)
    end proc:

    p(0) := 1;
    p(1) := 1;

    ModuleApply := proc(N)
        [1, 1, seq(p(), i=2..N)]
    end proc:
end module:


It works out to be slightly slower than f3; this may be because of the time taken in garbage-collecting the information from the cache.

> time(f4(N));
1.459

Next, I've tried to implement something like a cache, without actually using one. Here we have a utility procedure with two counters, which represent the last and second-last Fibonacci number respectively. The procedure updates the values and returns the appropriate one. We'll build the list with recursive calls to this procedure. Like the cache, the intermediate data structures used here are of bounded size, since there are just two of them.

f5 := module()
    local p, n1, n2, ModuleApply;

    n1 := 1;
    n2 := 1;

    p := proc(n)
        (n1, n2) := (n2, n1+n2);
        n2;
    end proc:

    ModuleApply := proc(N)
        [1, 1, seq(p(), i=2..N)]
    end proc:
end module:


It's about the same speed as using the cache, which is impressive since it involves a lot more procedure calls into the procedure body than the cache example, which mostly consisted of calls into the cache.

> time(f5(N));
1.439

Finally, mostly for completeness, we try to see we get by pre-allocating an Array and adding to that. Arrays can be modified inplace in O(1) time, but cannot be dynamically grown. Here we make an array of the appropriate size, fill it, then convert it to a list.

f6 := proc(N)
    local A, i;
    A := Array(0..N, {0=1,1=1});
    for i from 2 to N do
        A[i] := A[i-2] + A[i-1];
    end do:
    convert(A, 'list')
end proc:


It's about as fast as the previous two examples, and has the advantage of simplicity of design. However, building the array and converting that to a list is another instance of using large intermediate data structures, which we avoided with the previous two examples.

> time(f6(N));
1.469


So it looks like f2 is the winner in this very specialized contest. This probably has more to do with the quirks of this example than any difference in efficiency of the code concerned. And if minimizing intermediate memory use is the goal, then f4 or f5 would be preferable.