MaplePrimes Commons General Technical Discussions

Workaround for the problem of installing MapleCloud...

by: ecterrab 12724 Product: Maple 2020

September 24 2020

9 8

Hi,
Some people using the Windows platform have had problems installing MapleCloud packages, including the Maplesoft Physics Updates. This problem does not happen in Macintosh or Linux/Unix, also does not happen with all Windows computers but with some of them, and is not a problem of the MapleCloud packages themselves, but a problem of the installer of packages.

I understand that a solution to this problem will be presented within an upcoming Maple dot release.

Meantime, there is a solution by installing a helper library; after that, MapleCloud packages install without problems in all Windows machines. So whoever is having trouble installing MapleCloud packages in Windows and prefers not to wait until that dot release, instead wants to install this helper library, please email me at physics@maplesoft.com.

Edgardo S. Cheb-Terrab
Physics, Differential Equations and Mathematical Functions, Maplesoft

Physics updates

by: Jean-Michel 230 Product: Maple 2020

August 18 2020

5 5

Hello each and everyone,

I am still not able to install/update the Physics package and this since version 713.

I think the phenomenom happens when Maple 2020 was updated to version 2020.1

The 2 ways fail :

1) Physics:-Version(latest) ==> kernel crashes (lost connection)

2) via the Cloud : it's even worse... "installation" lasts forever (i usually break it after 15 (!) minutes)

I removed totally Maple 2020.1 and reinstalled it, the problem is still there.

I wonder when Maplesoft developpers will fix this problem whitch has been around for a long time now.

I do *not* want to do it by hand as one of this forum contributor suggested me a while ago.

I use the Physics paxkage intensively so i'm hampered in my work.

Kind regards to all.

Jean-Michel

integration example

by: acer 27705 Product: Maple

May 06 2020

5 2

There is an interesting preprint here, on a method for integration of some pseudo-elliptic integrals.

It was mentioned in a (sci.math.symbolic) usenet thread, which can be accessed via Google Groups here.

Inside the paper, the author mentions that the following is possible for some cases -- to first convert to RootOf form.

>

restart;

>	kernelopts(version);

>	ig:=((-3+x^2)(1-6x^2+x^4)^(-1/4))/(-1+x^2);

(x^2-3)/((x^4-6*x^2+1)^(1/4)*(x^2-1))

>	int(ig,x);

int((x^2-3)/((x^4-6*x^2+1)^(1/4)*(x^2-1)), x)

>	infolevel[int] := 2:

>	S:=simplify([allvalues(int(convert(ig,RootOf),x))],size):

Stage1: first-stage indefinite integration

Stage2: second-stage indefinite integration

Norman: enter Risch-Norman integrator

Norman: exit Risch-Norman integrator

int/algrisch/int: Risch/Trager's algorithm for algebraic function

int/algrisch/int: entering integrator at time 9.029

int/algrisch/int: function field has degree 4

int/algrisch/int: computation of an integral basis: start time 9.031

int/algrisch/int: computation of an integral basis: end time 9.038

int/algrisch/int: normalization at infinity: start time 9.039

int/algrisch/int: normalization at infinity: end time 9.048

int/algrisch/int: genus of the function field 3

int/algrisch/int: computation of the algebraic part: start time 9.059

int/algrisch/int: computation of the algebraic part: end time 9.060

int/algrisch/int: computation of the transcendental part: start time 9.063

int/algrisch/transcpar: computing a basis for the residues at time 9.068

int/algrisch/residues: computing a splitting field at time 9.068

int/algrisch/transcpar: basis for the residues computed at time 9.103

int/algrisch/transcpar: dimension is 2

int/algrisch/transcpar: building divisors at time 9.300

int/algrisch/transcpar: testing divisors for principality at time 9.605

int/algrisch/goodprime: searching for a good prime at time 9.606

int/algrisch/goodprime: good prime found at time 9.704

int/algrisch/goodprime: searching for a good prime at time 9.704

int/algrisch/goodprime: good prime found at time 9.762

int/algrisch/areprinc: the divisor is principal: time 10.084

int/algrisch/areprinc: the divisor is principal: time 11.833

int/algrisch/transcpar: divisors proven pincipal at time at time 11.833

int/algrisch/transcpar: generators computed at time 11.834

int/algrisch/transcpar: orders are [1 1]

int/algrisch/transcpar: check that the candidate is an actual antiderivative

int/algrisch/transcpar: the antiderivative is elementary

int/algrisch/transcpar: antiderivative is (1/2)*ln((RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^3*_z^3-RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^2*_z^4+RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^2*_z^2-3*_z^3*RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)+RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)*_z-5*_z^2+1)/((_z+1)*(_z-1)*_z^2))+(1/2)*RootOf(_Z^2+1)*ln((RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^2*RootOf(_Z^2+1)*_z^4+RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^3*_z^3-RootOf(_Z^2+1)*RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)^2*_z^2+3*_z^3*RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)-5*RootOf(_Z^2+1)*_z^2-RootOf(_Z^4*_z^4-_z^4+6*_z^2-1 index = 1)*_z+RootOf(_Z^2+1))/((_z+1)*(_z-1)*_z^2))

int/algrisch/int: computation of the transcendental part: end time 12.040

int/algrisch/int: exiting integrator for algebraic functions at time 12.041

>

S[1];

(1/2)*ln(-((x^4-6*x^2+1)^(3/4)*x+(x^4-6*x^2+1)^(1/2)*x^2+(x^4-6*x^2+1)^(1/4)*x^3+x^4-(x^4-6*x^2+1)^(1/2)-3*(x^4-6*x^2+1)^(1/4)*x-5*x^2)/((x+1)*(x-1)))+((1/2)*I)*ln((-(x^4-6*x^2+1)^(3/4)*x+(I*x^2-I)*(x^4-6*x^2+1)^(1/2)+(x^3-3*x)*(x^4-6*x^2+1)^(1/4)-I*x^2*(x^2-5))/((x+1)*(x-1)))

>

S[2];

(1/2)*ln(-((x^4-6*x^2+1)^(3/4)*x+(x^4-6*x^2+1)^(1/2)*x^2+(x^4-6*x^2+1)^(1/4)*x^3+x^4-(x^4-6*x^2+1)^(1/2)-3*(x^4-6*x^2+1)^(1/4)*x-5*x^2)/((x+1)*(x-1)))-((1/2)*I)*ln((-(x^4-6*x^2+1)^(3/4)*x+(-I*x^2+I)*(x^4-6*x^2+1)^(1/2)+(x^3-3*x)*(x^4-6*x^2+1)^(1/4)+I*x^2*(x^2-5))/((x+1)*(x-1)))

>	simplify( diff(S[1],x) - ig ), simplify( diff(S[2],x) - ig );

>

Download int_pe.mw

Maple Companion Update

by: Karishma 695 Product: Maple Calculator

May 06 2020

11 0

I’m extremely pleased to introduce the newest update to the Maple Companion. In this time of wide-spread remote learning, tools like the Maple Companion are more important than ever, and I’m happy that our efforts are helping students (and some of their parents!) with at least one small aspect of their life. Since we’ve added a lot of useful features since I last posted about this free mobile app, I wanted to share the ones I’m most excited about.

(If you haven’t heard about the Maple Companion app, you can read more about it here.)

If you use the app primarily to move math into Maple, you’ll be happy to hear that the automatic camera focus has gotten much better over the last couple of updates, and with this latest update, you can now turn on the flash if you need it. For me, these changes have virtually eliminated the need to fiddle with the camera to bring the math in focus, which sometimes happened in earlier versions.

If you use the app to get answers on your phone, that’s gotten much better, too. You can now see plots instantly as you enter your expression in the editor, and watch how the plot changes as you change the expression. You can also get results to many numerical problems results immediately, without having to switch to the results screen. This “calculator mode” is available even if you aren’t connected to the internet. Okay, so there aren’t a lot of students doing their homework on the bus right now, but someday!

Speaking of plots, you can also now view plots full-screen, so you can see more of plot at once without zooming and panning, squinting, or buying a bigger phone.

Finally, if English is not you or your students’ first language, note that the app was recently made available in Spanish, French, German, Russian, Danish, Japanese, and Simplified Chinese.

As always, we’d love you hear your feedback and suggestions. Please leave a comment, or use the feedback forms in the app or our web site.

Visit Maple Companion to learn more, find links to the app stores so you can download the app, and access the feedback form. If you already have it installed, you can get the new release simply by updating the app on your phone.

3D plot size option

by: acer 27705 Product: Maple 2019

January 23 2020

4 7

The ideas here are to allow 3D plotting commands such as plot3d to handle a `size` option similarly to how 2D plotting commands do so, and for the plots:-display command to also handle it for 3D plots.

The size denotes the dimensions of the inlined plotting window, and not the relative lengths of the three axes.

I'd be interested in any new problems introduced with this, eg. export, etc.

>

restart;

>

#
# Using ToInert/FromInert
#
# This might go in an initialzation file.
#
try
  __ver:=(u->:-parse(u[7..:-StringTools:-Search(",",u)-1]))(:-sprintf("%s",:-kernelopts(':-version')));
  if __ver>=18.0 and __ver<=2019.2 then
    __KO:=:-kernelopts(':-opaquemodules'=false);
    :-unprotect(:-Plot:-Options:-Verify:-ProcessParameters);
    __KK:=ToInert(eval(:-Plot:-Options:-Verify:-ProcessParameters)):
    __LL:=[:-op([1,2,1,..],__KK)]:
    __NN:=:-nops(:-remove(:-type,[:-op([1,..],__KK)],
                          ':-specfunc(:-_Inert_SET)'))
          +:-select(:-has,[:-seq([__i,__LL[__i]],
                                 __i=1..:-nops(__LL))],
                    "size")[1][1];
    if :-has(:-op([5,2,2,2,1],__KK),:-_Inert_PARAM(__NN)) then
      __KK:=:-subsop([5,2,2,2,1]
                     =:-subs([:-_Inert_PARAM(__NN)=:-NULL],
                              :-op([5,2,2,2,1],__KK)),__KK);
      :-Plot:-Options:-Verify:-ProcessParameters:=
      :-FromInert(:-subsop([5,2,2,3,1]
                  =:-subs([:-_Inert_STRING("size")=:-NULL],
                          :-op([5,2,2,3,1],__KK)),__KK));
      :-print("3D size patch done");
    else
      :-print("3D size patch not appropriate; possibly already done");
    end if;
  else
    :-print(sprintf("3D size patch not appropriate for version %a"),__ver);
  end if;
catch:
  :-print("3D size patch failed");
finally
  :-protect(:-Plot:-Options:-Verify:-ProcessParameters);
  :-kernelopts(':-opaquemodules'=__KO);
end try:

>

>	P := plot3d(sin(x)*y^2, x=-Pi..Pi, y=-1..1, size=[150,150], font=[Times,5], labels=["","",""]): P;

>	plots:-display(P, size=[300,300], font=[Times,10]);

>	# # inherited from the contourplot3d (the plot3d is unset). # plots:-display( plots:-contourplot3d(sin(x)y^2, x=-Pi..Pi, y=-1..1, thickness=3, contours=20, size=[800,800]), plot3d(sin(x)y^2, x=-Pi..Pi, y=-1..1, color="Gray", transparency=0.1, style=surface) );

>

# Some options should still act as 2D-plot-specific.
#
try plot3d(sin(x)*y^2, x=-Pi..Pi, y=-1..1, legend="Q");
print("Not OK");
catch:
if StringTools:-FormatMessage(lastexception[2..-1])
="the legend option is not available for 3-D plots"
then print("OK"); else print("Not OK"); error; end if; end try;

>

Download 3Dsize_hotedit.mw

If this works fine then it might be a candidate for inclusion in an initialization file, so that it's
automatically available.

Several contentious issues in the treatment of...

by: mmcdara 4555

December 18 2019

1 3

Hi,

This is more of an open discussion than a real question. Maybe it would gain to be displaced in the post section?

Working with discrete random variables I found several inconsistencies or errors.
In no particular order:

The support of a discrete RV is not defined correctly (a real range instead of a countable set)
The plot of the probability function (which, in my opinion, would gain to be renamed "Probability Mass Function, see https://en.wikipedia.org/wiki/Probability_mass_function) is not correct.
The ProbabiliytFunction of a discrte rv of EmpiricalDistribution can be computed at any point, but its formal expression doesn't exist (or at least is not accessible).
Defining the discrete rv "toss of a fair dice" with EmpiricalDistribution and DiscreteUniform gives different results.

The details are given in the attached file and I do hope that the companion text is clear enough to point the issues.
I believe there is no major issues here, but that Maple suffers of some lack of consistencies in the treatment of discrete (at least some) rvs. Nothing that could easily be fixed.

As I said above, if some think this question has no place here and ought to me moved to the post section, please feel free to do it.

Thanks for your attention.

>

restart:

>	with(Statistics):

Two alternate ways to define a discrete random variable on a finite set
of equally likely outcomes.

>	Universe := [$1..6]: toss_1_dice := RandomVariable(EmpiricalDistribution(Universe)); TOSS_1_DICE := RandomVariable(DiscreteUniform(1, 6));

(1)

Let's look to the ProbabilityFunction of each RV

>	ProbabilityFunction(toss_1_dice, x); ProbabilityFunction(TOSS_1_DICE, x);

piecewise(x < 1, 0, x <= 6, 1/6, 6 < x, 0)

(2)

It looks like the procedure ProbabilityFunction is not an attribute of RV with EmpiticalDistribution.
Let's verify

>	law := [attributes(toss_1_dice)][3]: lprint(exports(law))

Conditions, ParentName, Parameters, CDF, DiscreteValueMap, Mean, Median, Mode, ProbabilityFunction, Quantile, Specialize, Support, RandomSample, RandomVariate

Clearly ProbabilityFunction is an attribute of toss_1_dice.

In fact it appears the explanation of the difference of behaviours relies upon different definitions
of the set of outcomes of toss_1_dice and TOSS_1_DICE

>	LAW := [attributes(TOSS_1_DICE)][3]: exports(LAW): law:-Conditions; LAW:-Conditions;

[(Vector(6, {(1) = 1, (2) = 2, (3) = 3, (4) = 4, (5) = 5, (6) = 6}))::rtable]

(3)

From :-Conditions one can see that toss_1_dice is realy a discrete RV defined on a countable set of outcomes,
but that nothing is said about the set over which TOSS_1_DICE is defined.

The truly discrete definition of toss_1_dice is confirmed here :
(the second result is correct

ProbabilityFinction(toss_1_dice, x) = {0 if x < 1, 0 if x > 6, 1/6 if x::integer, 0 otherwise

>	ProbabilityFunction~(toss_1_dice, Universe); ProbabilityFunction~(toss_1_dice, [seq(0..7, 1/2)]);

(4)

One can also see that the Support of both of these RVs are wrong

(see for instance https://en.wikipedia.org/wiki/Discrete_uniform_distribution)

There should be {1, 2, 3, 4, 5, 6}, not a RealRange.

>	Support(toss_1_dice); Support(TOSS_1_DICE);

(5)

>

Now this is the surprising ProbabilityFunction of TOSS_1_DICE.
This obviously wrong result probably linked to the weak definition of the conditions for this RB.

>	# plot(ProbabilityFunction(TOSS_1_DICE, x), x=0..7); plot(ProbabilityFunction(TOSS_1_DICE, x), x=0..7, discont=true)

These differences of treatments raise a lot of questions :
    - Why is a DiscreteUniform RV not defined on a countable set?
    - Why does the ProbabilityFunction of an EmpiricalDistribution return no result
        if its second parameter is not set to one its outcomes.

All this without even mentioning the wrong plot shown above.

I believe something which would work like the module below would be much better than what is done

right now

>

EmpiricalRV := module()
export MassDensityFunction, PlotMassDensityFunction, Support:

MassDensityFunction := proc(rv, x)
  local u, v, N:
  u := [attributes(rv)][3]:
  if u:-ParentName = EmpiricalDistribution then
    v := op([1, 1], u:-Conditions);
    N := numelems(v):
    return piecewise(op(op~([seq([x=v[n], 1/N], n=1..N)])), 0)
  else
    error "The random variable does not have an EmpiricalDistribution"
  end if
end proc:

PlotMassDensityFunction := proc(rv, x1, x2)
  local u, v, a, b:
  u := [attributes(rv)][3]:
  if u:-ParentName = EmpiricalDistribution then
    v := op([1, 1], u:-Conditions);
    a := select[flatten](`>=`, v, x1);
    b := select[flatten](`<=`, a, x2);
    PLOT(seq(CURVES([[n, 0], [n, 1/numelems(v)]], COLOR(RGB, 0, 0, 1), THICKNESS(3)), n in b), VIEW(x1..x2, default))
  else
    error "The random variable does not have an EmpiricalDistribution"
  end if
end proc:

Support := proc(rv, x1, x2)
  local u, v, a, b:
  u := [attributes(rv)][3]:
  if u:-ParentName = EmpiricalDistribution then
    v := op([1, 1], u:-Conditions);
    return {entries(v, nolist)}
  else
    error "The random variable does not have an EmpiricalDistribution"
  end if
end proc:

end module:

>	EmpiricalRV:-MassDensityFunction(toss_1_dice, x);

piecewise(x = 1, 1/6, x = 2, 1/6, x = 3, 1/6, x = 4, 1/6, x = 5, 1/6, x = 6, 1/6, 0)

(6)

>	f := unapply(EmpiricalRV:-MassDensityFunction(toss_1_dice, x), x): f(2); f(5/2);

(7)

>	EmpiricalRV:-PlotMassDensityFunction(toss_1_dice, 0, 7);

>

Download Discrete_RV.mw

Are Maple's pseudo random number generators good...

by: mmcdara 4555

December 04 2019

2 3

I'm particularly interested in data analysis and more specifically in statistical analysis of computer code outputs.

One of the main activity of this very broad field is named Uncertainty Propagation. In a few words it consists in perturbing the inputs of a computational code in order to understand (and quantify) how these perturbations propagates through the outputs of this code.

At the core of uncertainty propagation is the ability to generate large numbers of "random" variations of the inputs. Knowing that these entries can be counted in tens, one sees that the first problem consists in generating "random" points in a space of potentially very large dimension.

Even among my mathematician colleagues an impressive number of them is completely ignorant of the way "random" numbers are generated. I guess that a lot of Mapleprimes' users are too. My purpose is not to give a course on this topic and the affording litterature is vast enough for everyone interested might find informations of any level of complexity.
Among those who have some knowledge about Pseudo Random Numbers Generators (PRNG), only a few of them know that a PRNG has to pass severe tests ("tests of randomness") before the streams of number it generates might be qualified as "reasonably random" and therefore this PRNG might be released.

One of most famous example of a bad PRNG is given by "randu" (IBM 1966, and probably used in Fortran libraries during more than 30 years), this same PRNG that Knuth qualified himself as the "infamous generator".

These tests of randomness are generally gathered in dedicated libraries and Diehard is probably tone of the most known of them.
Diehard has originally been developed by George Marsaglia more than twenty years ago and it's still widely ued today.

I recently decided, not because I have doubts about the quality of the work done by Maplesoft, to test the Maple's PRNG named "Mersenne Twister". First, because it can do no harm to publish quantitative information that allows everyone to know that it is using a proven PRNG; second, because the (very simple) approach used here can fill the gaps I have mentioned above.

Mersenne Twister (often dubbed mt19937) is considered as a very good PRNG; it is used in a lot of applications (including finance where it is not so rare to sample input spaces of dimensions larger than 1000... ok I know, mt19937 is often considered as a poor candidate for cryptography applications, but it's not my concern here).

I have thus decided to spend some time to run the Diehard suite of tests on a sequence of integers numbers generated by RandomTools[MersenneTwister].

>

restart:

DIEHARD tests suite for Pseudo Random Numbers Generators (PRNG)

Reference: http://webhome.phy.duke.edu/~rgb/General/dieharder.php

The installation procedure (Mac OSX) can be found here
    https://gist.github.com/blixt/9abfafdd0ada0f4f6f26
or here
    http://macappstore.org/dieharder/

For other operating systems, please search on the web pages.

dieharder [-h]   # for inline help
dieharder -l      # to get the lists all the avaliable tests

A description of the many tests can be found here:
    https://en.wikipedia.org/wiki/Diehard_tests
    https://sites.google.com/site/astudyofentropy/background-information/the-tests/dieharder-test-descriptions
    https://www.stata.com/support/cert/diehard/randnumb_mt64.out

General theory about PRNG testing can be found here (a reference among many):
    http://liu.diva-portal.org/smash/get/diva2:740158/FULLTEXT01.pdf

or here (more oriented to the NIST test suite)
    https://www.random.org/analysis/Analysis2005.pdf
    https://nvlpubs.nist.gov/nistpubs/legacy/sp/nistspecialpublication800-22r1a.pdf

In a terminal window execute the following commands for an exhaustive testing ("-a" option).
The "-g 202" option means that the generator is replaced by a text format input file
(use dieharder -h for more details).

cd //..../Desktop/DIEHARD

dieharder -g 202 -f SomeAsciiFile -a > //..../Desktop/DIEHARD/TheResultFile.txt

Be carefull, the complete testing takes several hours (about 5 on my computer)

__________________________________________________________________________________

Maple's Mersenne Twister Generator

Maple help page : RandomTools[MersenneTwister][GenerateInteger]
(see rincluded references to the Mersenne Twister PRNG).

Note: in the sequel this generator will be dubbed mt19937

The Mersenne Twister is implemented in many softwares.
It is higly likely that this PRNG (and the others these softwares propose) have been intensively
tested with one of the existing PRNG testing libraries.
Unfortunately only a few editors have made public the results of these tests (probably because
the implementation in itself is rarely questioned... but a code typo is always a possibility).

One exception is ths software STATA.
A summary of the results can be found here
https://www.stata.com/support/cert/diehard/.
A complete description of the results of the tests passed is given here
https://www.stata.com/support/cert/diehard/randnumb_mt64.out

The classical pattern of the performances of mt19937 can be found here

http://www2.ic.uff.br/~celso/artigos/pjo6.ps.

and the table below comes from it (P means "Passed", F means "Failed"):

____________________________________________________________________________

In the Maple code below, a sequence of N UnsignedInt32 numbers is generated from the
Maple's Mersenne Twister and the result is exported in an ASCII file.
The Seed is set to 1 (SetState(state=1)) to compare, with a small value of N (let's say N=10)
the sequence produced by Maple's mt19937 with the the sequence of the same length generated
by Diehard's mt19937.
To generate this later sequence and save it in file Diehard_mt19937, just run in a terminan window
the command (-S 1 means "seed = 1", -t 10 means "a sequence of length 10"):
dieharder -S 1 -B -o -t 10 > Diehard_mt19937

About the value of N:

In http://webhome.phy.duke.edu/~rgb/General/dieharder.php it's recommend that N be at least
equal to 2.5 million; STATA used N=3 million.
Other web sources say this value is too small.
For N=10 million the Maple's mt19937 doesn't pass the tests successfully.
I used here N=50 million (the resulting ASCII file has size 537 Mo).

Name of the input file.

The file generated by Maple is named Maple_mt19937_N=5e7.txt

One important thing is the preamble of a licit input file.

This preamble must have 6 lines (the value 10 right to count must be set to the value of N).
A licit preamble is of the form.

#==================================================================

# some text indicating the generator used

#==================================================================

type: d

count: 10

numbit: 32

As Maple_mt19937_N=5e7.txt is generated from an ExportMatrix command, this preamble is added
by hand.

Running multiple Diehard tests

To run the same tests used to qualify STATA's Mersenne Twister, open a terminal window,
go to the directory that contains input file Maple_mt19937_N=5e7.txt and run this script:

for i in {0,1,2,3,4,8,9,10,11,12,13,14,15,16}; do

dieharder -g 202 -f Maple_mt19937_N=5e7.txt -d $i >> Diehard___Maple_mt19937_N=5e7

done ;

The results are then forked in the ASCII file Diehard___Maple_mt19937_N=5e7

>	with(RandomTools[MersenneTwister]):

>	dir := cat("/", currentdir(), "Desktop/DIEHARD/"): InputFile := cat(dir, "Maple_mt19937_N=5e7.txt"):

>	SetState(state=1); N := 5*10^7: st := time(): S := convert([seq(GenerateUnsignedInt32(), i=1..N)], Matrix)^+; time()-st;

$S := Vector(4, {(1) = ` 50000000 x 1 `*Matrix, (2) = `Data Type: `*anything, (3) = `Storage: `*rectangular, (4) = `Order: `*Fortran_order})$

(1)

>	st := time(): ExportMatrix(InputFile, S, format=rectangular, mode=ascii); time()-st;

(2)

Diehard's results

Full test suite (about 5 hours of computational time)

Command :
dieharder -g 202 -f Maple_mt19937_N=5e7.txt -a > Diehard___ALL___Maple_mt19937_N=5e7

The results are compared to those obtained for Diehard's mt19937.
Two ways are used :

  - 1 - In a first stage one generates a stream of PRN and store it in an ASCII file (just as we did with Maple).
         The whole suite of tests is then run on this file.
         Commands (-g 013 codes for mt19937):

         dieharder -S 1 -g 013 -o -t 50000000 > Diehard_mt19937_N=5e7.txt
         dieharder -g 202 -f Diehard_mt19937_N=5e7.txt -a > Diehard___ALL___Diehard_mt19937_N=5e7

  - 2 - The whole suite is run by invoking directectly mt19937 "online"
         Commands :
         dieharder -S 1 -g 013 -t 50000000 -a > Diehard___ALL___Online

A UNIX diff command has been used to verify that the two files Maple_mt19937_N=5e7.txt and
Diehard_mt19937_N=5e7.txt were identical (thet were).

Note that the Diehard doens't responds identically depending on the stream of random numbers comes from a file
or is generated online (this last [- 2 -] situation seems to give better results).-

Résumé (114 tests):
   - * - Maple's and Diehard's mt19937 respond exactly the same way when the stream of random
          numbers is read from an ASCII file (8 tests failed (******) and 6 weak (**)).
   - * - Diehard's mt19937 fails 0 test and is weak on 4 tests when the stream is generated online

>

restart:

>

dir := currentdir():
FromMapleFile     := cat(dir, "Diehard___ALL___Maple_mt19937_N=5e7"):
FromDiehardFile   := cat(dir, "Diehard___ALL___diehard_mt19937_N=5e7"):
FromDiehardNoFile := cat(dir, "Diehard___ALL___Online"):

printf("                           ======================|======================|======================|\n"):
printf("                          |   From Maple's file | From Diehard's File | Diehard online test |\n"):
printf("==========================|======================|======================|======================|\n"):
printf("          test       ntup | p.value   Assessment | p.value   Assessment | p.value   Assessment |\n"):
printf("==========================|======================|======================|======================|\n"):

for k from 1 to 9 do
  LMF := readline(FromMapleFile):
  LDF := readline(FromDiehardFile):
  LDNF := readline(FromDiehardNoFile):
end do:

while LMF <> 0 do
  if StringTools:-Search("|", LMF) > 0 then
    res := StringTools:-StringSplit(LMF, "|")[[1, 2, 5, 6]];
    printf("%-20s %3d | %1.7f ", res[1], parse(res[2]), parse(res[3]));
      if StringTools:-Search("WEAK" , res[4]) > 0 then printf("    **     |")
    elif StringTools:-Search("FAILED", res[4]) > 0 then printf(" ******   |")
    else printf(" PASSED   |")
    end if:
  end if:
  LMF := readline(FromMapleFile):

  if StringTools:-Search("|", LDF) > 0 then
    res := StringTools:-StringSplit(LDF, "|")[[5, 6]];
    printf(" %1.7f ", parse(res[1]));
      if StringTools:-Search(" WEAK" , res[2]) > 0 then printf("     **    |")
    elif StringTools:-Search(" FAILED", res[2]) > 0 then printf("   ****** |")
    else printf("   PASSED |")
    end if:
  end if:
  LDF := readline(FromDiehardFile):

  if StringTools:-Search("|", LDNF) > 0 then
    res := StringTools:-StringSplit(LDNF, "|")[[5, 6]];
    printf(" %1.7f ", parse(res[1]));
      if StringTools:-Search("WEAK" , res[2]) > 0 then printf("     **    |")
    elif StringTools:-Search("FAILED", res[2]) > 0 then printf("   ******    |")
    else printf("   PASSED |")
    end if:
    printf("\n"):
  end if:
  LDNF := readline(FromDiehardNoFile):

end do:

>

Download DIEHARD_test_of_MAPLE_MersenneTwister.mw

A lot of supplementary details are given in the attached file.
I let the readers discover by themselves if Maple's implementation of the Mersenne Twister PRNG is correct or not.
Beyond this exercise, I hope this work will be useful to people who could be tempted to test their own generator.

maple.ini (Windows) and .mapleinit (Linux)

by: struckmeier 70 Product: Maple

November 19 2019

9 3

After updating to Maple2019.2, the user initialization files maple.ini (Windows) and .mapleinit (Linux), which are placed into the user's home dierectories, are no longer read in upon startup. The behavior of Maple2019.1 was correct, the files were read.

Maple and MapleSim 2019.2 updates

by: eithne 1927 Products: Maple 2019 MapleSim 2019

November 12 2019

3 14

We have just released updates to Maple and MapleSim.

Maple 2019.2 includes corrections and improvements to a variety of areas in the product, including a new “Go to page ____” option in print preview (that am personally quite pleased about), sections are expanded by default when printing or exporting, a fix to a problem using non-executable math with text in document mode that sometimes made it impossible to advance to a new line using Enter, improvements to VectorCalculus, select, abs and other math functions, support for macOS Catalina, and more. We recommend that all Maple 2019 users install these updates.

This update is available through Tools>Check for Updates in Maple, and is also available from our website on the Maple 2019.2 download page, where you can also find more details.

For MapleSim users, the MapleSim 2019.2 family of products includes enhancements in the areas of model development and toolchain connectivity, including substantial enhancements to the MapleSim CAD toolbox. For more details and download instructions, visit the MapleSim 2019.2 download page.

Preparing a parametrized surface for 3D printing...

by: Rouben Rostamian 7173 Product: Maple

August 15 2019

8 4

Although the graph of a parametrized surface can be viewed and manipulated on the computer screen as a surface in 3D, it is not quite suitable for printing on a 3D printer since such a surface has zero thickness, and thus it does not correspond to physical object.

To produce a 3D printout of a surface, it needs to be endowed with some "thickness". To do that, we move every point from the surface in the direction of that point's nomral vector by the amount ±T/2, where T is the desired thickness. The locus of the points thus obtained forms a thin shell of thickness T around the original surface, thus making it into a proper solid. The result then may be saved into a file in the STL format and be sent to a 3D printner for reproduction.

The worksheet attached to this post provides a facility for translating a parametrized surface into an STL file. It also provides a command for viewing the thickened object on the screen. The details are documented within that worksheet.

Here are a few samples. Each sample is shown twice—one as it appears within Maple, and another as viewed by loading the STL file into MeshLab which is a free mesh viewing/manipulation software.

Here is the worksheet that produced these: thicken.mw

Using Statistics:-Sample in the `Grid` environment...

by: Carl Love 25000 Product: Maple 2019

August 02 2019

5 3

I experienced a significant obstacle while trying to generate independent random samples with Statistics:-Sample on different nodes of a Grid multi-processing environment. After many hours of trial-and-error, I discovered an astonishing workaround, and I achieved excellent time and memory performance. Since this seems like a generally useful computation, I thought that it was worthy of a Post.

This Post is also worth reading to learn how to use Grid when you need to initialize a substantial environment on each node before using Grid:-Map or Grid:-Seq.

All remaining details are in the following worksheet.

How to use Statistics:-Sample in the `Grid` environment

Author: Carl Love <carl.j.love@gmail.com> 1 August 2019

I experienced a significant obstacle while trying to generate indenpendent random samples with Statistics:-Sample on the nodes of a multi-processor Grid (on a single computer). After several hours of trial-and-error, I discovered that two things are necessary to do this:

1.	The random number generator needs to be seeded differently in each node. (The reason for this is easy to understand.)

2.

The random variables generated by Statistics:-RandomVariable need to have different names in each node. This one is mind-boggling to me. Afterall, each node has its own kernel and so its own memory It's as if the names of random variables are stored in a disk file which all kernels access. And also the generator has been seeded differently in each node.

Once these things were done, the time and memory performance of the computation were excellent.

>

restart
:

>	Digits:= 15 :

>	#Specify the size of the computation: (n1,n2,n3):= (100, 100, 1000): # n1 = size of each random sample; # n2 = number of samples in a batch; # n3 = number of batches.

>

#
#Procedure to initialize needed globals on each node:
Init:= proc(n::posint)
local node:= Grid:-MyNode();
   #This is wrapped in parse so that it'll act globally. Otherwise, an environment
   #variable would be reset when this procedure ends.
   parse("Digits:= 15;", 'statement');

   randomize(randomize()+node); #Initialize independent RNG for this node.
   #If repeatability of results is desired, remove the inner randomize().

   (:-X,:-Y):= Array(1..n, 'datatype'= 'hfloat') $ 2;

   #Perhaps due to some oversight in the design of Statistics, it seems necessary that
   #r.v.s in different nodes **need different names** in order to be independent:
   N||node:= Statistics:-RandomVariable('Normal'(0,1));
   :-TRS:= (X::rtable)-> Statistics:-Sample(N||node, X);
   #To verify that different names are needed, change N||node to N in both lines.
   #Doing so, each node will generate identical samples!

   #Perform some computation. For the pedagogical purpose of this worksheet, all that
   #matters is that it's some numeric computation on some Arrays of random Samples.
   :-GG:= (X::Array, Y::Array)->
      evalhf(
         proc(X::Array, Y::Array, n::posint)
         local s, k, S:= 0, p:= 2*Pi;
            for k to n do
               s:= sin(p*X[k]);
               S:= S + X[k]^2*cos(p*Y[k])/sqrt(2-sin(s)) + Y[k]^2*s
            od
         end proc
         (X, Y, n)
      )
   ;
   #Perform a batch of the above computations, and somehow numerically consolidate the
   #results. Once again, pedagogically it doesn't matter how they're consolidated.
   :-TRX1:= (n::posint)-> add(GG(TRS(X), TRS(Y)), 1..n);

   #It doesn't matter much what's returned. Returning `node` lets us verify that we're
   #actually running this on a grid.
   return node
end proc
:

The procedure Init above uses the :- syntax to set variables globally for each node. The variables set are X, Y, N||node, TRS, GG, and TRX1. Names constructed by concatenation, such as N||node, are always global, so :- isn't needed for those.

>	# #Time the initialization: st:= time[real](): #Send Init to each node, but don't run it yet: Grid:-Set(Init) ; #Run Init on each node: Nodes:= Grid:-Run(Init, [n1], 'wait'); time__init_Grid:= time[real]() - st;

The only purpose of array Nodes is that it lets us count the nodes, and it lets us verify that Grid:-MyNode() returned a different value on each node.

>	num_nodes:= numelems(Nodes);

>	#Time the actual execution: st:= time[real](): R1:= [Grid:-Seq['tasksize'= iquo(n3, num_nodes)](TRX1(k), k= [n2 $ n3])]: time__run_Grid:= time[real]() - st

>	#Just for comparison, run it sequentially: st:= time[real](): Init(n1): time__init_noGrid:= time[real]() - st; st:= time[real](): R2:= [seq(TRX1(k), k= [n2 $ n3])]: time__run_noGrid:= time[real]() - st;

R1 and R2 will be different because different random numbers were used, but they should have similar histograms.

>	plots:-display( Statistics:-Histogram~( <R1 \| R2>, #side-by-side plots 'title'=~ <<"With Grid\n"> \| <"Without Grid\n">>, 'gridlines'= false ) );

(Plot output deleted because MaplePrimes cannot handle side-by-side plots!)

They look similar enough to me!

Let's try to quantify the benefit of using Grid:

>	speedup_factor:= time__run_noGrid / time__run_Grid;

Express that as a fraction of the theoretical maximum speedup:

>	efficiency:= speedup_factor / num_nodes;

I think that that's really good!

The memory usage of this code is insignificant, which can be verified from an external memory monitor such as Winodws Task Manager. It's just a little bit more than that needed to start a kernel on each node. It's also possible to measure the memory usage programmatically. Doing so for a Grid:-Seq computation is a little bit beyond the scope of this worksheet.

>

Download GridRandSample.mw

Here are the histograms: