mmcdara

A question of number theory...

Asked: mmcdara 7002 Product: Maple

January 08 2025

2 3

Does any one have any idea to demonstrate, using Maple, that for any couple (a, b) of strictly positive integers

(a*b)! / (a! * (b!)^a)

is an integer?

Delete questions if you like all but lea...

Asked: mmcdara 7002 Product: MaplePrimes

December 16 2024

0 4

It's becoming increasingly common for questions to be deleted, the most frequently evoked argument being that they duplicate earlier questions.
Usually it's the author of the question who complains, in this case it's the one who provided an answer.

I've always found this self-given freedom on the part of moderators somewhat inappropriate, especially given that these deletions are done on the sly, without even the courage to be explicitly claimed by their authors.

Couldn't these latter take the time to leave a message indicating what they've just done (and why) and where the deleted question, and any exchanges associated with it, can be found?
Otherwise, this attitude is more akin to contempt and censorship than to real management of this blog.

A few years ago, when I came to this site, these deletions were the exception and I can only deplore the fact that they have become commonplace.
They say that life is better elsewhere, I don't know, but what I do know is that I've never come across this on any Scilab, R, SageMath or OpenTurns user's blog.

Anyway, here's the reply I sent to @salim-barzani : Hirota_derivative_KdV_equation.mw

Details about Statistics:-Sample...

Asked: mmcdara 7002 Product: Maple

December 12 2024

0 0

Where can I found details about Statistics:-Sample(..., method=envelope).

It would be nice to have a link to a description of the envelope method Sample uses.
For instance does it share some features of the Cuba library for numeric integration? Does it use the same envelope method evalf/Int(..., method=_CubaSuave)) uses?

Thanks in advance.

A strange result which requires explanat...

Asked: mmcdara 7002 Product: Maple 2015

December 06 2024

1 3

Can anyone explain me the reason of the last result?
Thanks in advance

>

restart

>	kernelopts(version)

(1)

>	a/n^b; den := denom(%); print(cat(`_`$50)); 3/n^2; den := denom(%); print(cat(`_`$50)); 1.23/n^1.65; den := denom(%); num := numer(%%);

(2)

Download What-does-happen-here.mw

Numerical integration using Monte-Carlo ...

Asked: mmcdara 7002 Product: Maple

November 23 2024

2 5

I have a lot of questions about using evalf/Int with Monte-Carlo like methods.
They are red written in the attached file and concern several different points.
I would like you to answer each of them and not to focus on a specific one.
Thanks in advance.

method=_MonteCarlo uses the Fortran procedure d01gbc (maybe Maple rewritten?): the original NAG procedure is described here d01gbc

>

restart

>	kernelopts(version)

(1)

>	domain := 0.8..3.; f := x -> 1/(1 + sinh(2x)log(x)^2);

(2)

>

infolevel[`evalf/int`] := 4:

# I understand _CubaSuave used a random sample of size 134000 of the (x, y) integration domain.
# Am I right?

evalf(Int(f(x), [x=domain, y=0..1], method=_CubaSuave, epsilon=1e-4));
printf("\n%s\n", cat("-"$100));

# Did _CubaDivonne used a random sample of size 3256 ?

evalf(Int(f(x), [x=domain, y=0..1], 'method=_CubaDivonne', epsilon=1e-4));
printf("\n%s\n", cat("-"$100));

# What is the true number of points _MonteCarlo used?

evalf(Int(f(x), [x=domain, y=0..1], method=_MonteCarlo, epsilon=1e-4));

Control_multi: integrating on [.8, 0] .. [3., 1] the integrand

cuba: transformed original integrand

cuba: with lower bounds [.8, 0.] and upper bounds [3., 1.], to the following integrand to be integrated over the unit n-cube:

cuba: integration completed successfully
cuba: # of integrand evaluations: 134000
cuba: estimated (absolute) error: 6.73167e-05
cuba: chi-square probability that the error is not reliable: 1
cuba: number of regions that the domain was divided into: 134

----------------------------------------------------------------------------------------------------
Control_multi: integrating on [.8, 0] .. [3., 1] the integrand

cuba: transformed original integrand

cuba: with lower bounds [.8, 0.] and upper bounds [3., 1.], to the following integrand to be integrated over the unit n-cube:

cuba: integration completed successfully
cuba: # of integrand evaluations: 3256
cuba: estimated (absolute) error: 6.37672e-05
cuba: chi-square probability that the error is not reliable: 0
cuba: number of regions that the domain was divided into: 22

----------------------------------------------------------------------------------------------------
Control_multi: integrating on [.8, 0] .. [3., 1] the integrand

trying d01gbc (nag_multi_quad_monte_carlo)
d01gbc: epsrel=.1e-3; minpts=0; maxpts=500000000; method=2; cont=0
d01gbc: procedure for evaluation is:
proc (X) 1/(1.+sinh(2.*X[1])*ln(X[1])^2) end proc

d01gbc: result=.676840185506968228
d01gbc: relerr=.103631215397982221e-4; usedpts=1044
result=.676840185506968228

(3)

>	# By the way, why is infolevel[`evalf/int`] output discarded? evalf(Int(f(x), [x=domain, y=0..1], method=_CubaSuave, epsilon=1e-4)); printf("\n%s\n", cat("-"$100)); evalf(Int(f(x), [x=domain, y=0..1], method=_MonteCarlo, epsilon=1e-4));

----------------------------------------------------------------------------------------------------

(4)

>

# Let us suppose I don(t care of the accuracy of the result as I am capable to assess it
# by some other way:
#      How can I use a Monte-Carlo integration method with a given number of points?
#      Why there is no estimation of the integral returned?

infolevel[`evalf/int`] := 0:
infolevel[`evalf/int`] := 4:
evalf(
  Int(
    f(x), [x=domain, y=0..1]
    , method=_CubaVegas
    , methodoptions=[minimalpoints = 10^3, maximalpoints = 10^3]
  )
);

Control_multi: integrating on [.8, 0] .. [3., 1] the integrand

cuba: transformed original integrand

cuba: with lower bounds [.8, 0.] and upper bounds [3., 1.], to the following integrand to be integrated over the unit n-cube:

cuba: could not attain requested accuracy
cuba: # of integrand evaluations: 1000
cuba: estimated (absolute) error: 0.025423
cuba: chi-square probability that the error is not reliable: -999
evalf/int: error from Control_multi was:
"could not attain requested accuracy; try increasing epsilon or absepsilon or maximalpoints"

Int(Int(1/(1.+sinh(2.*x)*ln(x)^2), x = .8 .. 3.), y = 0. .. 1.)

(5)

>	infolevel[`evalf/int`] := 0: infolevel[`evalf/int`] := 4: evalf( Int( f(x), [x=domain, y=0..1] , method=_CubaSuave , methodoptions=[minimalpoints = 10^3, maximalpoints = 10^3] ) );

Control_multi: integrating on [.8, 0] .. [3., 1] the integrand

cuba: transformed original integrand

cuba: with lower bounds [.8, 0.] and upper bounds [3., 1.], to the following integrand to be integrated over the unit n-cube:

cuba: could not attain requested accuracy
cuba: # of integrand evaluations: 1000
cuba: estimated (absolute) error: 0.025423
cuba: chi-square probability that the error is not reliable: -999
cuba: number of regions that the domain was divided into: 1
evalf/int: error from Control_multi was:
"could not attain requested accuracy; try increasing epsilon or absepsilon or maximalpoints"

Int(Int(1/(1.+sinh(2.*x)*ln(x)^2), x = .8 .. 3.), y = 0. .. 1.)

(6)

>	infolevel[`evalf/int`] := 0: infolevel[`evalf/int`] := 4: evalf( Int( f(x), [x=domain, y=0..1] , method=_MonteCarlo , methodoptions=[minimalpoints = 10^3, maximalpoints = 10^3] ) );

Control_multi: integrating on [.8, 0] .. [3., 1] the integrand

evalf/int: error from Control_multi was:
"NAG d01gbc expects epsilon >= 0.5e-4, but received %1", .5000000000e-9

Int(Int(1/(1.+sinh(2.*x)*ln(x)^2), x = .8 .. 3.), y = 0. .. 1.)

(7)

>	# Example 8 of the reference given in the question text evalf(Int(4x1x3^2exp(2x1*x3)/(1+x2+x4)^2, [seq(x\|\|i=0..1, i=1..4)], method=_MonteCarlo, epsilon=1e-2))

(8)

>	# Why is not the seeed updated ? for j from 1 to 3 do Threads:-Sleep(10): evalf(Int(4x1x3^2exp(2x1*x3)/(1+x2+x4)^2, [seq(x\|\|i=0..1, i=1..4)], method=_MonteCarlo, epsilon=1e-2)) end do;

(9)

>

# Here a new seed is forced at each iteration but the estimations of the integral are always the same.
# This is impossible as these estimationsare the realizations of a random variable: so they cannot be identical.
#
# So, does evalf/Int always use the same internal seed?
# More importantly: does it really use a random generator?
for j from 1 to 3 do
  seed := randomize(rand()());
  J := evalf(Int(4*x1*x3^2*exp(2*x1*x3)/(1+x2+x4)^2, [seq(x||i=0..1, i=1..4)], method=_MonteCarlo, epsilon=1e-2)):
  print('seed' = seed, 'J' = J);
end do:

Control_multi: integrating on [0, 0, 0, 0] .. [1, 1, 1, 1] the integrand

4*x1*x3^2*exp(2*x1*x3)/(1+x2+x4)^2

trying d01gbc (nag_multi_quad_monte_carlo)
d01gbc: epsrel=.1e-1; minpts=0; maxpts=500000000; method=2; cont=0
d01gbc: procedure for evaluation is:
proc (X) 4.*X[1]*X[3]^2*exp(2.*X[1]*X[3])/(1.+X[2]+X[4])^2 end proc
d01gbc: result=.574090186993690188
d01gbc: relerr=.675204841605886539e-2; usedpts=8064
result=.574090186993690188

(10)

>	# This is what one expects from Monte-Carlo integration use Statistics in for i from 1 to 10 do seed := randomize(rand()()); S := Sample(Uniform(op(domain)), 100): print('seed' = seed, 'J' = Mean(f~(S)) * (- `-`(op(domain))) ); end do: end use: