@Mariusz Iwaniuk 

Is I'm not mistaken you do not solve the correct problem
 

f := 1+r^2*(1-sqrt(1+20*lambda*M/r^5))/(6*lambda);
func := eval(simplify(lambda*(2*(diff(f, r, r))+4*(diff(f, r))/r)-1), [M = 1, lambda = 1]);

# having written lambda*(2*(diff(f, r, r))+4*(diff(f, r))/r)-1  you must solve func=0, not func=1 (what you did probably given the roots you obtained)

# simpler if we do this transform :

Ufunc := subs(r=U^(1/5), func);

subs({M = 1, lambda = 1}, Ufunc);
solve(%=0,U);
op([2,1], %[1]);
RootFinding:-Isolate(%);

@Mariusz Iwaniuk 

Hi, we have had the same idea.
I think we can go a little bit further (look my file in my reply to mhdlynch).
Furthermore I don't think MMA provides the good solution.
If I'm not mistaken it should be 
] -2.217533520, -2.040737917 [   union   ] 0.6124468982, 1.177609717 [

@litun 

Thank you for the info

@litun 

Are you thinking to something like that?

Run.mw

RunRun.mw

Run.mw contains the code (please note the line writeto(terminal) before the "return' line in order to redirect the results to your active worksheet)/

An example of a double loop using DocumentTools:-RunWorksheet  is given in RunRun.mw

 

@Kitonum 

 [{2}$2] could be cosidered as a little cheating...
... but an astute one :-)

Thanks for the answer

@Joe Riel 

The double quotes are "ugly" indeed,  and your last use of map is definitively simpler.

Thanks for the answer

BTW: before coming here I didn't know about element-wise operators (funny thing is that I thought at this time that Maple suffered from lack when comparred to Matlab or R on this point ...). As I was reading more and more answers using "~" I gradually became a fan of it and began to set aside "map", at least when possible.
I guess the truth lies in a reasonable use of each of them.
 

@acer  @Carl Love

Thanks to your replies.

To acer: "Did you mean   unapply('P'(i,1,1,x), x)  so that a procedure P doen't get its call evaluated there?"
No, I didn't mean 'P' but simply P. So to understant the difference between '¨' and P I did this

restart:
P := proc(a, x)
a+x
end proc:

for k from 1 to 1 do 
f[k] := unapply(P(k,x), x);
g[k] := unapply('P'(k,x), x);
end do;
     f₁ := x -> 1 + x
     g₁ := x -> P(1,x)

f[1](x);
     1+x
g[1](x);
     1+x

Then, even if the final results are the same, using your 'P' instead my naïve P is definitively better, simply because
having g₁(x)= P(1,x)  is probably the thing one implicitely looks for when one writes the inloop assignment.

Good weekend to both of you

@Carl Love 

I send you three worksheets withe very slight modifications between them

• Observation 1 : for high values of elements of the matrix Maple can fail to verify if the product of the eigenvalues is equal to the determinant (generator = 0..10^(10^3),  eig1.mw
   
• Observation 2 : trying to simplify the formal expressions of the eigenvalues and of the determinant may lead to dramatic failures of the test of equality  eig2.mw
   
• Observation 3 : In  eig3.mwi the simplifications are made on the valued expressions of the product of the eigenvalues and of the determinant. The test of equality seems to perform well even for an upper bound of 10^(10^3),
   
• Observation 4 : An upper bound of 10^(10^4),​​​​​​ is beyond the capacities of my laptop.
  But !!! The time spent is mainly due to the equality test

eig1.mw
eig2.mw
eig3.mw

@Carl Love 

Maybe.
It depends on what kristavaldes uses as a generator.
There will be no problem if generator=a..b where a abd are two floats
If a and b are integers (let's say a=0 and b "large") problems may arise, but they are more problems from manipulating large numbers than really problems that could be solved by using expand :

restart:
with(LinearAlgebra):
N := 100:
A := RandomMatrix(3, 3, generator=0..N):
CodeTools:-Usage(Eigenvalues(A)):
memory used=6.37MiB, alloc change=32.00MiB, cpu time=134.00ms, real time=2.89s, gc time=7.14ms

N := 10^100:
A := RandomMatrix(3, 3, generator=0..N):
CodeTools:-Usage(Eigenvalues(A)):
memory used=11.06MiB, alloc change=32.00MiB, cpu time=215.00ms, real time=447.00ms, gc time=20.99ms

N := 10^1000:
A := RandomMatrix(3, 3, generator=0..N):
CodeTools:-Usage(Eigenvalues(A)):
memory used=47.43MiB, alloc change=0 bytes, cpu time=6.18s, real time=6.51s, gc time=0ns

On my laptop (Mac OS X, Maple 2015.2, 8 Gb of memory) N = 10^(10^4) generates a "connection to server lost"

@tomleslie 

Thanks for the advices about plottools:-rotate(), plottools:-scale() and plottools:-rotate().
I know very few about the plottools package beyond some functions. I didn't even know that plot displacement functions had been developped in Maple.
I see you use it to modify the tickmarks too, something I never find obvious.

I'm going to see right now how I can still improve your last plot.

Thanks again

@Preben Alsholm 

You're right, a well posed BC problem invoking a third order ODE cannot contain a third order derivative.
So  D[1,1,1](f)(6)=2 is probabibly a tipo error

In your previous reply you mention this error :

Such an approximate solution that verifies the boundary conditions is guess := eta -> eta+(1/72)*eta^4
Probably a poor guess for the error message now becomes (I don't understand it)

dsolve( {ode,  D(f)(0)=1, D[1,1](f)(0), D[1, 1, 1](f)(6)=2}, numeric, range=0..6, approxsoln=[f(eta)=guess(eta)]);
Error, (in dsolve/numeric/BVPSolve) unable to store '((D@@3)(HFloat(HFloat(undefined))))(6)-2.' when datatype=float[8]

For the bc (D@@2)(f)(6) = rhs(bc3) an initial guess could be  guess := eta -> eta+(1/18)*eta^3
I tried to modify the problem by setting f(eta) = g(eta)+guess(eta).
In this case g is the solution of an ODE with homogenous bcs  D(g)(0) =0, (D@@2)(g)(0)=0, (D@@2)(g)(6) = 0

I had hoped it could help but I obtain a

Error, (in dsolve/numeric/bvp) initial Newton iteration is not converging

error I don't kow how to circumvent


Last but not least(but you already noticed that), the same ODE is solved in  Numerical Solution of Difficult ODE Boundary Value Problems help page (equation 12) but with other bcs (equation 13)


Not sure that this reply can help you ...
 

@acer 

I'm really not familiar with subsindets.
I see this command is often used here and that it seems very powerfull. I guess I have to learn more about it and begin to understand how to use it.
Your last file will help me a lot.

Than you Acer

@acer 

Thank you Acer for all those details.

You write < Your question mentioned the possibility to "modify the ellipses", but without "ad hoc modifications" >. : I wanted to say that it is always possible to "extract" the content of a plot through plottools:-getdata, to remove the undesired part from the returned list (or to modify parts of it), and eventually using PLOT to obtain something that meets out expectations. 
But it can be lengthy and sometimes written in a hurry, to answer a specific issue (so the "ad hoc" term).
More of this,    P := DrawGraph(...):   MyPlot:=plottools:-getdata(P):  doesn't contain all the informations about P: for instance there is colouring information.
That is why modifying MyPlot is rather lengthy.

<  For example, do you mean that you would only want the ellipse re-jigged to the longest label if it could be done fully automatically and robustly? > ... the truth is that I'm interested in any idea that could help me to "adjust" the result of DrawGraph, to a reasonable extent.
Among these adjustements are the aspect ratios of the ellipses (maybe rectangles instead), the font (thank you for the tip), the linestyle of the edges/arcs (I'm not yet familiar with Maple 2018  but I remember that, even this last point, was not obvious with Maple 2015).


Thanks again

@Carl Love 

I had thought to that,  of course. 
But inuitively I was certain that it would not be very efficient because of this loop, you know sort of "flee the loop like the pleague".

After a few tests your solution appears to be faster than mine's.
More of this, a slight modification of your's enables computing the first hitting times for increasing calues of c in a very efficient way (obviously c2 > c1 ==> k2 > k1)

FirstHit:= proc(L::{Array,list}(float), C::realcons, n::posint)
   local c:= evalf(C), k;
   for k from n to numelems(L) do
      if L[k] > c then return k fi
   od;
   FAIL
end proc:

h := FirstHit(XC, 300, 1);
FirstHit(XC, 305, h);

Thank you Carl

@Joe Riel 

Thank you for the confirmation.
Unfortunately I don't have the possibility to test this at home (Mac OSX) but I have some notes in hand from which I can give you some more of the details.
In fact the problem with ExcelTools:-Export occured when we used cmaple with Windows 7 (we used Windows XP before that). With Windows XP Excel files are ".xls" files while they are ".xlsx" since Windows 7.
ExcelTools:-Export(MyMatrix, "MyFile.xls") runs correcty, but ExcelTools:-Export(MyMatrix, "MyFile.xlsx")  doesn't.

Then you're right, ExcelTools:-Export runs correctly when launched from cmaple, the "xlsx file problem" is another issue.

Thanks again