@Marvin Ray Burns 

And Chezaro (k+1)-transform faster than Chezaro k-transform if at least latter is sign-changed. The question could be wheather Langevin u-transform, which maple takes, better than all Chezaro ones (as i asked before)

At least i think so. You take exactly Chezaro 1-transform from covergent series.

@Joe Riel 

1) As i understand your buffers override wrap mode. Have in my .emacs
 '(word-wrap t) but in your buffers lines are still nonwrapped. So all time i do appropriate things in "Options/line wrapping in this buffer"

2) After reconnect 'showstat' buffer is empty

Need next features for better debugging:

1) mdc:-Reconnect() which reopens broken socket. After closing a couple buffers in emacs found that client

cannot debug anymore:

Error, (in Sockets:-Write) Unknown error.

2) I REALLY need finally now at least simply skip maple procedures that debugger  use. At least how to skip line

 body := pointto(disassemble(addressof(eval(p)))[6]);

(line 24 in `intrep/makeintrep` that called from line 2 of `codegen/packlocals`)

to see what happens next? Kernel crashes for now.

I did

showstat(IntegralTransforms:-Tools:-SimplifyPower);

Ctrl-C-Ctrl-V to editor, cut block with line numbers, patch, save to maple_patch.mpl and read it after kernel restart. The problem only, it's OK there because symbol is global. What to do if it local (=nonexported procedure under package) Suspect, i cannot simply reread it.

I did

showstat(IntegralTransforms:-Tools:-SimplifyPower);

Ctrl-C-Ctrl-V to editor, cut block with line numbers, patch, save to maple_patch.mpl and read it after kernel restart. The problem only, it's OK there because symbol is global. What to do if it local (=nonexported procedure under package) Suspect, i cannot simply reread it.

@Axel Vogt 

Not convex, but simply hull/tube around derivative, which, however, should be more coarsed than convex one.

Rather seems to be smth like variation of derivative but still not.

-----------

update:

(u,v)->|F'(u)|+Int(abs(F''(t)),y=u..v) satisfies all conditions.

Unfortunately algorythm is not need to hold with even k. So, x_n,2 is beyond law.

In example for F:=(x::algebraic)->(x^2-1)^2;
M:=(u::algebraic,v::algebraic)->4*max(abs(u), abs(v))*max(`if`(u*v

OneDimImplShifted(F, M, -2., 2.0,2); loses x=1.0, it jumps over it.

So, x_n,1, x_n,3, ... , x_n,2k+1,... are only possible as was proved in my 2nd post. Two questions remain unsolved.

1) With which k better run algorythm for given function and it majoant.Practically doesn't have big sense because

1.1) If even those cryterium exist it could be hard to check itin practice.

1.2) On practice one can parallelize by creating subtasks with different k's and return 1st better.

2) (i know, a hardest one). How to calculate majorant of derivative in general case? i.e. function M(F):{(u,v) \in R^2| uR with next properties

2.1 M(F)(u,v)>=sup {abs(F'(x)) | x \in [u,v]} - continuous

2.2 For v2>=v1:: M(F)(u,v2)>=M(F)(u,v1)

2.3 (an optional but desirable one): M(F)(u,u)=abs(F'(u))

wanna in parrticular test on Product(1-1/(x*k)^2,k=1..100) - should be not far away from sin(Pi/x)/x

And... should be "majorant analysis" created as derivative's one?
3) Minor improvements to remain quadratical covergence for roots of order>1.

@Markiyan Hirnyk 

Software regression detected :( I thought so, that vote is yours. But still want have rather ideal solution. Was really surprised, that can use x_n,2. It's essential speedup not only in this example but in half of general case too (in other half x_n,1 should be taken). Need counterexamples or proof. Still learning Taylor theorem with G(x) term. That M is monotonous by 2nd parameter. It could be essential.

@Markiyan Hirnyk 

which version of maple do you use? After waiting >30 min on my machine kernel crashes because all >1.5Gb available free of memory was wasted.

While my one still can be better. After taking x_(n+1)=x_n,2 i.e.

and with majorant

proc(u::algebraic,v::algebraic)
  local x:=1/u, y:=1/v, mx:=floor(2*x/Pi), my:=floor(2*y/Pi),dd;
  #
  if (mx=my) then
    if (mx mod 2=0) then
      exp(-y^2)*(x^2*abs(cos(y))+2*x^3*abs(sin(x)));
    else
      exp(-y^2)*(x^2*abs(cos(x))+2*x^3*abs(sin(y)));
    end if;
  elif (x-y    dd:=evalf(sin(max(abs(x-mx*Pi/2),abs(y-mx*Pi/2))));
    if (mx mod 2=0) then
      exp(-y^2)*(x^2+2*x^3*dd);
    else
      exp(-y^2)*(x^2*dd+2*x^3);
    end if;
  else
    exp(-y^2)*(x^2+2*x^3);
  end if;
end proc

i found all roots in ~40 sec. Unlike to odd ones i cannot say that choose of x_(n+1)=x_n,k with even k is legally but it works in this example. While with such majorant both approach to roots (from left) and escaping from them (from right) has near to quadratical convergency/divergency.

Ideally if limit(M(u,v), v->u)=|F'(u)| - then convergence will be really quadratical.

1dim_roots_best_fitt.mw

Thank you for question. Finally i'm ready to answer.

1dim_roots.zip

@Joe Riel 

It's OK, because i understand that debugger is relatively new one and nothing could work from very first time.

As i said, ssytem maple call is solution. And it shouldn't depend on operating system as in help written:

"The behavior of ssystem is exactly the same as that of system, except that in the case of ssystem, the result is returned to Maple so that it can be used in the session."

They also want scary about

"Invoking interactive commands via system or ssystem can result in undefined behavior."

but in case with emacs it's OK.

My OS is Windows 7 Home Edition.

Solution is ssystem instead of system.

@Pseudomodo 

It's partially OK. But it's not what Joe wants. Intermediate shell terminates while maple screen still open with message: "Exit Value: 0".Effect actually the same as if i would run simply system("<emac_path>").

But it rather right direction for solution. I mean, simply close maple screen after.

mdc_default['launch_emacs'] := true:

+
mdc_default['emacs'] := "C:/emacs/bin/emacs.exe":

in ini file doesnt work under windows.

Reason:

you tried to run it via maple system("start /b <emac_path>" ). It fails because "start" not seems to be a separate program, it's rather command in cmd shell.