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Offline Davide Carpi  
#1 Posted : 05 September 2012 05:08:15(UTC)
Davide Carpi


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Hi all,

This plugin contains some root-finding methods and optimization algorithms.

Password (Extensions Manager): NonlinearSolvers

Since the plugin may have big changes in future, I can't make a complete documentation but you can referrer to:
- Martin's handbook
- this album







PLEASE REPORT HERE ANY ISSUE


Best regards,

Davide


requirements: .Net Framework 2.0 / AnyCPU / SMath Studio 0.95.4594 (30 July 2012)

Edited by user 13 May 2016 12:55:27(UTC)  | Reason: EM password

File Attachment(s):
NonLinearSolvers Documentation.zip (63kb) downloaded 538 time(s).
BETA_testing sm files.zip (32kb) downloaded 264 time(s).
Davide Carpi attached the following image(s):
NLS.png
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Offline omorr  
#2 Posted : 05 September 2012 10:42:17(UTC)
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w3b5urf3r you are my hero Air kiss Biggrinrinks: Friends

Many things like this one would make me very happy Biggrinancing:

Regards,
Radovan

P.S. Do not take me wrong please, I am very happy that you've made this plugin but I think that my "home made" function does not deserve to be the reference to this rather welcome and appreciated plugin. I hope that your updates would base this plugin to some more trustworthy source.

Edited by user 05 September 2012 11:55:38(UTC)  | Reason: Not specified

When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
Offline omorr  
#3 Posted : 05 September 2012 12:36:37(UTC)
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Hello w3b5urf3r Good

Rather surprised and excited with this sys() thing regarding Broyden() Good

As I do not know how is this working yet, this is a minor observation at a first glance.

It seems that the vector indices play main role here.


Broyden(sys(el(x,1)-2*el(x,2)+5,el(x,1)+3*el(x,2)-1,2,1),sys(1,2,2,1),eps,maxiter)=mat(-2.6,1.2,2,1)

Broyden(sys(el(a,1)-el(2*b,2)+5,el(c,1)+3*el(d,2)-1,2,1),sys(1,2,2,1),eps,maxiter)=mat(-2.6,1.2,2,1)

Broyden(sys(el(a,1)-el(2*b,2)+el(e,1),el(c,1)+3*el(d,2)+el(f,1),2,1),sys(1,2,2,1),eps,maxiter)=mat(0,0,2,1)

This might be a confusion because vectors a,b,c,d,e,f are not defined and there is still a result returned. You managed to avoid mentioning vector of unknowns like in the roots(), and I think that vector indices are crucial things here to distinguish between unknowns, not sure. Am I right? If that is the case, I thing that this should be taken care of in the case of different vectors elements with the same indices in order what to refer to as vector of unknowns.

Best wishes,
Radovan
When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
Offline Davide Carpi  
#4 Posted : 05 September 2012 16:39:55(UTC)
Davide Carpi


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Originally Posted by: omorr Go to Quoted Post
w3b5urf3r you are my hero Air kiss Biggrinrinks: Friends

Many things like this one would make me very happy Biggrinancing:

Regards,
Radovan

P.S. Do not take me wrong please, I am very happy that you've made this plugin but I think that my "home made" function does not deserve to be the reference to this rather welcome and appreciated plugin. I hope that your updates would base this plugin to some more trustworthy source.

you're right, but I did not know where to start and that function was a good starting point. Biggrin
Currently, the plugin follows almost exactly that function, but if anyone has a more efficient version of the algorithm could certainly be implemented. Good

Originally Posted by: omorr Go to Quoted Post
Hello w3b5urf3r Good

Rather surprised and excited with this sys() thing regarding Broyden() Good

Yes, I have lost a bit of time on this, I wanted the function to accept both individual functions or systems defined in vectors or in the "multiple values" item. Good

Originally Posted by: omorr Go to Quoted Post
As I do not know how is this working yet, this is a minor observation at a first glance.

It seems that the vector indices play main role here.

Broyden(sys(el(x,1)-2*el(x,2)+5,el(x,1)+3*el(x,2)-1,2,1),sys(1,2,2,1),eps,maxiter)=mat(-2.6,1.2,2,1)

Broyden(sys(el(a,1)-el(2*b,2)+5,el(c,1)+3*el(d,2)-1,2,1),sys(1,2,2,1),eps,maxiter)=mat(-2.6,1.2,2,1)

Broyden(sys(el(a,1)-el(2*b,2)+el(e,1),el(c,1)+3*el(d,2)+el(f,1),2,1),sys(1,2,2,1),eps,maxiter)=mat(0,0,2,1)

This might be a confusion because vectors a,b,c,d,e,f are not defined and there is still a result returned. You managed to avoid mentioning vector of unknowns like in the roots(), and I think that vector indices are crucial things here to distinguish between unknowns, not sure. Am I right? If that is the case, I thing that this should be taken care of in the case of different vectors elements with the same indices in order what to refer to as vector of unknowns.

Best wishes,
Radovan

You're right, my parser check if there are undefined variables but it doesn't distinguish between them... Actually the assumption is that there is "one variable" (x,X,K,or something you want) or "a system of variables" (x[1],x[2],x[3],...), so actually a[1] and b[1] are both the same thing (the first element of the starting values vector)

I think I can easily add a control that returns an error on the number of vars...

If you want to use "multiple variables" (one varibale -> one vector element) I think we can do something, but with some limitations regarding the names (f.e. "i" and "e" are both constants).


regards,

w3b5urf3r

Edited by user 05 September 2012 16:59:48(UTC)  | Reason: Not specified

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Offline omorr  
#5 Posted : 05 September 2012 17:28:52(UTC)
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Hello w3b5urf3r,

Whatever you do regarding this plugin will be very welcome AFAIC Good

I have just to mention in the case you based your plugin on my SMath function, that I made it from a simple code found somewhere on the Net (even do not remember where from). If you are interested to spare some more of your time, you could try to find yourself some better algorithm and make a plugin from it Good .

Regards,
Radovan
When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
Offline omorr  
#6 Posted : 05 September 2012 21:38:49(UTC)
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Hello w3b5urf3r,

Here is a bit of testing the Broyden plugin. It is a clasical "driving crasy" situation with multiple solutions. It was mentioned by kilele in the link to the document http://en.smath.info/forum/yaf_postsm7678_solve---and-roots-----again.aspx#post7678

Your plugin failed (if I did not do something wrong) but do not worry about it - it is a common situation. The problem is why "Can not calculate" error returned. I do not know. My function returned another solution than few Newton-Raphson iterations or roots(). I also used another software that uses some kind of broyden algorithm and here is the result - the same as roots() returned
Code:
>function f(x):=[(x[1]+3)*(x[2]^3-7)+18,sin(x[2]*exp(x[1])-1)]
>x0:=[-0.5,1.4]
 [ -0.5  1.4 ]
>f(x0)
 [ 7.36  -0.150285529841 ]
>res:=broyden("f",x0)
 [ 0  1 ]
>f(res)
 [ 0  0 ]

Try, for instance to change the guess values (vector x0) from [-0.5,1.4] to [1,4].
Here is the picture and the sm file

Regards,
Radovan

Edited by user 05 September 2012 21:41:29(UTC)  | Reason: Not specified

File Attachment(s):
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Broyden_testings-1.png
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Offline kilele  
#7 Posted : 06 September 2012 02:10:18(UTC)
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maybe a good approach would be combine homotopy + broyden ?
or obtaining good approximations of roots with a generalized bisection method and then refine with broyden or other method, as explained at the conclusions of this document:
( see the link "pdf download" )
A Rapid Generalized Method of Bisection for Solving Systems of Non-linear Equations
Feel free to explore that strategy, as for me I'll take it easy with a first plugin for a one dimension bisection Biggrin

Edited by user 06 September 2012 02:17:01(UTC)  | Reason: Not specified

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Offline omorr  
#8 Posted : 06 September 2012 10:38:03(UTC)
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Hello kilele,

I hope you would not mind if I repeat some of my comments I've already posted here on the Forum.

All of you who are trying to make any SMath plugin have all my respect and appreciation. I really hope that you will enjoy this and the SMath users will benefited from that, no doubt. On the other hand, this will very likely take quite a long time until most of the basic numerical functions and procedures SMath needs would be realized, regarding that you are starting from scratch. In order not to "inventing the will" there must be some more pragmatical way to "fill up" the SMath with some already made, free available, mature and well tested numerical code (using them like black boxes and make plugins from them). Actually, I was told that this is not likely to be possible in SMath and I regret for that very much. I think this will, unfortunately, slow down the SMath progress and restrict the SMath user community. Many of the SMath users are (actual/former) Mathcad users as well, and I think that most of them would like that SMath had some Mathcad functionality comparing to even Mathcad 6 (originated from 1996).

Regarding,
Radovan
When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
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Offline kilele  
#9 Posted : 06 September 2012 13:09:47(UTC)
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Originally Posted by: w3b5urf3r_reloaded Go to Quoted Post

Currently, the plugin follows almost exactly that function, but if anyone has a more efficient version of the algorithm could certainly be implemented. Good


I've just read that this implementation is better for large problems (see pag 133 of this pdf)
and this is its matlab code

@Radovan
As for making use of third part libraries on SMath, I don't know if that is possible aside from license issues
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Offline omorr  
#10 Posted : 06 September 2012 14:36:24(UTC)
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Hello,
Originally Posted by: kilele Go to Quoted Post
Originally Posted by: w3b5urf3r_reloaded Go to Quoted Post

Currently, the plugin follows almost exactly that function, but if anyone has a more efficient version of the algorithm could certainly be implemented. Good


I've just read that this implementation is better for large problems (see pag 133 of this pdf)
and this is its matlab code

Thank you for this link Good . I do not know if w3b5urf3r has enough time to devote to the Broyden method explained there (or any other one) Thank you

This might be an example of how to make a plugin - as far as I understood this process. Ordinary user who understands Matlab could try to make the SMath function following this code. It might be more or less time consuming. A plugin developer could also use this Matlab code directly or the already made SMath function (do not know which one could make less trouble) and make a plugin by using some software tools. I have no idea how difficult might that be. On the other hand (have a vague idea about this) if there would be a plugin mechanism which will just include some kind of compiled Matlab (or C,C++,C#,F95) code where there should be defined input arguments output arguments and using a "black box" of this compiled function. This is working in many other software with the goal of reusing already made code. Otherwise, it would be a disaster to write everything from the beginning. I thought that the idea of "plugin" will follow this concept but as I was told this is not working in SMath this way. How is this actually working, I do not know.
Originally Posted by: kilele Go to Quoted Post

@Radovan
As for making use of third part libraries on SMath, I don't know if that is possible aside from license issues

I was thinking about the freely available libraries, open source software etc. I do not see then any license problems (am I right?) except the ones I was talking about.

Regards,
Radovan

Edited by user 06 September 2012 14:44:14(UTC)  | Reason: Not specified

When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
Offline Davide Carpi  
#11 Posted : 06 September 2012 16:09:21(UTC)
Davide Carpi


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Plugin updated, many issues fixed and bisection (not generalized) implemented...

an optional argument at the end of the function arguments, different from 0, can show you the number of iterations...

Originally Posted by: omorr Go to Quoted Post
Hello w3b5urf3r,

Here is a bit of testing the Broyden plugin. It is a clasical "driving crasy" situation with multiple solutions. It was mentioned by kilele in the link to the document http://en.smath.info/forum/yaf_postsm7678_solve---and-roots-----again.aspx#post7678

Your plugin failed (if I did not do something wrong) but do not worry about it - it is a common situation. The problem is why "Can not calculate" error returned. I do not know. My function returned another solution than few Newton-Raphson iterations or roots(). I also used another software that uses some kind of broyden algorithm and here is the result - the same as roots() returned
Code:
>function f(x):=[(x[1]+3)*(x[2]^3-7)+18,sin(x[2]*exp(x[1])-1)]
>x0:=[-0.5,1.4]
 [ -0.5  1.4 ]

>f(x0)
 [ 7.36  -0.150285529841 ]
>res:=broyden("f",x0)
 [ 0  1 ]
>f(res)
 [ 0  0 ]

Try, for instance to change the guess values (vector x0) from [-0.5,1.4] to [1,4].
Here is the picture and the sm file

Regards,
Radovan


The function as in the *.sm file doesn't work because the first approximated inverse jacobian ( B ) is an Identity matrix... i've seen that is a common issue (f.e. root finding methods)... in SMath this could be fixed using the Jacob() function on the user input solution (see the attachment).

Unfortunately the function Jacob() (as well as other functions such as error() function) are not in the 4 standard libraries; I will try to see if importing the library on which it was made ​​by Andrey can be implemented.


Originally Posted by: kilele Go to Quoted Post
Originally Posted by: w3b5urf3r_reloaded Go to Quoted Post

Currently, the plugin follows almost exactly that function, but if anyone has a more efficient version of the algorithm could certainly be implemented. Good


I've just read that this implementation is better for large problems (see pag 133 of this pdf)
and this is its matlab code

@Radovan
As for making use of third part libraries on SMath, I don't know if that is possible aside from license issues


Thank you, is very useful; Happy
i've implemented this script in a Smath sheet and it's more readable, unfortunately the issue related to the first Jacob approximation still remains Grabli

regards,

w3b5urf3r

Edited by user 06 September 2012 16:48:47(UTC)  | Reason: Not specified

File Attachment(s):
Broyden_testings-2.sm (48kb) downloaded 59 time(s).
Davide Carpi attached the following image(s):
broyden.png
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Offline omorr  
#12 Posted : 06 September 2012 18:53:05(UTC)
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Hello w3b5urf3r,

Just keep up this way Good

I would suggest you to strictly avoid using Jacob() inside Broyden - it will get new troubles. I really do not know what to use instead as a first Jacobian approximation. If someone have any idea, please help.

When you have the time, put some examples using Bisection() as well in the next update.

If I understood you well, you are trying to use Broyden with units (Dimensional analysis check failed...). Am I right? My suggestion is to skip this as well at the moment.

Best wishes,
Radovan

Edited by user 06 September 2012 18:54:57(UTC)  | Reason: Not specified

When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
Offline kilele  
#13 Posted : 06 September 2012 21:59:50(UTC)
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wow, have you used bisection as an initialization of broyden ?
also i've searched information about the broyden method over time and it seems the most advanced/recent version is a modification of Johnson's by Baran :
http://arxiv.org/abs/0805.4446

As for the Jacob() function, I thought that Broyden didn't use Jacob directly but an approximation..

EDIT:

Never mind, the above paper refers to a gpl program based on other libraries.
There you have the historial of Broyden method,
http://www.unedf.org/con.../Day2/PackForest_JJM.pdf
it seems to me that the "compact" version of 1994 corresponds to the matlab code that I linked above.

Edited by user 06 September 2012 22:17:09(UTC)  | Reason: Not specified

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Offline Davide Carpi  
#14 Posted : 07 September 2012 02:51:21(UTC)
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Plugin updated. Broyden now use this solution strategy

1) check for results with the "bad" Broyden method
- if solutions works -> output the results
- if solutions are not closest to zero -> 2

2) check for results with the "good" Broyden method
- if solutions works -> output the results
- if solutions are not closest to zero -> output a can not solve error

the algorithm is similar to that shown in the attachment (replace the B update definition to see the different behavior).

Added a SMath file with examples for the Bisection() function.

Originally Posted by: kilele Go to Quoted Post
wow, have you used bisection as an initialization of broyden ?
also i've searched information about the broyden method over time and it seems the most advanced/recent version is a modification of Johnson's by Baran :
http://arxiv.org/abs/0805.4446

As for the Jacob() function, I thought that Broyden didn't use Jacob directly but an approximation..

EDIT:

Never mind, the above paper refers to a gpl program based on other libraries.
There you have the historial of Broyden method,
http://www.unedf.org/con.../Day2/PackForest_JJM.pdf
it seems to me that the "compact" version of 1994 corresponds to the matlab code that I linked above.


Very interesting articles, thank you Good


regards,

w3b5urfer

Edited by user 07 September 2012 02:59:48(UTC)  | Reason: Not specified

File Attachment(s):
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Offline kilele  
#15 Posted : 07 September 2012 03:01:45(UTC)
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Another matlab code to reduce memory requirements on this quite recent pdf pag.211 Appendix B
Thanks for your work!!

Edit:
Again since Broyden is not globally convergent, bisection might be interesting as an initial approaching to the roots

Edited by user 07 September 2012 03:16:27(UTC)  | Reason: Not specified

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Offline kilele  
#16 Posted : 07 September 2012 13:22:44(UTC)
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Originally Posted by: omorr Go to Quoted Post

I was thinking about the freely available libraries, open source software etc. I do not see then any license problems (am I right?) except the ones I was talking about.


since SMath is not a commercial software i guess we can be not too strict about licenses Biggrin
anyway some licenses like gpl don't allow to publish opensource plugins for closed programs (unlike lgpl)
as for distributing compiled matlab scripts, I think you are right! see this link
don't know how this black boxes could be used in SMAth though, however the approach of making plugins out of matlab, fortran, pseudo-code from scientific papers.. could be more convenient due to compatibility and dll size
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Offline omorr  
#17 Posted : 07 September 2012 16:21:25(UTC)
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Hello,

This is one of my working problems I was already mentioned Primer 65a and struggled in SMath with. I reported there the problems of solve() and roots() solutions.

I used it again for testing and attached it to this post. It seems the problem is a bit numerically "tricky" for solving in SMath. The original file is not working in the recent SMath version, neither solve() nor roots() (error message about T as undefined variable - not quite sure why). Here I attached the revised file - just removed eval() I was talking about in the previous post. Fortunately, solve() will now get the more accepted result Good , roots() will fail, Broyden() will give the answer if the guess value is quite close to the solution, Bisection() will fail as well. I hope I did not make any serious mistake here.

Regarding this problem, I am satisfied that at least one of the possible tool in SMath can give the result.

Regards,
Radovan

Edited by user 07 September 2012 16:29:03(UTC)  | Reason: Not specified

File Attachment(s):
Primer65aa.sm (42kb) downloaded 47 time(s).
omorr attached the following image(s):
Primer65aa.png
When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
Offline kilele  
#18 Posted : 07 September 2012 19:18:39(UTC)
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According to what I've read, there seems not to be robust methods in multidimensions:
Without insight into the nature of the problem, root finding is more or less blind search, this is why I suggested a multidimensional bisection method for a initial guess with just a few iterations.
With some insight into the nature of the root, Newton’s method using derivative information, and Broyden’s method using only function information seem good candidates.
Some information about methods followed by Octave:
fsolve in Octave implements the multidimensional Newton method
fzero in Octave uses Brent’s method for one dimension (bisection+secant)
All these info is nearly copy-pasted of this doc Biggrin

I think also that there are some interesting advice on this pdf on page 20:Final Considerations, about how to rephrase your function so that all these algorithms work better.

As a final suggestion (sorry for so many links to documents) it could be a good idea to make plugins for several methods: Brent (one dimension); Newton, Broyden and Homotopy methods (multidimension); and It could be even better combine some of these methods in a row or according to the type of function entered by the user (I assume that's how Smath Solve() works)

Edited by user 07 September 2012 20:30:37(UTC)  | Reason: Not specified

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Offline Davide Carpi  
#19 Posted : 07 September 2012 20:08:33(UTC)
Davide Carpi


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Originally Posted by: omorr Go to Quoted Post
Hello,

This is one of my working problems I was already mentioned Primer 65a and struggled in SMath with. I reported there the problems of solve() and roots() solutions.

I used it again for testing and attached it to this post. It seems the problem is a bit numerically "tricky" for solving in SMath. The original file is not working in the recent SMath version, neither solve() nor roots() (error message about T as undefined variable - not quite sure why). Here I attached the revised file - just removed eval() I was talking about in the previous post. Fortunately, solve() will now get the more accepted result Good , roots() will fail, Broyden() will give the answer if the guess value is quite close to the solution, Bisection() will fail as well. I hope I did not make any serious mistake here.

Regarding this problem, I am satisfied that at least one of the possible tool in SMath can give the result.

Regards,
Radovan



I omorr, i'll check this issue ASAP, especially because in this case the script seems to work while the plugin don't work...

about eval(), you cannot use it where there are undefined variables because the function "force" the numeric evaluation.


best regards,

w3b5urf3r

Edited by user 18 September 2012 01:29:23(UTC)  | Reason: Not specified

Davide Carpi attached the following image(s):
Primer65aa.PNG
If you like my plugins please consider a donation to SMath Studio; for personal contributions to me: paypal.me/dcprojects
thanks 1 user thanked Davide Carpi for this useful post.
on 07/09/2012(UTC)
Offline omorr  
#20 Posted : 07 September 2012 22:05:00(UTC)
omorr


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Originally Posted by: kilele Go to Quoted Post
As a final suggestion (sorry for so many links to documents) it could be a good idea to make plugins for several methods: Brent (one dimension); Newton, Broyden and Homotopy methods (multidimension); and It could be even better combine some of these methods in a row or according to the type of function entered by the user (I assume that's how Smath Solve() works)

I agree with you completely Good I also agree about the mentioned observation on the 20-page of your reference. This is up to the user. A simple advice is to rearrange the function f(x)= 0 by avoiding division of terms when denominator can be close to zero.

@w3b5urf3r Based on the picture in your previous post (many methods in a plugin), did I say that you are my hero? Friends

Regards,
Radovan

Edited by user 07 September 2012 22:15:32(UTC)  | Reason: Not specified

When Sisyphus climbed to the top of a hill, they said: "Wrong boulder!"
thanks 1 user thanked omorr for this useful post.
on 08/09/2012(UTC)
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