Jacobian of a Matrix
05-15-2015, 08:20 PM
Post: #1 salvomic Senior Member Posts: 1,396 Joined: Jan 2015
Jacobian of a Matrix
hi all,
here there is a CAS program to calc the Jacobian of a Matrix.

Input: [f(1), f(2), ...], [x,y,...]

Enjoy!

Salvo Micciché

Code:
 #cas jacob(args):= // Jacobian Matrix by Salvo Micciché // input vectorial expression, vector of variables BEGIN local argv, argc, mat, f, var, fn, j, k,  gr, vd; argv:=[args]; argc:=size(argv); IF argc !=2 THEN return "Input:[f1(x),f1(y),f1(z)...], [x,y,z,...]";  ELSE f:=argv(1); var:=argv(2); fn:=size(f); vd:=size(var); mat:=makemat(0,fn,vd); FOR j FROM 1 TO fn DO // gradients gr:=grad(f(j),var); FOR k FROM 1 TO vd DO // items mat[j,k]:=gr(k); END; // for k END; // for j return mat; END; // if-else END; #end

∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C - DM42, DM41X - WP34s Prime Soft. Lib
03-25-2017, 02:51 PM
Post: #2 Rudi Junior Member Posts: 15 Joined: May 2015
RE: Jacobian of a Matrix
hi all,
I've improved the Code of the Jacobian Matrix by salvomic - 15.05.2015 21:20

Enjoy!

Rudi Steeger

For Example:

The same Example with the new Feature in this Code:
jacob2([e^(x*y^2)*sin(z)],[x,y,z])
==>
[[[y^4*e^(x*y^2)*sin(z)],[2*y*(x*y^2+1)*e^(x*y^2)*sin(z)],[y^2*cos(z)*e^(x*y^2)]],[[2*y*(x*y^2+1)*e^(x*y^2)*sin(z)],[2*x*(2*x*y^2+1)*e^(x*y^2)*sin(z)],[2*x*y*cos(z)*e^(x*y^2)]],[[y^2*cos(z)*e^(x*y^2)],[2*x*y*cos(z)*e^(x*y^2)],[-e^(x*y^2)*sin(z)]]];

Code:
#cas
jacob2(args):=
// Jacobian Matrix by Salvo Micciché
// input vectorial expression, vector of variables
BEGIN
local argv, argc, mat, f, var, fn, fg, j, k, gr, vd;
argv:=args;
argc:=size(argv);
IF argc !=2 THEN
return "Input:[f1(x),f1(y),f1(z)...], [x,y,z,...]";
ELSE
f:=argv(1);
var:=argv(2);
fn:=size(f);
vd:=size(var);
IF fn:=1 THEN
f:=fg;
fn:=size(f);
END;
mat:=makemat(0,fn,vd);
FOR j FROM 1 TO fn DO // gradients
FOR k FROM 1 TO vd DO // items
mat[j,k]:=factor(gr(k));
END; // for k
END; // for j
return mat;
END; // if-else
END;
#end
03-25-2017, 03:23 PM
Post: #3 Han Senior Member Posts: 1,882 Joined: Dec 2013
RE: Jacobian of a Matrix
A slightly shorter program:

Code:
 #cas jacob(args):= begin local argv, argc, mat, f, var, fn, j, k,  gr, vd; argv:=[args]; argc:=size(argv); IF argc !=2 THEN return "Input:[f1(x1,...,xn),...,fm(x1,...,xn)], [x1,...,xn]";  ELSE return transpose(diff(argv,argv)); END; end; #end

Basically, the Jacobian is: transpose(diff([f1,f2,...,fm], [x1,x2,...,xn]))

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03-26-2017, 11:36 AM
Post: #4 Rudi Junior Member Posts: 15 Joined: May 2015
RE: Jacobian of a Matrix
hi all,
Wow very nice, the Jacobi matrix contains only the first derivatives. However, the jaboc function calculates the 2nd derivatives. Corresponds essentially to the Hessian matrix. For me it was important to understand the Jacob function in connection with matrices. I am therefore able to write similar functions.
Thank you very much.

P.S.

Enjoy!

Rudi Steeger
04-09-2018, 12:49 PM
Post: #5
 sitomix Junior Member Posts: 1 Joined: Dec 2014
RE: Jacobian of a Matrix
Can you make a new function to evaluate jacobian in one point, for example: jacob( [x*y,y*y],[x,y],[x=2,y=5]) ?
04-09-2018, 03:00 PM
Post: #6 salvomic Senior Member Posts: 1,396 Joined: Jan 2015
RE: Jacobian of a Matrix
(04-09-2018 12:49 PM)sitomix Wrote:  Can you make a new function to evaluate jacobian in one point, for example: jacob( [x*y,y*y],[x,y],[x=2,y=5]) ?

I'll thing about it, thanks. As soon as I'll have some spare time...

Salvo

∫aL√0mic (IT9CLU) :: HP Prime 50g 41CX 71b 42s 39s 35s 12C 15C - DM42, DM41X - WP34s Prime Soft. Lib
04-09-2018, 10:16 PM (This post was last modified: 04-09-2018 10:26 PM by Arno K.)
Post: #7
 Arno K Senior Member Posts: 469 Joined: Mar 2015
RE: Jacobian of a Matrix
I improved Han's program a little bit, now you can enter what you desire, with and without substitution:
Code:
#cas  jacob(args):=  begin  local argv,argc,mat,f;  LOCAL var,fn,j,k,gr,vd;  argv:=[args];  argc:=size(argv);  IF argc == 3 THEN    return subst(transpose(diff(argv,argv)),argv);   END;  IF argc == 2 THEN    return transpose(diff(argv,argv));   END;  return "Input:[f1(x1,...,xn),...,fm(x1,...,xn)], [x1,...,xn][,[x1=a1,...xn=an]]";   end; #end
For its usage see the following picture.
Hope that helps
Arno

Attached File(s) Thumbnail(s) 04-15-2018, 10:38 PM
Post: #8
 Arno K Senior Member Posts: 469 Joined: Mar 2015
RE: Jacobian of a Matrix
(04-09-2018 12:49 PM)sitomix Wrote:  Can you make a new function to evaluate jacobian in one point, for example: jacob( [x*y,y*y],[x,y],[x=2,y=5]) ?

I am deeply impressed how people wanting our ( that is everybody really involved in things like improving programs, for example) help, finally appreciate this, a tiny "thank you" by sitomix would have been nice, this is one thing my parents taught me, when I was young.
Arno
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