[MUG] differental operators

[MUG] differental operators
Author: Maple User Group    Posted: Mon, 17 Feb 2003 22:05:23 -0500
>> From: Maple User Group /> -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Thu, 13 Feb 2003 09:41:53 -0500 (EST) From: Fred Chapman /> To: /> Subject: differental operators On Tue, 11 Feb 2003, Richard Patterson wrote: > >> From: Richard Patterson /> > > I want to define differential operators and test them on > a polynomial f(x,y): > > f:=(x,y)->a*x^2+b*x*y+d*y^2; > 2 2 > f := (x, y) -> a x + b x y + d y > > If the differential operator has constant coefficients > > > op1:=2*D[1]+3*D[2]; > > op1 := 2 D[1] + 3 D[2] > > then everything works fine: > > > op1(f)(x,y); > > 4 a x + 2 b y + 3 b x + 6 d y > > How do I accomplish the same thing if the coefficients are > unspecified constants? The simplest reliable solution that I found for your problem is to define `c` and `d` to be constant functions: > f:=(x,y)->a*x^2+b*x*y+d*y^2; 2 2 f := (x, y) -> a x + b x y + d y > op2:=c*D[1]+d*D[2]; op2 := c D[1] + d D[2] > op2(f)(x,y); c(f)(x, y) (2 a x + b y) + d(f)(x, y) (b x + 2 d y) > c := (f::anything) -> c; c := f::anything -> c > d := (f::anything) -> d; d := f::anything -> d > op2(f)(x,y); c (2 a x + b y) + d (b x + 2 d y) If you like, you can assign numerical values to `c` and `d` afterwards: > c := 2: d := 3: op2(f)(x,y); 4 a x + 2 b y + 3 b x + 18 y With best wishes, Fred Chapman NOTE: In anticipation of obtaining my Ph.D. this June, I'm now using my new, permanent, lifetime e-mail address: /> ---------------- http://www.scg.uwaterloo.ca/~fwchapma/ ---------------- Frederick W. Chapman, Ph.D. Student UW Office: Math & Computer 5162 Department of Applied Mathematics UW Phone: (519) 888-4567 x6672 University of Waterloo Waterloo, Ontario, N2L 3G1, Canada Curriculum Vitae: see homepage -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Thu, 13 Feb 2003 22:02:23 +0100 From: Helmut Kahovec /> To: /> Subject: differental operators Richard Patterson wrote: >| I want to define differential operators and test them on >| a polynomial f(x,y): >| >| f:=(x,y)->a*x^2+b*x*y+d*y^2; >| 2 2 >| f := (x, y) -> a x + b x y + d y >| >| If the differential operator has constant coefficients >| >| > op1:=2*D[1]+3*D[2]; >| >| op1 := 2 D[1] + 3 D[2] >| >| then everything works fine: >| >| > op1(f)(x,y); >| >| 4 a x + 2 b y + 3 b x + 6 d y >| >| How do I accomplish the same thing if the coefficients are >| unspecified constants? >| >| > op2:=c*D[1]+d*D[2]; >| >| op2 := c D[1] + d D[2] >| >| > op2(f)(x,y); >| >| c(f)(x, y) (2 a x + b y) + d(f)(x, y) (b x + 2 d y) Well, you may use an extension to evalapply() as shown below. > restart; > type(1,procedure); false > 1(f); 1 > 1(f)(x,y); 1 > type(u,procedure),type(v,procedure); false, false > u(f),v(f); u(f), v(f) > u(f)(x,y),v(f)(x,y); u(f)(x, y), v(f)(x, y) > ConstOperator:=proc() op(0,args[1]) end proc: > `evalapply/u`:=eval(ConstOperator): > `evalapply/v`:=eval(ConstOperator): > type(u,procedure),type(v,procedure); false, false > u(f),v(f); u(f), v(f) > u(f)(x,y),v(f)(x,y); u, v <===!!!=== > f:=(x,y)->a*x^2+b*x*y+c*y^2; 2 2 f := (x, y) -> a x + b x y + c y > op1:=2*D[1]+3*D[2]; op1 := 2 D[1] + 3 D[2] > op1(f)(x,y); 4 a x + 2 b y + 3 b x + 6 c y > op2:=u*D[1]+v*D[2]; op2 := u D[1] + v D[2] > op2(f)(x,y); u (2 a x + b y) + v (b x + 2 c y) > eval(%,{u=2,v=3}); 4 a x + 2 b y + 3 b x + 6 c y Kind regards, Helmut -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Sat, 15 Feb 2003 04:25:19 -0500 (EST) From: Carl Devore /> To: /> Subject: differental operators On Tue, 11 Feb 2003, Richard Patterson wrote: > I want to define differential operators and test them on > a polynomial f(x,y): > f:=(x,y)->a*x^2+b*x*y+d*y^2; > If the differential operator has constant coefficients > > op1:=2*D[1]+3*D[2]; > then everything works fine: > > op1(f)(x,y); > 4 a x + 2 b y + 3 b x + 6 d y > > How do I accomplish the same thing if the coefficients are > unspecified constants? > > op2:=c*D[1]+d*D[2]; > > op2(f)(x,y); > c(f)(x, y) (2 a x + b y) + d(f)(x, y) (b x + 2 d y) You need to make the coefficients operators at the same level as the D's. Numbers are automatically operators, but names are not. > op2:= (()->()->c)*D[1] + (()->()->d)*D[2]; -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- From: "Dr Francis J. Wright" /> To: /> Subject: differental operators Date: Sat, 15 Feb 2003 18:43:50 -0000 From: "Richard Patterson" /> To: /> Sent: Tuesday, February 11, 2003 3:26 PM Subject: [MUG] differental operators > I want to define differential operators and test them on > a polynomial f(x,y): > > f:=(x,y)->a*x^2+b*x*y+d*y^2; > 2 2 > f := (x, y) -> a x + b x y + d y > > If the differential operator has constant coefficients > > > op1:=2*D[1]+3*D[2]; > > op1 := 2 D[1] + 3 D[2] > > then everything works fine: > > > op1(f)(x,y); > > 4 a x + 2 b y + 3 b x + 6 d y > > How do I accomplish the same thing if the coefficients are > unspecified constants? > > > op2:=c*D[1]+d*D[2]; > > op2 := c D[1] + d D[2] > > > op2(f)(x,y); > > c(f)(x, y) (2 a x + b y) + d(f)(x, y) (b x + 2 d y) Maple is treating c and d as mappings, so one solution is explicitly to define them to be constant mappings, i.e. mappings that evaluate to their own names when they are applied, regardless of their arguments. This is essentially what Maple does automatically for numbers, which is why your explicit example worked. > c := proc() 'procname' end proc; c := proc() 'procname' end proc > d := eval(c); d := proc() 'procname' end proc > op2(f)(x,y); c (2 a x + b y) + d (b x + 2 d y) Francis -=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=-=*=- Date: Mon, 17 Feb 2003 17:01:43 -0500 (EST) From: Carl Devore /> To: /> Subject: differential operators Please CC all articles to the mailing-list. If that is unacceptable to you, then this is my last article on this subject. Let others benefit from these writings. On Mon, 17 Feb 2003, Richard Patterson wrote: > > f:=(x,y)->a*x^2+b*x*y+c*y^2; > > g:=(x,y)->p*x^3+q*x^2*y+r*x*y^2+s*y^3; > > Procedure to change the coefficients of polynomials to constant functions. > > > Change:=proc(f) local C: > > for C in coeffs(f(x,y),[x,y]) do assign(C,unapply(C,f)) end do; > > end proc; > > This leaves me uneasy as it affects f even though the procedure returns > nothing. Worse than that, it affects the coefficients. Here is how to get it to returned the modified polynomial without any side effects: Change:= (f, V)-> unapply(subsindets(f(V[]), name, N-> `if`(N in V, N, unapply(N))), V[]) ; And the call becomes F:= change(f, [x,y]); I chose to use a different name, F, but you re-use f if you want. It is best not to rely on there being 2 variables, so I pass in the variable names. We can get around that if you insist on not passing in the variable names. The above uses the Maple 8 feature `in`. You can retrofit it to Maple 7 by using member. susbindets is a feature of Maple 7. We can go back further than that if you need. > However the Changed polynomials can still be multiplied, > evaluated, differentiated just like ordinary polynomials. diff works on them, but there is a bug that prevents D from working on them. The coefficient functions are changed to simply t. This affects at least Maple 6 and Maple 8.

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