cas-function: locuscurve

[Arno K]

This small program provides on input of a set of function f(x,k) and an additional parameter p a list containing the equations (as strings to be easily usable in plots) of extreme points (p=1) or turning points (p=2). As I did not want to bother about used variables x and k must be used. f may also be an expression.

Code:

locuscurve(x*e^(-k*x^2),1)

provides the output:

Code:

{"y:=x^2/(abs(x)*e^(1/2))","y:=-x^2/(abs(x)*e^(1/2))"}

Here you must clearly see for yourself that the first part is for max-points and the second one is for mins, putting this together to only "y:=x*e^(-1/2)" would have caused a lot of different cases to be taken care of, so I omitted that.

Code:

// calculates the curve through locuspoints of a parametric function which must be dependent from x and k
// p=1: extreme points, p=2: turning points
// f can be function or symbolic 
#cas
locuscurve(f,p):=
BEGIN
LOCAL g,x,k,z;
LOCAL l1,l2,j;
purge(x);
purge(k);
IF type(f)≠DOM_FUNC THEN
f(x,k):=f;//transform to function
END;

g:=diff(f(x,k),x,p);
l1:=zeros(g);//x-vals
l2:=apply(z→f(z,k),l1);//y-vals
l1:=x.-l1;
l1:=apply(z→solve(z=0,k),l1);
FOR j FROM 1 TO size(l1) DO
l1[j]:=l1[j][1];
l2[j]:="f"+j+"(x):="+simplify(subst(l2[j],k=l1[j]));//string for convenience
END;


  return convert(l2,23);// convert matrix to list
END;
#end

Arno

[Arno K]

Slight changes because it does not work on turning points when they are unique, i.e. independent from k. In the moment solve returns a vector containing 0 as first entry as I tried on some examples, so I made a quick and dirty fix for this. When someone encounters problems, pm me.
Another thing to be mentioned, CAS settings, Complex on and Use i on increase speed noticably.
Arno

Code:


#cas
locuscurve(f,p):=
BEGIN
LOCAL g,x,k,z;
LOCAL l1,l2,j;
purge(x);
purge(k);
IF type(f)≠DOM_FUNC THEN
f(x,k):=f;
END;

g:=diff(f(x,k),x,p);
l1:=zeros(g);
IF l1[1]==0 THEN
l1:=tail(l1);
END;


l2:=apply(z→f(z,k),l1);
l1:=x.-l1;
l1:=apply(z→solve(z=0,k),l1);
FOR j FROM 1 TO size(l1) DO
l1[j]:=l1[j][1];
l2[j]:="f"+j+"(x):="+simplify(subst(l2[j],k=l1[j]));
END;


  return convert(l2,23);
END;
#end