8.32 collecting radicals (20.4.01)

8.32.1 Ken Monks

Is there a built in Maple command that collects the coefficients of like numeric radicals in an expression. For example, a command that would convert this:

2*a*sqrt(2)+2*b*sqrt(3)+c*sqrt(2)+7*d*sqrt(3)+5 
 
#to this: 
 
(2*a+c)*sqrt(2)+(2*b+7*d)*sqrt(3)+5
                                                                                  
                                                                                  
 

8.32.2 Allan Wittkopf (20.4.01)

You could use frontend and collect:

> xpr:=2*a*sqrt(2)+2*b*sqrt(3)+c*sqrt(2)+7*d*sqrt(3)+5; 
 
                          1/2        1/2      1/2        1/2 
              xpr := 2 a 2    + 2 b 3    + c 2    + 7 d 3    + 5 
 
> frontend(collect,[xpr,indets(xpr,`^`)],[{`+`,`*`,`set`},{}]); 
 
                                  1/2              1/2 
                     (2 b + 7 d) 3    + (2 a + c) 2    + 5
 

8.32.3 David Harrington (20.4.01)

Not very elegant but

subs(rt2=sqrt(2),rt3=sqrt(3),collect(subs(sqrt(2)=rt2,sqrt(3)=rt3,x),{rt2,rt3}));
 

will do it.

8.32.4 Robert Israel (20.4.01)

I don’t think so. But it’s not too hard to write one. More generally, here’s a command that collects with respect to all subexpressions of a given type.

> collectype:= proc(f::algebraic,T::type) local fp,x,X,S,q; 
   X:= indets(f,T); 
   S:= [seq(x = `tools/genglobal`(q),x=X)]; 
   fp:= subs(S,f); 
   subs(map(t -> rhs(t)=lhs(t),S),collect(fp,map(rhs,S),distributed)); 
   end;
 

In your case:

> collectype(2*a*sqrt(2)+2*b*sqrt(3)+c*sqrt(2)+7*d*sqrt(3)+5, radical); 
 
       (c + 2 a) sqrt(2) + (2 b + 7 d) sqrt(3) + 5
 

Or, for example:

> S:= a[1]*x*y+a[1]*x^3*y^3+a[1]*x*y^2+a[1]*x^3*y^2+a[2]*x^2*y 
    +a[2]*x^2*y^3+2*a[2]*x^2*y^2+a[3]*x^3*y 
    +a[3]*x*y^3+a[3]*x^3*y^2+a[3]*x*y^2; 
 
> collectype(S, indexed); 
 
      2    3  2          3  3           2      2  3      2  2 
  (x y  + x  y  + x y + x  y ) a[1] + (x  y + x  y  + 2 x  y ) a[2] 
               3    3  2      2    3 
         + (x y  + x  y  + x y  + x  y) a[3]
 

8.32.5 Helmut Kahovec (20.4.01)

Either of the two solutions shown below is a bit clumsy. You may either introduce dummy variables like sqrt2 and sqrt3

> e1:=2*a*sqrt(2)+2*b*sqrt(3)+c*sqrt(2)+7*d*sqrt(3)+5: 
 
> r:=sqrt(2)=sqrt2,sqrt(3)=sqrt3: 
> rinv:=sqrt2=sqrt(2),sqrt3=sqrt(3): 
 
> subs({rinv},collect(subs({r},e1),[sqrt2,sqrt3])); 
 
             (2 a + c) sqrt(2) + (2 b + 7 d) sqrt(3) + 5
 

or define new types like t1 and t2

> lprint(e1); 
2*a*2^(1/2)+2*b*3^(1/2)+c*2^(1/2)+7*d*3^(1/2)+5 
 
> t1:=identical(2)^identical(1/2); 
 
                                     identical(1/2) 
                   t1 := identical(2) 
 
> t2:=identical(3)^identical(1/2); 
 
                                     identical(1/2) 
                   t2 := identical(3) 
 
> factor(select(hastype,e1,t1)) 
>   +factor(select(hastype,e1,t2)) 
>   +remove(hastype,e1,{t1,t2}); 
             (2 a + c) sqrt(2) + (2 b + 7 d) sqrt(3) + 5
 

8.32.6 Preben Alsholm (20.4.01)

I think not. The best I could come up with was to use frontend with collect twice. One single use of frontend didn’t work as expected.

> u:=2*a*sqrt(2)+2*b*sqrt(3)+c*sqrt(2)+7*d*sqrt(3)+5: 
> u1:=frontend(collect,[u,sqrt(2)]); 
 
       u1 := (c + 2 a) sqrt(2) + 5 + 2 b sqrt(3) + 7 d sqrt(3) 
 
> frontend(collect,[u1,sqrt(3)]); 
 
             (2 b + 7 d) sqrt(3) + (c + 2 a) sqrt(2) + 5 
 
> frontend(collect,[u,[sqrt(2),sqrt(3)]]); #Doesn't work, why not? 
 
       2 a sqrt(2) + 2 b sqrt(3) + c sqrt(2) + 7 d sqrt(3) + 5