Substitution, change of variables: EXEC.

The EXEC programs checks the object type at stack level 1 and performs the corresponding action:

• one algebraic substitution:
  If stack 1 is an equation ('objA=objB'), replace objA by objB in stack2. The syntax is 'old_name=expression' EXEC.
  oldname may be a global name, an expression (in this case, the first global name in this expression will be isolated) or a user-defined function.
  Examples:
  • 'X^2+2*X+5' 'X=1' EXEC: evaluate an expression at x=1.
  • 'X=Y^2' EXEC: change of variables, works in integrals too
  • '2*Z(X)-X*d1Z(X)' 'Z(X)=X^2 EXEC: in a differential equation, replace the function z(x) by x².
  • 'Z(X)+d1Z(X)' 'Z(X)=EXP(-X)*Y(X)' EXEC:
    change of function in a differential equation.
  • 'X^2+X*COS(X)' 'X^2=1-Y' EXEC: replace x² by 1-y and replace x by $\sqrt{1-y}$.

• multiple substitutions:
  If stack 1 and 2 are lists, replace each object of list2 in stack level 3 by the corresponding object of list1. The syntax is
  { old_name_1 ... old_name_n } { expr_1 ... expr_n } EXEC
  Note that here EXEC does only substitutions.
  Examples:
  • 'SIN(X)^2+SIN(X)*COS(X)' { 'SIN(X)^2' } { '1-COS(X)^2' } EXEC: replace $\sin(x)^2$ by $1-\cos(x)^2$ but does not replace $\sin(x)$ by $\sqrt{1-\sin(x)^2}$.

  • 'COS(X)+i*SIN(X)' 
    { SIN COS }
    { << i * EXP DUP INV - i 2 * / >>
      << i * EXP DUP INV + 2 / >>
    }
    EXEC
    

    replace $\sin$ and $\cos$ by complex exponentials. If you call EXPA after you get e^ix.

• variable isolation:
  If stack 1 is a symbolic but not an equality, EXEC tries to isolate stack level 1 in stack level 2. Example:
  

  'X^2-5'
  X
  EXEC
  

  returns 'X='$\sqrt{5}$.

• doall function:
  If stack 1 is a program, EXEC executes program at stack level 1 recursively on the components of a list object at stack level 2. Example: { 1 2 3 } << NEG >> EXEC
  is the same as { 1 2 3 } CHS