#4799 taylor contagion

[Barton Willis]

Here are two examples where Taylor contagion fails:

(%i1)   xxx : exp(-(x-1)^2) - exp(-(x+1)^2);
(xxx)   %e^(-(x-1)^2)-%e^(-(x+1)^2)

(%i2)   substitute(x=taylor(x,x,inf,2),%);
Maxima encountered a Lisp error:

 The variable #:G530 is unbound.

Automatically continuing.

To enable the Lisp debugger set *debugger-hook* to nil.

And a similar case with an asksign:

(%i1)   zzz : exp(-(x-a)^2) - exp(-(x+a)^2);
(zzz)   %e^(-(x-a)^2)-%e^(-(x+a)^2)
(%i2)   substitute(x=taylor(x,x,inf,2),zzz);
"Is "a" positive, negative or zero?"z;
debugger invoked on a SIMPLE-CONDITION in thread

#<THREAD tid=8388 "main thread" RUNNING {1100038003}>:

  bad singular datum

Type HELP for debugger help, or (SB-EXT:EXIT) to exit from SBCL.

restarts (invokable by number or by possibly-abbreviated name):

  0: [CONTINUE    ] Return from COMMON-LISP:BREAK.

  1: [MACSYMA-QUIT] Maxima top-level

  2:                Ignore runtime option --eval "(progn (load (maxima::$sconcat (namestring (pathname (maxima::maxima-getenv \"MAXIMA_PREFIX\"))) \"/bin/win_signals.lisp\")) (cl-user::run))".

  3: [ABORT       ] Skip rest of --eval and --load options.

  4:                Skip to toplevel READ/EVAL/PRINT loop.

  5: [EXIT        ] Exit SBCL (calling #'EXIT, killing the process).

(ADJOIN-SING-DATUM (((MEXPT) $%E ((MTIMES RATSIMP) $A $X)) ((2 . 1)) $ZEROA NIL))

   error finding frame source: Bogus form-number: the source file has probably

                               changed too much to cope with.

   source: NIL

0] 

Maxima 5.49 + SBCL 2.6.0.

[Stavros Macrakis]

Bizarrely, it seems to be the negation that is responsible for the problem:

tt: subst(taylor(x,x,inf,2)$

subst(tt,x, %e^-(x+1)^2 ) => +(+%e^-1*(%e^-x)^2)*%e^-x^2
subst(tt,x, -%e^-(x+1)^2 ) => ERROR, only difference is initial "-"

subst(tt,x, %e^-x^2 ) => +%e^-x^2   <<< OK
subst(tt,x, -%e^-x^2 ) => -1+...   <<< ???, only difference is initial "-"

Even simpler case (getting 20 taylor terms to make sure it's not a truncation issue):

subst(taylor(x,x,inf,20),x,exp(-x)) => +%e^-x
subst(taylor(x,x,inf,20),x,-exp(-x)) => +(-1)  <<< !!!

And more:

taylor(exp(x),x,inf,5) => +1/%e^-x
subst(taylor(exp(x),x,inf,5),x,x) => +1/%e^-x   << OK
subst(taylor(exp(x),x,inf,5),x,x^2) => +1  << !!!
taylor(exp(x),x,inf,5)^2 => +1   << !!!

Tested in Maxima 5.49.0 SBCL 2.6.0

[David Scherfgen]

It's any multiplication, not just negation. And we don't need subst:

(%i1) tt : taylor(x, x, inf, 2)$

(%i2) %e^tt;
(%o2) +1/%e^-x

(%i3) 2*%e^tt;
(%o3) +2

[David Scherfgen]

It works correctly when we use a finite expansion point, e.g. taylor(x, x, 0, 2).

[David Scherfgen]

The same bug exists for addition:

(%i1) tt : taylor(x, x, inf, 2)$

(%i2) %e^tt;
(%o2) +1/%e^-x

(%i3) %e^tt + 10;
(%o3) +11

[David Scherfgen]

This is not a bug in the simplifier (simptimes, simplus), but in the Taylor code:

(%i1) xx : taylor(x, x, inf, 2)$
(%i2) eexx : %e^xx$
(%i3) :lisp (setq $ans ($taylor (list '(mtimes) 2 $eexx)))
(%i3) ans;
(%o3) +2
(%i4) :lisp (setq $ans ($taylor (list '(mplus) 10 $eexx)))
(%i4) ans;
(%o4) +11