Conference rusure::math

    I don't think it's difficult at all - it's a very well
    behaved integrand with no singularities on the interval, so
    that even simple-minded Romberg integration (extrapolation to the
    limit via the asymptotic error term from trapezoidal integration)
    will converge, as shown by the following sequence of approximations
    in VAX H precision:

 subdiv fncalls    approx                           error estimate
 ------ ---- -----------------------------------    --------------
    2     3  0.626126137655402601064255906824330    0.247226      
    3     5  0.873352304294445890170144033002492    1.832444E-0003
    4     9  0.875184748795601088023570336288850    2.266590E-0004
    5    17  0.874958089701637284335865903154165    8.968199E-0007
    6    33  0.874957192881641013304750745339498    5.892269E-0009
    7    65  0.874957198773910750800636856996653    6.474081E-0012
    8   129  0.874957198780384832026717863989211    8.015067E-0016
    9   257  0.874957198780384030520000990857232    2.814897E-0019
   10   513  0.874957198780384030238511216019147    3.944041E-0024
   11  1025  0.874957198780384030238507271977611    4.882136E-0029
   12  2049  0.874957198780384030238507272026433    9.629649E-0035

Math'a text: 0.8749571987803839994388329808353131568824

MapleV:	     0.8749571987803840302385072720264325057257

   Mathematica is not to be trusted for any serious work in my opinion -
   I've heard too many stories about bad answers coming out of it.

   Which is scary, actually - to hear educators talk, students don't even
   know how to do simple arithmetic.  How are they going to sanity check
   the stuff coming out of some algebra system???

    - Jim

    By the way, is the base note in keeping with the first of April?

    (like the posting about the world's smallest prime number recently)

i think that is a good question, but it applies to software in general.

May be we can use similar idea as in hardware fault tolerant architecture, 
send the input through 3 modules, compare the output from the 3,
if one is different from the other two output, you know something is most
    likely wrong with that one that produced a different output, replace that 
    module.

this is what Jim did in .1, checked the output from maple and from 
a third independent program, found the maple and the VAX output to 
be closest to each other, then there is a good chance that mma output is
the wrong one. 

/nasser

    

   Re:         <<< Note 1590.1 by GAUSS::ROTH "Geometry is the real life!" >>>

>   Which is scary, actually - to hear educators talk, students don't even
>   know how to do simple arithmetic.  How are they going to sanity check
>   the stuff coming out of some algebra system???

answer:

   You must examine your assumption that somehow these things
   will be tools to help non-initiates do mathematics.  They are
   not and will not be magic boxes that eliminate the need for
   education.  Worse (for the egalitarians), they put more
   (intellectual) power in the hands of those who already have
   it.  The prognosis is for widening the breach between the
   have's and have-not's in the intellectual sphere.  Basically,
   the need for students to know their math is greater, not
   less, in the computer age.
   
Dick

   ps.
   
   Note that it is no longer the 1st of April.  I'm very serious
   about this.

    It just occurred to me that it was brain dead to try and numerically
    integrate this problem when all you really have to do is freshman
    calculus...

    Change the variable x in the integrand to u = log(x)

	sin(exp(x)) dx = sin(u)/u du

    and it's the integral from 1 to e.

    Expanding this in a series and integrating term by term we find

	f(u) = u - u^3/(3*3!) + u^5/(5*5!) - ...

    so the integral is f(e)-f(1) as the following program shows.

    - Jim

	PROGRAM APRIL_FOOLS

	IMPLICIT REAL*16 (A-H,O-Z)

	X = 1
	CALL MTH$HEXP(E, X)
	CALL SINX(X, S1)
	CALL SINX(E, SE)

	TYPE *, S1, SE, SE-S1

	END

	SUBROUTINE SINX(X, S)

	IMPLICIT REAL*16 (A-H,O-Z)

	S = 0
	T = 1
	X2 = -X*X
	DO I = 1,100000,2
	  STMP = S
	  S = S+T/I
	  IF (STMP.EQ.S) GOTO 100
	  T = T*X2/((I+1)*(I+2))
	ENDDO
100	CONTINUE

	S = S*X

	RETURN

	END