Re: Mathematica Weirdness
- To: mathgroup at smc.vnet.net
- Subject: [mg116730] Re: Mathematica Weirdness
- From: Gary Wardall <gwardall at gmail.com>
- Date: Fri, 25 Feb 2011 06:36:10 -0500 (EST)
- References: <ik5feh$sm2$1@smc.vnet.net>
On Feb 24, 5:29=C2 am, Bob Hanlon <hanl... at cox.net> wrote:
> You need to use more precision
>
> N[Integrate[1000000*Exp[x^2 - 12*x]*x^14, {x, 0, 1}], 20]
>
> 2.7482124237345664276
>
> NIntegrate[1000000*Exp[x^2 - 12*x]*x^14, {x, 0, 1}]
>
> 2.74821
>
> Bob Hanlon
>
> ---- Steve Heston <shes... at rhsmith.umd.edu> wrote:
>
> =============
> My question is why I get a negative integral of a positive
> function?
>
> Integrate[1000000*Exp[x^2-12*x]*x^14,{x,0,1}]//N
> Integrate[1000000*Exp[x^2-12*x]*x^14,{x,0.,1}]//N
> NIntegrate[1000000*Exp[x^2-12*x]*x^14,{x,0,1}]
>
> The first line gives a negative answer, while the second two lines give
> identical positive answers. =C2 Something is strange here.
>
> Steve
>
> Steven L. Heston
> Associate Professor
> Finance Department
> Robert H. Smith School of Business
> 4447 Van Munching Hall Van Munching Hall
> University of Maryland
> College Park, MD 20742-1815
> 301-405-9686 TEL
> 301-405-0359 FAX
> shes... at rhsmith.umd.eduhttp://www.rhsmith.umd.edu
I made a slight error yesterday. I checked the derivative of the
antiderivative of
1000000*Exp[x^2 - 12*x]*x^14
Mathematica 8 yields:
1000000 E^((-12 + x) x) x^14
Which the same as:
1000000*Exp[x^2 - 12*x]*x^14
In other words Mathematica 8 did the symbolic calculations correctly.
The problem must be in the numerical evaluations. Both the
antiderivative and the derivative of the antiderivative do not render
accurate results near x=0.
Gary Wardall
***************************
See below the steps below
Define:
F[x_] := Integrate[1000000*Exp[t^2 - 12*t]*t^14, {t, 0, x}]
F[x]=15625/2 (-174657787110 + 2065912371945 DawsonF[6] + =E2=80=A8 E^((-12 +
x) x) (174657787110 + =E2=80=A8 x (29981073375 + =E2=80=A8 2 x
(2614326570 + =E2=80=A8 x (463615155 + =E2=80=A8 2x
(41841090 + =E2=80=A8 x (7695585 + =E2=80=A8 2 x
(722070 + =E2=80=A8 x (138465 + =E2=80=A8 =
2 x
(13590 + =E2=80=A8 x (2735 + 2 x (282 + x (59 + 2 x (6 +
x)))))))))))) - =E2=80=A8 2065912371945 DawsonF[6 - x]))
Compute:
D[15625/2 (-174657787110 + 2065912371945 DawsonF[6] + =E2=80=A8 E^((-12 +=
x)
x) (174657787110 + =E2=80=A8 x (29981073375 + =E2=80=A8 2 x (2=
614326570
+ =E2=80=A8 x (463615155 + =E2=80=A8 2 x (41841090=
+
x (7695585 + =E2=80=A8 2 x (722070 +
x (138465 + =E2=80=A8 2 x (13590 +
x (2735 + 2 x (282 + x (59 + 2 x (6 +
x)))))))))))) - =E2=80=A8 2065912371945 DawsonF[6 - x])),x]
We get:
15625/2 (E^((-12 + x) x) (-12 + 2 x) (174657787110 + =E2=80=A8 x
(29981073375 + =E2=80=A8 2 x (2614326570 + =E2=80=A8 x (=
463615155
+ =E2=80=A8 2 x (41841090 + =E2=80=A8 x (769=
5585 +
2 x (722070 + =E2=80=A8 x (138465 +
2 x (13590 + =E2=80=A8 x (2735 + 2 x
(282 + x (59 + 2 x (6 + x)))))))))))) - =E2=80=A8 2065912371945 Dawson=
F[6
- x]) + =E2=80=A8 E^((-12 + x) x) (29981073375 + =E2=80=A8 2 x (2614=
326570 +
x (463615155 + =E2=80=A8 2 x (41841090 + =E2=80=A8 =
x
(7695585 + =E2=80=A8 2 x (722070 + =E2=80=A8 =
x
(138465 + =E2=80=A8 2 x (13590 + =E2=80=A8 =
x
(2735 + 2 x (282 + x (59 + 2 x (6 + x))))))))))) + =E2=80=A8 x (2 x
(463615155 + =E2=80=A8 2 x (41841090 + =E2=80=A8 x=
(7695585
+ =E2=80=A8 2 x (722070 + =E2=80=A8 x (=
138465 +
2 x (13590 + =E2=80=A8 x (2735 + 2 x
(282 + x (59 + 2 x (6 + x))))))))) + =E2=80=A8 x (2 x (7695585 +
2 x (722070 + =E2=80=A8 x (138465 +
2 x (13590 + =E2=80=A8 x (2735 + 2 x
(282 + x (59 + 2 x (6 + x))))))) + =E2=80=A8 x (2 x (13846=
5
+ =E2=80=A8 2 x (13590 + =E2=80=A8 x =
(2735 + 2 x
(282 + x (59 + 2 x (6 + x))))) + =E2=80=A8 x (2 x (2735 =
+ 2
x (282 + x (59 + 2 x (6 + x))) + =E2=80=A8 x (2 x (59 + =
2 x
(6 + x) + x (2 x + 2 (6 + x))) + =E2=80=A8 2 (282 + x (5=
9 +
2 x (6 + x))))) + =E2=80=A8 2 (13590 +
x (2735 + 2 x (282 + x (59 + 2 x (6 + x))))))) +
2 (722070 + =E2=80=A8 x (138465 + =E2=80=A8 =
2 x
(13590 + =E2=80=A8 x (2735 + 2 x (282 + x (59 + 2 x (6 +
x))))))))) + =E2=80=A8 2 (41841090 + =E2=80=A8 =
x
(7695585 + =E2=80=A8 2 x (722070 + =E2=80=A8 =
x
(138465 + =E2=80=A8 2 x (13590 + =E2=80=A8 =
x
(2735 + 2 x (282 + x (59 + 2 x (6 + x))))))))))) + =E2=80=A8 2
(2614326570 + =E2=80=A8 x (463615155 + =E2=80=A8 2=
x
(41841090 + =E2=80=A8 x (7695585 + =E2=80=A8 =
2 x
(722070 + =E2=80=A8 x (138465 + =E2=80=A8 =
2 x
(13590 + =E2=80=A8 x (2735 + 2 x (282 + x (59 + 2 x (6 +
x)))))))))))) + =E2=80=A8 2065912371945 (1 - 2 (6 - x) DawsonF[6 - x])=
))
Simplify[%]
The result is:
1000000 E^((-12 + x) x) x^14
Which is the same as:
1000000*Exp[x^2 - 12*x]*x^14