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Re: How to evaluate the following expression in Mathematica?

• To: mathgroup at smc.vnet.net
• Subject: [mg64136] Re: [mg64105] How to evaluate the following expression in Mathematica?
• From: Daniel Lichtblau <danl at wolfram.com>
• Date: Thu, 2 Feb 2006 00:06:34 -0500 (EST)
• References: <200602010934.EAA23069@smc.vnet.net>
• Sender: owner-wri-mathgroup at wolfram.com

comtech wrote:
> It had been running on my PC for 1 day now, and it still did not get
> any results...
> 
> What's wrong?
> 
> Cell[BoxData[
>     RowBox[{"N", "[",
>       RowBox[{
>         SubsuperscriptBox["\[Integral]",
>           RowBox[{"-", "\[Infinity]"}], "\[Infinity]"],
>         RowBox[{
>           SubsuperscriptBox["\[Integral]",
>             RowBox[{"-", "\[Infinity]"}], "\[Infinity]"],
>           RowBox[{
>             FractionBox["1",
>               SqrtBox[
>                 RowBox[{"2", "\[Pi]"}]]],
>             RowBox[{"Exp", "[",
>               RowBox[{"-",
>                 FractionBox[
>                   RowBox[{
>                     SuperscriptBox["u", "2"], "+",
>                     SuperscriptBox[
>                       RowBox[{"(",
>                         RowBox[{"v", "-", "1"}], ")"}], "2"]}],
>                   "2"]}], "]"}],
>             RowBox[{"N", "[",
>               RowBox[{
>                 FractionBox[
>                   SqrtBox["5"],
>                   SqrtBox[
>                     RowBox[{"2", "\[Pi]"}]]],
>                 RowBox[{
>                   SubsuperscriptBox["\[Integral]",
>                     RowBox[{"-", "\[Infinity]"}],
>                     RowBox[{"+", "\[Infinity]"}]],
>                   RowBox[{
>                     SubsuperscriptBox["\[Integral]",
>                       RowBox[{
>                         FractionBox[
>                           SqrtBox["2"], "2"],
>                         RowBox[{"Abs", "[",
>                           RowBox[{"u", "-", "v"}], "]"}]}],
>                       "\[Infinity]"],
>                     RowBox[{
>                       RowBox[{"Exp", "[",
>                         RowBox[{"-",
>                           FractionBox[
>                             RowBox[{"(",
>                               RowBox[{
>                                 SuperscriptBox["x", "2"], "+",
>                                 SuperscriptBox["y", "2"]}], ")"}],
>                             RowBox[{"2", "/", "5"}]]}], "]"}],
>                       RowBox[{"\[DifferentialD]", "x"}],
>                       RowBox[{"\[DifferentialD]", "y"}]}]}]}]}],
>               "]"}],
>             RowBox[{"\[DifferentialD]", "u"}],
>             RowBox[{"\[DifferentialD]", "v"}]}]}]}], "]"}]], "Input"]
> 
> 
> 
> 
> 
> 
> 
> Notebook[{
> Cell[BoxData[
>     \(N[\[Integral]\_\(-\[Infinity]\)\%\[Infinity]\(\[Integral]\_\(-\
> \[Infinity]\)\%\[Infinity]\( 1\/\@\(2  \[Pi]\)\)
>             Exp[\(-\(\(u\^2 + \((v - 1)\)\^2\)\/2\)\)]
>             N[\(\@5\/\@\(2  \[Pi]\)\) \(\[Integral]\_\(-\[Infinity]\)\
> \%\(+\[Infinity]\)\(\[Integral]\_\(\(\@2\/2\) Abs[u - v]\)\%\
> \[Infinity] Exp[\(-\(\((x\^2 + y\^2)\)\/\(2/
>                                 5\)\)\)] \[DifferentialD]x \
> \[DifferentialD]y\)\)] \[DifferentialD]u \[DifferentialD]v\)]\)], \
> "Input"]
> },
> FrontEndVersion->"5.2 for Microsoft Windows",
> ScreenRectangle->{{0, 800}, {0, 543}},
> WindowSize->{665, 494},
> WindowMargins->{{0, Automatic}, {Automatic, 0}}
> ]
> [...]


Would be nice if this were readable. That can be achieved by selecting 
the cell and using Cell > Convert To > InputForm before cutting and 
pasting into email.

As for how to do this, you can either use quadrature throughout, or 
integrate the inner thing symbolically. The latter approach turns out to 
be less troublesome but they give consistent results. Also I would guess 
  the first method can be improved upon by sage use of NIntegrate options.

In[3]:= (Sqrt[5]/Sqrt[2*Pi])*NIntegrate[
   (1/Sqrt[2*Pi])*Exp[-((u^2 + (v - 1)^2)/2)]*
     Exp[-((x^2 + y^2)/(2/5))],
   {v, -Infinity, Infinity},
   {u, -Infinity, Infinity},
   {y, -Infinity, Infinity},
   {x, (Sqrt[2]/2)*Abs[u - v], Infinity}]

During evaluation of In[3]:=
NIntegrate::"slwcon": "Numerical integration converging too slowly; 
suspect \
one of the following: singularity, value of the integration being 0, \
oscillatory integrand, or insufficient WorkingPrecision. If your 
integrand is \
oscillatory try using the option Method->Oscillatory in NIntegrate. More\
\[Ellipsis]"

During evaluation of In[3]:=
NIntegrate::"eincr": "The global error of the strategy GlobalAdaptive has \
increased more than 1000 times. The global error is expected to decrease \
monotonically after a number of integrand evaluations. Suspect one of the \
following: the difference between the values of PrecisionGoal and \
WorkingPrecision is too small; the integrand is highly oscillatory or it 
is \
not a (piecewise) smooth function; the true value of the integral is 0. \
Increasing the value of the GlobalAdaptive option 
MaxNumberOfErrorIncreases \
might lead to a convergent numerical integration."

Out[3]= 0.301417

In[4]:= ii=Integrate[Exp[-((x^2 + y^2)/(2/5))],
   {y, -Infinity, Infinity},
   {x, (Sqrt[2]/2)*Abs[u - v], Infinity}]

Out[4]= (1/5)*Pi*Erfc[(1/2)*Sqrt[5]*Abs[u - v]]

In[5]:= (Sqrt[5]/Sqrt[2*Pi])*NIntegrate[
   (1/Sqrt[2*Pi])*Exp[-((u^2 + (v - 1)^2)/2)]*ii,
   {v, -Infinity, Infinity},
    {u, -Infinity, Infinity}]

Out[5]= 0.301412


Daniel Lichtblau
Wolfram Research

• References:
  • How to evaluate the following expression in Mathematica?
    • From: "comtech" <comtech.usa@gmail.com>