In structural dynamics analysis (SDA) of continuous systems with random space and time inputs, complex numerical integrations are required. The inputs data are given in tables and the outputs are needed in table for continued analysis for further assessment analysis. The purpose of the needed computations is to use Mathematica as a tool to accomplish SDA results.
I think the integral you want can be integrated explicitly:
L=.;
a=.;
f=.;
U=.;
c2 = Exp[-a(2\[Pi] f \[Xi])/U ] * Cos[(2 \[Pi] f \[Xi])/U]
integral = Integrate[c2^2 \[Xi],{\[Xi],0,L}]
with the following result
E^(-((2 a f \[Pi] \[Xi])/U)) Cos[(2 f \[Pi] \[Xi])/U]
(U (a^2 (-1+a^2) U+(1+a^2)^2 U-E^(-((4 a f L \[Pi])/U)) ((1+a^2)^2 (4 a f L \[Pi]+U)+a^2 (4 a (1+a^2) f L \[Pi]+(-1+a^2) U) Cos[(4 f L \[Pi])/U]-2 a^2 (2 (1+a^2) f L \[Pi]+a U) Sin[(4 f L \[Pi])/U])))/(32 a^2 (1+a^2)^2 f^2 \[Pi]^2)
Then a table could be constructed with (2*integral)^0.5.
Thanks for the reply, Jim
The integrand in practice is given a “table”; the exponential function I provide is an example to illustrate.
Look at the end of the nb, it has an example of the type of table I’m trying to Integrate. It’s this type of table I’m trying to integrate to get the required table.I’m trying to get around using Simpson’s rule or the trapezoidal rule without a number of iterations. I was hoping this problem had be solved already.
How about something like either the two ways below? One uses Map and the other uses Table.
L = 100
f[z_] := NIntegrate[z, {x, 0, L}]
t = {{1, Exp[-0.1 x] Cos[2.3 x]}, {2, Exp[-0.2 x] Cos[4.3 x]}}
Map[f, t[[All, 2]]]
Table[NIntegrate[t[[i, 2]], {x, 0, L}], {i, Length[t]}]
You don’t need to use Map or Table. NIntegrate handles vector and matrix integrands:
In[358]:= NIntegrate[t[[All, 2]], {x, 0, L}]
Out[358]= {0.0188565, 0.0107933}
This is a reply to all —Thanks
I provided equations as examples; do not want to apply equations in the integral.
My input (integrand) to the integral is a “Table”; see the Table at the end of the example in nb. I want to input a similar table into the integral; the table could have 500 (X, Y) terms; don’t want to do this term by term. The “X” part is not effected by the integration; only the “Y” part. After integration only a table of (X,Y) terms (numbers) will remain. I’m going to do this about 150 times.
My customer provides me a the “table” which is a cross power spectral density and this integrated result is a generalize force.
May be I do not understand the question. NIntegrate handles vector and matrix integrands. This is what I get with the table C2 at the end of the notebook in the original message:
In[16]:= NIntegrate[C2, {\[Xi], 0, 1}]
Out[16]= {{0., 1.}, {5., 0.99994}, {10., 0.999877}, {15.,
0.999813}, {20., 0.999747}, {25., 0.999679}, {30., 0.999609}, {35.,
0.999537}, {40., 0.999464}, {45., 0.999388}, {50., 0.999311}, {55.,
0.999232}, {60., 0.999151}, {65., 0.999068}, {70., 0.998983}, {75.,
0.998896}, {80., 0.998807}, {85., 0.998717}, {90., 0.998624}, {95.,
0.99853}, {100., 0.998434}, {105., 0.998336}, {110.,
0.998236}, {115., 0.998134}, {120., 0.998031}, {125.,
0.997925}, {130., 0.997818}, {135., 0.997709}, {140.,
0.997598}, {145., 0.997485}, {150., 0.99737}, {155.,
0.997253}, {160., 0.997135}, {165., 0.997014}, {170.,
0.996892}, {175., 0.996768}, {180., 0.996642}, {185.,
0.996514}, {190., 0.996384}, {195., 0.996253}, {200.,
0.996119}, {205., 0.995984}, {210., 0.995847}, {215.,
0.995708}, {220., 0.995567}, {225., 0.995424}, {230.,
0.99528}, {235., 0.995133}, {240., 0.994985}, {245.,
0.994835}, {250., 0.994683}, {255., 0.994529}, {260.,
0.994373}, {265., 0.994216}, {270., 0.994056}, {275.,
0.993895}, {280., 0.993732}, {285., 0.993567}, {290.,
0.993401}, {295., 0.993232}, {300., 0.993061}, {305.,
0.992889}, {310., 0.992715}, {315., 0.992539}, {320.,
0.992361}, {325., 0.992182}, {330., 0.992}, {335., 0.991817}, {340.,
0.991632}, {345., 0.991445}, {350., 0.991256}, {355.,
0.991065}, {360., 0.990873}, {365., 0.990678}, {370.,
0.990482}, {375., 0.990284}, {380., 0.990084}, {385.,
0.989883}, {390., 0.989679}, {395., 0.989474}, {400.,
0.989267}, {405., 0.989058}, {410., 0.988847}, {415.,
0.988635}, {420., 0.98842}, {425., 0.988204}, {430.,
0.987986}, {435., 0.987766}, {440., 0.987544}, {445.,
0.987321}, {450., 0.987096}, {455., 0.986869}, {460.,
0.98664}, {465., 0.986409}, {470., 0.986176}, {475.,
0.985942}, {480., 0.985706}, {485., 0.985468}, {490.,
0.985228}, {495., 0.984987}, {500., 0.984743}, {505.,
0.984498}, {510., 0.984251}, {515., 0.984002}, {520.,
0.983752}, {525., 0.983499}, {530., 0.983245}, {535.,
0.982989}, {540., 0.982731}, {545., 0.982472}, {550.,
0.982211}, {555., 0.981947}, {560., 0.981682}, {565.,
0.981416}, {570., 0.981147}, {575., 0.980877}, {580.,
0.980605}, {585., 0.980331}, {590., 0.980055}, {595.,
0.979778}, {600., 0.979499}, {605., 0.979218}, {610.,
0.978935}, {615., 0.978651}, {620., 0.978364}, {625.,
0.978076}, {630., 0.977786}, {635., 0.977495}, {640.,
0.977201}, {645., 0.976906}, {650., 0.976609}, {655.,
0.976311}, {660., 0.97601}, {665., 0.975708}, {670.,
0.975404}, {675., 0.975098}, {680., 0.974791}, {685.,
0.974481}, {690., 0.97417}, {695., 0.973858}, {700.,
0.973543}, {705., 0.973227}, {710., 0.972909}, {715.,
0.972589}, {720., 0.972268}, {725., 0.971944}, {730.,
0.971619}, {735., 0.971293}, {740., 0.970964}, {745.,
0.970634}, {750., 0.970302}, {755., 0.969968}, {760.,
0.969633}, {765., 0.969295}, {770., 0.968956}, {775.,
0.968616}, {780., 0.968273}, {785., 0.967929}, {790.,
0.967583}, {795., 0.967236}, {800., 0.966887}, {805.,
0.966536}, {810., 0.966183}, {815., 0.965828}, {820.,
0.965472}, {825., 0.965114}, {830., 0.964755}, {835.,
0.964393}, {840., 0.96403}, {845., 0.963665}, {850.,
0.963299}, {855., 0.962931}, {860., 0.962561}, {865.,
0.962189}, {870., 0.961816}, {875., 0.961441}, {880.,
0.961064}, {885., 0.960686}, {890., 0.960306}, {895.,
0.959924}, {900., 0.95954}, {905., 0.959155}, {910.,
0.958768}, {915., 0.958379}, {920., 0.957989}, {925.,
0.957597}, {930., 0.957204}, {935., 0.956808}, {940.,
0.956411}, {945., 0.956013}, {950., 0.955612}, {955.,
0.95521}, {960., 0.954806}, {965., 0.954401}, {970.,
0.953994}, {975., 0.953585}, {980., 0.953175}, {985.,
0.952763}, {990., 0.952349}, {995., 0.951933}, {1000., 0.951516}}
Anton --Thanks, I appreciate your help!!!
Note:
I shortened C2 and called it C3. (shortened the calculations)
In29 and Out29 below deal with that.
In36 deals with NIntegrate and shows a problem so I change to Integrate In30 and got your result Out30.
When I changed the limit from 1 to 100; see “In and out 33”; the “X” value increases where it should stay the same as the “X” in C3.
In34 is the actual integral I’m using and the first term in Out34 is correct; but the rest of the “X” are not correct; should be the same as C3. The integrated “Y” values may be correct; it’s the “X”'s.
luke
In[29]:=
!(C3 = {{0,
1}, {5, [ExponentialE]^\(-0.00011899972172688612`\ ?)\ Cos[(?\
?)/1320]}, {10, [ExponentialE]^\(-0.00023799944345377223`\ ?)\
Cos[?\ ?/660]}, {15, [ExponentialE]^\(-0.00035699916518065836`\
?)\ Cos[?\ ?/440]}, {20,
[ExponentialE]^\(-0.00047599888690754447`\ ?)\ Cos[?\ ?/330]},
{25, [ExponentialE]^\(-0.0005949986086344305`\ ?)\ Cos[(?\
?)/264]}, {30, [ExponentialE]^\(-0.0007139983303613167`\ ?)\ Cos[(?
\ ?)/220]}, {35, [ExponentialE]^\(-0.0008329980520882028`\ ?)\
Cos[7\ ?\ ?/1320]}, {40, [ExponentialE]^\(-0.0009519977738150889`\
?)\ Cos[?\ ?/165]}, {45, [ExponentialE]^\(-0.001070997495541975`\
?)\ Cos[3\ ?\ ?/440]}, {50, [ExponentialE]^((-0.001189997217268861\ \)\ ?\)\ Cos[\(?\ ?\)\/132]}, {55, \ \[ExponentialE]\^\(\(-0.0013089969389957472)\ ?)\ Cos[?\ ?/120]},
{60, [ExponentialE]^\(-0.0014279966607226335`\ ?)\ Cos[(?\
?)/110]}, {65, [ExponentialE]^\(-0.0015469963824495195`\ ?)\
Cos[13\ ?\ ?/1320]}, {70, [ExponentialE]^\(-0.0016659961041764057`
\ ?)\ Cos[7\ ?\ ?/660]}, {75,
[ExponentialE]^\(-0.0017849958259032917`\ ?)\ Cos[?\ ?/88]}, {80,
[ExponentialE]^\(-0.0019039955476301779`\ ?)\ Cos[2\ ?\ ?/165]},
{85, [ExponentialE]^\(-0.002022995269357064`\ ?)\ Cos[(17\ ?\
?)/1320]}, {90, [ExponentialE]^\(-0.00214199499108395`\ ?)\ Cos[(3\
?\ ?)/220]}, {95, [ExponentialE]^\(-0.002260994712810836`\ ?)\
Cos[19\ ?\ ?/1320]}, {100, [ExponentialE]^\(-0.002379994434537722`
\ ?)\ Cos[?\ ?/66]}, {105, [ExponentialE]^((-0.0024989941562646085`
)\ ?)\ Cos[7\ ?\ ?/440]}})
Out[29]=
!{{0, 1}, {5, \[ExponentialE]\^\(\(-0.00011899972172688612`\ ?)\
Cos[?\ ?/1320]}, {10, [ExponentialE]^\(-0.00023799944345377223`\ ?
)\ Cos[?\ ?/660]}, {15, [ExponentialE]^\(-0.00035699916518065836`
\ ?)\ Cos[?\ ?/440]}, {20,
[ExponentialE]^\(-0.00047599888690754447`\ ?)\
Cos[?\ ?/330]}, {25,
[ExponentialE]^\(-0.0005949986086344304`\ ?)\ Cos[?\ ?/264]},
{30, [ExponentialE]^\(-0.0007139983303613166`\ ?)\ Cos[(?\
?)/220]}, {35, [ExponentialE]^\(-0.0008329980520882028`\ ?)\ Cos[(7
\ ?\ ?)/1320]}, {40, [ExponentialE]^\(-0.0009519977738150888`\ ?)\
Cos[?\ ?/165]}, {45,
[ExponentialE]^\(-0.001070997495541975`\ ?)\ Cos[3\ ?\ ?/440]},
{50, [ExponentialE]^\(-0.001189997217268861`\ ?)\
Cos[?\ ?/132]}, {55,
[ExponentialE]^\(-0.0013089969389957472`\ ?)\ Cos[?\ ?/120]},
{60, [ExponentialE]^\(-0.0014279966607226335`\ ?)\ Cos[(?\
?)/110]}, {65, [ExponentialE]^\(-0.0015469963824495195`\ ?)\
Cos[(13\ ?\
?)/1320]}, {70,
[ExponentialE]^\(-0.0016659961041764057`\ ?)\ Cos[7\ ?\ ?/660]},
{75, [ExponentialE]^\(-0.0017849958259032917`\ ?)\ Cos[?\ ?/88]},
{80, [ExponentialE]^\(-0.0019039955476301779`\ ?)\ Cos[(2\ ?\
?)/165]}, {85, [ExponentialE]^\(-0.002022995269357064`\ ?)\ Cos[(17
\ ?\ ?)/1320]}, {90, [ExponentialE]^\(-0.00214199499108395`\ ?)\
Cos[3\ ?\ ?/220]}, {95, [ExponentialE]^\(-0.002260994712810836`\ ?
)\ Cos[(19\ ?\
?)/1320]}, {100,
[ExponentialE]^\(-0.002379994434537722`\ ?)\ Cos[(?\
?)/66]}, {105, [ExponentialE]^\(-0.0024989941562646085`\
?)\ Cos[7\ ?\ ?/440]}})
In[36]:=
NIntegrate[(C3),{?,0,1}]
From In[36]:=
!(*
RowBox[{(NIntegrate::"inum
"), \(:\ ), “<"Integrand \!\([LeftSkeleton]
1 [RightSkeleton]\) is not numerical at \!\({?}\) =
\!\({0.5`}\). \!\(\*ButtonBox[\"More
\",
ButtonStyle->\"RefGuideLinkText\", ButtonFrame->None,
ButtonData:>\"NIntegrate::inum\"]\)">”}])
Out[36]=
NIntegrate[C3,{?,0,1}]
In[30]:=
Integrate[(C3),{?,0,1}]
Out[30]=
!{{0, 1}, {5, \(\(0.9999395585214899`\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {10, \(\(0.9998772341476005)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {15, \(\(0.9998130273929275)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {20, \(\(0.9997469387783866)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {25, \(\(0.9996789688312022)\[InvisibleSpace]) -
2.4256423570287898*^-20\ \[ImaginaryI]}, {30, \(\(0.9996091180849116)(
[InvisibleSpace])) - 2.477051560095065*^-20\ \[ImaginaryI]}, {35, \ \(\(0.9995373870793505)\[InvisibleSpace]) + 0.\ \[ImaginaryI]}, { 40, \(\(0.9994637763606591)\[InvisibleSpace]) -
2.5798304177902294*^-20\ \[ImaginaryI]}, {45, \(\(0.9993882864812746)(
[InvisibleSpace])) -
2.6311997854715805*^-20\ \[ImaginaryI]}, {50, \ \(\(0.99931091799993)\[InvisibleSpace]) - 2.6825555878650602*^-20\ \ \[ImaginaryI]}, {55, \(\(0.9992316714816498)\[InvisibleSpace]) -
2.7338976816811725*^-20\ \[ImaginaryI]}, { 60, \(\(0.9991505474977442)\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {65, \(\(0.9990675466258099)\[InvisibleSpace]) -
2.8365401707937054*^-20\ \[ImaginaryI]}, { 70, \(\(0.9989826694497241)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {75, \(\(0.9988959165596416)\[InvisibleSpace]) -
2.9391261079888114*^-20\ \[ImaginaryI]}, {80, \ \(\(0.9988072885519903)\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {85, \(\(0.9987167860294703)\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {90, \(\(0.9986244096010474)\[InvisibleSpace]) -
3.092896477640538*^-20\ \[ImaginaryI]}, {95, \(\(0.9985301598819512)(
[InvisibleSpace])) -
3.144123753447079*^-20\ \[ImaginaryI]}, {100, \ \(\(0.9984340374936708)\[InvisibleSpace]) - 3.195336034482304*^-20\ \ \[ImaginaryI]}, {105, \(\(0.9983360430639522)\[InvisibleSpace]) -
3.246533178224931`*^-20\ [ImaginaryI]}})
In[33]:=
Integrate[(C3),{?,0,100}]
Out[33]=
!{{0, 100}, {500, \(\(98.47432186732775`\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, { 1000, \(\(95.1516218739715)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {1500, \(\(90.17147438682308)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, { 2000, \(\(83.72532790153666)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {2500, \(\(76.04801583417462)\[InvisibleSpace]) -
2.36814499273618*^-17\ \[ImaginaryI]}, {3000, \(\(67.40775875992027)(
[InvisibleSpace])) - 2.6345632087437145*^-17\ \[ImaginaryI]}, {3500, \ \(\(58.095116058842834)\[InvisibleSpace]) + 0.\ \[ImaginaryI]}, { 4000, \(\(48.41137871044244)\[InvisibleSpace]) -
2.9550667609301643*^-17\ \[ImaginaryI]}, {4500, \(\(38.65690592278126)(
[InvisibleSpace])) -
3.008098528664352*^-17\ \[ImaginaryI]}, {5000, \ \(\(29.11989647335106)\[InvisibleSpace]) - 2.9930188967289435*^-17\ \ \[ImaginaryI]}, {5500, \(\(20.06605221903229)\[InvisibleSpace]) -
2.915125934892944*^-17\ \[ImaginaryI]}, { 6000, \(\(11.729538319052729)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, { 6500, \(\(4.305575275687835)\[InvisibleSpace]) -
2.6011131052702112*^-17\ \[ImaginaryI]}, {7000, \(-2.0550844119812233) +
0.\ \[ImaginaryI]}, { 7500, \(-7.249633214176027) - 2.1391685235662612*^-17\ \ \[ImaginaryI]}, {8000, \(-11.224572699093214) +
0.\ \[ImaginaryI]}, {8500, \(-13.974993453098536) + 0.\ \ \[ImaginaryI]}, {9000, \(-15.541889926583302) -
1.3517813048945516*^-17\ \[ImaginaryI]}, { 9500, \(-16.007711255679638) - 1.1038212145295681*^-17\ \ \[ImaginaryI]}, {10000, \(-15.490418957862715) - 8.769264381390492*^-18\ \ \[ImaginaryI]}, { 10500, \(-14.136377377680587) - 6.7730419377129346`*^-18\
[ImaginaryI]}})
In[34]:=
!\((2*Integrate[\((C3)^ .5, {?, 0, 100}]))^ .5)
Out[34]=
!({{0, 14.142135623730951}, {21.147425268811283,
\(14.087759680463986`\[InvisibleSpace]) + 0.\ \[ImaginaryI]}, \ {25.14866859365871, \(13.966018979649741`\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {27.831576837137405,
\(13.773943380696016`\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {29.906975624424412, \(13.504825224204863`(
[InvisibleSpace])) +
0.\ \[ImaginaryI]}, {31.622776601683793,
\(13.144614394528887`\[InvisibleSpace]) -
1.0811161353745691*^-15\ \[ImaginaryI]}, {33.097509196468735,
\(12.658581855965714`\[InvisibleSpace]) + 1.1226261264412287*^-15\ \ \[ImaginaryI]}, {34.397906282503925, \(11.896032641194601`(
[InvisibleSpace])) + 0.0948355593921728\ \[ImaginaryI]}, \ {35.56558820077846, \(11.142261172590315`\[InvisibleSpace]) +
0.5759427561557676\ \[ImaginaryI]}, { 36.628415014847064, \(10.559267746178637`\[InvisibleSpace]) +
1.1989359199429737\ \[ImaginaryI]}, {37.60603093086394,
\(10.126447984679059`\[InvisibleSpace]) +
1.8682423768453593\ \[ImaginaryI]}, {38.51285106843081,
\(9.8204630896133`\[InvisibleSpace]) + 2.528209140548143\ \ \[ImaginaryI]}, {39.35979342530861, \(9.613578993086847`(
[InvisibleSpace])) +
3.1425125345707876\ \[ImaginaryI]}, {40.155342728704355,
\(9.47691206352986`\[InvisibleSpace]) + 3.689986626433017\ \ \[ImaginaryI]}, {40.90623489235047, \(9.384361955617077`(
[InvisibleSpace])) +
4.1608833011487105\ \[ImaginaryI]}, {41.61791450287817,
\(9.314626709614732`\[InvisibleSpace]) + 4.552566077990048\ \ \[ImaginaryI]}, {42.294850537622565, \(9.25121850616666`(
[InvisibleSpace])) +
4.86578943226933\ \[ImaginaryI]}, {42.940760257109766,
\(9.181355713740524`\[InvisibleSpace]) + 5.1019476310179\ \ \[ImaginaryI]}, {43.55877174692863, \(9.094257655321282`(
[InvisibleSpace])) +
5.260720316074253\ \[ImaginaryI]}, {44.151544355342686,
\(8.978156270895685`\[InvisibleSpace]) + 5.336035968645732\ \ \[ImaginaryI]}, {44.721359549995796, \(8.813577568914233`(
[InvisibleSpace])) + 5.281427433646056\ \[ImaginaryI]}, \ {45.270190558378694, \(8.744387269338645`\[InvisibleSpace]) +
5.069730409921358`\ [ImaginaryI]}})
Would you mind re-posting the Mathematica code in your last reply either as a notebook or using the “Code Sample” formatting option?
I’m apparently still not understanding the issue described from “I want to input a similar table into the integral”. While I will make unsophisticated responses questions posted here (e.g., not using Map when I should and using it when I don’t have to), the best part of this forum is to get answers from experts who are so much kinder and diplomatic than on StackExchange. Making it easier on those experts pays tremendous dividends.
Jim,
Thanks, I have attached a file that I hope clarified the definition of the problem. In addition I have completed more calculation to show various
features to save others from having to do that. I don’t think this should have a “vote of 1” yet. I don’t feel that the concerns are understood.
thanks luke
Luke, I think we are using different versions of Mathematica. What is your version?
I’m using 5.2
Anton --Thanks, I appreciate your help!!!
Note:
I shortened C2 and called it C3. (shortened the calculations)
In29 and Out29 below deal with that.
In36 deals with NIntegrate and shows a problem so I change to Integrate In30 and got your result Out30.
When I changed the limit from 1 to 100; see “In and out 33”; the “X” value increases where it should stay the same as the “X” in C3.
In34 is the actual integral I’m using and the first term in Out34 is correct; but the rest of the “X” are not correct; should be the same as C3. The integrated “Y” values may be correct; it’s the “X”'s.
luke
In[29]:=
!(C3 = {{0,
1}, {5, [ExponentialE]^\(-0.00011899972172688612`\ ?)\ Cos[(?\
?)/1320]}, {10, [ExponentialE]^\(-0.00023799944345377223`\ ?)\
Cos[?\ ?/660]}, {15, [ExponentialE]^\(-0.00035699916518065836`\
?)\ Cos[?\ ?/440]}, {20,
[ExponentialE]^\(-0.00047599888690754447`\ ?)\ Cos[?\ ?/330]},
{25, [ExponentialE]^\(-0.0005949986086344305`\ ?)\ Cos[(?\
?)/264]}, {30, [ExponentialE]^\(-0.0007139983303613167`\ ?)\ Cos[(?
\ ?)/220]}, {35, [ExponentialE]^\(-0.0008329980520882028`\ ?)\
Cos[7\ ?\ ?/1320]}, {40, [ExponentialE]^\(-0.0009519977738150889`\
?)\ Cos[?\ ?/165]}, {45, [ExponentialE]^\(-0.001070997495541975`\
?)\ Cos[3\ ?\ ?/440]}, {50, [ExponentialE]^((-0.001189997217268861\ \)\ ?\)\ Cos[\(?\ ?\)\/132]}, {55, \ \[ExponentialE]\^\(\(-0.0013089969389957472)\ ?)\ Cos[?\ ?/120]},
{60, [ExponentialE]^\(-0.0014279966607226335`\ ?)\ Cos[(?\
?)/110]}, {65, [ExponentialE]^\(-0.0015469963824495195`\ ?)\
Cos[13\ ?\ ?/1320]}, {70, [ExponentialE]^\(-0.0016659961041764057`
\ ?)\ Cos[7\ ?\ ?/660]}, {75,
[ExponentialE]^\(-0.0017849958259032917`\ ?)\ Cos[?\ ?/88]}, {80,
[ExponentialE]^\(-0.0019039955476301779`\ ?)\ Cos[2\ ?\ ?/165]},
{85, [ExponentialE]^\(-0.002022995269357064`\ ?)\ Cos[(17\ ?\
?)/1320]}, {90, [ExponentialE]^\(-0.00214199499108395`\ ?)\ Cos[(3\
?\ ?)/220]}, {95, [ExponentialE]^\(-0.002260994712810836`\ ?)\
Cos[19\ ?\ ?/1320]}, {100, [ExponentialE]^\(-0.002379994434537722`
\ ?)\ Cos[?\ ?/66]}, {105, [ExponentialE]^((-0.0024989941562646085`
)\ ?)\ Cos[7\ ?\ ?/440]}})
Out[29]=
!{{0, 1}, {5, \[ExponentialE]\^\(\(-0.00011899972172688612`\ ?)\
Cos[?\ ?/1320]}, {10, [ExponentialE]^\(-0.00023799944345377223`\ ?
)\ Cos[?\ ?/660]}, {15, [ExponentialE]^\(-0.00035699916518065836`
\ ?)\ Cos[?\ ?/440]}, {20,
[ExponentialE]^\(-0.00047599888690754447`\ ?)\
Cos[?\ ?/330]}, {25,
[ExponentialE]^\(-0.0005949986086344304`\ ?)\ Cos[?\ ?/264]},
{30, [ExponentialE]^\(-0.0007139983303613166`\ ?)\ Cos[(?\
?)/220]}, {35, [ExponentialE]^\(-0.0008329980520882028`\ ?)\ Cos[(7
\ ?\ ?)/1320]}, {40, [ExponentialE]^\(-0.0009519977738150888`\ ?)\
Cos[?\ ?/165]}, {45,
[ExponentialE]^\(-0.001070997495541975`\ ?)\ Cos[3\ ?\ ?/440]},
{50, [ExponentialE]^\(-0.001189997217268861`\ ?)\
Cos[?\ ?/132]}, {55,
[ExponentialE]^\(-0.0013089969389957472`\ ?)\ Cos[?\ ?/120]},
{60, [ExponentialE]^\(-0.0014279966607226335`\ ?)\ Cos[(?\
?)/110]}, {65, [ExponentialE]^\(-0.0015469963824495195`\ ?)\
Cos[(13\ ?\
?)/1320]}, {70,
[ExponentialE]^\(-0.0016659961041764057`\ ?)\ Cos[7\ ?\ ?/660]},
{75, [ExponentialE]^\(-0.0017849958259032917`\ ?)\ Cos[?\ ?/88]},
{80, [ExponentialE]^\(-0.0019039955476301779`\ ?)\ Cos[(2\ ?\
?)/165]}, {85, [ExponentialE]^\(-0.002022995269357064`\ ?)\ Cos[(17
\ ?\ ?)/1320]}, {90, [ExponentialE]^\(-0.00214199499108395`\ ?)\
Cos[3\ ?\ ?/220]}, {95, [ExponentialE]^\(-0.002260994712810836`\ ?
)\ Cos[(19\ ?\
?)/1320]}, {100,
[ExponentialE]^\(-0.002379994434537722`\ ?)\ Cos[(?\
?)/66]}, {105, [ExponentialE]^\(-0.0024989941562646085`\
?)\ Cos[7\ ?\ ?/440]}})
In[36]:=
NIntegrate[(C3),{?,0,1}]
From In[36]:=
!(*
RowBox[{(NIntegrate::"inum
"), \(:\ ), “<"Integrand \!\([LeftSkeleton]
1 [RightSkeleton]\) is not numerical at \!\({?}\) =
\!\({0.5`}\). \!\(\*ButtonBox[\"More
\",
ButtonStyle->\"RefGuideLinkText\", ButtonFrame->None,
ButtonData:>\"NIntegrate::inum\"]\)">”}])
Out[36]=
NIntegrate[C3,{?,0,1}]
In[30]:=
Integrate[(C3),{?,0,1}]
Out[30]=
!{{0, 1}, {5, \(\(0.9999395585214899`\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {10, \(\(0.9998772341476005)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {15, \(\(0.9998130273929275)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {20, \(\(0.9997469387783866)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {25, \(\(0.9996789688312022)\[InvisibleSpace]) -
2.4256423570287898*^-20\ \[ImaginaryI]}, {30, \(\(0.9996091180849116)(
[InvisibleSpace])) - 2.477051560095065*^-20\ \[ImaginaryI]}, {35, \ \(\(0.9995373870793505)\[InvisibleSpace]) + 0.\ \[ImaginaryI]}, { 40, \(\(0.9994637763606591)\[InvisibleSpace]) -
2.5798304177902294*^-20\ \[ImaginaryI]}, {45, \(\(0.9993882864812746)(
[InvisibleSpace])) -
2.6311997854715805*^-20\ \[ImaginaryI]}, {50, \ \(\(0.99931091799993)\[InvisibleSpace]) - 2.6825555878650602*^-20\ \ \[ImaginaryI]}, {55, \(\(0.9992316714816498)\[InvisibleSpace]) -
2.7338976816811725*^-20\ \[ImaginaryI]}, { 60, \(\(0.9991505474977442)\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {65, \(\(0.9990675466258099)\[InvisibleSpace]) -
2.8365401707937054*^-20\ \[ImaginaryI]}, { 70, \(\(0.9989826694497241)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {75, \(\(0.9988959165596416)\[InvisibleSpace]) -
2.9391261079888114*^-20\ \[ImaginaryI]}, {80, \ \(\(0.9988072885519903)\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {85, \(\(0.9987167860294703)\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {90, \(\(0.9986244096010474)\[InvisibleSpace]) -
3.092896477640538*^-20\ \[ImaginaryI]}, {95, \(\(0.9985301598819512)(
[InvisibleSpace])) -
3.144123753447079*^-20\ \[ImaginaryI]}, {100, \ \(\(0.9984340374936708)\[InvisibleSpace]) - 3.195336034482304*^-20\ \ \[ImaginaryI]}, {105, \(\(0.9983360430639522)\[InvisibleSpace]) -
3.246533178224931`*^-20\ [ImaginaryI]}})
In[33]:=
Integrate[(C3),{?,0,100}]
Out[33]=
!{{0, 100}, {500, \(\(98.47432186732775`\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, { 1000, \(\(95.1516218739715)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {1500, \(\(90.17147438682308)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, { 2000, \(\(83.72532790153666)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, {2500, \(\(76.04801583417462)\[InvisibleSpace]) -
2.36814499273618*^-17\ \[ImaginaryI]}, {3000, \(\(67.40775875992027)(
[InvisibleSpace])) - 2.6345632087437145*^-17\ \[ImaginaryI]}, {3500, \ \(\(58.095116058842834)\[InvisibleSpace]) + 0.\ \[ImaginaryI]}, { 4000, \(\(48.41137871044244)\[InvisibleSpace]) -
2.9550667609301643*^-17\ \[ImaginaryI]}, {4500, \(\(38.65690592278126)(
[InvisibleSpace])) -
3.008098528664352*^-17\ \[ImaginaryI]}, {5000, \ \(\(29.11989647335106)\[InvisibleSpace]) - 2.9930188967289435*^-17\ \ \[ImaginaryI]}, {5500, \(\(20.06605221903229)\[InvisibleSpace]) -
2.915125934892944*^-17\ \[ImaginaryI]}, { 6000, \(\(11.729538319052729)\[InvisibleSpace]) + 0.\ \ \[ImaginaryI]}, { 6500, \(\(4.305575275687835)\[InvisibleSpace]) -
2.6011131052702112*^-17\ \[ImaginaryI]}, {7000, \(-2.0550844119812233) +
0.\ \[ImaginaryI]}, { 7500, \(-7.249633214176027) - 2.1391685235662612*^-17\ \ \[ImaginaryI]}, {8000, \(-11.224572699093214) +
0.\ \[ImaginaryI]}, {8500, \(-13.974993453098536) + 0.\ \ \[ImaginaryI]}, {9000, \(-15.541889926583302) -
1.3517813048945516*^-17\ \[ImaginaryI]}, { 9500, \(-16.007711255679638) - 1.1038212145295681*^-17\ \ \[ImaginaryI]}, {10000, \(-15.490418957862715) - 8.769264381390492*^-18\ \ \[ImaginaryI]}, { 10500, \(-14.136377377680587) - 6.7730419377129346`*^-18\
[ImaginaryI]}})
In[34]:=
!\((2*Integrate[\((C3)^ .5, {?, 0, 100}]))^ .5)
Out[34]=
!({{0, 14.142135623730951}, {21.147425268811283,
\(14.087759680463986`\[InvisibleSpace]) + 0.\ \[ImaginaryI]}, \ {25.14866859365871, \(13.966018979649741`\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {27.831576837137405,
\(13.773943380696016`\[InvisibleSpace]) +
0.\ \[ImaginaryI]}, {29.906975624424412, \(13.504825224204863`(
[InvisibleSpace])) +
0.\ \[ImaginaryI]}, {31.622776601683793,
\(13.144614394528887`\[InvisibleSpace]) -
1.0811161353745691*^-15\ \[ImaginaryI]}, {33.097509196468735,
\(12.658581855965714`\[InvisibleSpace]) + 1.1226261264412287*^-15\ \ \[ImaginaryI]}, {34.397906282503925, \(11.896032641194601`(
[InvisibleSpace])) + 0.0948355593921728\ \[ImaginaryI]}, \ {35.56558820077846, \(11.142261172590315`\[InvisibleSpace]) +
0.5759427561557676\ \[ImaginaryI]}, { 36.628415014847064, \(10.559267746178637`\[InvisibleSpace]) +
1.1989359199429737\ \[ImaginaryI]}, {37.60603093086394,
\(10.126447984679059`\[InvisibleSpace]) +
1.8682423768453593\ \[ImaginaryI]}, {38.51285106843081,
\(9.8204630896133`\[InvisibleSpace]) + 2.528209140548143\ \ \[ImaginaryI]}, {39.35979342530861, \(9.613578993086847`(
[InvisibleSpace])) +
3.1425125345707876\ \[ImaginaryI]}, {40.155342728704355,
\(9.47691206352986`\[InvisibleSpace]) + 3.689986626433017\ \ \[ImaginaryI]}, {40.90623489235047, \(9.384361955617077`(
[InvisibleSpace])) +
4.1608833011487105\ \[ImaginaryI]}, {41.61791450287817,
\(9.314626709614732`\[InvisibleSpace]) + 4.552566077990048\ \ \[ImaginaryI]}, {42.294850537622565, \(9.25121850616666`(
[InvisibleSpace])) +
4.86578943226933\ \[ImaginaryI]}, {42.940760257109766,
\(9.181355713740524`\[InvisibleSpace]) + 5.1019476310179\ \ \[ImaginaryI]}, {43.55877174692863, \(9.094257655321282`(
[InvisibleSpace])) +
5.260720316074253\ \[ImaginaryI]}, {44.151544355342686,
\(8.978156270895685`\[InvisibleSpace]) + 5.336035968645732\ \ \[ImaginaryI]}, {44.721359549995796, \(8.813577568914233`(
[InvisibleSpace])) + 5.281427433646056\ \[ImaginaryI]}, \ {45.270190558378694, \(8.744387269338645`\[InvisibleSpace]) +
5.069730409921358`\ [ImaginaryI]}})
Luke, you are using a very old version of Mathematica. Version 5.2 still uses the old NIntegrate. (Version 6.0 uses the current NIntegrate framework.) It will be quite hard for me to help you since I do not have access to V5.2.
That said, I think the singularity handler manifests in your code. Please repeat your computations with SingularityDepth->1000 and experiment using higher values for MinRecursion and MaxRecursion.
Anton ,
I have installed 10.01 and transferred my nb that is in V5.2 to the 10.01version. I have included that nb.
luke
PS basically I’m trying to integrate a table of functions and then plot table of (x,y) numbers. During this process the x values
must remain the same.
Is the following what you want? You’re given a table of pairs {f,C2} and you want to end up a table of pairs {f,(2*Integrate[C2])^0.5} ? If so, the following should work:
(* Create an example table where there are pairs {f,C1} *)
a = 5/100;
U = 13200;
L = 100;
C1 = Exp[-a (2 \[Pi] f \[Xi])/U ] * Cos[(2 \[Pi] f \[Xi])/U]
C2 = Table[{f, C1}, {f, 0, 1000, 5}]
(* Take the table of pairs of {f,C1} and create a table of pairs \
{f,(2*Integrate[C1^0.5])^0.5} *)
C3 = Table[{C2[[i,
1]], (2 NIntegrate[C2[[i, 2]]^0.5, {\[Xi], 0, L}])^0.5}, {i,
Length[C2]}]
(When you say in your Notebook that the X’s aren’t right, is that because the X’s (or rather your values for f) were integrated, too?)
(When you say in your Notebook that the X’s aren’t right, is that because the X’s (or rather your values for f) were integrated, too?) Yes
Thanks; everything seemed to work ok except the calculation takes a long time and below is a comparison of the accuracies
(* Comparison of data*)
f = 0; 14.142135623730951`; compare to 14.142135623730951`(recent calcs)
f = 5; 13.927347535477212`; compared to 14.087759680463982``(recent calcs)
f = 70 9.448983612771324`;compared to 9.384308379490255`(recent calcs)`
can check the nb to see results.
Thanks ---- luke
Anton ,
I have installed 10.01 and transferred my nb that is in V5.2 to the 10.01version. I have included that nb.
luke
PS basically I’m trying to integrate a table of functions and then plot table of (x,y) numbers. During this process the x values
must remain the same.