Unable to find a minima for a symbolic integral

Question

RB 2017-5-19

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/341058-unable-to-find-a-minima-for-a-symbolic-integral

关闭： MATLAB Answer Bot 2021-8-20

Hello! I have the following code. Actually, I am looking to maximize w.r.t. x1 but since it appears with a negative sign it boils down to minima.

n=35; 
T=15; 
g=.111;
K=1/g;
d=2; 
rbar=0.04;
mubar=0; 
R=[1, 0; 0, 0];
M=[0, 0; 0, 1];
X012=0.002;
X0=[0.01, X012; X012, 0.001];
H=[-0.5, 0.4; 0.007, -0.008];
Q=[0.06 -0.0006; -0.06, 0.006];
Scheck7=0;
beta=3;
SIGMA=[0.006811791307233, -0.000407806990090; -0.000407806990090, 0.00039291243623];
PSICURL=[0.011526035149236, 0.758273970303934; 0.013935191202735, 0.955423605940771];   
S0i=zeros(n,1);
S0i2=zeros(n,1);
Yn0=0; 
Yi=zeros(n,1);
S1MC=zeros(n,1);
POW=zeros(n,1);
SPOW=0;
SPHI=0;
SPSI=0;
delta=0.75; 
PSIi = cell(1, n);
PHIi=zeros(n,1);
% Prameters for the symbolic sum
syms Gamma1 ik
for i=2:n;
         APHNEW=expm([(i-1).*H,(i-1).*(2*(Q'*Q));(i-1).*(R+M),(i-1).*(-(H'))]);
         APHNEW11=APHNEW(1:2,1:2); %That is good
         APHNEW21=APHNEW(3:4,1:2);
         APHNEW12=APHNEW(1:2,3:4);
         APHNEW22=APHNEW(3:4,3:4);
         % Computation of PSIi and PHIi  
         BPHNEW22=inv(APHNEW22);
         PSIi{i}=APHNEW22\APHNEW21;
         M7=PSIi{i};
         SPSI=SPSI+PSIi{i};
         PHIi(i)=beta*(log(det(APHNEW22))+(i-1)*trace(H'))/2; 
         S0i(i)=exp(-((rbar+mubar)*(i-1)+PHIi(i)));
         S0i2(i)=(-((rbar+mubar)*(i-1)+PHIi(i))); 
         Yn0=Yn0+S0i2(i);
end
SIGMAi=inv(SIGMA);
THETA1=SIGMAi*(PSICURL'*X0*PSICURL);
% i stands for iota
syms x1
fun0=@(Gamma) exp(trace(1i*(THETA1*inv(eye(2)-2i.*SIGMA*Gamma)*SIGMA*Gamma)))/((det(eye(2)-2i.*SIGMA*Gamma))^(beta/2));
%fun3 = matlabFunction((exp((1i*Gamma1+delta)*Yn0)*(symsum(S0i(ik)*fun0(1i*PSIi{ik}-(Gamma1-1i*delta)*SPSI), ik, 2, n)-(K-1)*fun0(-(Gamma1-1i*delta)*SPSI)))/(1i+Gamma1));
fun3a = arrayfun(@(idx) S0i(idx) * fun0(1i*PSIi{idx}-(Gamma1-1i*delta).*SPSI), 2:n, 'Uniform', 0);
sum3a = sum([fun3a{:}]);
fun3_formula = ((exp(-((1i*Gamma1)*x1)))*exp((1i*Gamma1+delta)*Yn0)*(sum3a-(K-1)*fun0(-(Gamma1-1i*delta).*SPSI)))/(1i+Gamma1);
a = vpa( subs(fun3_formula, Gamma1, 1) )
%q = quadgk(fun3,0,Inf)
q_formula = int(fun3_formula, Gamma1, 0, inf);
q = matlabFunction(q_formula, 'vars', x1);
%q1= exp(-(delta*x1))*max(q/pi, 0)
q1 = @(x1_x1)  max(exp(-(delta * x1_x1)) * q(x1_x1)/pi, 0);
%x = fminbnd(q1,-Inf,+Inf)
%initial_guess = rand;
%x_theta1 = fminsearch(q1, initial_guess);
%x = fminbnd(q1,-Inf,+Inf)
[x,fval]=fminbnd(q1,-Inf,+Inf)

My aim is to minimize q1 w.r.t. x1. However MATLAB does not give me any answer.

Can anyone please help?

Thanks.

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Answer 1

Walter Roberson 2017-5-20

2
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/341058-unable-to-find-a-minima-for-a-symbolic-integral#answer_267756

After adjusting your code to allow computations to proceed, I find that your q1 values are mostly complex. However, there are values at which the imaginary part vanishes; there appear to be a series of such points towards positive x1 values, with real magnitude growing smaller and smaller (damping.) The negative x1 at which the imaginary components vanish are approximately

-192.141514316517 -188.655787231716 -174.349836830908 -164.117505872502 -159.453121627949 -155.267512530917 -148.325029411257 -134.466528151559 -128.757615075388 -120.467541617748 -113.873014115027 -112.710435738364 -102.681758746963 -89.6122555789685 -80.9922253336114 -75.3528553255843 -69.9030673982657 -66.9348134318309 -65.0450650466323 -62.8009792161248 -59.9236072335582 -57.4428794570977 -55.2754416634021 -53.3054688314303 -51.3143402109365 -49.3375806679832 -47.326990485515 -46.5858590341201 -30.1886651033708 -29.0337559483385 -27.498965552452 -25.8680259013049 -24.0692753628916 -22.0290261987157 -19.9586548105586 -17.9076174690997 -15.3127142047397 -12.4586653906846 -11.0638473820923 -8.23092022558657 -3.48183500381984

Note: to 16 digits of precision, some of those locations have a significant imaginary component. That is because the function is so steep at those points that the adjacent representable value has an imaginary component of opposite sign and larger magnitude: the above are the closest representations in binary floating point.

In order to minimize the function, we must select the minimum value that it achieves at the points where the imaginary component vanishes. The function is effectively not continuous for your purposes, so you have to find the points where it exists and then take the minimum among those points. The real components corresponding to the above locations are

-1.54114175976339e+48 5.66125848286782e+46 -4.43941266035521e+42 9.32268918449569e+38 -3.44960496385963e+37 1.19438990077474e+36 -1.10285019640586e+34 2.79224578549348e+29 -5.3872782404782e+26 1.31888812894348e+25 -1.6629378087883e+22 5.44482392809658e+21 -1.00225388086346e+19 747319448309669 -1414346152497.49 13447342311.9391 -56267167.2873899 9465813.2535106 543870.090637689 352535.261711927 20688.0050475019 7617.06546618484 881.274400895307 423.751022552601 78.9506814740713 27.408464913681 7.01669839359559 4.83341055220347 -1.57164698806938e-05 -5.56618491318963e-06 -1.6584846542248e-06 -3.51523264324361e-07 -1.07468533983336e-07 -1.28720508948511e-08 -4.62935878887861e-09 -3.23320760267986e-10 -1.17979676353903e-10 -6.2872110686137e-12 -3.62149260857601e-12 4.92853608204376e-13 -4.59671900947997e-14

You can observe the dampening that I mentioned earlier. The minimum of those is the first of them, -1.54114175976339e+48, at x1 = -192.141514316517

As you indicated earlier that you are expecting your q values to be real, I would tend to suspect that you still have an error in how you set up your computations. Or, possibly your expectation that the q values are real is mistaken.

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Walter Roberson 2017-5-31

fweb8.m

Here is the altered code for the original delta. This code happens to rely on R2016b or later, in a minor way; if you have an earlier version then move the function WB at the end of the file into a separate file WB.m .

On the other hand, it also uses vpaintegral() for accurate integration at a reasonable speed, and vpaintegral() is also new in R2016b. If you try to replace vpaintegral(expression) by vpa(int(expression)) then the int will take a long time and will not succeed anyhow, so if you are using a release before R2016b then that line would need to be rewritten a little.

Anyhow, when you execute the code, you will get a useless x and an fval of NaN coming out of the fminbnd() . That reflects the fact that your q1 function is complex-valued.

If you plot the imaginary component of q1 as x1 varies, you can see that the imaginary component crosses 0 at some locations. Those are the places at which the function can be said to "exist".

I calculated q1 over a linspace of x1 values, took the imaginary component, and looked for sign changes in that. I took the x1 values on both sides of the sign changes and used those values as the bracketing values for fsolve() of q1 in order to find the best points possible in floating point closest to where the imaginary component is 0. Then, q1() evaluated at those crossings are (as best possible in floating point) the set of possible values for the function. Take the minimum among those to determine what the minimum value of the function is.

I am still executing on doing the same thing with the revised delta.

Walter Roberson 2017-6-1

在 MATLAB Online 中打开

When I execute with the revised delta, I find that the function starts out at effectively 0, and its real part is otherwise strictly positive and increasing; by about roughly +20-ish the real magnitude overflows double precision. I did not attempt to calculate symbolically to figure out whether there were locations where it might decrease from there.

The q1 values are mostly complex, but there is a bit of a wave to the imaginary part, so the imaginary part crosses 0 about 5 times, all for negative x1 (it might possibly have crossed again in more negative areas). The function therefore exists only at those 5+ locations. As the real component is strictly increasing and non-negative, the minima of the function is the one furthest left, closest to 0, where the value is down around 1E-200. If there are further crossings of the imaginary component, then they would be associated with real values closer to 0, perhaps on the order of 1E-350 or less.

Such values are, of course, essentially numeric noise compared to the uncertainty in the coefficients. For example, you have g=.111 and we do not know whether that means 111/1000 or 1/9 or 7998392938210001/72057594037927936, and the differences between those are going to have noticeable effect on exactly what the minima is.

As you do not appear to expect that q1 will be generating mostly complex values, I would suggest to you that either your equations are incorrect or that your implementation is incorrect or that your expectations are wrong.

By the way, you may wish to comment out

BPHNEW22 = inv(APHNEW22);

as you do not use the result of that computation.

RB 2017-6-14

编辑：RB 2017-6-14

check7.m

Hello Walter Sir. I have finally got MATLAB 2016b. I have made few changes in the parameter values of SIGMA and PSICURL. I also make some changes in the program and now only integrate the real part of the function. I am attaching the program.

However I am still stuck up with one change I want to make. If you say I can ask that as a separate question.

In line 78 I NEED TO DEFINE the argument of fun0 in the fun3a differently

for k=2:n,

it would be defined as fun0((1i-(1./(n-1)).*(Gamma1-1i*delta))PSI{kdx} for the kth component of the sum

and

-(1./(n-1)).*(Gamma1-1i*delta))PSI{k'dx} whenever k' is not equal to k. When the outer k and the argument k match PSI{kdx} is to be multiplied by a different coefficient.

I want to attach an indicator to 1i which takes he value 1 when the two k's match and o otherwise.

IN A WAY THIS FUN ITSELF BREAKDOWN INTO TWO FUNCTIONS.

Mathematically writing the argument of fun0 to be used is -1i.*e_{k}+(gamma1-1i*delta)/(n-1).*w (in defining fun3a)

where e_{k} is the kth element of the canonical basis i.e. e_{k}={0,0,...,1,0,...,0} i.e. 1 at the kth position and w=(1,1,...,1), i.e. vector of ones.

I am actually trying to make the point that 1 at the kth position will appear depending on the index of the outer sum.

I would be grateful if you can help.

Thanks and regards, RB

RB 2017-6-16

编辑：Walter Roberson 2017-6-17

在 MATLAB Online 中打开

MOREBlessings7.m

Hello Walter Sir! Even if I keep the same program I always get a NAN answer

a =

real(exp(-x1*1.0i)*(0.75943101243872985951205713777525 + 2.1976661532219400420125628728784i))

q1(x1) =

(exp(-(3*x1)/4)*vpaintegral(real((exp(- (Gamma1*8054717210969435543i)/7656119366529843200 - 24164151632908306629/30624477466119372800)*exp(-Gamma1*x1*1i)*((exp(-230567373719136843/140737488355328000)*exp(-(1.0*(Gamma1*1.0329815929729743604700473384972e86i + 1.6765078623734534328608629970877e84*Gamma1^2 + Gamma1^3*1.0780318937778291795601163550056e83i + 1.5863831808884602107557503573376e81*Gamma1^4 + Gamma1^5*4.3684168722114149618094477974258e79i + 6.1712208961864093957537400603178e77*Gamma1^6 + Gamma1^7*8.5991814149927599008453273539918e75i + 1.1857215511099378371784246839875e74*Gamma1^8 + Gamma1^9*8.2495336327316538962580933753009e71i + 1.1187980531997824531534175582746e70*Gamma1^10 + Gamma1^11*3.0964336010111026102191203786472e67i + 4.1482827355588000578638295217891e65*Gamma1^12 + 2.1822819387885352810351599760711e86))/(6.6363341255449211491993956377153e84*Gamma1^2 + 3.318235837023957353972456579736e81*Gamma1^4 + 8.6140777844393491797400287388507e77*Gamma1^6 + 1.2282988425856020736559905470718e74*Gamma1^8 + 9.1510829296196987195954021939518e69*Gamma1^10 + 2.7910157008965129256960452044554e65*Gamma1^12 + 5.3754255285041449103050715229169e87)))/(34*(Gamma1*0.03202777296534638223127988958106i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.066009087273428461770154800745)^(3/2)) - (82061027611601696916992681239449003143812732116138125132270354902795627792039962962826633409505789753426816171769856*exp(-(Gamma1*1341278620012886930112312050705346157018090730920584404222763475561995949499088811360719562861745230026612547562241382383131554413580299712649352491399606389432409761311941283285958242592240465164107776i + 19191724434469919898783546353393138867212404190953932419752727408469979217943227283729243850360554769804016985151361906653332465712300837880601720236440348736838559062673628542971068008024607732716640*Gamma1^2 + Gamma1^3*385650316364212456686182040345212307485569021494079084463693126150456717654211346287611264004257526654308558880782092829524397689747024998124996995027127993243648766228079179400762852304664959909888i + 5166545631122402068296717272990417071064954803633896761276211381896086289143999746722314734375843214603121466908956034156113994249888887409330172218849209549339919042499982551875829666714850970368*Gamma1^4 + 1016589979049509778074605877796322964810003159471051238251778272749909048448096878987252424452738514827274763283649208752558527508929538850717986994700621798358920375299697866605975906408647676196957115)/(15387058114967613197924658046000514599133823107072000*(2133020007780744177528917181658317755445864832862929330528989757874410605462702316243763243686016556482568181620809599433430542790836277078718206752*Gamma1^2 + 225911638767354844823673280626496428920139212408012509844508623080761992498822719871438666862870691015631431971574844374647552513488781902745856*Gamma1^4 + 4369558635059319222086544295263970793352985140735528049789141125406244101148201677928593678577835330888651078287750869256596828785716288056750876810081))))/(111*(Gamma1*64188102548952022979662410808778633835137640872169966138009649443419197992i - 475301629249632222238141155462927871942673013524673072861239379425059184*Gamma1^2 + 2090348926629073384108029764565449882744956861872235361588599558625906263759)^(3/2)) + (exp(-(1.0*(Gamma1*1.0265245425492754331545788631625e86i + 1.66141937557519186634729365374e84*Gamma1^2 + Gamma1^3*1.0732425410859030157751030983628e83i + 1.5778079581944338376636165139127e81*Gamma1^4 + Gamma1^5*4.3555653357524718379368376036676e79i + 6.150562877525448190652506297328e77*Gamma1^6 + Gamma1^7*8.5842379718602555505841889427687e75i + 1.1834568744729639804413955914266e74*Gamma1^8 + Gamma1^9*8.2431392429561570476808657111523e71i + 1.1178643917521869846252124209714e70*Gamma1^10 + Gamma1^11*3.0964336010111026102191203786473e67i + 4.1482827355588000578638295217891e65*Gamma1^12 + 2.1106205673560165846874346767372e86))/(6.5969235462864929318331888117299e84*Gamma1^2 + 3.3042108161578118051013135625652e81*Gamma1^4 + 8.5896964674839841877648805872288e77*Gamma1^6 + 1.226216361693721712249579330494e74*Gamma1^8 + 9.1440644797404447196437405735224e69*Gamma1^10 + 2.7910157008965129256960452044554e65*Gamma1^12 + 5.3325434217044062971764830025211e87))*exp(-8516656400399249/5629499534213120))/(34*(Gamma1*0.032001167692367106155424852691434i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0638501991252423475445245111441)^(3/2)) + (exp(-(1.0*(Gamma1*3.0231205718655231732821322010059e87i + 4.7528829432714127779195046721384e85*Gamma1^2 + Gamma1^3*3.2161071233825379149894153681579e84i + 4.6810030347188917045058615935989e82*Gamma1^4 + Gamma1^5*1.3240139511371373214601271328384e81i + 1.8618500438774192167647490682916e79*Gamma1^6 + Gamma1^7*2.6393288814735042718430109380343e77i + 3.6322471454985312655701937912683e75*Gamma1^8 + Gamma1^9*2.5574428509161158063894565183731e73i + 3.4660738570463819270454033720129e71*Gamma1^10 + Gamma1^11*9.6763550031596956569347511832726e68i + 1.2963383548621250180824467255591e67*Gamma1^12 + 4.4990331850039595202549430478367e87))/(1.9470860186731674785520195842126e86*Gamma1^2 + 9.9162796897606315254329616355965e82*Gamma1^4 + 2.612847087899083380294928905136e79*Gamma1^6 + 3.770738799705371135784737460618e75*Gamma1^8 + 2.8368521264729020505791921052363e71*Gamma1^10 + 8.7219240653016028928001412639231e66*Gamma1^12 + 1.5427973389439478621827898047778e89))*exp(-56886336528070163/140737488355328000))/(34*(Gamma1*0.031747693062674549153556260940568i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0432920760167893157882773201646)^(3/2)) + (exp(-51028454327606229/35184372088832000)*exp(-(1.0*(Gamma1*1.0232747342877580616922990124253e86i + 1.6538327422762412809444063887048e84*Gamma1^2 + Gamma1^3*1.0708300599001110026774474738702e83i + 1.5734905845585001736503223391805e81*Gamma1^4 + Gamma1^5*4.34908818002030540178950006078e79i + 6.1401543174940250433564338024908e77*Gamma1^6 + Gamma1^7*8.5767035408247725352146049318282e75i + 1.1823152518641995053706948953344e74*Gamma1^8 + Gamma1^9*8.2399142889743293928193985574756e71i + 1.1173935706078050136442873037417e70*Gamma1^10 + Gamma1^11*3.0964336010111026102191203786472e67i + 4.1482827355588000578638295217891e65*Gamma1^12 + 2.0747054550794129278765149392533e86))/(6.5770837135667683194491238445949e84*Gamma1^2 + 3.297145452111292239973043420896e81*Gamma1^4 + 8.5774076602459073477466092587477e77*Gamma1^6 + 1.2251663351254840293154671172607e74*Gamma1^8 + 9.1405245922804164726821775086468e69*Gamma1^10 + 2.7910157008965129256960452044554e65*Gamma1^12 + 5.3109780964860548256010243226521e87)))/(34*(Gamma1*0.031987727775179785433288018449724i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0627596155697624894419014385389)^(3/2)) + (exp(-23319365616740131/17592186044416000)*exp(-(1.0*(Gamma1*1.0167350154383472316564652403871e86i + 1.6385805740998411550002935131025e84*Gamma1^2 + Gamma1^3*1.0659712030211615496951933351716e83i + 1.5647994011239688078058709273809e81*Gamma1^4 + Gamma1^5*4.3360354977356961091904191042021e79i + 6.1191851725852615331008581685237e77*Gamma1^6 + Gamma1^7*8.5615142212086336916533732937977e75i + 1.1800141874744571867024676047062e74*Gamma1^8 + Gamma1^9*8.2334109369177102203274058790643e71i + 1.1164442488577693301905669633542e70*Gamma1^10 + Gamma1^11*3.0964336010111026102191203786472e67i + 4.1482827355588000578638295217891e65*Gamma1^12 + 2.0027389532181636950057005480914e86))/(6.537149323674884426336732376857e84*Gamma1^2 + 3.2829138442868862086524200039132e81*Gamma1^4 + 8.5526417855429477334462411107345e77*Gamma1^6 + 1.22304937358518073668373722653e74*Gamma1^8 + 9.1333856710566214412695789468432e69*Gamma1^10 + 2.7910157008965129256960452044554e65*Gamma1^12 + 5.267615796103985844624405980646e87)))/(34*(Gamma1*0.031960580197478839359150451582788i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0605567232510035527675327716061)^(3/2)) + (exp(-12559496557454751/14073748835532800)*exp(-(1.0*(Gamma1*1.3492961589824018518941369672182e200i + 2.7095874324484676861505491248312e198*Gamma1^2 + Gamma1^3*3.6800320749645330656855385928489e196i + 7.5360073317981897299083284630961e194*Gamma1^4 + Gamma1^5*2.4796824956596509634366217782796e192i + 5.2845408740430355809807324737466e190*Gamma1^6 + 2.4419547581928526530935440636101e200))/(4.5247152879063454443521492404428e201*Gamma1^2 + 8.7227583476935479371666622113156e197*Gamma1^4 + 5.3460266345371206314531619971808e193*Gamma1^6 + 7.2363945740656984509068426958995e204))*exp(-(1.0*(Gamma1*4.7401872112383427910492266248266e203i + 6.3269983652011038452132241205712e201*Gamma1^2 + Gamma1^3*2.2586807267428120978007657864388e200i + 2.8703140440523992782890999895444e198*Gamma1^4 + Gamma1^5*2.3311211941517500740584567012758e196i + 2.9226558066679280959336246616031e194*Gamma1^6 + 8.559982913607007561711776898571e203))/(2.2623576439531727221760746202214e203*Gamma1^2 + 4.3613791738467739685833311056578e199*Gamma1^4 + 2.6730133172685603157265809985904e195*Gamma1^6 + 3.6181972870328492254534213479498e206))*exp(-(1.0*(Gamma1*3.9678466999891490960445584036823e200i + 5.291696060859623677060843135147e198*Gamma1^2 + Gamma1^3*1.8906646354707971861297654096649e197i + 2.4017903776105846253657910336226e195*Gamma1^4 + Gamma1^5*1.9513026332957644799521100324772e193i + 2.4459340888265602126040129847319e191*Gamma1^6 + 7.094711468789233823175095181605e200))/(1.8098861151625381777408596961771e199*Gamma1^2 + 3.4891033390774191748666648845263e195*Gamma1^4 + 2.1384106538148482525812647988723e191*Gamma1^6 + 2.8945578296262793803627370783598e202))*exp(-(1.0*(Gamma1*3.1774482426756686287578645640572e198i + 6.351607795413577608879668324604e196*Gamma1^2 + Gamma1^3*8.6669915361696824195120370096116e194i + 1.7691025738547606148034341863167e193*Gamma1^4 + Gamma1^5*5.8407617780214561075184082859134e190i + 1.2411409911080115051304938412914e189*Gamma1^6 + 5.2886538439909066105583482903913e198))/(4.5247152879063454443521492404428e197*Gamma1^2 + 8.7227583476935479371666622113156e193*Gamma1^4 + 5.3460266345371206314531619971808e189*Gamma1^6 + 7.2363945740656984509068426958995e200)))/(34*(Gamma1*0.031862922343727218850836960347101i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0526323541961611172873369130806)^(3/2)) + (exp(-(1.0*(Gamma1*9.4857041353434596576019451128745e200i + 1.2338010365863332897314350303192e199*Gamma1^2 + Gamma1^3*4.6036801839202486464088472843649e197i + 5.7871332074341029924325994411356e195*Gamma1^4 + Gamma1^5*4.7931444105664442793150426357609e193i + 5.9715187715492192690527660760055e191*Gamma1^6 + 1.1818361749619796941278706830466e201))/(4.3577390086025612640899906837408e199*Gamma1^2 + 8.4703831654076851152412283698911e195*Gamma1^4 + 5.2207291352901568666534785128719e191*Gamma1^6 + 6.8734918530881495984186540677981e202))*exp(-(1.0*(Gamma1*1.5920611363330527173312074887732e199i + 3.1934179465792934612237748591518e197*Gamma1^2 + Gamma1^3*4.1969865889333450820465932009697e195i + 8.803829850070848661613833880286e193*Gamma1^4 + Gamma1^5*2.7058246319410052178394152869394e191i + 6.0602587456445874273949894594307e189*Gamma1^6 + 1.4434367734556503649106940903375e199))/(2.1788695043012806320449953418704e198*Gamma1^2 + 4.2351915827038425576206141849455e194*Gamma1^4 + 2.6103645676450784333267392564359e190*Gamma1^6 + 3.436745926544074799209327033899e201))*exp(-(1.0*(Gamma1*2.2533276420003533655656847724089e203i + 3.0024726237967064778801453325481e201*Gamma1^2 + Gamma1^3*1.093582984364048890759009074549e200i + 1.3886280525015427425429698997929e198*Gamma1^4 + Gamma1^5*1.1385796378572916208840255624511e196i + 1.4270780305995742655925901667984e194*Gamma1^6 + 3.9354235028697238720817633188897e203))/(1.0894347521506403160224976709352e203*Gamma1^2 + 2.1175957913519212788103070924728e199*Gamma1^4 + 1.305182283822539216663369628218e195*Gamma1^6 + 1.7183729632720373996046635169495e206))*exp(-45884262366509229/281474976710656000)*exp(-(1.0*(Gamma1*5.3868091142731872094302948535704e199i + 1.0926554575820497787092287149937e198*Gamma1^2 + Gamma1^3*1.4195396979476968851985553635517e196i + 3.0019647188839996455256009735909e194*Gamma1^4 + Gamma1^5*9.1472721080586635196527017714386e191i + 2.0642737789230607738205986225572e190*Gamma1^6 + 6.7727483066810753761583787757145e199))/(1.7430956034410245056359962734963e201*Gamma1^2 + 3.3881532661630740460964913479564e197*Gamma1^4 + 2.0882916541160627466613914051488e193*Gamma1^6 + 2.7493967412352598393674616271192e204)))/(34*(Gamma1*0.0316844087499607602823827903741i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.038460665276222270633655088426)^(3/2)) + (exp(-26953980483409361/562949953421312000)*exp(-(1.0*(Gamma1*1.8773206622117686958733095052716e202i + 2.4080298873072765817654115580014e200*Gamma1^2 + Gamma1^3*9.1531237877019161817377669456338e198i + 1.1466930749166331759767590600804e197*Gamma1^4 + Gamma1^5*9.5905422669546946430168355030963e194i + 1.1943037543098438538105532152011e193*Gamma1^6 + 1.6537016960235737819356325557033e202))/(8.5992195980857562645220284847066e200*Gamma1^2 + 1.6823214083608870358164317156984e197*Gamma1^4 + 1.0441458270580313733306957025744e193*Gamma1^6 + 1.3490813575716592917832734833771e204))*exp(-(1.0*(Gamma1*4.2762596488874321929654344546617e202i + 8.5995016890742024334761798252137e200*Gamma1^2 + Gamma1^3*1.1327993141175121467156961417081e199i + 2.3854232559328062755875345789334e197*Gamma1^4 + Gamma1^5*7.3312257547619643529261553162292e194i + 1.6514190231384486190564788980459e193*Gamma1^6 + 3.7911723025743976451583317593063e202))/(1.3758751356937210023235245575531e204*Gamma1^2 + 2.6917142533774192573062907451175e200*Gamma1^4 + 1.670633323292850197329113124119e196*Gamma1^6 + 2.1585301721146548668532375734034e207))*exp(-(1.0*(Gamma1*1.783024475181156776516583832846e202i + 2.347805235035730970120745584033e200*Gamma1^2 + Gamma1^3*8.6931969130416826963997124022413e198i + 1.1009611856573285758527091356999e197*Gamma1^4 + Gamma1^5*9.1085533024313640074646795377391e194i + 1.1416624244796594124740721334387e193*Gamma1^6 + 2.52248379217254365453969464406e202))/(8.5992195980857562645220284847066e201*Gamma1^2 + 1.6823214083608870358164317156984e198*Gamma1^4 + 1.0441458270580313733306957025744e194*Gamma1^6 + 1.3490813575716592917832734833771e205))*exp(-(1.0*(Gamma1*2.0118460977855156401938763888248e202i + 4.0350829743207225070187406406767e200*Gamma1^2 + Gamma1^3*5.3299462227710959142357299054774e198i + 1.1202131986729595282966797632803e197*Gamma1^4 + Gamma1^5*3.4498445022547072087968508262436e194i + 7.7571311944250719070655865080708e192*Gamma1^6 + 1.6180232503450128857243575873427e202))/(2.7517502713874420046470491151061e201*Gamma1^2 + 5.3834285067548385146125814902349e197*Gamma1^4 + 3.341266646585700394658226248238e193*Gamma1^6 + 4.3170603442293097337064751468067e204)))/(34*(Gamma1*0.03159019295190926028086521230677i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0327654318174153650603690777816)^(3/2)) + (exp(-(1.0*(Gamma1*3.9934303926562140113950086139668e197i + 7.9754199118010976461240858186136e195*Gamma1^2 + Gamma1^3*1.0818363734948675145411179246024e194i + 2.2190453607592175932819400198192e192*Gamma1^4 + Gamma1^5*7.2285192180618246889227340584988e189i + 1.5514262388850143814131173016143e188*Gamma1^6 + 5.6235207011770723801179789142146e197))/(5.6321563271813647085790273100601e196*Gamma1^2 + 1.0885586598001392343656583068528e193*Gamma1^4 + 6.682533293171400789316452496476e188*Gamma1^6 + 8.9666282645681049785802112203344e199))*exp(-(1.0*(Gamma1*1.8353461332399711001475843549654e201i + 2.4500803151524855994079204916572e199*Gamma1^2 + Gamma1^3*8.7991299027751508966057194176714e197i + 1.1180992076457914661672602004875e196*Gamma1^4 + Gamma1^5*9.1065804471086493473207623173288e193i + 1.1416624244796594124740721334387e192*Gamma1^6 + 3.3113985640487773195101703869914e201))/(8.8002442612208823571547301719689e200*Gamma1^2 + 1.7008729059377175536963411044575e197*Gamma1^4 + 1.0441458270580313733306957025744e193*Gamma1^6 + 1.4010356663387664029031580031772e204))*exp(-182228629708701923/281474976710656000)*exp(-(1.0*(Gamma1*8.453707308465437741068840130803e200i + 1.7047051715726331091310891717579e199*Gamma1^2 + Gamma1^3*2.2895933171167011996749566844056e197i + 4.7286958294885203084097923371589e195*Gamma1^4 + Gamma1^5*1.5293765212792619951905821163791e193i + 3.3028380462768972381129577960917e191*Gamma1^6 + 1.4484259544396131016859640132719e201))/(2.81607816359068235428951365503e202*Gamma1^2 + 5.442793299000696171828291534264e198*Gamma1^4 + 3.341266646585700394658226248238e194*Gamma1^6 + 4.4833141322840524892901056101672e205))*exp(-(1.0*(Gamma1*3.0760292511692895007521585378631e198i + 4.0848965997588920518775625139386e196*Gamma1^2 + Gamma1^3*1.4747345962579272050554843355796e195i + 1.8697919154572159631287090321871e193*Gamma1^4 + Gamma1^5*1.5262660073739408860994678703451e191i + 1.9108860068957501660968851443218e189*Gamma1^6 + 5.2091089054482509124028366701373e198))/(1.408039081795341177144756827515e197*Gamma1^2 + 2.721396649500348085914145767132e193*Gamma1^4 + 1.670633323292850197329113124119e189*Gamma1^6 + 2.2416570661420262446450528050836e200)))/(34*(Gamma1*0.031805525557062547264661108423329i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0479755982389169916825056772086)^(3/2)) + (exp(-(1.0*(Gamma1*1.8047911606811661389793205259912e196i + 3.6231945040043073835005373935683e194*Gamma1^2 + Gamma1^3*5.0617817329266027496392865544212e192i + 1.0129972327279145556345063576418e191*Gamma1^4 + Gamma1^5*3.5272872700325121879483649384828e188i + 7.206015155344337012360272841178e186*Gamma1^6 + 4.9985117410804145559631406300887e196))/(2.674151939551475357488610598575e195*Gamma1^2 + 5.0998675849108096654827844729112e191*Gamma1^4 + 3.1038817689002121680460633251319e187*Gamma1^6 + 4.3579129218992666176446447149419e198))*exp(-(1.0*(Gamma1*1.4277923388068804307977182664984e201i + 1.9046114012060879844822206603153e199*Gamma1^2 + Gamma1^3*6.6327500332854075615876875581496e197i + 8.4314674861976931567992507083489e195*Gamma1^4 + Gamma1^5*6.7652461207277799335225216919979e193i + 8.4844115969059112104196799452936e191*Gamma1^6 + 2.5871240243218828498435041460295e201))/(6.6853798488786883937215264964376e200*Gamma1^2 + 1.2749668962277024163706961182278e197*Gamma1^4 + 7.7597044222505304201151583128298e192*Gamma1^6 + 1.0894782304748166544111611787355e204))*exp(-35484847207211259/17592186044416000)*exp(-(1.0*(Gamma1*1.5522942142340081184310690467233e199i + 3.0618202891221212200274173974362e197*Gamma1^2 + Gamma1^3*4.3547855644138700167737653046794e195i + 8.6098596323149330923251693359332e193*Gamma1^4 + Gamma1^5*3.0355697590562191511630689484682e191i + 6.1363667625536883882895321716923e189*Gamma1^6 + 3.4177836396864383618739457153528e199))/(5.3483038791029507149772211971501e200*Gamma1^2 + 1.0199735169821619330965568945822e197*Gamma1^4 + 6.2077635378004243360921266502638e192*Gamma1^6 + 8.7158258437985332352892894298838e203))*exp(-(1.0*(Gamma1*1.1897718594891862175940866320646e198i + 1.6134359497507293890109693294462e196*Gamma1^2 + Gamma1^3*5.5269800739841317292541438190644e194i + 7.0760699691199125925453000336422e192*Gamma1^4 + Gamma1^5*5.6373598313141222031748046498169e190i + 7.1004979447672048383504947396023e188*Gamma1^6 + 2.584671940573904482506798823395e198))/(5.3483038791029507149772211971501e196*Gamma1^2 + 1.0199735169821619330965568945822e193*Gamma1^4 + 6.2077635378004243360921266502638e188*Gamma1^6 + 8.7158258437985332352892894298838e199)))/(34*(Gamma1*0.032105304692730621212007927235884i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.072300409596612287458469552444)^(3/2)) + (exp(-(1.0*(Gamma1*7.6515907783447445368126019238455e198i + 1.5336452375618221428287566874314e197*Gamma1^2 + Gamma1^3*2.1237299201957304450765740982868e195i + 4.2818022381188249618410162888258e193*Gamma1^4 + Gamma1^5*1.4618098860599083975439009325582e191i + 3.0301293728222937136974947297153e189*Gamma1^6 + 1.7990103714241116867851282887338e199))/(1.1170163612006989144529339029413e198*Gamma1^2 + 2.138874507008380006596703011881e194*Gamma1^4 + 1.305182283822539216663369628218e190*Gamma1^6 + 1.8077164451128433145618768526298e201))*exp(-(1.0*(Gamma1*5.2402941776932677173769517202758e198i + 1.04065905111415481004950908226e197*Gamma1^2 + Gamma1^3*1.4547053999573495170959268425538e195i + 2.9141243226652640555890177448322e193*Gamma1^4 + Gamma1^5*1.0015038096454447844867102886297e191i + 2.0642737789230607738205986225572e189*Gamma1^6 + 1.0767419723339296620479398900996e199))/(1.7872261779211182631246942447062e200*Gamma1^2 + 3.4221992112134080105547248190096e196*Gamma1^4 + 2.0882916541160627466613914051488e192*Gamma1^6 + 2.8923463121805493032990029642077e203))*exp(-(1.0*(Gamma1*2.3687753550953951937689975795986e202i + 3.1605418163051591254828941092429e200*Gamma1^2 + Gamma1^3*1.110820873648944636582759312852e199i + 1.4118985242135454145416319276979e197*Gamma1^4 + Gamma1^5*1.1380372841145442556807665617543e195i + 1.4270780305995742655925901667984e193*Gamma1^6 + 4.2867286044947968072861977490055e202))/(1.1170163612006989144529339029413e202*Gamma1^2 + 2.138874507008380006596703011881e198*Gamma1^4 + 1.305182283822539216663369628218e194*Gamma1^6 + 1.8077164451128433145618768526298e205))*exp(-13858393563289549/8796093022208000)*exp(-(1.0*(Gamma1*9.8861512270037404658762128287208e200i + 1.3324231121301715594750862993513e199*Gamma1^2 + Gamma1^3*4.636012344493229918545356442719e197i + 5.9181583733709127841577144786045e195*Gamma1^4 + Gamma1^5*4.7495853825306819638653653201255e193i + 5.9715187715492192690527660760056e191*Gamma1^6 + 2.0051594166927278415988471130529e201))/(4.4680654448027956578117356117654e199*Gamma1^2 + 8.5554980280335200263868120475239e195*Gamma1^4 + 5.2207291352901568666534785128719e191*Gamma1^6 + 7.2308657804513732582475074105193e202)))/(34*(Gamma1*0.032014516524782127958765103143603i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0649333917329232012219417420629)^(3/2)) + (exp(-(1.0*(Gamma1*7.9389569386867025234269554384532e197i + 1.5934271464075424985747215755997e196*Gamma1^2 + Gamma1^3*2.2233453820866750066531786928611e194i + 4.4541292698300665993726982862188e192*Gamma1^4 + Gamma1^5*1.5466935423458446636332403836513e190i + 3.16617599772451914574105571758e188*Gamma1^6 + 2.1521194984125836258100074082344e198))/(1.1738754428598960982339336596362e197*Gamma1^2 + 2.2399552678600130714368677730275e193*Gamma1^4 + 1.3637823047288573039421331625461e189*Gamma1^6 + 1.911151210403685846010396435066e200))*exp(-137478419701786319/70368744177664000)*exp(-(1.0*(Gamma1*7.8267997888193687261015350148897e199i + 1.0440923105750833126008987347995e198*Gamma1^2 + Gamma1^3*3.6406698028980990260640471520612e196i + 4.6278989172571538953894900018013e194*Gamma1^4 + Gamma1^5*3.7156948802992159387429223724985e192i + 4.6598466305292220917309066670967e190*Gamma1^6 + 1.4179458885978193777581292616603e200))/(3.6683607589371753069810426863633e199*Gamma1^2 + 6.9998602120625408482402117907109e195*Gamma1^4 + 4.2618197022776790748191661329566e191*Gamma1^6 + 5.9723475325115182687824888595813e202))*exp(-(1.0*(Gamma1*1.3047147195008001937501346417731e201i + 1.7677991232270860290349062299034e199*Gamma1^2 + Gamma1^3*6.0688786932964413042433083486442e197i + 7.7667022012285224442576263815e195*Gamma1^4 + Gamma1^5*6.1939144036249817633774052033075e193i + 7.7995347220234700657015720176399e191*Gamma1^6 + 2.8080720079311789478830567018707e201))/(5.8693772142994804911696682981812e199*Gamma1^2 + 1.1199776339300065357184338865137e196*Gamma1^4 + 6.8189115236442865197106658127306e191*Gamma1^6 + 9.5557560520184292300519821753301e202))*exp(-(1.0*(Gamma1*8.5297919276127440118060174074907e197i + 1.6840609210541464137493095905115e196*Gamma1^2 + Gamma1^3*2.3894232737085276461954695725321e194i + 4.7328013837133389543130294693869e192*Gamma1^4 + Gamma1^5*1.6627267482627668489896553745065e190i + 3.3702429043641808552173038735629e188*Gamma1^6 + 1.8605198832175830403389691455605e198))/(2.9346886071497402455848341490906e199*Gamma1^2 + 5.5998881696500326785921694325687e195*Gamma1^4 + 3.4094557618221432598553329063653e191*Gamma1^6 + 4.7778780260092146150259910876651e202)))/(34*(Gamma1*0.032092627172359071080517653706451i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.071271690498876208774629910698)^(3/2)) + (exp(-(1.0*(Gamma1*1.9495514619169829966957821538733e199i + 3.9111882578781426652574747416473e197*Gamma1^2 + Gamma1^3*5.4437477896511019601752055170938e195i + 1.0928623149574468679111226997216e194*Gamma1^4 + Gamma1^5*3.7739100806188718058753499410068e191i + 7.7571311944250719070655865080712e189*Gamma1^6 + 5.0537914950917841584554860064893e199))/(2.8705875670587435936106136473074e198*Gamma1^2 + 5.4838192467134238958642442835697e194*Gamma1^4 + 3.341266646585700394658226248238e190*Gamma1^6 + 4.6643665020468703965923728432063e201))*exp(-(1.0*(Gamma1*4.1837630772784235350429220727832e199i + 8.2760166000846023331297056502113e197*Gamma1^2 + Gamma1^3*1.168503799017772866697592876497e196i + 2.3230912355697042329149236489276e194*Gamma1^4 + Gamma1^5*8.1029188104361096482199862617671e191i + 1.6514190231384486190564788980459e190*Gamma1^6 + 8.9517020470122807788602250366819e199))/(1.4352937835293717968053068236537e201*Gamma1^2 + 2.7419096233567119479321221417848e197*Gamma1^4 + 1.670633323292850197329113124119e193*Gamma1^6 + 2.3321832510234351982961864216031e204))*exp(-51431334729405579/28147497671065600)*exp(-(1.0*(Gamma1*1.5281181390785747263317415843459e203i + 2.03863138726424402404953874384e201*Gamma1^2 + Gamma1^3*7.1270054221039092150432941623907e199i + 9.0593187228204991129755848707907e197*Gamma1^4 + Gamma1^5*7.2829841766667204794952744754636e195i + 9.1332993958372752997925770675095e193*Gamma1^6 + 2.7674321886887240356934377669209e203))/(7.1764689176468589840265341182685e202*Gamma1^2 + 1.3709548116783559739660610708924e199*Gamma1^4 + 8.353166616464250986645565620595e194*Gamma1^6 + 1.1660916255117175991480932108016e206))*exp(-(1.0*(Gamma1*6.3714161319675999330936012448938e201i + 8.6178858789284050505662034820823e199*Gamma1^2 + Gamma1^3*2.9715454920947833257101490207718e198i + 3.799742696265408461495050956367e196*Gamma1^4 + Gamma1^5*3.0365669675890858260575374495146e194i + 3.8217720137915003321937702886435e192*Gamma1^6 + 1.3453170468108492756926215152749e202))/(2.8705875670587435936106136473074e200*Gamma1^2 + 5.4838192467134238958642442835697e196*Gamma1^4 + 3.341266646585700394658226248238e192*Gamma1^6 + 4.6643665020468703965923728432063e203)))/(34*(Gamma1*0.032066975649856631587598012244763i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0691901942648205863297614106797)^(3/2)) + (exp(-(1.0*(Gamma1*2.6977820177694852753215911328545e198i + 5.4457757512957467470128842849117e196*Gamma1^2 + Gamma1^3*7.2926720336003257802776861511868e194i + 1.5095166391750406470225377943187e193*Gamma1^4 + Gamma1^5*4.8595286399355649248203103922529e190i + 1.0532009076138065172554074604884e189*Gamma1^6 + 4.5506545899399141257395432175959e198))/(8.9703367006565893830556058674786e199*Gamma1^2 + 1.7348694237810619171750959452753e196*Gamma1^4 + 1.0654549255694197687047915332392e192*Gamma1^6 + 1.4264706629501070400859227303052e203))*exp(-(1.0*(Gamma1*4.6715741469508432922754714668535e200i + 6.2364675162538060786000587818126e198*Gamma1^2 + Gamma1^3*2.2431522581504679395043021051207e197i + 2.8502956696364841902212569512205e195*Gamma1^4 + Gamma1^5*2.3231505313079739154100073894443e193i + 2.9124041440807638073318166669355e191*Gamma1^6 + 8.4263312306144099020867993669685e200))/(2.2425841751641473457639014668696e200*Gamma1^2 + 4.3371735594526547929377398631882e196*Gamma1^4 + 2.6636373139235494217619788330979e192*Gamma1^6 + 3.5661766573752676002148068257631e203))*exp(-(1.0*(Gamma1*9.7899062391557331012266166883154e199i + 1.2985342156402997536237279861449e198*Gamma1^2 + Gamma1^3*4.7008468139261250937617458550457e196i + 5.9569469228956708554380083173111e194*Gamma1^4 + Gamma1^5*4.8685052592124045320498668670999e192i + 6.0933865015808359888293531387812e190*Gamma1^6 + 1.6323785769544367009933439931878e200))/(4.4851683503282946915278029337393e198*Gamma1^2 + 8.6743471189053095858754797263765e194*Gamma1^4 + 5.3272746278470988435239576661958e190*Gamma1^6 + 7.1323533147505352004296136515262e201))*exp(-1031679393350023/1759218604441600)*exp(-(1.0*(Gamma1*6.3765720390978630910802163674817e198i + 1.2732733788000045375751354447727e197*Gamma1^2 + Gamma1^3*1.7241977884441293800830451679217e195i + 3.5414607261264699906719509345676e193*Gamma1^4 + Gamma1^5*1.1493350671969160846409309456955e191i + 2.4735749982222805826101997793593e189*Gamma1^6 + 8.5691657007161093811375875580578e198))/(8.9703367006565893830556058674786e197*Gamma1^2 + 1.7348694237810619171750959452753e194*Gamma1^4 + 1.0654549255694197687047915332392e190*Gamma1^6 + 1.4264706629501070400859227303052e201)))/(34*(Gamma1*0.031791059444968356585068049435312i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0468023700424099675462512142925)^(3/2)) + (exp(-(1.0*(Gamma1*5.2593617269957292405442603103644e198i + 1.0518208637898193816312915234556e197*Gamma1^2 + Gamma1^3*1.4439081482469072235189300718871e195i + 2.93272972566518894653142237362e193*Gamma1^4 + Gamma1^5*9.8085823755470351586207858207272e190i + 2.0642737789230607738205986225572e189*Gamma1^6 + 9.9974333422289837160933049583028e198))/(1.7747769260494775609776476025823e200*Gamma1^2 + 3.4128286743798514074550037553709e196*Gamma1^4 + 2.0882916541160627466613914051488e192*Gamma1^6 + 2.8509994378183072839449794229675e203))*exp(-(1.0*(Gamma1*2.3346092821235590202553776834177e202i + 3.1156952023610704635115326658891e200*Gamma1^2 + Gamma1^3*1.1058115511615628516639403876899e199i + 1.405359623166008328020212207269e197*Gamma1^4 + Gamma1^5*1.1381663189401113886096705099202e195i + 1.4270780305995742655925901667984e193*Gamma1^6 + 4.2192722769096799374248538866112e202))/(1.1092355787809234756110297516139e202*Gamma1^2 + 2.1330179214874071296593773471068e198*Gamma1^4 + 1.305182283822539216663369628218e194*Gamma1^6 + 1.7818746486364420524656121393547e205))*exp(-(1.0*(Gamma1*9.7608442468664476925897651567285e199i + 1.3069148389616844621320382617303e198*Gamma1^2 + Gamma1^3*4.6233134457060637999917785323448e196i + 5.883903980828932738474731311413e194*Gamma1^4 + Gamma1^5*4.7585813274293895179535825103031e192i + 5.9715187715492192690527660760056e190*Gamma1^6 + 1.8324862275193165773543136118867e200))/(4.4369423151236939024441190064557e198*Gamma1^2 + 8.5320716859496285186375093884272e194*Gamma1^4 + 5.2207291352901568666534785128719e190*Gamma1^6 + 7.1274985945457682098624485574187e201))*exp(-(1.0*(Gamma1*1.5435630799799050899326122243762e198i + 3.0886033680705309844438353975595e196*Gamma1^2 + Gamma1^3*4.2376620086539092576474538885749e194i + 8.6103218592980813599531574933229e192*Gamma1^4 + Gamma1^5*2.8786400476262790246467159142283e190i + 6.0602587456445874273949894594306e188*Gamma1^6 + 2.9600519738392072829223912924712e198))/(2.2184711575618469512220595032278e197*Gamma1^2 + 4.2660358429748142593187546942136e193*Gamma1^4 + 2.610364567645078433326739256436e189*Gamma1^6 + 3.5637492972728841049312242787093e200))*exp(-80166902739265123/70368744177664000))/(34*(Gamma1*0.031919211927133157764560220979853i - 0.00023270864079412549142054964921625*Gamma1^2 + 1.0571998983963591239960330216363)^(3/2)) + (exp(-195326621885201857/140737488355328000)*exp(-(1.0*(Gamma1*2.354291980945736497638182745442e203i + 3.141531110490761069525189632533e201*Gamma1^2 + Gamma1^3*1.1086969844884752845100972194301e200i + 1.4091263124137440463657900956362e198*Gamma1^4 + Gamma1^5*1.1380918603957396387449591327077e196i + 1.4270780305995742655925901667984e194*Gamma1^6 + 4.2581196945547434397896234040024e203))/(1.1137178236210534470911686394178e203*Gamma1^2 + 2.1363915351544320031267060554425e199*Gamma1^4 + 1.305182283822539216663369628218e195*Gamma1^6 + 1.7967619534593339508854646775337e206))*exp(-(1.0*(Gamma1*5.2483831400211302634304284705164e199i + 1.0453838162475523054043933177159e198*Gamma1^2 + Gamma1^3*1.450142991292204058236452104802e196i + 2.9219992206801152997169205565233e194*Gamma1^4 + Gamma1^5*9.9277160465418349421589893608514e191i + 2.0642737789230607738205986225572e190*Gamma1^6 + 1.0442189354295762853854321897853e200))/(1.7819485177936855153458698230685e201*Gamma1^2 + 3.418226456247091205002729688708e197*Gamma1^4 + 2.0882916541160627466613914051488e193*Gamma1^6 + 2.874819125534934321416743484054e204))*exp(-(1.0*(Gamma1*1.9666155394052122793064920236587e200i + 2.6432187003791902541095454886121e198*Gamma1^2 + Gamma1^3*9.2612572083568056565654112432254e196i + 1.1807286906579344141867148630124e195*Gamma1^4 + Gamma1^5*9.5067794995791877516240770415245e192i + 1.1943037543098438538105532152011e191*Gamma1^6 + 3.8631882927417285548970125769717e200))/(8.9097425889684275767293491153425e198*Gamma1^2 + 1.709113228123545602501364844354e195*Gamma1^4 + 1.0441458270580313733306957025744e191*Gamma1^6 + 1.437409562767467160708371742027e202))*exp(-(1.0*(Gamma1*3.0718068813644364679642052140308e199i + 6.15257440031583952413395145228... Output truncated. Text exceeds maximum line length of 25,000 characters for Command Window display.

x =

NaN

fval =

NaN

LB7 =

NaN % code end

I would be grateful if you can help.

Walter Roberson 2017-6-18

在 MATLAB Online 中打开

I noticed by the way that you had

symfum( real( simplify(EXPRESSION) ) )

and thought to myself, "Ah, it would make more sense to simplify() the real() instead of real() of the simplify() in order to improve the code execution.

Unfortunately that did not work out at all! simplify(real(EXPRESSION)) was taking so long that I ended up having to kill the mupad kernel and kill the MATLAB process itself :(

Walter Roberson 2017-6-19

在 MATLAB Online 中打开

In your fun3_formula, part of the expression is

exp(-((1i*Gamma1)*x1)

Here Gamma1 >= 0 (because the later integral is over 0 to inf), and x1 is real valued (because you later fminbnd over -inf to +inf).

There is a problem here: this is not defined as x1 approaches +/- infinity. If you rewrite it in terms of sin and cos you end up with sin(infinity) and cos(infinity) terms, and those are not defined.

This is your only term involving x1 in fun3_formula, and as x1 grows larger in magnitude (towards +/- infinity) any uncertainty in Gamma1 becomes more and more important, to the point where the value of the exp() term becomes something random-ish in the range +/- 1 according to the noise in the value of Gamma1 and the round-off error of the calculation. My tests suggest that below x = -32 the oscillations this drives the integral to are too large to be compensated for and vpaintegral and integral will fail.

Answer 2

John D'Errico 2017-5-20

0
链接

此回答的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/341058-unable-to-find-a-minima-for-a-symbolic-integral#answer_267733

fminbnd and fminsearch do NOT apply to symbolic functions. Nor does quadgk.

ALL of those tools are NUMERICAL tools. They do NOT operate on functions with symbolic variables in them.

Sorry, but I won't even try to wander through this mess of code, as you are doing many confusing things, half of which are commented out.

If you want to do integration, then use int for symbolic expressions.

If you want to find a minimum, then go back to basic calculus. Differentiate, set the result equal to zero, and solve. Decide which solution from that process results in a MINIMUM.