Is there anyway to increase the calculation speed of this sqrt integral2 without loosing accuracy?
3 vues (au cours des 30 derniers jours)
Afficher commentaires plus anciens
Any suggestion to increase the calculation speed of this integral2?
g = 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integral2(g,-1,1,-1,1)
8 commentaires
Paul
le 12 Déc 2023
I was wondering the same thing, i.e. split integral2 into real/imag, thanks for showing that. Any idea why doing so is so much more efficient?
Réponse acceptée
Steven Lord
le 12 Déc 2023
Let's look at what the surface you're trying to integrate looks like.
g = 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figure
fsurf(@(xx,yy)real(g(xx,yy)), [-1,1,-1,1])
title("Real part of g")
Okay, not so bad.
figure
fsurf(@(xx,yy)imag(g(xx,yy)),[-1,1,-1,1])
title("Imaginary part of g")
I might consider integrating over each of those peaks separately to see if that helps. Rotating the plot of the real part it looks like the peaks are around x = 0.4 and x = 0.6. So let's integrate once for the flat part and once for each peak.
R = @(xx, yy) real(g(xx, yy));
tic
part1 = integral2(R, -1, 0.3, -1, 1);
toc
tic
part2 = integral2(R, 0.3, 0.5, -1, 1);
toc
tic
part3 = integral2(R, 0.5, 1, -1, 1);
toc
sol = part1+part2+part3
That agrees with Torsten's answer pretty closely.
Focusing even more closely at the "flat" part of the surface, it looks like there's another much smaller ridge at x = -0.4.
figure
fsurf(R, [-1, 0.3, -1, 1])
zlim([0, 1e-5])
title("Flat part of real(g)")
If we integrate that region in parts:
tic
part1a = integral2(R, -1, -0.5, -1, 1) + ...
integral2(R, -0.5, -0.3, -1, 1) + ...
integral2(R, -0.3, 0.3, -1, 1);
toc
part1-part1a
That matches the result of the integration from x = -1 to x = 0.3 pretty closely while taking less time than is required to compute the integration over the whole region.
2 commentaires
Torsten
le 12 Déc 2023
And the next step is to enable "integral2" to choose these regions automatically :-)
Paul
le 13 Déc 2023
Modifié(e) : Paul
le 13 Déc 2023
All kidding aside, shouldn't the adpative step size control of integral2 essentially find those regions? I wouldn't expect integral2 to do it as well as a person carefully inspecting the surface and then dividing and conquering, but I'm surprised that ~2.5 seconds by-hand is so much better than the 20-seconds time that @Torsten showed for integrating the real part.
Plus de réponses (2)
Torsten
le 13 Déc 2023
Modifié(e) : Torsten
le 13 Déc 2023
The solution of the 2d-integral via an ode problem is adaptive and fast.
It's somehow like applying iterated "trapz", but adaptive.
tic
% Integrate the x-integrals in y-direction
yspan = [-1 1];
Gy0 = 0;
[~,G] = ode45(@fun,yspan,Gy0,odeset('RelTol',1e-10,'AbsTol',1e-10));
G(end)
toc
function dGydy = fun(y,Gy)
% Compute the x-integrals at y = y
xspan = [-1 1];
Gx0 = 0;
[~,Gx] = ode45(@(x,~)g(x,y),xspan,Gx0,odeset('RelTol',1e-10,'AbsTol',1e-10));
dGydy = Gx(end);
end
function value = g(xx,yy)
value = 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end
4 commentaires
Torsten
le 14 Déc 2023
Modifié(e) : Torsten
le 14 Déc 2023
If your g is symbolic, use "matlabFunction" first to turn it into a numeric function.
Note that the code below can be used for any function g(x,y) and any rectangular x-y-region to be integrated over.
tic
g =@(xx,yy) 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.*yy.^10.*2.610329616357622e+2-xx.^14.*yy.^9.*9.995500470504359e+1-xx.^15.*yy.^8.*2.556878311939802e+2-xx.^16.*yy.^7.*3.961883526311578e+1-xx.^17.*yy.^6.*1.173608324930028e+2-xx.^18.*yy.^5.*3.536053671430259-xx.^6.*yy.^18.*9.220149238483281e-4-xx.^7.*yy.^17.*1.014690476686303e-1-xx.^8.*yy.^16.*2.473189356253753-xx.^9.*yy.^15.*5.594697728012737e-1-xx.^10.*yy.^14.*5.664508747841497-xx.^11.*yy.^13.*3.965075348636596-xx.^12.*yy.^12.*9.783286253951298e+1-xx.^13.*yy.^11.*9.179583293626597-xx.^14.*yy.^10.*1.573007989970698e+2-xx.^15.*yy.^9.*5.235877836375165-xx.^16.*yy.^8.*9.009099148888656e+1-xx.^17.*yy.^7.*8.544704478967668e-1-xx.^18.*yy.^6.*2.674768472208937e+1+xx.^7.*yy.^18.*8.789660503979215e-3+xx.^8.*yy.^17.*5.154726443207739e-1+xx.^9.*yy.^16.*4.575501924068617+xx.^10.*yy.^15.*3.519392032862595+xx.^11.*yy.^14.*7.244553726468354+xx.^12.*yy.^13.*2.129145940801576e+1+xx.^13.*yy.^12.*9.652573649437794e+1+xx.^14.*yy.^11.*5.089115632054402e+1+xx.^15.*yy.^10.*9.845644712361996e+1+xx.^16.*yy.^9.*3.474525767319002e+1+xx.^17.*yy.^8.*3.73992298363824e+1+xx.^18.*yy.^7.*5.405894441477231+xx.^8.*yy.^18.*2.917787272284469e-3+xx.^9.*yy.^17.*2.572393452106676e-1+xx.^10.*yy.^16.*4.733102430610986+xx.^11.*yy.^15.*8.181721343591682e-1+xx.^12.*yy.^14.*6.193203199800339+xx.^13.*yy.^13.*2.853719379455096+xx.^14.*yy.^12.*5.718940055267839e+1+xx.^15.*yy.^11.*3.241086639512342+xx.^16.*yy.^10.*4.931962357420912e+1+xx.^17.*yy.^9.*6.747402570722794e-1+xx.^18.*yy.^8.*1.000453477214209e+1-xx.^9.*yy.^18.*2.770194324088346e-2-xx.^10.*yy.^17.*1.060768626920755-xx.^11.*yy.^16.*6.063659935013136-xx.^12.*yy.^15.*4.298729062320957-xx.^13.*yy.^14.*5.188023946675464-xx.^14.*yy.^13.*1.408951687993052e+1-xx.^15.*yy.^12.*3.658477349173497e+1-xx.^16.*yy.^11.*1.763694828691533e+1-xx.^17.*yy.^10.*1.546810666592484e+1-xx.^18.*yy.^9.*4.825340323808244-xx.^10.*yy.^18.*4.375685387384132e-3-xx.^11.*yy.^17.*3.713267341545781e-1-xx.^12.*yy.^16.*5.390084222106306-xx.^13.*yy.^15.*6.738582988011823e-1-xx.^14.*yy.^14.*3.859376170867516-xx.^15.*yy.^13.*1.037177872154021-xx.^16.*yy.^12.*1.775828191771638e+1-xx.^17.*yy.^11.*4.29668961753619e-1-xx.^18.*yy.^10.*6.377690249692412+xx.^11.*yy.^18.*5.061696088482958e-2+xx.^12.*yy.^17.*1.300627194368403+xx.^13.*yy.^16.*4.682344463791613+xx.^14.*yy.^15.*3.094039065852764+xx.^15.*yy.^14.*2.03153119138446+xx.^16.*yy.^13.*4.946110325242825+xx.^17.*yy.^12.*5.777902331673031+xx.^18.*yy.^11.*2.424963328716446+xx.^12.*yy.^18.*3.183114231013491e-3+xx.^13.*yy.^17.*3.092145947884248e-1+xx.^14.*yy.^16.*3.629527888134607+xx.^15.*yy.^15.*2.949491814312678e-1+xx.^16.*yy.^14.*1.2714196413809+xx.^17.*yy.^13.*1.440966441698496e-1+xx.^18.*yy.^12.*2.241647785000835-xx.^13.*yy.^18.*5.349374321560299e-2-xx.^14.*yy.^17.*9.33783004746455e-1-xx.^15.*yy.^16.*1.957315224018444-xx.^16.*yy.^15.*1.199954149395863-xx.^17.*yy.^14.*3.355010021173755e-1-xx.^18.*yy.^13.*6.934728168026845e-1-xx.^14.*yy.^18.*9.373212372286827e-4-xx.^15.*yy.^17.*1.387400101197578e-1-xx.^16.*yy.^16.*1.336780615484007-xx.^17.*yy.^15.*5.351190538689366e-2-xx.^18.*yy.^14.*1.702723527279216e-1+xx.^15.*yy.^18.*3.026462613229377e-2+xx.^16.*yy.^17.*3.601327004082595e-1+xx.^17.*yy.^16.*3.430316844923598e-1+xx.^18.*yy.^15.*1.915157980525296e-1+xx.^16.*yy.^18.*2.672769955889301e-5+xx.^17.*yy.^17.*2.60341252165812e-2+xx.^18.*yy.^16.*2.078858690727986e-1-xx.^17.*yy.^18.*7.079740141201421e-3-xx.^18.*yy.^17.*5.710577672108429e-2+xx.^18.*yy.^18.*1.167810588291038e-6-xx.*yy.*3.075366976262442e-2-xx.*yy.^2.*3.526499241348434-xx.^2.*yy.*8.22091599289123e-2+xx.*yy.^3.*1.279267114308256e-1+xx.^3.*yy.*2.422755366776858e-1+xx.*yy.^4.*5.274972053599984+xx.^4.*yy.*9.838541330816038e-1-xx.*yy.^5.*2.057588038303203e-1-xx.^5.*yy.*4.953532667643909e-1-xx.*yy.^6.*3.667311777247011-xx.^6.*yy.*4.373943336369689+xx.*yy.^7.*1.591772639121506e-1-xx.^7.*yy.*4.450755114932008e-1+xx.*yy.^8.*1.241212300797983+xx.^8.*yy.*1.066815031820515e+1-xx.*yy.^9.*6.133207109844262e-2+xx.^9.*yy.*3.188597989877851-xx.*yy.^10.*3.006491051813672e-1-xx.^10.*yy.*1.585254055500515e+1+xx.*yy.^11.*1.34537384760043e-2-xx.^11.*yy.*5.153376996842015+xx.*yy.^12.*9.083199337053412e-2+xx.^12.*yy.*1.452649282223464e+1-xx.*yy.^13.*3.117573298460605e-3+xx.^13.*yy.*3.98291350637101-xx.*yy.^14.*6.304899409665063e-3-xx.^14.*yy.*7.82968931460302+xx.*yy.^15.*4.110970690634906e-4-xx.^15.*yy.*1.512490222340712+xx.*yy.^16.*2.003209230634498e-3+xx.^16.*yy.*2.180160034236967-xx.*yy.^17.*6.692898196101731e-6+xx.^17.*yy.*2.232626342763958e-1+xx.*yy.^18.*3.833781977872521e-7-xx.^18.*yy.*2.184421777805796e-1+xx.^2.*8.858362630041713e-1-xx.^3.*1.82382173556976e+1-xx.^4.*1.957558756949681e+1+xx.^5.*1.312418883880755e+2+xx.^6.*1.305923371426672e+2-xx.^7.*4.615843547867497e+2-xx.^8.*3.929721561768615e+2+xx.^9.*9.056915732081774e+2+xx.^10.*6.372469002766035e+2-xx.^11.*1.043360695274391e+3-xx.^12.*5.955720645853718e+2+xx.^13.*7.011815709552087e+2+xx.^14.*3.203552367072348e+2-xx.^15.*2.540433444872568e+2-xx.^16.*9.173269575880823e+1+xx.^17.*3.82198342698243e+1+xx.^1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% Integrate the x-integrals in y-direction
xspan = [-1 1];
yspan = [-1 1];
Gy0 = 0;
[~,G] = ode45(@(y,Gy)fun(y,Gy,g,xspan),yspan,Gy0,odeset('RelTol',1e-10,'AbsTol',1e-10));
G(end)
toc
function dGydy = fun(y,Gy,g,xspan)
% Compute the x-integrals at y = y
Gx0 = 0;
[~,Gx] = ode45(@(x,~)g(x,y),xspan,Gx0,odeset('RelTol',1e-10,'AbsTol',1e-10));
dGydy = Gx(end);
end
Paul
le 14 Déc 2023
The iterated method works much faster as compared to auto (which I assume goes to tiled). Unclear why iterated integral2 is slower than @Torsten's ode45 approach.
g = 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tic,integral2(g,-1,1,-1,1,'Method','iterated'),toc
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