Is there anyway to increase the calculation speed of this sqrt integral2 without loosing accuracy?

Any suggestion to increase the calculation speed of this integral2?

g = 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integral2(g,-1,1,-1,1)

[Torsten]

A function partly real-valued, partly complex-valued - very unpleasant.

g = 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+1+xx.^18.*1.077219415605556e+1+yy.^2.*6.121374088507809e-5+yy.^3.*7.566679105625542e-3-yy.^4.*2.616937320847826e-4-yy.^5.*1.204865172584651e-2+yy.^6.*4.468547430148061e-4+yy.^7.*9.176084750846805e-3-yy.^8.*3.785602968701871e-4-yy.^9.*3.422992169154646e-3+yy.^10.*1.62999960348087e-4+yy.^11.*6.876030356471064e-4-yy.^12.*3.352881389468098e-5-yy.^13.*1.388009800634545e-4+yy.^14.*3.333314996184432e-6+yy.^15.*1.360695926359456e-5-yy.^16.*1.655644414705918e-7-yy.^17.*7.64905319289316e-7+yy.^18.*1.674878549087182e-9-4.550268315834584e-7).^2.*(-8.845979223770551e-5));
g = @(xx,yy)abs(g(xx,yy));
fcontour(g,[-1 1 -1 1],'Fill','on')
colorbar

[Paul]

Did you try changing the 'Method' of integral2? tiled might be faster than iterated (or is it the other way around?). Or maybe the tolerances?

The real and imaginary parts of g looked smooth to the eye (well, my eye), maybe trapz will get a good approximation? I don't know because integral2 was taking too long to run here on Answers.

g = 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+1+xx.^18.*1.077219415605556e+1+yy.^2.*6.121374088507809e-5+yy.^3.*7.566679105625542e-3-yy.^4.*2.616937320847826e-4-yy.^5.*1.204865172584651e-2+yy.^6.*4.468547430148061e-4+yy.^7.*9.176084750846805e-3-yy.^8.*3.785602968701871e-4-yy.^9.*3.422992169154646e-3+yy.^10.*1.62999960348087e-4+yy.^11.*6.876030356471064e-4-yy.^12.*3.352881389468098e-5-yy.^13.*1.388009800634545e-4+yy.^14.*3.333314996184432e-6+yy.^15.*1.360695926359456e-5-yy.^16.*1.655644414705918e-7-yy.^17.*7.64905319289316e-7+yy.^18.*1.674878549087182e-9-4.550268315834584e-7).^2.*(-8.845979223770551e-5));
tic,
[X,Y] = meshgrid(-1:.01:1);
G = g(X,Y);
trapz(trapz(G))*0.01^2

ans = 5.7964e-04 + 9.5367e-04i

toc

Elapsed time is 1.826327 seconds.

%integral2(g,-1,1,-1,1)

Is g in its most simple form? If this came for the Sympbolic Math Toolbox, maybe it can be simplified further to reduce the number of operations (no idea if that really causes integral2 to slow down, just a thought).

[Walter Roberson]

(This was a failed attempt at a solution, but it is being left here because I need to report a bug about it)

version

ans = '23.2.0.2436196 (R2023b) Update 4'

g = 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g = 

%{
syms xx yy
start = tic; result1 = int(int(g, xx, -1, 1), yy, -1, 1); t1 = toc(start);

G = matlabfunction(g, 'vars', {'xx', 'yy'});
start = tic; result2 = integral2(G, -1, 1, -1, 1); t2 = toc(start);

start = tic; result3 = vpaintegral(int(g, xx, -1, 1, 'hold',true), -1, 1); t3 = toc(start);

format long g
[t1; t2; t3]
[double(result1); result2; double(result3)]
%}

[Torsten]

@Paul

Do you have a result from integral2 to compare with the trapz result ? I suspect it took to long for MATLAB online ?

[Paul]

I didn't try it on Matlab online.

I just ran it locally:

>> tic,I = integral2(g,-1,1,-1,1),toc

Warning: Reached the maximum number of function evaluations (10000). The result passes the global error test.

> In integral2Calc>integral2t (line 136)

In integral2Calc (line 9)

In integral2 (line 105)

I =

5.7963e-04 + 9.5392e-04i

Elapsed time is 262.605441 seconds.

>>

Hmm. Reached the maximum number of evaluations, which sounds bad, but passes the global error test, which sounds good. Don't know what to make of that.

So trapz using a fixed 0.01 step size may be good enough, or maybe not. If going down the trapz route, I suspect it would be better to more carefully examine g(x,y) and consider modifying the mesh to use a smaller step size where the g(x,y) is changing (second derivative) rapidly. If needed, consider trapz-ing the real and imaginary parts separately. Don't know if this approach meets the needs of the OP, was just throwing something out there for consideration.

[Torsten]

If needed, consider trapz-ing the real and imaginary parts separately.

Seems this is a good idea for integral2, too:

g = 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tic
integral2(@(xx,yy)real(g(xx,yy)),-1,1,-1,1)

ans = 5.7963e-04

toc

Elapsed time is 20.647107 seconds.

tic
integral2(@(xx,yy)imag(g(xx,yy)),-1,1,-1,1)

ans = 9.5393e-04

toc

Elapsed time is 7.545205 seconds.

[Paul]

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?

[Torsten]

Any idea why doing so is so much more efficient?

To be honest: no.

Answer 1

[Steven Lord]

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

Elapsed time is 2.387720 seconds.

tic
part2 = integral2(R, 0.3, 0.5, -1, 1);
toc

Elapsed time is 0.229699 seconds.

tic
part3 = integral2(R, 0.5, 1, -1, 1);
toc

Elapsed time is 0.471389 seconds.

sol = part1+part2+part3

sol = 5.7963e-04

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

Elapsed time is 0.486807 seconds.

part1-part1a

ans = 1.6351e-11

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.

[Torsten]

@Steven Lord

And the next step is to enable "integral2" to choose these regions automatically :-)

[Paul]

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.

Answer 2

[Torsten]

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)

ans = 5.7963e-04 + 9.5394e-04i

toc

Elapsed time is 1.574866 seconds.

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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.*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.^18.*1.077219415605556e+1+yy.^2.*6.121374088507809e-5+yy.^3.*7.566679105625542e-3-yy.^4.*2.616937320847826e-4-yy.^5.*1.204865172584651e-2+yy.^6.*4.468547430148061e-4+yy.^7.*9.176084750846805e-3-yy.^8.*3.785602968701871e-4-yy.^9.*3.422992169154646e-3+yy.^10.*1.62999960348087e-4+yy.^11.*6.876030356471064e-4-yy.^12.*3.352881389468098e-5-yy.^13.*1.388009800634545e-4+yy.^14.*3.333314996184432e-6+yy.^15.*1.360695926359456e-5-yy.^16.*1.655644414705918e-7-yy.^17.*7.64905319289316e-7+yy.^18.*1.674878549087182e-9-4.550268315834584e-7).^2.*(-8.845979223770551e-5));
end

[Mehdi]

Is it possible not to define functions fun and g since in my orginal codes the g is produced in a sophisticated loop.

g =@(xx,yy) 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xspan = [-1 1];
Gx0 = 0;
syms xx yy
[~,Gx] = ode45(@(xx,~)g(xx,yy),xspan,Gx0,odeset('RelTol',1e-10,'AbsTol',1e-10));

Error using superiorfloat
Inputs must be floats, namely single or double.

Error in odearguments (line 114)
dataType = superiorfloat(t0,y0,f0);

Error in ode45 (line 104)
odearguments(odeIsFuncHandle,odeTreatAsMFile, solver_name, ode, tspan, y0, options, varargin);

dGydy = Gx(end);
yspan = [-1 1];
Gy0 = 0;
[~,G] = ode45(@Gx,yspan,Gy0,odeset('RelTol',1e-10,'AbsTol',1e-10));
G(end)

[Walter Roberson]

Is it possible not to define functions fun and g

Not while using numeric computing.

You declared syms yy and you are passing gg(xx,yy) into ode45. The result of computing gg(xx,yy) with symbolic yy is going to be symbolic in yy. It is not clear what you would want ode45 to do with the resulting symbolic expression if it could handle symbolic expressions (which it cannot)

Note: the first parameter to the ode function is always time (or equivalent) and the second is boundary conditions. Passing @(xx,~) in to ode45 means to ignore the boundary conditions

[Torsten]

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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% 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)

ans = 5.7963e-04 + 9.5394e-04i

toc

Elapsed time is 1.421408 seconds.

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

[Mehdi]

Nice! So thanks for your suggested approach.

Answer 3

[Paul]

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

ans = 5.7963e-04 + 9.5393e-04i

Elapsed time is 6.408831 seconds.