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Incorrectly calculated integral. - hp-zl - 05-20-2023 06:00 PM
Hello
There is an elliptic integral EllipticF[80*Pi/180,(2+Sqrt[3])/4] whose correct result is: 2.1338951423213... (according to Wolfram Alpha).
On the other hand, HP-Prime CAS Mode after pressing Shift+Enter gets the results as in the attached image.
In the former action, the amplitude = 80.*π/180 (with a dot).
In the second = = 80*π/180 (without the dot).
The results are decidedly different: 2.13... vs. 7.96...
Hence my question is it just like that with me or is it a bug?
Could someone check this.
Greetings.
PS.
Firmware 2.1.14730 (2023 04 13).
After the purge(x) command it is the same.
[attachment=12133][attachment=12134][attachment=12135]
RE: Incorrectly calculated integral. - parisse - 05-20-2023 08:00 PM
If you want an approx answer, it is always best to introduce an approx boundary because there is no attempt to find an antiderivative.
Without dot, the CAS tries to compute a symbolic antiderivative. It transforms
f:=integrate(1/sqrt(1-(2+sqrt(3))/4*sin(x)^2),x,0,80*pi/180)
to integration(2/(sqrt(sin(x)^2+4*sqrt(3)-8)*(i*sqrt(2)/2-i*sqrt(6)/2)),x,0,80*pi/180), with a rewrite of the sqrt that unfortunately introduces complex numbers. After that when you press shift-Enter, the new integral is evaluated numerically, but the choice of sqrt branch is not the same as for the initial integrand.
RE: Incorrectly calculated integral. - hp-zl - 05-21-2023 07:25 AM
Thank you for the clarification.
I thought that pressing Shift+Enter reduces the arguments to floating point form ( in this case: limits of integration and subintegral function), then....
Sorry, this is just such my musings.
By the way.
Maybe it would be worth considering whether to introduce elliptic integrals (F, E, Π) into HP-Prime in the next software update.
Maybe also Jacobi functions (even if only the three basic ones: sn, cn ,dn)?
Greetings.
RE: Incorrectly calculated integral. - Albert Chan - 05-21-2023 01:56 PM
(05-20-2023 08:00 PM)parisse Wrote: Without dot, the CAS tries to compute a symbolic antiderivative. It transforms
f:=integrate(1/sqrt(1-(2+sqrt(3))/4*sin(x)^2),x,0,80*pi/180)
to integration(2/(sqrt(sin(x)^2+4*sqrt(3)-8)*(i*sqrt(2)/2-i*sqrt(6)/2)),x,0,80*pi/180) ...
This is a true bug, not just sqrt picking wrong branch.
Integand denominator should have +i*sqrt(6), not -i*sqrt(6)
m = (2 + √3)/4 ≈ 0.933 → √m = (√2 + √6)/4 ≈ 0.966
s = sin(x)
√(1-m*s^2) = ±√(-m) * √(s^2-1/m) = ±√(m)*i * √(s^2-1/m)
1-m*s^2 > 1-m > 0. Even if we pick the wrong sign, we should have correct size result.
Cas> m := (2+sqrt(3))/4
Cas> approx(int(1/(sqrt(m)*i * sqrt(sin(x)^2-1/m)), x, 0, 80*pi/180))
−2.13389514232
The bug is when Cas simplify √(-m).
Note: XCas 1.9.0-31 had the same bug
Cas> simplify(sqrt(-(2+sqrt(3))/4)) → (i*√2+(−i)*√6)/4 ≈ -0.259*i ???
(05-20-2023 08:00 PM)parisse Wrote: ... the choice of sqrt branch is not the same as for the initial integrand.
Another way is to stop integrand rewrite
Cas> quote(int(1/sqrt(1-(2+sqrt(3))/4*sin(x)^2), x, 0, 80/180*pi))
Cas> float(Ans)
2.13389514232
RE: Incorrectly calculated integral. - parisse - 05-22-2023 07:26 PM
@Chan: you're right in the sense it is not a (pure) sqrt branch choice issue, but it *is* really an algebraic branch choice issue. The CAS always tries to simplify sqrt (this is necessary for some operations, like symbolic integration), using algebraic equations (here it involves a polynomial of degree 4, cfsolve((4x^2+2)^2-3)), which means sqrt(z) may mean any of the 2 solutions x^2=z if z is negative or complex, there is code to choose the right solution only for the largest root.
@hp-zl: adding new special functions would require a large coding effort, because that would not be restricted to output these functions for a few selected integral inputs, it would also mean add support for numeric evaluation (in the complex plane or a part of), series expansions and limits (including at infinities), integration by part/change of variables. It's not on my roadmap (but of course I'm not against any contribution to Giac codebase). Therefore on the Prime, keep in mind to always enter one approx data in your integral if you want a numeric answer, in order to call the numeric algorithm (which is I believe robust).
RE: Incorrectly calculated integral. - Albert Chan - 05-22-2023 08:35 PM
(05-21-2023 01:56 PM)Albert Chan Wrote: Cas> simplify(sqrt(-(2+sqrt(3))/4)) → (i*√2+(−i)*√6)/4 ≈ -0.259*i ???
Hi, parisse
It seems sqrt of positive argument is correctly simplified.
If argument is negative, perhaps simplify sqrt patch is simply pull out i ?
Cas> simplify(sqrt((2+sqrt(3))/4)) * i → (√2+√6)/4 * i ✔
RE: Incorrectly calculated integral. - parisse - 05-23-2023 05:13 AM
Indeed, for this example, I can fix it, thanks!