Jacobi elliptic functions and complex arguments

[rtoy]

Imported from SourceForge on 2024-07-05 22:36:59 Created by vladutzanu on 2016-09-14 15:01:11 Original: https://sourceforge.net/p/maxima/bugs/3215

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Hello

(This is a following from maxima-discuss: "Jacobi elliptic functions, maxima 5.38.1")

Given the following input:

kill(all)$ [wp, ws, Ap, As]:[1, 1.1, 1, 15]$
k  : wp / ws, numer$
k1 : (10^(Ap / 10) - 1) / (10^(As / 10) - 1), numer$
K  : elliptic_kc(k)$
K1 : elliptic_kc(k1)$
K_ : elliptic_kc(1-k)$
K1_: elliptic_kc(1-k1)$
N  : ceiling(K * K1_ / (K_ * K1)), numer;
lg : log(10), numer$
q0 : ((1 - sqrt(sqrt(1-k))) / (1 + sqrt(sqrt(1-k)))) / 2, numer$
q0sq : q0^4, numer$
q : q0 * (q0sq * (q0sq * (150 * q0sq + 15) + 2) + 1), numer$
As : 10 * log(0.0625 / q^N * (10^(Ap * 0.1) - 1) + 1) / lg, numer;
k1 : (10^(Ap * 0.1) - 1) / (10^(As * 0.1) - 1), numer$
q : (k1 / 16)^(1 / N), numer$
q_SQ : q^2, numer$
k : (q_SQ * (q_SQ * (q_SQ * (q_SQ * (q_SQ^2 + 2) + 2) + 1) + 2) + 1) /
    (q * (q * (q_SQ * (q * (4 * q_SQ * q + 8) + 4) + 4) + 4) + 1) * 4 * sqrt(q), numer$
K1 : elliptic_kc(k1), numer$
K  : elliptic_kc(k^2), numer$
R(x) := jacobi_cd(rectform(N * K1 / K * inverse_jacobi_cd(x / wp, k^2)), k1)$
H(x) := 1 / sqrt(1 + (10^(Ap / 10) - 1) * R(x)^2)$
R2(xi, x) := ((sqrt(1 - 1 / xi^2) + 1) * x^2 - 1) / ((sqrt(1 - 1 / xi^2) - 1) * x^2 + 1)$
p:1/k;
H2(x) := 1 / sqrt(1 + (10^(Ap / 10) - 1) * R2(R2(p, p), R2(p, x))^2)$
plot2d([H2(x), realpart(H(x)), imagpart(H(x))], [x, 0, 3], grid2d)$

R(x) can only be plotted with rectform() encompassing its argument, otherwise it ouputs these: BIGFLOAT: unable to convert 3.15105041004758 (2.257205326820854-6.664001874625061e-8%i) to a CL or BIGFLOAT number.

It should plot the frequency response of an elliptic/Cauer filter. The imagpart(H(x)) should be, ideally, zero.

[rtoy]

Imported from SourceForge on 2024-07-05 22:37:00 Created by robert_dodier on 2016-09-16 05:15:53 Original: https://sourceforge.net/p/maxima/bugs/3215/#9eaf

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Thanks for the bug report. Can you please state specifically what is the expression which triggers the error. I can't guess what it is.

[rtoy]

Imported from SourceForge on 2024-07-05 22:37:03 Created by vladutzanu on 2016-09-17 05:47:20 Original: https://sourceforge.net/p/maxima/bugs/3215/#9eaf/bc90

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R(x) is the culprit, 6th line from the bottom. If declared like this:

R(x) := jacobi_cd(N * K1 / K * inverse_jacobi_cd(x / wp, k^2), k1)$

It won't work, messages about BIGFLOAT and whatnot keep flooding the screen. The argument must be encompassed in rectform() to work, as it is now (a solution suggested in the mailing list). I think it's the inverse functions that behave abnormally.

[rtoy]

Imported from SourceForge on 2024-07-05 22:37:07 Created by robert_dodier on 2016-09-17 20:43:57 Original: https://sourceforge.net/p/maxima/bugs/3215/#9eaf/bc90/a753

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OK, I get it now.

If I replace the definition of R(x) as you say, I get the same plot; I don't see any errors about BIGFLOAT.

I am working with:

Maxima version: "branch_5_38_base_223_gcf9cbb2"
Maxima build date: "2016-08-24 16:33:57"
Host type: "i686-pc-linux-gnu"
Lisp implementation type: "CLISP"
Lisp implementation version: "2.49 (2010-07-07) (built 3605610186) (memory 3681070445)"

Note that I made some bug fixes to the elliptical functions code on 2016-07-26 (commits 80cab10, 133408b, and 372582e). Maybe you can try your problem with current source code.

These bug fixes and others will appear in Maxima 5.39, which should be on the horizon. But I'm not making releases these days, so I don't know for sure.

[rtoy]

Imported from SourceForge on 2024-07-05 22:37:10 Created by robert_dodier on 2016-09-17 20:45:05 Original: https://sourceforge.net/p/maxima/bugs/3215/#8f6f

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• labels: --> elliptic functions

[rtoy]

Imported from SourceForge on 2024-07-05 22:37:14 Created by vladutzanu on 2016-09-20 20:00:46 Original: https://sourceforge.net/p/maxima/bugs/3215/#9eaf/bc90/a753/a58c

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These bug fixes and others will appear in Maxima 5.39, which should be on the horizon. But I'm not making releases these days, so I don't know for sure.

If you say the changes were made, then that's the solution. I'll try to see how the changes cope with my current install (archlinux x64), if not, I'll wat, patiently, for the new release. The rectform() workaround will hold until then. Thank you for the fix.