Timeline for Intersection Solutions of four nonlinear equations
Current License: CC BY-SA 4.0
15 events
| when toggle format | what | by | license | comment | |
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| Dec 26, 2018 at 19:59 | comment | added | user64494 | Let us continue this discussion in chat. | |
| Dec 26, 2018 at 19:57 | comment | added | Iosif Pinelis | @user64494 : As follows immediately from my answer, the "trivial" solutions $(\alpha,\beta,\gamma,\delta)$ of the original system are of the form $(x^2,x^3,x^2,x^3)$ for complex $x$. The representative given in your file dropbox.com/s/ul0u4wxzxuyv2jx/modifiedsys.pdf?dl=0 is in this set of solutions, up to some rounding -- corresponding to $x\approx-0.76577$. Is there still something in my answer that makes you not quite happy? | |
| Dec 26, 2018 at 19:33 | comment | added | user64494 | Can you kindly describe the solutions of the system from the question which correspond your "the set of "trivial" solutions of the form $(x,x)$ for complex $x$ ", basing your claim? | |
| Dec 26, 2018 at 18:52 | comment | added | Iosif Pinelis | @user64494 : I certainly did not claim that the set of all solutions is finite. I only claimed that the set of all "nontrivial" solutions is finite, and this was proved. On the other hand, the set of "trivial" solutions of the form $(x,x)$ for complex $x$ is obviously infinite. As for numerical solvers, one cannot trust any part of their output without rigorous additional examination -- basically, you still need a rigorous proof. So, it may actually be much easier to re-check the proof already given in my answer. | |
| Dec 26, 2018 at 18:15 | comment | added | user64494 | If I correctly understand it, you claim a finite set of the solutions whereas the NSolve command dropbox.com/s/ul0u4wxzxuyv2jx/modifiedsys.pdf?dl=0 claims an infinite set of the solutions. Hope I am clear. | |
| Dec 26, 2018 at 18:03 | comment | added | Iosif Pinelis | @user64494 : Why? The claim in my answer was that we have a complete set of solutions, and a detailed proof of that claim was given. Do you see any defect in the proof? | |
| Dec 26, 2018 at 17:34 | comment | added | user64494 | Sorry, I don't see any complete solution in your answer. | |
| Dec 26, 2018 at 15:55 | comment | added | Iosif Pinelis | @user64494 : The many obvious questions about numerical solutions, one of them stated in my previous comment, still remain. | |
| Dec 26, 2018 at 10:55 | comment | added | user64494 | Solving numerically the system from the edited question dropbox.com/s/mkvvskm0i0m5mtx/screen26.12.2018.docx?dl=0, I obtain that the solution set is infinite (see dropbox.com/s/ul0u4wxzxuyv2jx/modifiedsys.pdf?dl=0 ). | |
| Dec 26, 2018 at 5:13 | comment | added | user64494 | Here dropbox.com/s/6exiv40yikocwg5/sys.pdf?dl=0 is the result obtained with Mathematica and here dropbox.com/s/aqdh72m3cg8lwtg/sys%20in%20maple.pdf?dl=0 is the result in Maple. The DirectSearch is one of the best numerical solvers in the world. | |
| Dec 26, 2018 at 0:23 | comment | added | Iosif Pinelis | @user64494 : How can solving numerically ensure that, among other things, you got all the solutions? | |
| Dec 25, 2018 at 21:48 | comment | added | user64494 | Solving numerically the original system, both Maple and Mathematica produce 13 different real solutions. Taking into account multiplicity, Mathematica counts 18 solutions (in particular, zero solution is of multiplicity 4 and the solution $(2.45653, -3.85019, 0., 0.)$ is of multiplicity 2 and the solution $(3., 5.19615, 0., 0.)$ is of multiplicity 2). | |
| Dec 25, 2018 at 20:59 | vote | accept | Sara yaqob | ||
| Dec 25, 2018 at 20:46 | comment | added | Sara yaqob | You mean that the solution is (0,0,3, 3 (3)^(1/2))......I don't know hwo I can thank you....Thanks a lot sir for your help | |
| Dec 25, 2018 at 20:21 | history | answered | Iosif Pinelis | CC BY-SA 4.0 |