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@Domen Thanks for the info. I was wondering where people were accessing 14.2

Why do we have this tag when we already have calculus-and-analysis?

Is it appropriate to answer this question with my comment here? I feel like my answer will just be an exact copy of this answer here so I feel like my answer would add no new information to the site

I don't know if anyone would be interested, but are there any shortcuts here to give OP a better answer?

I put in an answer for this question which the OP wants to find the saddle point equations for eigenvalues of hermitean matrices. I turned it into a minimization problem but I feel like there's got to be some linear algebra thereoms regarding Hermetian matrices that could make this easier (but I never took a college level linear algebra course)...(cont'd)

sorry haha *github

@Istvan Zachar I have used ConvexOptimization with Gurobi when I had a license (from my school) but this was 2 years ago on a different laptop. I believe it involved using GurobiLink from git

@tush there used to be FunctionApproximations ListIntegrate[](reference.wolfram.com/language/FunctionApproximations/ref/…) but it is now obsolete. The method proposed by @kirma is the suggested method by the ListIntegrate documentation now: Integrate[Interpolation[f, InterpolationOrder -> 1][x], {x, 0, 1}]

    graph = Graphics3D[{Yellow, Cuboid[]}];
plot = Plot3D[Sqrt[x^2 + y^2], {x, -2, 2}, {y, -2, 2},
   PlotStyle -> Opacity[0.5, Blue], Mesh -> None];
Show[plot, graph, PlotRange -> {{-2, 2}, {-2, 2}, {0, Sqrt[8]}}]

I think i am not understanding what you want to do, sorry. But if I wanted to plot a Cuboid on some 3d plot I would just use Show:

@user10478 I'm having trouble understanding exactly what you want? Maybe post in more detail as a question? Do you know about Show? reference.wolfram.com/language/ref/Show.html

Same default seeding for NetInitialize

Yes sorry SeedRandom

It is probably best practice though to not assign a RandomSeed[] value though I guess? Unless you want your prng to be reproducible which I guess you would have to specify a seed

I am also probably just going to far with the lag all the way out to `Length[rands]-1` also. I don't really use autocorrelation tests ever so I could be doing something meaningless by going out that far in the lags. Another seed that is poor is 12345. The automatic seedings appear to do much better with the autocorrelation tests

    SeedRandom[1234, Method -> "ExtendedCA"];
rands = RandomReal[{0, 1}, 1000];
Tally[AutocorrelationTest[rands, #, "ShortTestConclusion"] & /@
  Range[999]]
(*{{"Do not reject",473},{"Reject",526}}*)

Some seeds can produce suspicious autocorrelation tests, but I am not sure how important this is and I don't really know a lot about prng testing

except I forgot to include definition of err in that code block err[inp_] := Abs[inp . pTab - E];

Haha! I did it

maxPower = 6;
maxInteger = 9;
pTab = Table[ Pi^-i, {i, 0, maxPower}];
soln = x /.
   QuadraticOptimization[
    Norm[Inactive[Plus][{pTab} .
       x, -{E}]], {-maxInteger \[VectorLessEqual]
      x \[VectorLessEqual] maxInteger}, x, Integers,
    Tolerance -> 10^(-20), PerformanceGoal -> "Quality"];
representation = pTab . soln
error = err[soln] // N

@b3m2a1 ok thanks. I will try it now as I wrote a version that uses QuadraticOptimization instead of a brute force search for finding coefficients. It is fast but does not always find the optimal values.

Also apparently I don't know how to format code blocks in chat... :(

@MichaelE2 thanks. And yes @Nasser I have the same problem

Is there a better way to copy long code blocks from a question other than highlighting it with your cursor?

Wish I had more to say but I'm still only on question 4 so far. I wish the Q, A, and comments were all together for each number, because I am reading the answers one at a time, reading the commentary, and then I have to go back to re-read what the last question was...but I probably need to loosen up because this is just a fun read

@Nasser this was enjoyable to read

But the only thing I changed is I removed the newlines and put A and B in different cells. But now they have the frame line between them since Frame->All and I don't know how to tell Frame to put Frame lines everywhere except between the cells containing A and B

I should've mentioned this in the above comment, but in the matrices I'm looking at the constant scalars $k$, $b$ are always positive real-valued and $k > b$

I feel like this could be a really good opportunity to pose this as a graph problem where the adjacency matrix is related to the input matrix, and find the highest weight cycle. The problem is I'm having issues figuring out what to do for the diagonal elements (I.e. you have the option of "staying" at a given vertex instead of moving to the next one)

Sorry I've been away for so long. Have you considered writing your sum function in C? And then adding the gradient descent method also in C? I think mma is just too slow to do this for high dimensional stuff. It already took 40 mins for 1000 points, and the computation time required grows like the square of the number of points.

It took ~40 mins. wolframcloud.com/env/1b2e1cb6-8b95-43ef-b0eb-0a16f68cf504 There are artifacts near the endpoints, but otherwise it is a good fit. My guess is that these artifacts come from the fact that I reduced the size of ns to get this to run in an appreciable amount of time. I can run it with the full sized ns=Floor[n/2] tonight to see (will take ~3hrs). I will let you know. I pasted the resulting reconstructed x, along with the y generated from the reconstructed x at the bottom so you can look at them without having to re-run the code

'val' is never assigned a value, it is just used as an indexing variable in xIndexed which is the variable list sent into FindRoot (xIndexed looks like {val[1],val[2],val[3],...} and then when FindRoot runs it returns a list of replacement rules {val[1]->number1,val[2]->number2,...} where {number1,number2,...} is the solution. As for adding bounds to my original gradient descent, I am not sure if I can implement this, but one thing you could do is at each step, if the step takes below 0, just clip the value to 0. I don't know if this will effect the quality of the solution however.

No need to implement BFGS. FindRoot works fine and returns a good solution if you specify a small lower bound that isn't exactly zero (like 10^-10) wolframcloud.com/env/dtrimas/Published/fasterGradDescent.nb Also if you use FindMinimum instead and select "QuasiNewton" method it actually uses BFGS anyways (convergence is quite a bit slower however with QuasiNewton compared to Newton). I didn't use this though because FindRoot worked fine once I changed the lower bound from 0 to 10^-10

It looks my method using FindRoot with constraints actually does not produce a good solution, so I may have to look at that.

After a little bit of reading through the documentation, I was able to get FindRoot to work with the constraint $x_j\geq0$ and I added it to the bottom of this notebook: wolframcloud.com/env/dtrimas/Published/fasterGradDescent.nb It's kind of messy the way I did it and probably can be done in a better way. I tested it with n=50 and ns=25 because it only takes 2 s to run 10 iterations with that. I am running it now with n=400 and with ns=100 to see how long it takes.

Ok, I made a new version with 'ns=100` instead of 200 and it is basically the same, but 4 times faster. I also realized I was calculating the sum function fp multiple times for the same input value with the way I defined grad and vec as separate functions, so this is quite a bit faster to run: wolframcloud.com/env/dtrimas/Published/fasterGradDescent.nb

I wrote this just in case you need it (but I'm sure you're familar with gradient descent) but this is a gradient descent algorithm for your problem, using $\frac{\pi}{4}y^{-2}$ as an initial condition. You can optimize the step size on each descent step if you like. It's not an analytical solution, so I again did not want to post this as an answer wolframcloud.com/obj/dtrimas/Published/gradDescent.nb

The approximation $𝑦_j \approx \frac{1-e^{-x_j}}{x_j}$ only works for large $x_j$ though

I didn't want to put this as answer because it's not a general solution: for large valued x, $y_j \approx \frac{1 - e^{-x_j}}{x_j}$ Solving for $x_j$ yields $x_j \approx \frac{1 + y_j \mathrm{ProductLog}[-\frac{e^\frac{-1}{y_j}}{y_j}]}{y_j}$ you can test this by multiplying x by a large value: inv[yj_] = First@SolveValues[(1 - Exp[-xj])/xj == yj, xj]; xBig = 10^6 x; yBig = Table[RT[xBig, i, ns], {i, Length@xBig}] // Quiet; xnBig = inv[yBig]; Norm[xnBig - xBig]/Norm[xBig] (about 10^-7) Ifyou plot xnBig, you'll see it captures all of the behavior of xBig

Just a comment on my previous comment, I actually was testing this with smaller n (n=100) and realized they aren't perfectly equal when n=400. The relative difference though Norm[time2 - fln]/Norm[fln] is on the order of $10^{-7}$ however so they are almost the same.

interestingly, fln is the same (up to a constant) as applying RT again to time and squaring it: time2 = Pi/4*(Table[RT[time, i, ns], {i, Length@time}]^2); Norm[time2 - fln] // Chop (*outputs 0*)

@Max which answer?

@Max wolframcloud.com/obj/dtrimas/Published/PolyLogProof.nb

@MichaelE2 I got the same answer using SeriesCoefficient expanded at $z=0$ and then setting $z=-1/4$. I was hoping FindGeneratingFunction would find a non-Hypogeometric generating function for the series coefficients but it returned the original $_5F_4$ expression. Do you think there is another way to force FindGeneratingFunction to only search in the elementary function space?

@Max PolyLog[4, (1 - Sqrt[5])/2] + PolyLog[4, (-1 + Sqrt[5])/2] can be simplified to 1/8 PolyLog[4, 3/2 - Sqrt[5]/2] But the PolyLog Part still remains. I will see if I can express it as a product of a rational with an irrational