@AlexShura Email response (below) does not contain the polynomial in a(n). Please advise how you got that.

Greetings from The On-Line Encyclopedia of Integer Sequences! https://oeis.org/

> lookup 1 2 4 5 8 12 14 16 28 32 37 64 94 106 128 144 232 256 289 320

> 512 560 704 760 838 1024 1328 1536 1944 2048 2329 3104 3328 4096 4864

> 6266 6802 7168 8192 11952 15360 16384 16428 19149 28928 32768

a(n) = 1, 2, 4, 5, 8, 12, 14, 16, 28, 32, 37, 64, 94, 106, 128, 144, 232, 256, 289, 320, 512, 560, 704, 760, 838, 1024, 1328, 1536, 1944, 2048, 2329, 3104, 3328, 4096, 4864, 6266, 6802

# Direct matches

These sequences match the query directly.

oeis.org/search?q=1,2,4,5,8,12,14,16,28,32,37,64,94,106,128,144,232,256,289,320,512,560,704,760,838,1024,1328,1536,1944,2048,2329,3104,3328,4096,4864,6266,6802

oeis.org/A035001 Sorted elements of table (A035002) of a(m,n) =

                 sum(a(m-k,n), k=1..m-1)+sum(a(m,n-k), k=1..n-1).

    <1, 2, 4, 5, 8, 12, 14, 16, 28, 32, 37, 64, 94, 106, 128, 144, 232,

    256, 289, 320, 512, 560, 704, 760, 838, 1024, 1328, 1536, 1944, 2048,

    2329, 3104, 3328, 4096, 4864, 6266, 6802>, 7168, 8192, 11952, 15360,

    16384, 16428, 19149, 28928, 32768, 37120, 42168

# Transformations

These sequences match transformations of the original query.

T001 a(n) itself

 = 1, 2, 4, 5, 8, 12, 14, 16, 28, 32, 37, 64, 94, 106, 128, 144, 232, 256, 289, 320, 512, 560, 704, 760, 838, 1024, 1328, 1536, 1944, 2048, 2329, 3104, 3328, 4096, 4864, 6266, 6802

oeis.org/A035001 Sorted elements of table (A035002) of a(m,n) =

                 sum(a(m-k,n), k=1..m-1)+sum(a(m,n-k), k=1..n-1).

    <1, 2, 4, 5, 8, 12, 14, 16, 28, 32, 37, 64, 94, 106, 128, 144, 232,

    256, 289, 320, 512, 560, 704, 760, 838, 1024, 1328, 1536, 1944, 2048,

    2329, 3104, 3328, 4096, 4864, 6266, 6802>, 7168, 8192, 11952, 15360,

    16384, 16428, 19149, 28928, 32768, 37120, 42168

# Transformations as Deltas

The deltas of these sequences match transformations of the original query.

T018 a(n+1) - a(n)

 = 1, 2, 1, 3, 4, 2, 2, 12, 4, 5, 27, 30, 12, 22, 16, 88, 24, 33, 31, 192, 48, 144, 56, 78, 186, 304, 208, 408, 104, 281, 775, 224, 768, 768, 1402, 536 (as deltas)

oeis.org/A035001 Sorted elements of table (A035002) of a(m,n) =

                 sum(a(m-k,n), k=1..m-1)+sum(a(m,n-k), k=1..n-1).

    <1, 2, 4, 5, 8, 12, 14, 16, 28, 32, 37, 64, 94, 106, 128, 144, 232,

    256, 289, 320, 512, 560, 704, 760, 838, 1024, 1328, 1536, 1944, 2048,

    2329, 3104, 3328, 4096, 4864, 6266, 6802>, 7168, 8192, 11952, 15360,

    16384, 16428, 19149, 28928, 32768, 37120, 42168

In transformation descriptions,

Sn(z) denotes the ordinary generating function with coefficients a(n), and

En(z) denotes the exponential generating function with coefficients a(n).

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Perhaps I'm not understanding what you did, but I can't reproduce this.

A035001.mw

So you know that the line

printf("After iterations %d, A = %f and B = %f\n",i,b(1),b(2));

was the problem because only the i was printed out as 1, and not the b(1). So you can try print(b(1)) and see what the value was that wasn't a number.

b and c are 2x1 Matrices since you generated then differently (extra inner <>) than for Vector P and t. So b should really be indexed  b[1,1] or b[2,1]. Or just make them Vectors.

Use square brackets for indexing Matrices, just as you did with the Vectors, i.e. use b[2,1] not b(1) and J[r,2] not J(r,2). The round brackets are for advanced users.

Since your calculation is all in floating point, you can initialize b and c and t in floating point, e.g, 1.0 not 1. Then calculations that have some floating point numbers will give floating point results - you don't then need the evalfs in the for loop or the while condition.

@abdulganiy I understand from your question and clarification that you want the exact same plot as the following line of code produces

Error_plot_y1 := pointplot3d(time_t, exy1_lst, err1, style = point, symbol = box, color = orange,labels = [`h=1/4`,`Exact`, typeset((`Absolute Errors`))]);

except that you want to use the plot3d command and not pointplot3d. If so, that is not possible because pointplot3d allows for specifying 3 vectors, but plot3d does not. If this is not what you want then you need to be more specific.

The link in your post works, but I agree that clicking on the author of that post leads to the front page. Presumably, the author has deleted their account.

@philroe For run to run consistency you can use sort on the list. It's not always clear what order you get, but for the same list elements your should get the same order each time.

@Gharhoud That's correct. (As in @vv's first response.)

This has been improved in Maple 2023, where ExtremePoints(f(x)); returns [-1,0,2]

@JimAxon 

So this is just multiplication of each term by exp(k*2*Pi*I*(degree of that term)). So the procedure (checked against your outputs) is below. So here k=0 would give sheet1 unchanged, k=1 would give sheet2, k=2 would give sheet3. 

conjugatePuiseux:=proc(generator,k::nonnegint);
  local var,deg;
  # find indeterminate
  var:=indets(generator,'name');
  if nops(var)<>1 then
      error "unable to determine variable"
  else
      var:=var[];
  end if;
  # deg finds degree of a term
  deg:=term->diff(term,var)/term*var;
  if generator::`+` then
    map(term->term*exp(k*2*Pi*I*deg(term)),generator);
  else
    generator*exp(k*2*Pi*I*deg(generator)) # if only one term
  end if;
end proc:

puiseuxPlotsVer2.mw

You could just pass the variable through as a second argument, which will be more efficient if you are doing it many times.

@2cUniverse The aspect ratio does seem to be slightly different from the one exported with the context menu.

Edit: I added scaling=constrained to the display and then both the context menu export and the exportplot command gave the same aspect ratio.

@22117147 Happy to help. I added the new conjecture at the end. Keeping solutions in symbolic form removes the doubt that may exist with numerical solutions.

solvecartesian2.mw

@NIMA112 That's why it is a reply and not an answer. As I said "I don't know precisely what you want, since you do not show any calculation that didn't work." Your title mentions a numerical error but your worksheet does not show any explicit error. What is your question? What exactly do you think the "singularity problem" is? I am guessing that the physical problem might have a singularity and therefore no amount of mathematics will make it go away.

I don't know precisely what you want, since you do not show any calculation that didn't work. But visually there does seem to be a singularity in the physical problem at (-1,1.5) where the contour lines converge. So I guess you can go closer by using a higher setting of Digits. Note that you have mispelled Digits so it is not actually at 30.

@Nicole Sharp I don't use the Maple add-on in Excel so don't have an answer for all the inconsistencies you see. So just two comments that might be helpful

Since Excel works in hardware precision (about 15 digits), if Maple passes it a result with 32 digit precision Excel must truncate that in further calculations, so how the extra ones are displayed or if they are just removed is a moot point.

9*10^9 is exact and not a floating point number with finite precision, so displays as "9000000000". evalf converts it to a floating point number. But 9.*10^9 has an explicit decimal point and so is a floating point number. In Maple, numbers created with the "e" or "E" exponent notation are always floating point, even if there is no explicit decimal point.

@SHIVAS As in my last reply, pdsolve can't find an exact solution, so I don't know how to find Theta_exact. Do you have some reason to believe there is an analytical solution? If so, you can perhaps use hints or other methods to help pdsolve it, but you would need to know or guess something about the solution.