Perhaps only grainy images of solutions or suggestions could be offered in response.

@bliengme I think that it's still unclear what you want, and what is wrong with the following.

Do you accept what Maple's plots:-semilogplot gives you, for the case of a semilog plot with logarithmic scaling on the horizontal axis and linear scaling on the vertical axis? If so, then what's wrong with the reverse of that, as in the plots:-logplot behavior?

(Naturally, the labels could be anything you want.)

Or, perhaps it is the spacing of the subticks (the minor gridlines) that you want different?

Please state exactly what you want, not what some others think of definitions.

Download logplot.mw

@bliengme In a semilogarithmic plot, one axis has a logarithmic scale and the other axis has a linear scale.

From the description of the logplot command, "semi-logarithmic plot of functions where the vertical axis has a logarithmic scale." And the example shows that the horizontal axis has the usual linear scale.

So, how is the logplot command not supplying what you're asking for?

Anyway, you can also control the mode of either axis, separately. As described in the other link I gave.

Using dualaxisplot for this seems silly.

@eslamelidy You try a modestly high-order series solution, for various values of eq(z).

eslam_acc_complex.mw

@xinwmath What special form of generally invalid result from combine(..,symbolic) or simplify(...,symbolic) are you looking for?

Download symbolic_sick.mw

@vv That is good and simple. Vote up.

But if the "polynomial entries in a single variable" have floating-point coefficients then it does not treat the concept of rank in the same manner as the Rank command (or Matlab's, etc, via singular values which is pretty standard). I mean, following substitution of numeric values for the unknown b, of course.

floatRank.mw

@Kitonum Another kick at the can, for Joe's query. (If `solve` returns an inquality, or non-equality, then it'll be more difficult still.)

Of course, this is the complement.

restart;
with(LinearAlgebra):

A := <b,1,3 | 4,b,6>:
B := <b,1,3 | 4,b+1,6>:

GE1 := GaussianElimination(A):
S1 := [solve({`*`(seq(GE1[i,i], i=1..2))=0}, b)]:
select(u->Rank(eval(A, u))<2, S1);
                           [{b = 2}]

GE2 := GaussianElimination(B):
S2 := [solve({`*`(seq(GE2[i,i], i=1..2))=0}, b)]:
select(u->Rank(eval(A, u))<2, S2);

                               []

@Kitonum Thanks for that observation. How about this amendment?

restart;
A := <b,1,3 | 4,2,6>:
GE := LinearAlgebra:-GaussianElimination(A):

solve({`*`(seq(GE[i,i], i=1..2))<>0}, b);

                            {b <> 2}

@Joe Riel Another way would be to perform Gaussian elimination, and solve for the set of restrictions than all diagonal elements are nonzero.

For example,

restart;

A := <1, 1, -2, -3|-1, -9, b, 11|-1, 3, 2, -1|b, -10, -4, 6>:

cols := [1,2,3]:
GE := LinearAlgebra:-GaussianElimination(A[..,cols]):
solve({seq(GE[i,i]<>0, i=1..nops(cols))}, b);

                            {b <> 2}

solve({seq(Or(GE[i,i]>0, GE[i,i]<0), i=1..nops(cols))}, b);

                        {2 < b}, {b < 2}

cols := [1,2]:
GE := LinearAlgebra:-GaussianElimination(A[..,cols]):
solve({seq(GE[i,i]<>0, i=1..nops(cols))}, b);

                            {b = b}

The help pages for topic   DocumentTools/Components/Plot has, as its very last example a 3D animation that is embedded with a custom scaling (zoom). That's the only way I know to currently get a 3D plot shown automatically with non-square viewing window as well as a nice zoom.  The point is that there is no irritating white space that one often gets at top and bottom.

I have a procedure (somewhere, on some hard-drive) that takes an constrained 3D plot and applies a transformation to get custom aspect ratios of the axes (eg, x vs z, and y vs z).  The point here is that currently Maple offers a unconstrained 3D plot and display its axes as a cube, or it matches the axes to the numeric ranges of the data. But there's nothing to allow arbitrary, pleasing axes-ratios in the case that the x- or y-data are on a completely different scale than the z-data.

That is not too hard to set up with numeric integration. You can even make a "black box" procedure which accepts a numeric value for c and returns the float approximation of the integral. That could be plotted as a function of c, etc.

(If taking a numeric approach then I'd recommend trying either one of the Monte-Carlo or the _CubaCuhre methods for evalf(Int(...)) since the integrand will be discontinuous at the boundary, because outside the region it will be zero. Most other methods rely on smoothness, and the cost of splitting at the implicit boundary will not be nice, especially if nesting a 1D integrator.)

Or are you really hoping for an explicit formula? Whether that can be accomplished will depend on the example. In general it won't be possible.

@Mac Dude Yes, UseHardwareFloats=true will cause some computations to be done faster. But preventing many instances of software float computation will break far more computations.

The UseHardwareFloats environment variable was introduced in Maple 6, and for several major releases almost the only effect it had was on whether datatype=float acted like datatype=sfloat or datatype=float[8] for rtables.

A few years later the scalar HFloat was devised, and a few people sought out ways to make fast scalar floating-point computations easier to accomplish. (There will always be some people whose wish for a silver bullet defies cold logic. HFloats are not immediate and still need memory management, and do not bring the same degree of performance benefits as evalhf, let alone the Compiler.) The option hfloat for procedures arose around the same time, and allowed more flexibility than evalhf even if not as much performance benefit.) Then UseHardwareFloats was used to also control default HFloat creation upon extraction of scalars from float[8] rtables, and in modern Maple it can plot a role similar to option hfloat, but at the top-level. Alas, UseHardwareFloats documentation is thin.

The reason I'm describing some history is that it's important to realize that a very large portion of the Maple Library (many 100s of thousands of lines of code) was designed and existed for decades under the scheme that increasing Digits would normally allow more accurate floating-point computation. This aspect is still relied upon in many places in the Library.

But if you set UseHardwareFloats to true then in modern Maple that will strictly prevent higher software precision computation and thus also more accurate results from being attained in quite a few routines, some of them key.

And there are additional, important nuances, aside from just high working precision. The hardware float environment has restricted range (roughly +-10^308 down to about +-10^(-308), as I reckon you know). But with software floats much larger or smaller exponents can be used, even with default Digits=10. There are Library commands which rely on that in order to function as designed.

Consider the expression exp(750.1 - x) where x is in the range 760 .. 765. This produces values which are not implausible for an underlying physical setting or model. But if one happens to expand that symbolically, then under forced hardware floating-point the result becomes Float(infinity)/exp(x) which will bring no joy for x in the stated range. So, here, with UseHardwareFloats=true a reasonable problem has suddenly become intractable and requires considerably more care and effort to handle. Here are a few examples, but note that many more problematic cases can arise. Float computations can be problematic under all settings, but this hardware float setting introduces a lot of issues which Maple's software floating-point arena handles nicely.

You wrote, "If it isn't working or useful, then why is the option even there?"  Now, I most certainly did not say or imply that UseHardwareFloats=true "isn't working or useful! I don't know how you managed to make that non sequitur. My opinion that UseHardwareFloats=true is not a good top-level, initialized setting does not at all imply that it is never useful.

Just as you can set option hfloat on a procedure of your own devising, you can also set UseHardwareFloats=true. Within a procedure, or for a limited kind of top-level calculation (pure float linear-algebra, say) it can indeed work and be useful. As Joe, mentioned, as an environment variable its value is inherited by child procedure calls, but setting it does not affect the parent.

So it can be useful, in a targeted, specific computational subprocess like a custom procedure for some task you might have. But setting it at the top-level, as a blanket setting, is going to break stuff.

When a procedure is generated by Localize it affects numeric output from the original Sol returned by dsolve.

If the procedure generated by Localize is ever used to change/set its own parameter then it affects Sol in different ways, according to whether the new parameter value was previously utilized in a call to Localize or not.

A call to query the current parameters, made to the result from Localize, contains the global rather than local names.

In other words, idiosyncratic (but not outright wrong or unrxplainable) behaviour follows if the Localize results are ever used to change their parameter value, or if the original solution from dsolve is utilized. And the remember table of Localize needs clearing if memory is to be recovered, even following unassignment of the Localization procedures and gc. All in all, I think that this not manageable by the common man. And the current behavior might change in the future.

foobar.mw

@bliengme You used = instead of := and so did not actually assign the result from solve.

Are you really using Maple version 16, or is it perhaps version 2016?