Well, it would have to depend upon what you know about the qualitative aspects of the given problem. If there were a reliable way to do it, then it wouldn't have to be an optional parameter. Isn't it obvious that it has to be an option since the exact extension possibilities are limitless, the precision of computation is variable, and the data is only approximate to a finite number of digits?

Or, with one less name,

h2:=subs(-1.751336299=a1,-0.5457363063736358=a2,
0.2467414433=a4,-0.3611166740=-a1*a5,-0.2061949348454455=a5,f);

((X1*(X3/X1+(X1*(1+X4+X1)/X4*X2/X3*a1-X1+a2)*X2+X1+X3)/X4*
(X4*a1+X2*a1)-X1+a2)*X2+X1)*(a4-X3*a1*a5-X1)/(X4+a5)

Is there a nicer way to get Maple to notice relationships like that amongst a1,a3,a5 ? (with convert/rational?)

Or, with one less name,

h2:=subs(-1.751336299=a1,-0.5457363063736358=a2,
0.2467414433=a4,-0.3611166740=-a1*a5,-0.2061949348454455=a5,f);

((X1*(X3/X1+(X1*(1+X4+X1)/X4*X2/X3*a1-X1+a2)*X2+X1+X3)/X4*
(X4*a1+X2*a1)-X1+a2)*X2+X1)*(a4-X3*a1*a5-X1)/(X4+a5)

Is there a nicer way to get Maple to notice relationships like that amongst a1,a3,a5 ? (with convert/rational?)

The closed form solutions for cubics and quartics may be particularly bad for numeric root approximation. So they aren't used in the best software.

You mentioned "the highest accuracy possible" but that in itself needs explaining. Matlab works at fixed hardware double precision, while Maple, Mathematica, etc, can compute in pretty much arbitrary higher precision.

You asked about the best software for computing roots of a quadratic. Is that what you meant, or did you also want to consider higher degrees?

Are the extremes of the ranges for the coefficients the most interesting challenge to such software, in practice? Or is bad conditioning and multiple roots more to the point?

Do you think that a good root-finder would necessarily show the accuracy of its results? What should we think of a routine that did not report on the accuracy (or, say, have the accuracy be implicit at say 1 ulp in the returned result)?

Are you considering polynomials with exactly known coefficients (no error, and not even float approximations, but rather exact rationals) or with floating-point (and thus approximate) coefficients? For exact coefficients maybe look at Maple's RootFinding:-Isolate routine. For inexact coefficients maybe look at Zeng's work.

You posted your original questions in one of the "Student Help" forums, and not in one of the non-Student-related "Get Help" or "General Discussion" forums.

You posted your original questions in one of the "Student Help" forums, and not in one of the non-Student-related "Get Help" or "General Discussion" forums.

In other responses in this thread below you mention int, simplify, series, limit, rsolve and assume as the sort of important symbolic algebra topics that need work. But these do not immediately or necessarily relate to what Jacubi has listed immediately above. So the continued existence of those items listed by Jacubi do not offer evidence that the corporate resources devoted to core symbolics (like int, limit, series) is relatively very small.

You have elsewhere indicated that you have not worked at Maplesoft for many years. Do you work at one of these mentioned research labs? If not, how can you state, except by hearsay, what relative proportion of work is done within the company on various items?

You broadly generalize that little core research is done at these research labs on core topics, and then in omitting mention of the company itself imply that nobody there may be doing work on it either. How can you know?

This might be of significantly more interest if it were available on platforms other than Windows.

The Maplesoft webstore doesn't (yet) appear to have details of this new version 2.

The cost of the first version with country of origin as Canada was (5 min ago) $399.00 if entered as commercial/govenment, $99.00 if student, and $199.00 if entered as academic. For that amount, if would be more appealing if the webstore also provided demo worksheets. Are there demos for it somewhere?

How can one judge the difference between the "Mechanics of Materials" MapleConnect 3rd party add-on and the "Structural Mechanics" add-on? (With country of origin as Canada, the Structural Mechanics add-on was $139.00 academic, $99.00 student and $199.00 commercial, 5 mins ago in the webstore.)

How about,

simplify(A) assuming x>0;

How about,

simplify(A) assuming x>0;

It helps if you ask a question, which you haven't yet done.

For an example of a linear programming problem being solved in Maple, look at the help for Optimization[LPSolve] .

It helps if you ask a question, which you haven't yet done.

For an example of a linear programming problem being solved in Maple, look at the help for Optimization[LPSolve] .

I think that your points are good ones. In order to use Maple really well it is necessary to know a lots of things that are quite under-documented. This affects the new user, sometimes by a great deal. Some days I wonder how new users manage at all.

Giving Maple a fancy wysiwyg gui only helps the new user up to the point that simpler examples work OK. But it's no substitute for better documentation.

> ?scoping
Help error, help for `scoping` not found

> # ?lexical just brings up the superdense help page for proc.

Another example: ?spec_eval brings up the Special Evaluation Rules help page. So, where is the Normal Evaluation Rules help page, that describes how top-level evaluation happens, how arguments to procedures are evaluated up front, and how paramaters and locals are evaluated within procedures? I don't think that the help page ?proc explains all that -- at least not in a way that anyone but a computer scientist or programmer would understand. And the bundled User's Guide doesn't explain it all. And the Advanced Programming Guide is not bundled in.

I think that your points are good ones. In order to use Maple really well it is necessary to know a lots of things that are quite under-documented. This affects the new user, sometimes by a great deal. Some days I wonder how new users manage at all.

Giving Maple a fancy wysiwyg gui only helps the new user up to the point that simpler examples work OK. But it's no substitute for better documentation.

> ?scoping
Help error, help for `scoping` not found

> # ?lexical just brings up the superdense help page for proc.

Another example: ?spec_eval brings up the Special Evaluation Rules help page. So, where is the Normal Evaluation Rules help page, that describes how top-level evaluation happens, how arguments to procedures are evaluated up front, and how paramaters and locals are evaluated within procedures? I don't think that the help page ?proc explains all that -- at least not in a way that anyone but a computer scientist or programmer would understand. And the bundled User's Guide doesn't explain it all. And the Advanced Programming Guide is not bundled in.

Assumptions can be tricky. When you place assumptions on a name in Maple then what happens is that Maple creates a new instance of that name. But it doesn't always affect any other instance of the name. This is one of those situations.

Once you make those assumptions on a and b, then you have to be more careful if you subsequently want to substitute for instance of a and b which don't have them.

The a and b in your definition of f do not have assumptions on them, per se. Sure, when you call f then maple uses its lexical scoping functionality to get them from an outer level. That's how it makes your convenient simplification of sqrt(b^2) work out. But in order to substitute into eval(f) you would need to change the original a and b.

> restart:

> f := x -> (a*b)/sqrt((a*cos(x))^2 + (b*sin(x))^2):
> assume (a > 0, b > 0):

> subs([a=14, b=20], eval(f));
                                        a b
                      x -> -----------------------------
                                 2       2    2       2
                           sqrt(a  cos(x)  + b  sin(x) )

> subs([':-a'=14, ':-b'=20], eval(f));
                                        280
                     x -> -------------------------------
                                         2             2
                          sqrt(196 cos(x)  + 400 sin(x) )

And the nice simplification that you wanted from f will still obtain.

> eq2 := 20.3 = f(Pi/2);
                               eq2 := 20.3 = a~

You can then plot it.

plot(subs([':-a'=14, ':-b'=20], eval(f)), 0..2*Pi, coords = polar);