@Markiyan Hirnyk

It is easier to define independence precisely. Assume that the equations have all been put in the form where the right sides are 0. They are independent if there exists an assignment of numerical values to all of the unknowns such that

1. none of the evaluated left sides are 0
2. none of the evaluated left sides are equal to any other evaluated left side.

PS: Upon sleeping on it, I woke up realizing the above definition is not adequate. I withdraw it, I'll need to think about it some more.

@Markiyan Hirnyk

It is easier to define independence precisely. Assume that the equations have all been put in the form where the right sides are 0. They are independent if there exists an assignment of numerical values to all of the unknowns such that

1. none of the evaluated left sides are 0
2. none of the evaluated left sides are equal to any other evaluated left side.

PS: Upon sleeping on it, I woke up realizing the above definition is not adequate. I withdraw it, I'll need to think about it some more.

Maple chose two of the five unknowns to eliminate. In this case, it is almost obvious that it will pick d and s, but we can't rely on that in general.

Maple chose two of the five unknowns to eliminate. In this case, it is almost obvious that it will pick d and s, but we can't rely on that in general.

@Markiyan Hirnyk 

A set of equations is independent if none of the equations can be derived from the others. I'm just extending the concept of "linear independence" to arbitrary equations.

The reason that I'm asking is that so far (in this thread) it seems that solve simply refuses to give any solution when the number of independent equations is greater than the number of variables for which solutions are requested. If that is the case, then solve should just give an error message telling the user to use eliminate instead. Returning NULL is not very helpful.

@Markiyan Hirnyk 

A set of equations is independent if none of the equations can be derived from the others. I'm just extending the concept of "linear independence" to arbitrary equations.

The reason that I'm asking is that so far (in this thread) it seems that solve simply refuses to give any solution when the number of independent equations is greater than the number of variables for which solutions are requested. If that is the case, then solve should just give an error message telling the user to use eliminate instead. Returning NULL is not very helpful.

@Markiyan Hirnyk

I would consider any algebraic response meaningful in this context. By saying "meaningful," I simply wanted to exclude responses such as warnings, error messages, communication via environment variable (like SolutionsMayBeLost), etc. You're right, I should've used a different word. Just change "meaningful" to "algebraic".

@Markiyan Hirnyk

I would consider any algebraic response meaningful in this context. By saying "meaningful," I simply wanted to exclude responses such as warnings, error messages, communication via environment variable (like SolutionsMayBeLost), etc. You're right, I should've used a different word. Just change "meaningful" to "algebraic".

Is there any example of solve giving a meaningful non-NULL answer when the number of independent equations is greater than the number of variables for which solutions are requested?

Is there any example of solve giving a meaningful non-NULL answer when the number of independent equations is greater than the number of variables for which solutions are requested?

@lisa1301 I just editted the piecewise command, so look again. See also my comment above about the bug in the MaplePrimes editor.

If you really want tapered natural-looking ends, any function f(t) can be used for the radius multplier (candy, in my example) that satisfies f(0) = 0, f(1) = 0, 0 < f(t) <= 1 for 0 < t < 1. A good example is 4*x*(1-x).

@lisa1301 I just editted the piecewise command, so look again. See also my comment above about the bug in the MaplePrimes editor.

If you really want tapered natural-looking ends, any function f(t) can be used for the radius multplier (candy, in my example) that satisfies f(0) = 0, f(1) = 0, 0 < f(t) <= 1 for 0 < t < 1. A good example is 4*x*(1-x).

There's the darnedest bug in the MaplePrimes editor in my web browser. This has happened to me before. If I edit a post that has a less-than sign, it removes the rest of that line, including the sign, as if it were expecting some HTML-type code enclosed in angle brackets. I have to manually put back in the rest of the line. Have you heard of this? I'll edit it now.

There's the darnedest bug in the MaplePrimes editor in my web browser. This has happened to me before. If I edit a post that has a less-than sign, it removes the rest of that line, including the sign, as if it were expecting some HTML-type code enclosed in angle brackets. I have to manually put back in the rest of the line. Have you heard of this? I'll edit it now.

Bravo, Joe. It is a pleasure to read code written so clearly. One question: What is reset? Obviously it is a command to reset the iterator to the beginner, but I can't find documentation on it.