Solving linear systems -- carefully!

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rouben - 26 Oct 2008 02:44 GMT

The solution of the equation a*x=0, where a and x are scalars,
depends on the value of the coefficient a. If a is nonzero,
then x=0 is the only solution. If a is zero, then any x is a
solution.

The Maple command

solve(a*x=0, x);

produces the x=0 answer. Is there a way to tell Maple to
produce a flag/warning/whatever, to let the user know that
solutions may have been lost?

A more interesting situation occurs when solving the
following systems 3 equations in the 2 unknowns x and y:

sys := [
x + a*y = 0,
a*x + b^2*y = 0,
a^2*x + a*b^2*y = 0
];

solve(sys, [x,y]);

this produces the solution [x=0, y=0] which is correct if
b^2-a^2 is nonzero.

If b^2-a^2 is zero, there is an infinite family of solutions
which is being ignored by Maple. Again, is there a way to ask
Maple to let the user know that solutions may have been lost?

As a related question: Are there other CAS that can produce
the complete families of solutions in these examples? I am
thinking of Macsyma, which I used to use back in the '80s,
and which I haven't used since. Macsyma used to stop in the
middle of a calculation and ask the user if b^2-a^2 was nonzero
before it attempted to divide by it.

Signature

Rouben Rostamian

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A N Niel - 26 Oct 2008 11:43 GMT

> As a related question: Are there other CAS that can produce
> the complete families of solutions in these examples? I am
> thinking of Macsyma, which I used to use back in the '80s,
> and which I haven't used since. Macsyma used to stop in the
> middle of a calculation and ask the user if b^2-a^2 was nonzero
> before it attempted to divide by it.

There was a system called Milo (which I haven't used for many years)
that would split into cases in situations like this. So you tell it
solve a*x=0 for x, and it gives you two side-by-side boxes,
one with case a <> 0 , the other with case a = 0.

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Robert Israel - 26 Oct 2008 20:57 GMT

> The solution of the equation a*x=0, where a and x are scalars,
> depends on the value of the coefficient a. If a is nonzero,
[quoted text clipped - 8 lines]
> produce a flag/warning/whatever, to let the user know that
> solutions may have been lost?

If you tell Maple to solve for x, it's in effect working in a ring of rational
functions of a, and in that sense x=0 is the only solution. If you want the
solution a=0, you must solve for both x and a:

solve(a*x=0, {x,a});

{a = 0, x = x}, {a = a, x = 0}

> A more interesting situation occurs when solving the
> following systems 3 equations in the 2 unknowns x and y:
[quoted text clipped - 13 lines]
> which is being ignored by Maple. Again, is there a way to ask
> Maple to let the user know that solutions may have been lost?

solve(sys, [x,y,a,b]);

[[x = 0, y = 0, a = a, b = b], [x = -b*y, y = y, a = b, b = b],
[x = b*y, y = y, a = -b, b = b]]

Signature

Robert Israel israel@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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Axel Vogt - 26 Oct 2008 21:04 GMT

>> The solution of the equation a*x=0, where a and x are scalars,
>> depends on the value of the coefficient a. If a is nonzero,
[quoted text clipped - 16 lines]
>
> {a = 0, x = x}, {a = a, x = 0}

...

working in a ring of rational *functions* ... is there a hint on that
in the help?

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Jon McLoone - 28 Oct 2008 10:17 GMT

<snip>

> A more interesting situation occurs when solving the
> following systems 3 equations in the 2 unknowns x and y:
[quoted text clipped - 4 lines]
> a^2*x + a*b^2*y = 0
> ];

<snip>

> If b^2-a^2 is zero, there is an infinite family of solutions
> which is being ignored by Maple. Again, is there a way to ask
> Maple to let the user know that solutions may have been lost?
>
> As a related question: Are there other CAS that can produce
> the complete families of solutions in these examples?

Mathematica gives:

In[1]:= Reduce[{x + a*y == 0, a*x + b^2*y == 0,
a^2*x + a*b^2*y == 0}, {x, y}]

Out[1]= (b == 0 && a == 0 && x == 0) || ((a == -b || a == b) &&
a != 0 &&
y == -(x/a)) || (a == 0 && x == 0 && y == 0 && b != 0) ||
(a^2 - b^2 != 0 && x == 0 && a != 0 && y == 0)

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Rouben Rostamian - 28 Oct 2008 23:07 GMT

In article <NeednXp-yZYOVp7UnZ2dnUVZ_gidnZ2d@comcast.com>, I asked:

>sys := [
> x + a*y = 0,
[quoted text clipped - 10 lines]
>middle of a calculation and ask the user if b^2-a^2 was nonzero
>before it attempted to divide by it.

Robert Israel and Jon McLoone responded with commands in
Maple and Mathematica for obtaining the full set of solutions.
Thanks! A N Niel pointed to a CAS called Milo which would
also handle the required computations. In searching for Milo,
it appears that it is no longer available. It also appears
that it was a Mac-specific software, although I may be wrong
on that.

Thanks again for all your comments.

Signature

Rouben Rostamian

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Math Forum / Math Software / Maple / October 2008

Solving linear systems -- carefully!