Hi James:
When you ask Axiom to solve
solve([a=3+x,b=1-x,x=2],[a,b])
I believe (note, this is an educated guess, but simple
problems in Axiom can often have deep reasons, so much so
it is difficult to understand whether things are bugs or
features) that x is treated as a parameter, or as lying in
the coefficient domain (usually an integral domain). All
polynomial equations are converted into polynomial ideals
over the coefficient domain before using the solver. Thus
an equation like x=2 is converted to x-2 (=0), but x-2 is
a non-zero element of the coefficient domain when x is an
indeterminate. To solve the equation in a,b, the
coefficient domain is extended to its field of fractions,
and x-2 is therefore a unit in the polynomial ring with a,
b as indeterminates. The polynomial ideal is then a unit
ideal, hence there is no solution.
Of course, this is not what the user wants. The user wants
to solve for a,b,x but just wants the projected part a, b
of the solution. However, the "equation" x=2 does not
involve a or b. If it is replaced by one that does,
provided it forms a consistent system in the variables a,
b with the rest (thus no single equation in x alone should
be resulted through elimination), then the system will
solved.
solve([a=b+x,b=a-x,x=a-3],[a,b])
[[a= x + 3,b= 3]]
If x is intended to be solved (not a parameter), then it
should be considered as an indeterminate like a, b.
Otherwise, if it is intended to substitute x with a value
like 2, then it should be done after the system in a, b
has been solved.
By distinguishing the two cases when the variables to be
solved is [a,b] or [a,b,x], Axiom allows the user to
indicate his intention.
I can't do much with the tan problem, since we have:
solve([tan(b) = a],[b])
Error detected within library code:
No identity element for reduce of empty list using
operation
append
(You mileage may differ, since I am using a very old
version of Axiom).
This is clearly a bug, for the old version.
William
On Fri, 02 Oct 2009 13:29:29 -0600
James <[email protected]> wrote:
I'm new to axiom, using it for calculations. There seems
to be odd behaviour
when using "solve" on a list of equations. This is with
the most recent axiom
binary package, Version Axiom (May 2009), on Ubuntu 8.10
"intrepid". For
instance:
(8) -> solve([a=3+x,b=1-x,x=2],[a,b])
...
(8) []
Type: List List Equation
Fraction Polynomial Integer
So, the answer is "the empty list"?! That's not very
useful, and seems not
correct either. What am I missing here?
Then instead:
(9) -> solve([a=3+x,b=1-x,x=2],[a,b,x])
(9) ->
(9) [[a= 5,b= - 1,x= 2]]
Type: List List Equation
Fraction Polynomial Integer
That works. Why?
Similarly, axiom seems confused about finding
substitutions. For instance:
(12) -> solve([ tan(bt)=(a*r)/(s*(r-d)), x=100*cos(bt),
y=d*sin(bt)],[x,y,bt])
(12) ->
(12) [[]]
Type: List List
Equation Expression Integer
Here, "the empty list" again - why?
If, instead, I "spell it out" for axiom, by taking the
arctan instead, then
(13) ->
solve([bt=atan((a*r)/(s*(r-d))),x=100*cos(bt),y=d*sin(bt)],[x,y,bt])
(13) ->
(13)
[
(100r - 100d)s a
d r
[x= ----------------------------, y=
----------------------------,
+-------------------------+
+-------------------------+
| 2 2 2 2 2 | 2
2 2 2 2
\|(r - 2d r + d )s + a r \|(r - 2d r
+ d )s + a r
a r
bt= atan(--------)]
(r - d)s
]
Type: List List
Equation Expression Integer
Axiom doesn't know about substituting inverse trig
functions by itself?
But then, again,
(14) ->
solve([bt=atan((a*r)/(s*(r-d))),x=100*cos(bt),y=d*sin(bt)],[x,y])
(14) ->
(14) [[]]
Type: List List
Equation Expression Integer
Arrrrgh! Ok, why is that again, returning "the empty
list" when using "solve"
with the truncated list of variables?
On anther topic, "Floats" in "solve", where this works,
mixing integers and
floats:
(16) -> solve([a=3+x,b=1-x,x=2.0],[a,b,x])
...
(16) [[a= 5.0,b= - 1.0,x= 2.0]]
Type: List List Equation
Fraction Polynomial Float
and this works:
(17) -> solve([a=3+x,b=1-x,x=2],0.001)
...
(17) [[x= 2.0,a= 5.0,b= - 1.0]]
Type: List List
Equation Polynomial Float
doing this, using "x=2.0" instead of "x=2":
(18) -> solve([a=3+x,b=1-x,x=2.0],0.001)
...
There are 20 exposed and 3 unexposed library
operations named solve
having 2 argument(s) but none was determined to be
applicable.
...
Cannot find a definition or applicable library
operation named solve
with argument type(s)
List Equation Polynomial Float
Float
Arrrrgh! - the dreaded "none was determined to be
applicable"!
Does that make sense, that?
Thanks in advance for any clues! Are these bugs?
James
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William Sit, Professor Emeritus
Mathematics, City College of New York
Office: R6/202C Tel: 212-650-5179
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/
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