"empty list" results, inverse trig substitutions, "Floats" in "solve"

[James-326]

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-2]

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 <james@...> 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
>
>
>
>_______________________________________________
>Axiom-developer mailing list
>Axiom-developer@...
>http://lists.nongnu.org/mailman/listinfo/axiom-developer

William Sit, Professor Emeritus
Mathematics, City College of New York                  
Office: R6/202C Tel: [phone]
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/


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[Bill Page-7]

On Sat, Oct 3, 2009 at 2:20 PM, William Sit <wyscc@...> wrote:

>
> 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.
>

Both Axiom (February 2009) and OpenAxiom (current) return

(1) -> solve([tan(b) = a],[b])

   (1)  []
                                  Type: List List Equation Expression Integer

But FriCAS gives:

wspage@debian:~$ fricas -nox

                 FriCAS (AXIOM fork) Computer Algebra System
                         Version: FriCAS 2009-09-17
            Timestamp: Wednesday September 30, 2009 at 22:17:16
-----------------------------------------------------------------------------
   Issue )copyright to view copyright notices.
   Issue )summary for a summary of useful system commands.
   Issue )quit to leave FriCAS and return to shell.
-----------------------------------------------------------------------------

(1) ->
(1) -> solve([tan(b) = a],[b])

   (1)  [[b= 0]]
                              Type: List(List(Equation(Expression(Integer))))
(2) ->

Regards,
Bill Page.


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[William Sit-2]

Hi James:

solve([a=3+x,b=1-x,x=2.0],0.001)

is an "incorrectly" constructed command. The "solve"
function that was applicable in

solve([a=3+x,b=1-x,x=2],0.001)

is available for the domain List Equation Fraction
Polynomial Integer (from the package FLOATRP) but is not
available when Integer is replaced by Float.

According to documentation from FLOATRP, "solve(leq, eps)"
finds all the real solutions of the system leq of
equations of *rational* functions with respect to all the
variables appearing in lp [sic:leq], with precision eps.

Thus, despite the package is called with FLOATRP(FLOAT),
the leq must be from List equation Fraction Polynomial
Integer.  I think the reason is that the package uses
Groebner basis. See numsolve.spad for source: the
restriction is given near the top, in package
InnerNumericFloatingSolver (K specifies the coefficient
field of the input polynomials and must be either Fraction
Integer or Complex(Fraction Integer).

William


On Fri, 02 Oct 2009 13:29:29 -0600
  James <james@...> 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
>
>
>
>_______________________________________________
>Axiom-developer mailing list
>Axiom-developer@...
>http://lists.nongnu.org/mailman/listinfo/axiom-developer

William Sit, Professor Emeritus
Mathematics, City College of New York                  
Office: R6/202C Tel: [phone]
Home Page: http://scisun.sci.ccny.cuny.edu/~wyscc/


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