I'm just learning how to do some computations with Axiom, but I've a problem with complex numbers. Why this evalutate to zero ? fricas A: Complex Polynomial Integer Type: Voidfricas A*conjugate A - A^2
Type: Complex(Polynomial(Integer))Definitely wrong. It looks like the source of the problem is: fricas conjugate(a)
Of course I agree that it is wrong. But what would you expect the answer to: conjugate(a) to be? It could return unevaluated as just: conjugate(a) but then what should the Type: be? Is such an expression still necessarily a Complex Polynomial Integer? This is dangerously close to the discussion that we had some months ago about the meaning of "indeterminants", objects which have a type but no specific value. For example, I can say: A:Integer but if I write: A+1 If you answered "yes" to the idea that Perhaps evaluating the expression: conjugate(a) should also complain about the lack of a value? A: Complex Polynomial Integer tells the system that
Thus, conjugate(A) has no meaning as If axiom could work with so that conjugate worked on the type
then axiom could work at some sort of an Tim Tim,I know that this is opening up the whole big subject again, but I do think that Axiom is already "two-faced" about this. Consider for example that we can write: (7) -> A: Complex Polynomial Integer Type: Void (8) -> B: Complex Polynomial Integer Type: Void (9) -> A+B (9) B + A Type: Complex Polynomial Integer Neither A or B "has a value" but Axiom has no trouble agreeing that A+B is still of type Complex Polynomial Integer. I do not see any essential difference between this and (10) -> A:Integer Type: Void (11) -> B:Integer Type: Void (12) -> A+B A is declared as being in Integer but has not been given a value. Regards, Bill Page. Clearly you're right and I agree with you.The problem, as I see it, is that there are subtle degrees of meaning that are easily stepped around when you work on paper but must be clarified in computational mathematics. A: Integer might mean that 1) A+1 is an error since or perhaps that 2) A+1 is another indeterminant integer, where or perhaps that 3) A+1 is a polynomial with 2 integers, representing a constant, where
or perhaps that 4) A+1 is a polynomial in A with integer coefficients ignoring 'A's type
where or perhaps that 5) A+1 is The exact interpretation chosen appears to be dictated by the underlying code and is not the same everywhere. Axiom is the product of many people, some of whom have chosen different interpretations. Indeed, some of the interpretations didn't exist before the computational aspects of mathematics came into play. There are quite a few areas of research that could be followed.
Indeed, working out the implications of the several meanings of
Axiom's main strength has always been as a research platform where it is possible to work out these ideas and reduce them to practice. Unfortunately, research funding seems nowhere to be found. Tim root [daly@idsi.net]? wrote:Clearly you're right and I agree with you. Why cannot we allow ## The error seems to be in the Polynomial domain:(16) -> c:Complex Integer Type: Void (17) -> conjugate(c) c is declared as being in Complex Integer but has not been given a value. (17) -> c:Complex Polynomial Integer Type: Void (18) -> conjugate(c) (18) c Type: Complex Polynomial Integer ## Adding the Polynomial to its domain causes this to go wrong...(19) -> c:Polynomial Complex Integer Type: Void (20) -> conjugate(c) There are 4 exposed and 1 unexposed library operations named conjugate having 1 argument(s) but none was determined to be ... I'm surprised this doesn't work. A Polynomial on a Ring is still a member of that ring and should inherit its functions. (in this case, conjugate) It also strikes me that Polynomial Complex Integer is the proper type here, not Complex Polynomial Integer...clearly they are inequivalent. A: Integer This is the approach taken by Integer, Complex, and Float it seems. or perhaps that 2) I cannot seem to construct an example of type (3). Given: c:Integer and
I want to construct a Polynomial Integer containing or perhaps that 4) The type of or perhaps that 5) Now this truly is a research project. ;) The exact interpretation chosen appears to be dictated by the underlying code and is not the same everywhere. I think this ```
There are quite a few areas of research that could be followed.
Indeed, working out the implications of the several meanings of
```
This does not seem so hard, but maybe I am being naive. Cheers, Bob McElrath? [Univ. of California at Davis, Department of Physics]? [Complex]? is domain constructor which takes a CommutativeRing? as an argument. Some properties of the [Complex]? domain seem to be built-in to the interpreter. For exampleMartin Baker wrote:
Date: Thu, Mar 18, 2010 at 1:53 PM Subject: [fricas-devel] Hypercomplex numbers How would one go about building a set of domains to represent hypercomplex numbers without defining each one individually as a separate domain? For example: complex numbers, double numbers, dual numbers, dual complex, dual quaternion, dual vectors, split complex, split quaternion and so on. Ref: http://en.wikipedia.org/wiki/Hypercomplex_number What I had in mind was 3 domains which all have one real dimension and a second dimension which squares as follows: Complex: i^2 = -1 Double D^2 = +1 Dual e^2 =0 Then these types could be combined in various combinations, for example: myType:Complex Complex Dual Integer down to any depth to produce the required hypercomplex type (even if the algebra gets degraded). The problem is that complex numbers are defined over CommutativeRing? and even types like this are not valid. fricas myType:Complex Complex Integer Even if this was accepted as a valid type we would need to be able to define how the product of the two algebras would be calculated, not necessarily direct product but semi-direct product or Cayley-Dickson doubling process. There would also be practical issues like if the outer Complex uses There are other cases like this, for instance Clifford algebras over Clifford algebras? Would I be right in assuming that FriCAS can't do this sort of thing at the moment? Martin |

complex numbers--root, Wed, 29 Dec 2004 00:47:32 -0600 reply