Re: Types in Mathematica, a practical example

*To*: mathgroup at smc.vnet.net*Subject*: [mg62937] Re: [mg62800] Types in Mathematica, a practical example*From*: "Ingolf Dahl" <ingolf.dahl at telia.com>*Date*: Fri, 9 Dec 2005 05:10:16 -0500 (EST)*Reply-to*: <ingolf.dahl at telia.com>*Sender*: owner-wri-mathgroup at wolfram.com

Thanks for all answers. In my previous submission I asked for a way to define/declare list with undefined elements. All the answers that I have got indicate that it is difficult in the present versions of Mathematica. Maybe such a construction would be awkward, definitely unnecessary and a way to see Mathematica from the wrong point of view. But once upon a time "zero" was unknown as concept and symbol. Once, but somewhat later, I did some programming in a language named ASYST (a dialect of Forth as PostScript also is), where zero-length empty strings were not allowed. This single design flaw I considered as the largest drawback of the ASYST language. From my point of view I see these shortfalls as symptoms of the same syndrome. I would not really enjoy having to explain to a newbie what error he had done if she/he had encountered the same problem as I stated initially. I would have to explain that in Mathematica it is necessary to choose between two ways to define matrices and similar structures: Either from bottom and up, e.g. by using Array, or from top and down, e.g. by using Table. In the first case, if you give the whole structure a name, it is not possible to use the same name as is used for assigning values to the individual elements. What I would prefer is a new Mathematica word: "Undefined". Then I could define a list x as x = {1,Undefined,2}; x[[1]] should evaluate to 1 as usual, but for this word "Undefined" the list part x[[2]] should evaluate to x[[2]], precisely as for an undefined unindexed symbol. x[[4]] should give the usual error message: "Part::partw: Part 4 of {1, Undefined, 2} does not exist." x[[3]] =. and Clear[x[[3]]] should redefine x as {1,Undefined,Undefined}, and x = Undefined should be interpreted as x =. Could this be implemented by operator overloading or by unprotecting and redefining Part? I believe that this could be implemented in Mathematica without destroying compatibility with existing code, since it would be an added-on feature. And my intuition (if that is of any value) tells me that such an addition would have benefits I fail to see. As newbie I was curious if I could do something according to the following idea: Define a triangle by t = triangle[{3,4,Undefined}, {Undefined, Undefined, Pi/2}] with a list of the sides as the first argument and a list of the angles as the second. Then I wanted to write a function, which would give the following output when applied to t: In[]= AngleSumTheorem[t] Out[]= {t[[2,1]]+t[[2,2}]]+Pi/2==Pi} Then I also should be able to write LawOfSines and LawOfCosines functions, solve the equations and feed back the values into t. Unfortunately I cannot do this in Mathematica in this elegant way. Is this to demand too much from Mathematica? In the answers I have got, there are some different work-arounds suggested to my initial problem, to correct the following code (see my first letter below): a = {{1, 2}, {3, 4}}; x - a /. {x -> a} The first work-around is to use xx = Array[x, Dimensions[a]] I know that this is a way to get around the problem in Mathematica, but it is a clumsy way. I have to have two different entities in the air; x for assignments of new values to the matrix, and xx as a short form for the matrix, reading out the values. It is like "My name is Bill but call me Tom". And a clumsy way leads the thinking astray. I have to choose every time I want to define a matrix: should I use a simple variable definition or a double notations as you suggest? It would be much better if the same notation could be used for the matrix and the elements consistently, without having to choose from the beginning if I am going to work with the entity or with the elements. The second work-around is to redefine or manipulate the operator that is applied to the list or matrix I am working with. For matrices there is a whole bunch of functions that has to be rewritten, and I do not find this idea very attractive. I feel it as a round-about way to solve the problem. Andrzej Kozlowski, who suggests such a solution, thinks that I am looking at Mathematica from the wrong view point. It is my strong belief that one should be able, especially when mathematics is involved, to observe from as many different view points as possible. An item is not really beautiful, or properly designed, if it is not beautiful from all view points. The same should apply to Mathematica, and I have also understood that Mathematica was intentionally designed to be well-behaved when seen from different view points. Or? Kristen W Carlson suggests that I should initialize the matrix to have zero in all element. No, a = {{1, 2}, {3, 4}}; x = {{0, 0}, {0, 0}}; x - a /. {x -> a} will clearly not work for me. But a simple way to solve my initial problem is of course to apply the operations in the right order: a = {{1, 2}, {3, 4}}; (x /. {x -> a}) - a Best regards Ingolf Dahl Sweden -----Original Message----- From: Ingolf Dahl [mailto:ingolf.dahl at telia.com] To: mathgroup at smc.vnet.net Subject: [mg62937] [mg62800] Types in Mathematica, a practical example To MathGroup, I am not an advocate for strong typing in Mathematica, but consider the following simple example: I want to see if two matrices are equal. One of them was the result from some equation, and is given inside a rule. Then I write some code similar to this: a = {{1, 2}, {3, 4}}; x - a /. {x -> a} I of course hope to get a matrix filled by zeroes, but if x is undefined, the following is returned: {{{{0, 1}, {2, 3}}, {{-1, 0}, {1, 2}}}, {{{-2, -1}, {0, 1}}, {{-3, -2}, {-1, 0}}}} First x was assumed to be a number, and (x - a) was evaluated. Then x was substituted by the matrix a. No bug in Mathematica, but it was not what I wanted as user. It is easy to make such a mistake in the programming. Of course there are many ways to get around this problem, but is there any reasonably simple way to "type" x to be a list of lists without specifying the elements, in such a way that the above example works? I could do ReleaseHold[Hold[x - a] /. {x -> a}] but then we are not in the "typing business" any longer. I think this question illuminates one aspect of the typing issue in Mathematica. I remember that I as a newbie looked for ways to declare matrices, in such a way that I later could specify matrix elements one-by-one, without initializing them first. I soon learned that there are other ways to achieve similar results, but still I do not see any good reason why I cannot force Mathematica to give the following response from x-a, if x in some way is declared to be a 2x2 list of lists: {{x[[1,1]] - 1, x[[1,2]] - 2},{x[[2,1]] - 3, x[[2,2]] - 4}} I am not allowed to Unset or Clear any part of a list either. Why not? Ingolf Dahl Sweden