Parse coefficients from a Resolve output

Question

I have a huge output from Resolve (quantifier elimination) that I must parse to extract the coefficients as numbers. Take for instance something that looks like this:

(c | x | y | z) \[Element] Reals && (c <= 1 && 
    c > -1 && -x - y + z == 0 && -2 c + 2 x >= -2)

As you can see, it outputs the basis and domain, followed by the expressions for the region the solution defines. (In this case, for instance, c <= 1 would be what I call an "expression", as well as -x - y + z == 0.)

I now must extract the coefficients from each expression of this result. A reasonable output for the example above would be to build the list

{{1, 0, 0, 0, "<=", 1}, {1, 0, 0, 0, ">", -1},
{0, -1, -1, 1, "==", 0}, {-2, 2, 0, 0, ">=", -2}}

but anything reasonable (in the sense it'd allow me to get the extract coefficients of each expression as numbers, while also storing the relation type) would do.

The closest hint I found in the documentation was the Coefficient function, but I couldn't adapt it to work in this case. Is it possible to do so? If not, are there other built-in options?

Also, I've seen a few related questions (such as this) that do something similar using pattern matching, but for a single expression. Would this approach be a extensible to my case? Do you have any hints on how should I proceed to apply it to my situation?

I'm quite new to Wolfram Language so I appreciate any help whatsoever.

For clarity, please provide the output that you expect for this case. — Bob Hanlon, Oct 08 '20 at 14:37
Your question is not too clear. To you mean to set all numeric coefficients to zero? Easy: expy/. x_/;NumericQ[x] ->0. Of course this may result in new coefficients != zero in this case use "//." instead of "/." — Daniel Huber, Oct 08 '20 at 15:06
@DanielHuber My question wasn't clear enough. Following the suggestion above, I updated it with a desired output.
Your answer helped me anyway. Do you have any tips on how would I use this type of pattern matching to, instead of substitute the values, extract them? More than that: how would I extract the ones in each expression, while settings the missing ones (i.e.: present in the basis but not in a given expression) to zero? — cab20, Oct 09 '20 at 00:37

score 2 · Accepted Answer · answered Oct 09 '20 at 09:03

You may extract the different comparisons using patterns by e.g.:

ex = (c | x | y | z) \[Element] 
    Reals && (c <= 1 && c > -1 && -x - y + z == 0 && -2 c + 2 x >= -2);
comp=Cases[ex, (Less | Greater | LessEqual | GreaterEqual | Equal)[__]  ]
({c <= 1, c > -1, -x - y + z == 0, -2 c + 2 x >= -2})

The coefficients of the variables you can get e.g. by:

coef = CoefficientRules[comp[[All, 1]], {c, x, y, z}]
({{{1, 0, 0, 0} -> 1}, {{1, 0, 0, 0} -> 
   1}, {{0, 1, 0, 0} -> -1, {0, 0, 1, 0} -> -1, {0, 0, 0, 1} -> 
   1}, {{1, 0, 0, 0} -> -2, {0, 1, 0, 0} -> 2}})

Or in another form:

coef /. Rule[x1_, x2_] :> {Pick[vars, x1 /. 1 -> True][[1]], x2}
({{{c, 1}}, {{c, 1}}, {{x, -1}, {y, -1}, {z, 1}}, {{c, -2}, {x, 2}}})

The right sides of the comparisons:

rhs=comp[[All, 2]]
({1, -1, 0, -2})

Parse coefficients from a Resolve output

1 Answers1