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petit loup

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Do you know an effective Gröbner's metho...

petit loup 40 Maple 2017
polynomial real
July 17 2018
2 29
I would like to use Gröbner's method to study polynomial systems (with equality or (and) inequalities) in the case where the variables are REAL. It is known that in general the problem is much more complicated than in the complex case; in particular it is necessary to use gradient methods.
In Maple, we can use the patch "Raglib" (Lip6 laboratory). However "with (RAG)" does not work very well, even for "simple problems" like this one: the $ 9 $ real unknowns are $ X = [x_ {i, j}] \ in M_3 (\ mathbb {R} $. The  system to satify is $ X ^ TX = I_3, x [1,1] <1 / 2,3 / 10 <x [2,3] $, that is, $6$ polynomial equations and $2$ inequalities; clearly, a particular solution is a permutation of the canonical basis. The "HasRealSolutions" command does not provide any result after 2 hours 15 minutes of calculation. The "PointsPerComponents" command indicates that there are no solutions... 
It seems to me that we can also use "RegularChains" but I am not familiar with this library.
  Have you any ideas on these questions? Thank you in advance.
 

How to obtain the parameters used in the...

petit loup 40 Maple 2017
solve substitution
December 10 2017
2 2

Hi...  I use the "solve" command to solve an algebraic  system. Sometimes, the solution is given using some unknowns as parameters.

A toy example: x+y+z=1,x-y+3z=7 gives x=-2z+4,y=z-3 as a function of the parameter z (in the RHS).

Can I obtain directly (without going through the block-by-block solution) the set of parameters used in the given solution?

Thanks in advance.

How to numerically solve a system of equ...

petit loup 40 Maple 2017
polynomial rootfinding
December 01 2017
1 3

Hi.... I'd want to numerically solve a system of  n  polynomial equations of degree 2 with respect to n unknowns. It seems to me that there is a software in Maple (which deals only with the equations of degree 2) that solves this problem. I cannot find it anymore. Do you know it ? Thanks in advance.

Groebner basis and polynomial ideals...

petit loup 40 Maple 18
groebner polynomialideals
November 12 2016
0 1

Hi, I have a big system with 27 polynomial equations in 16 unknowns: f_1=...=f_27=0.  I can store these equations but I cannot calculate a Grobner basis of the ideal  J generated by my polynomials (allocation problem) - I use the library "with(FGb)"-  What interests me is whether my system is minimal in the following sense.

If, for example,  I remove f_1, is the ideal generated by (f_2,...f_27)  J again ? That is to say, is f_1 in the ideal generated by f_2,...,f_27 ? I would like to get an answer "yes" or "no" for each removed  f_i.

My question: can we solve the problem above  without calculating a Grobner basis of J?

Thanks in advance.

 

 

 

 

 

Speed of calculation of the inverse of a...

petit loup 40 Maple 18
linear-algebra complexity
November 03 2016
2 6

I consider  100 .100 real matrices A,B=Matrix(100,100,(i,j)->rand()) (with 12 significant digits).  In general, ConditionNumber(A) is <10^5; also I choose Digits:=17.Theoretically, the complexity of the calculations of Determinant(A), CharacteristicPolynomial(A,x), A.B and MatrixInverse(A) are similar (~n^3). Yet, the times of these calculations are respectively: 0"13, 0"67, 0"60 and, what surprises me, 75" (moreover, I don't display any result).

My question: concerning the calculation of the inverse, where does this factor 100 come from ? Would Matlab  be 100 times faster ? I do not see why this would be the case; in particular, the standard methods for the calculation of the inverse are  easily programmable.

Thanks in advance.

 

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