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petit loup

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Unstability of the eigenvectors of some ...

petit loup 35 Maple 18
eigenvectors
August 21 2020
2 7
 

 Good morning all.
Consider $A=[a_{i,j}]$, a $18\times 18$ matrix (for example); the integer entries are randomly chosen in $[[-5,5]]$ (for example). In general, $A$ has distinct eigenvalues.

I seek the eigen-elements of $B=Transpose(A^{-1})A$ with $10$ significand digits (for example); in general, $B$ is diagonalizable ($P^{-1}BP=D$, a diagonal complex matrix). I use the command $evalf(Eigenvectors(?))$.

Many randomized tests require working with hundreds of Digits. The worst one requires $629$ digits!! Moreover (in this test), when $Digits:=400$, the condition number of $P$ is $10^{118}$ and , with $Digits:=619$, the condition number drops to $376$.

I am surprised by this instability. In particular, this method seems to be unusable when $n=100$.
Does there exist a method (using maple) which allows to solve the problem without dragging behind me a multitude of digits ? (perhaps with iterations...)
Thanks in advance.

 

Decomposition fields of polynomials and ...

petit loup 35 Maple 2017
polynomial galois
July 23 2018
0 1

Hi all. I have two questions about polynomials over Q.

i) Let f,g be polynomials over Q. How to show (with Maple) that the decomposition field of f is included in the decomposition field of g?

ii) Let f be a polynomial of degree n  over Q with Galois group C_n, the cyclic group of order n. Then, there is a polynomial P with coefficients in Q, of degree n-1, s.t. the iteration: u0=some root of f, u_{k+1}=P(u_k) gives all the roots of f; how to find P (with Maple)?

Thanks in advance.

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

petit loup 35 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 35 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 35 Maple 2017
system 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.

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