9 years, 41 days

## Unstability of the eigenvectors of some ...

Maple 18

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...)

## Decomposition fields of polynomials and ...

Maple 2017

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)?

Maple 2017

## How to obtain the parameters used in the...

Maple 2017

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?