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Here

https://stackoverflow.com/questions/41857938/how-can-i-find-linear-dependencys-when-one-or-more-coefficients-are-fixed

I asked how I can find linear dependency's with PARI/GP, if one or more coefficients are fixed, for example

How can I find integers $a,b$ with $x^2+ax+b\approx 0$ ?

I know how to find such dependency's with PARI/GP , but not with fixed coefficients. I know the lin-dep and the qflll-command. Does anyone know modifications solving my problem, or does anyone know a mathematical trick that helps ?

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    is $x$ fixed or do you need for all $x$? – 2017-01-25
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    $x$ is given and I want $a$ and $b$. The goal is to find a representation $x\approx u-\sqrt{v}$ or $x\approx \sqrt{u}-v$ (We can assume $0 – 2017-01-25
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    Define $\approx$ Are you hoping to get a sequence of $a_n$ and a sequence $ b_n$ that allow you to get arbitrarily close to $0$ in the $\epsilon – \delta$ sense? – 2017-01-25
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    @BrevanEllefsen Finally yes. I want approximations similar to the continued fraction convergents. – 2017-01-25
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    [See here](https://en.wikipedia.org/wiki/Lenstra–Lenstra–Lovász_lattice_basis_reduction_algorithm). – 2017-01-25
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    @CountIblis But does this work also for fixed coefficients ? – 2017-01-25

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I don't know PARI/GP but here is an algorithm that works for any affine function that does not represent a horizontal or vertical line. In particular it can be used with $$\ell\begin{pmatrix}a\\b\end{pmatrix} = x^2 + a x+b$$ for $x\neq 0$. Start with the matrix $$P=\left(p_1\,p_2 \, p_3\right) = \begin{pmatrix}0&1&0\\0&0&1\end{pmatrix}$$ and repeat the three steps below. Then after each round there will be increasingly accurate approximations in the quadruple of points $(p_1, p_1+p_2, p_1+p_3, p_1+p_2+p_3)$.

  1. Find $\lambda \in \mathbb{Z}$ such that $$\ell P\begin{pmatrix}1\\0\\\lambda\end{pmatrix} \textrm{ and } \ell P\begin{pmatrix}1\\0\\\lambda+1\end{pmatrix}$$ have different signs.
  2. Find $\mu \in \mathbb{Z}$ such that $$\ell P\begin{pmatrix}1\\1\\\mu\end{pmatrix} \textrm{ and } \ell P\begin{pmatrix}1\\1\\\mu+1\end{pmatrix}$$ have different signs.
  3. Substitute $$P \leftarrow P \begin{pmatrix}1&0&0\\0&0&1\\\lambda&1&\mu-\lambda\end{pmatrix}$$

For example for $x = \pi$ the matrices after the first few rounds are $$ \begin{pmatrix}0&0&1\\-10&1&-4\end{pmatrix}, \begin{pmatrix}-1&1&2\\-6&-4&-7\end{pmatrix}, \begin{pmatrix}1&2&-3\\-13&-7&10\end{pmatrix}, \ldots $$ The best approximations found along the way are: $$ \pi^2 + \pi - 13 \textrm{ after one round}$$ $$ \pi^2 + 8 \pi - 35 \textrm{ after eight rounds}$$ $$ \pi^2 - 24393 \pi + 76623 \textrm{ after nine rounds}$$ $$ \ldots $$

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    Seems to be a very good approach! I will try it! – 2017-01-25
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    Are the $\lambda$ and $\mu$ unique in each step ? – 2017-01-25
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    @Peter Yes. It can happen that you find a solution to the affine equation (an exact zero) in which case you can terminate the algorithm. The values for $\lambda$ and $\mu$ can be easily computed, a bit like the usual euclidean algorithm. – 2017-01-25
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    Can you show a simple example how ? – 2017-01-25
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    @Peter If you want to know: $$\lambda = -\left \lceil \frac{x^2 + P_{11} x + P_{21}}{P_{13} x + P_{23}} \right \rceil$$ and $$\mu = -\left \lceil \frac{x^2 + (P_{11} + P_{12}) x + P_{21} + P_{22}}{P_{13} x + P_{23}} \right \rceil$$ where the brackets are the ceiling function. – 2017-01-25
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    With this information, I could program it! Great! Do we know in advance which of the four approximations is best ? – 2017-01-25