# How to find real positive roots of high order polynomials with large coefficients? Can I scale it down while conserving the zeros?

I'm trying to solve high order polynomials (~100) with really large coefficients. In my earlier post, I actually confirmed that these specific sets that I'm working with can only have one real positive root. But the root finding algorithm on MATLAB usually gives multiple real positive roots for orders >40 and larger coefficients. I was wondering if there is a way to scale down the polynomial with larger coefficients while conserving the real positive root? Thanks in advance!

Here's an example of such a dataset. http://www.filedropper.com/polynomials

• Set $R=\min(c, \sqrt{\frac cb},\sqrt[m]{\frac ca})$ as an upper bound for the positive real root and apply a bracketing method like fzero starting from the interval $[0,R]$. Try this out, and also plot the graph close to the root to see how well the floating-point evaluation works. May 7, 2020 at 21:18
• It would be a good addition to your question to include a link to a .mat file with some problematic polynomials. May 7, 2020 at 21:49
• @CarlChristian Good idea, made the edits. May 9, 2020 at 19:45
• @LutzLehmann Thanks for the suggestion. It does seem to work better than before this way. I can go to higher orders but it does go to inf at some point. May 9, 2020 at 20:53
• Your file download is useless without the program. Could you just directly put its contents into a code block? May 10, 2020 at 6:21

Consider $$p(x) = a_0 + a_1x +a_2 x^2 ....+a_nx^n$$. If $$\alpha$$ is a root,
$$a_0+a_1\alpha +a_2 \alpha^2....+a_n \alpha^n =0 \\\implies \frac 1C (a_0 +a_1 \alpha +a_2 \alpha^2 ...+a_n \alpha^n)=0$$
You can make $$C$$ arbitrarily large to scale it down.