Python: my polynomial coefficients are off by a factor of 10

Question

I am learning scientific computing with python. In the exercise, I am supposed to generate a polynomial by using its roots with this formula:

Here is my implementation:

def poly(x,roots):            #Pass real and/or complex roots

    x = symbols(x)
    f = 1            
    for r in roots:
        f = f*(x - r)

    return expand(f)

When I test it:

from sympy import expand
poly('x',[(-1/2), 5, (21/5), (-7/2) + (1/2)*sqrt(73), (-7/2) - (1/2)*sqrt(73)])

I get:

x**5 - 1.7*x**4 - 50.5*x**3 + 177.5*x**2 - 24.8999999999999*x - 63.0

But I should get:

10*x**5 - 17.0*x**4 - 505.0*x**3 + 1775.0*x**2 - 248.999999999999*x - 630.0

Hence, everything is off by a factor of 10. If I set f = 10, it works, but I don't see why I should do that. Am I making an obvious mistake? Thank you!

You are constructing polynomial with specific roots. If you multiply it's coeffs to any number, the roots will remains the same. — mingaleg, Nov 17 '16 at 01:44
@matiaselgart Hi! It's supposed to make Python treat 'x' as an algebraic variable. That's how I was shown, but is there a better way? — Johnathan, Nov 17 '16 at 01:52
@Jonathan you shouldn't be getting 10*x**5, the above answer is correct and you can verify this by plugging in all the roots and seeing that for each `x`, `p(x)=0`. While this will also be true with `10 * p(x)`, there's no reason to do this. — Eli Sadoff, Nov 17 '16 at 01:54
@EliSadoff OH! I see now: I don't know why I didn't realize this obvious mistake. The authors chose those coefficients, but I have the form with the most reduced coefficients. Thank you!!! — Johnathan, Nov 17 '16 at 02:00
No worries. I'm a scientific computing guy, so I like helping out with this stuff. — Eli Sadoff, Nov 17 '16 at 02:01

score 1 · Accepted Answer · answered Nov 17 '16 at 02:03

1

While 10x**5 + ... is correct, that is 10 * p(x), which isn't really what is needed. The answer you are getting is fine right now as well and you can test that as for each r in roots, p(r) is 0.

answered Nov 17 '16 at 02:03

Eli Sadoff

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I tested each root. As the book warns me, due to floating-point rounding errors, I get very small numbers close to zero. If please possible, do you have a preferred technique to correct this? – Johnathan Nov 17 '16 at 02:21
1

@Johnathan You'll generally get floating point errors. I generally set a tolerance of `1e-12` and `abs(p(r)) < 1e-12`, you should be fine. Read more about it [here](http://stackoverflow.com/questions/588004/is-floating-point-math-broken?rq=1). – Eli Sadoff Nov 17 '16 at 02:22

score 0 · Answer 2 · answered Nov 17 '16 at 01:54

0

x**5 is correct.

As you can see in the formula, there's no coefficient for it.

answered Nov 17 '16 at 01:54

Karoly Horvath

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Thank you very much for your help! – Johnathan Nov 17 '16 at 02:01

Python: my polynomial coefficients are off by a factor of 10

2 Answers2