Relative error between a real and complex number.

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SUMMARY

The discussion centers on calculating the relative error when comparing a real number and a complex number derived from perturbing a polynomial's coefficients. The formula for relative error, traditionally defined as rel.err. = |x1 - x2| / x1, is confirmed to be applicable in the complex plane. Participants agree that this approach effectively measures the distance between the two values, maintaining its validity despite the complexity of the second value.

PREREQUISITES
  • Understanding of polynomial functions and their roots.
  • Familiarity with complex numbers and their properties.
  • Knowledge of relative error calculations in numerical analysis.
  • Basic skills in mathematical notation and equations.
NEXT STEPS
  • Explore the implications of perturbation theory in polynomial equations.
  • Learn about complex analysis and its applications in error measurement.
  • Investigate numerical methods for root finding in polynomials.
  • Study the concept of distance in the complex plane and its relevance to error analysis.
USEFUL FOR

Mathematicians, students studying numerical analysis, and anyone involved in polynomial computations or complex number theory will benefit from this discussion.

Pacopag
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Homework Statement


So I've got a polynomial with real roots. It turns out that if you perturb one of the coefficients by a small amount, two of the roots end up in the complex plane. Now I want to find the relative error in my computation. For two real numbers, say x1 and x2, I would take
rel.err. = {{x1-x2}\over{x1}}.
But what do I do if x1 is real and x2 is complex?


Homework Equations





The Attempt at a Solution

 
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Pacopag said:

Homework Statement


So I've got a polynomial with real roots. It turns out that if you perturb one of the coefficients by a small amount, two of the roots end up in the complex plane. Now I want to find the relative error in my computation. For two real numbers, say x1 and x2, I would take
rel.err. = {{x1-x2}\over{x1}}.
But what do I do if x1 is real and x2 is complex?


Homework Equations





The Attempt at a Solution


mabye try
<br /> \frac{|x_1-x_2|}{x_1}<br />
 
Yes. That makes sense. It is the distance between the two values. Works in the complex plane too. Thanks.
 

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