Large powers of complex numbers

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Homework Help Overview

The discussion revolves around the behavior of complex numbers when raised to large powers, specifically examining the numerical stability and error associated with different forms of representation (rectangular vs. polar). The original poster poses a question regarding the impact of these representations on computational accuracy.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • Participants consider the implications of using analytical versus numerical methods for computing large powers of complex numbers. There is a focus on the potential for numerical error depending on the form of the complex number used in calculations.

Discussion Status

Some participants express a preference for analytical computation to mitigate numerical errors, while others are exploring the differences in error rates between the two forms of representation. The discussion is ongoing with various perspectives being shared.

Contextual Notes

The original poster mentions specific large powers (n ~ 50, 500, one million) and raises concerns about numerical error, suggesting that assumptions about computational methods and their accuracy are being examined.

bjnartowt
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Homework Statement



Suppose you raise a complex number to a very large power, z^n, where z = a + ib, and n ~ 50, 500, one million, etc. On raising to such a large power, the argument will shift by n*ArcTan[b/a] mod 2*Pi, and this is easy to see analytically. However, is there less numerical error when z remains in rectangular form, or less when it is converted to rectangular form?


Homework Equations





The Attempt at a Solution

 
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What do you think?
 
Honestly, I think I'll have to compute Z analytically, and not leave it to a machine to do it.
 
The problem is about where the error is greater, assuming either method is done numerically.
 

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