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RE: Nonlinear equation
From: |
Bill Kreamer |
Subject: |
RE: Nonlinear equation |
Date: |
Sun, 20 Apr 2003 13:09:19 -0400 |
The equation x = a*y*log(b) - a*y*log(y) + a*y is non-linear, but it is
well behaved for positive numbers. I used Octave to apply the Newton-Raphson
formula to find y for various values of given x, with excellent results.
First, let's simplify the equation. Step one is to define w = x/a .
The equation becomes w = log(b)*y - y*log(y) + y
Then let K = log(b) + 1, so that the equation becomes w = K*y - y*log(y).
Remembering that the limit of y*log(y) as y approaches zero, is zero, the
picture of a plot comes into mind. You can plot this equation with Octave,
but start at a small positive number instead of 0, because Octave will
produce NaN for y*log(y) at y=0.
As y increases from zero, the function starts at zero, increasing to a
maximum, and then decreasing and crossing zero, and decreasing without
limit.
The Newton-Raphson formula is new y = y - ((function of y)/(first
derivative of function of y).
The function of y is K*y - y*log(y) - w where w is a given value for
which you are finding y.
The first derivative is (K - 1) - log(y)
The start value of y should be near where the solution should be. The plot
will provide the clues.
Newton - Raphson converges rapidly for most solutions, but more slowly for
solutions near the function's maximum. If you happen to enter a value of w
that is greater than the maximum, Octave will dutifully converge to a
complex number that satisfies the equation!
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