Today, I will show some examples of its use.
Example 1: f(x) = x5 - 3x - 1 in the interval [-2, +2]
The Sturm Chain for this polynomial is:
f0 = x5 - 3x - 1
f1 = 5x4 - 3
f2 = 12x + 5
f3 = 1
For x=-2, there are 3 sign changes.
For x=-1, there are 2 sign changes
For x=0, there is 1 sign change
For x=1, there is 1 sign change
For x=2, there is 0 sign changes
So, between -2 and -1, there is 3-2=1 real zero
Between -1 and 0, there is 2-1=1 real zero
Between 0 and 1, there are no 1-1=0 real zeros.
Between 1 and 2, there is 1-0=1 real zero.
In summary, between -2 and +2, there are 3-0=3 real zeros.
Example 2: x5 -ax -b when a,b are positive and 44a5 is greater than 55b4
The Sturm Chain for this polynomial is:
f0 = x5 -ax -b
f1 = 5x4 - a
f2 = 4ax + 5b
f3 = 44a5 - 55b4
For this example, let's look at the interval between -∞ and +∞
For -∞, there are 3 sign changes.
For +∞, there are 0 sign changes.
So, all equations of this form have 3-0=3 real roots.
References
- Heinrich Dorrie (Translated by David Antin), 100 Great Problems of Elementary Mathematics (Dover, 1965)
3 comments:
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thank you
my blog is www.eceblog.co.nr
I have favorited your blog. I plan on reading ALL the articles you've posted. Thanks so much for doing this, and I look forward to more.
Larry F'n Freeman. Hit me back with an email.
Lou Coronado
internetmotion@gmail.com
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