Saturday, February 07, 2009

Sturm's Theorem: Examples

In a previous entry, I posted the proof of Sturm's Theorem. This is a method for determining the number of real roots in a given interval.

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.




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