10.4 Scaling

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	Java Number Cruncher: The Java Programmer's Guide to Numerical Computing By Ronald  Mak
	Table of Contents
	Chapter  10.   Solving Systems of Linear Equations

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10.4 Scaling

Even partial pivoting is not always enough, however. The following is another simple system:

The approximate answer is x ₁ = 1.0002 and x ₂ = 0.9997, or, with three significant digits, x ₁ = 1.00 and x ₂ = 1.00.

Forward elimination gives us (with three significant digits)

and back substitution produces the inaccurate solution

So we failed, even though the pivot element was the largest possible. After forward elimination, we still have magnitude errors from the greatly different coefficient values.

Scaling helps with this problem. To scale an equation, divide through by the coefficient with the largest absolute value in that equation. Thus, we divide the first equation through by 10,000 and the second equation through by 2.00:

Now we must partially pivot:

Forward elimination gives us

And, finally, back substitution produces

We can improve the Gaussian elimination algorithm with a combination of scaling and partial pivoting.

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