Re: [Phys-l] Significant figures -- again
For those that want a reference; I first ran across this in "Numerical Recipes"
by Press, Teukolsky, Flannery, and Vetterling.
In their discussion (section 5.5 on Quadratic and Cubic Equations of the
original volume) they also mention a not so widely known alternative to the
standard quadratic formula that is typically taught in middle school: (-b +-
Root[b^2 - 4ac])/(2a)
Namely:
(2c)/(-b +- Root[b^2 - 4ac])
Which also suffers from the same numerical instability as the more widely known
formula.
Joel R.
| -----Original Message-----
| From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
| bounces@carnot.physics.buffalo.edu] On Behalf Of David Bowman
| Sent: Friday, March 30, 2012 1:21 PM
| To: Forum for Physics Educators
| Subject: Re: [Phys-l] Significant figures -- again
|
| Regarding:
|
| > One of my favorite problems, before I retired, was the "drop a
| > stone down a well and listen for the splash." Calculate the
| > well depth from the time delay. Following the normal sig figs
| > rule eliminates the answer. Students did not know what to do
| > next!
| >
| > "Expand" the square root (from the quadratic formula). This
| > shows precisely what is happening. The "big" terms cancel out
| > - exactly - leaving the answer in the (quickly converging)
| > series. Sig figs takes on a whole new meaning in this problem
| > after the expanding algebra is complete.
| >
| > Do students learn how to expand a function any more? Will a
| > computer do this for them??
| >
| > Oren Q.
|
| Any problem such as this one where a quadratic equation has one of its
| roots nearly cancel out from a subtraction involving the square root
| hardly requires an expansion of the square root to get a easily
| calculable result without the subtraction error. All that is needed is
| to rewrite the "small" root in a way that does not involve the
| subtraction. In particular, consider the quadratic equation:
|
| x^2 - 2*B*x = C
|
| where the magnitude of C is multiple orders of magnitude smaller than
| B^2. The way to write the 2 solutions to this equation so no
| subtraction of nearly cancelling terms is done is to write and evaluate
| them as:
|
| x_1 = B + sgn(B)*sqrt(B^2 + C) and
|
| x_2 = -C/(B + sgn(B)*sqrt(B^2 + C)) = -C/X1.
|
| In this case the x_2 root is the "small" root and no subtraction of
| nearly equal terms is done to evaluate it, nor is an expansion of the
| square root is needed to accurately evaluate it.
|
| Of course sometimes it *is* necessary to expand a function to preserve
| significance in the difference involving a subtraction of nearly equal
| quantities. It is just that in the case of the quadratic equation
| example, such a drastic step is not really needed, and it is easier to
| just evaluate the expression in a sensible way that avoids the
| subtraction error. In general, when faced with a problem that has a
| significant loss of significance due to subtraction, I would recommend
| that one first look for ways rewrite the expression to avoid the
| subtraction. I would recommend the series expansion method only
| *after* one has decided that there is no discernible way to rewrite
| things to avoid the subtraction.
|
| David Bowman
|
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