I am trying to locate a numerical technique to find the intersection of
2 curves. These are defined by tables of data rather than equations.
The data can be fitted with a 2nd degree polynomial, and only intersect
at one point. Both curves have positive slopes, though one has a
larger slope than the other.


Appreciate any suggestions or references.

dan

john2

curve a(x) is a0 + a1.x + a2.x^2
curve b(x) is b0 + b1.x + b2.x^2

Where they cross, a(x) = b(x)
So solving the quadratic
(a0-b0) + (a1-b1).x + (a2-b2).x^2 = 0

should do the trick

john2

I am trying to locate a numerical technique to find the intersection of
2 curves. These are defined by tables of data rather than equations.
The data can be fitted with a 2nd degree polynomial, and only intersect
at one point. Both curves have positive slopes, though one has a
larger slope than the other.


Appreciate any suggestions or references.

dan

julian v. noble

Here's how I would do it in a programming language:

1. Define functions f(x) and g(x) with the tables built-in
(either directly or by direct reference to an array),
using your favorite interpolation method--Newton, Lagrange,
whatever--to evaluate the tabulated function at any x in
the range.

2. Then solve the equation f(x)-g(x) = 0 using any of the
umpty-bazillion 1-dimensional root-finders in circulation.
My favorite happens to be the hybrid regula falsi/binary
search, but any will do (except Newton--you don't want to
have to calculate derivatives!).

3. Solve. Be happy.

Any standard programming language will do. You can probably even use
(ugh!) Maple's limited programming capability. (I do when I have
to.)

--
Julian V. Noble
Professor Emeritus of Physics

http://galileo.phys.virginia.edu/~jvn/

"For there was never yet philosopher that could endure the
toothache patiently."

-- Wm. Shakespeare, Much Ado about Nothing. Act v. Sc. 1.

spellucci

In article <1142249770.527787.102430@v46g2000cwv.googlegroups .com>,
dtshedd@yahoo.com writes:


you could do two independent least squares fits and then the intersection.
but in order to have only one intersection point the leading coefficients must
be equal, otherwise you get a quadratic equation for the intersection point,
which with one real solution must also have a second one.
hth
peter

israel

If dan's words are to be taken literally, the leading coefficients
must be 0 (otherwise the curves can't always have positive slopes).
But presumably what he means is that for x in the interval covered by
the data, the curves have positive slopes and one intersection point.
Then the leading coefficients need not be equal.

So least-squares fit the two curves to quadratics, take the difference and
solve the quadratic equation, taking the root that is in the correct
interval. Or if the x values for the two curves are the same, you could
just do a single least-squares fit for the difference.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

zzzjon

If the y values of each curve are all tabulated at the same x values,
just tabulate the differences. The curves intersect where the
difference changes sign. You could then interpolate between the
differences either side of zero to get a better estimate. If your
curves only have a small quadratic term, a linear interpolation might
be OK; if not, 2nd degree polynomial.

Jon