In article <firstname.lastname@example.org>, "junoexpress" <MTBrenneman@gmail.com> wrote:
> I am trying to optimize a function of a single variable which has a > very simple form. > > F(x) = A*[cos(x) - r*cos(c)]^2 + B*[sin(x)-r*sin(c)]^2 > > where A,B,r,c are all known constants with A,B, and r all positive. > (I know from experience with this problem that numerically A and B > have about the same value, r is close to 1, and x should be close to > c).
For a close to b, the form
((a + b) (1 + r^2) + (a - b) (Cos[2x] + r^2 Cos[2c]) - 4 a r Cos[c] Cos[x] - 4 b r Sin[c] Sin[x])/2
may be useful in that (a - b) is small. If a == b, then this reduces to
b (1 + r^2 - 2 r Cos[c - x])
and if c == x, then one obtains
b (r - 1)^2
Perhaps series expansion about a == b, x == c, r == 1, would be useful? The leading terms (dependent on the order of expansion) of this are
(r - 1)^2 (b + (a - b) Cos[c]^2) + (a - b) (r - 1)(x - c) Sin[2 c] + ...
_______________________________________________________________________ Paul Abbott Phone: 61 8 6488 2734 School of Physics, M013 Fax: +61 8 6488 1014 The University of Western Australia (CRICOS Provider No 00126G) AUSTRALIA http://physics.uwa.edu.au/~paul