Yang, Shuli a écrit : > Here is the problem, First, please do include a [math] prefix in the subject line when you post a message related to commons-math to this list. It is shared by several sub-projects and it helps people filtering their mail.
> > Function F(x) does not have a formula to specify the relation between x > and f(x), but for each x, I have got the value of f(x) by a measurement > machine. > > I am interested in the point xi at which the second derivative of f(xi) > is zero. > > Can anybody help me how to do that? If you do not have an explicit formula, you should use either polynomial interpolation or finite differences to get the derivatives. Polynomial interpolation is better suited if you have a sample of points f(xi) with large intervals between each xi and no way to reduce this (i.e. you cannot redo as many measurements as you want). In this case, you can fit a polynomial to a sub-sample of your points, compute the derivative of the polynomial from the coefficients, and solve the equation p''(x) = 0. If you can find a root in your sample, this is good. If the root is far from your sample, it is probably not good as the polynomial is a good approximation only near the sample. Beware not to use a too high degree polynomial as they tend to oscillate a lot between points (it is Gibbs phenomenon). If you can perform as many measurement as you want and can use small intervals between each xi, then finite differences is probably better suited. You should choose a small step size h and evaluate the function at several locations around x, for example f(x-2h), f(x-h), f(x), f(x+h), f(x+2h) ... You can combine these values to compute an approximation of the derivative. The more points you use and the smaller the step size is, the better the approximation will be. Here are some finite differences rules for second derivative, using the notation pkh(x) = f(x + k h) + f(x - k h) 3 points: f''(x) ≈ [ p1h - 2 f(x) ] / h^2 5 points: f''(x) ≈ [ - p2h + 16 p1h - 30 f(x) ] / 12h^2 7 points: f''(x) ≈ [ 2 p3h - 27 p2h + 270 p1h - 490 f(x) ] / 180h^2 Here are some finite differences rules for first derivative, using the notation mkh = f(x + k h) - f(x - k h) 2 points: f'(x) ≈ m1h / 2h 4 points: f'(x) ≈ [ - m2h + 8 m1h ] / 12h 6 points: f'(x) ≈ [ m3h - 9 m2h + 45 m1h ] / 60h 8 points: f'(x) ≈ [ -3 m4h + 32 m3h - 168 m2h + 672 m1h ] / 840h Luc > > > > Thanks, > > > > Shuli > > ********************************************************************* > This message and any attachments are solely for the > intended recipient. If you are not the intended recipient, > disclosure, copying, use or distribution of the information > included in this message is prohibited -- Please > immediately and permanently delete. > --------------------------------------------------------------------- To unsubscribe, e-mail: [EMAIL PROTECTED] For additional commands, e-mail: [EMAIL PROTECTED]
