Yunfei duan Hui Pan. A parabola through three points f(a) f(b) f(c) for the derivation of this formula is from denominator should not be zero.

10.3/4 PARABOLIC INTERPOLATION AND BRENT’S

METHOD IN ONE DIMENSION AND ONE-DIMENSIONAL SEARCH WITH FIRST

DERIVATIVES

Yunfei duan Hui Pan

 Golden section search combined with parabolic interpolation

Formula for the abscissa x

A parabola through three points f(a) f(b) f(c)

for the derivation of this formula is from

denominator should not be zero

Brent’s method

six function points a, b, u, v, w and x, The minimum is bracketed between a

and b; x is the point with the very least function

value found w is the point with the second least

function value; v u is the previous value of w;

Xm is the value of the f

Brent’s method

Brent’s method

Brent's Method: This method takes

advantage of the fact that, near a minimum, the function is approximately quadratic. It conisists of the following steps

Find the minimum of the 2nd order polynomial defined by the three bracket points

Use this minimum to define the new point x

Re-define the bracket Of course, there also need to

be checks on this algorithm, when it is implemented in practice, to ensure that:

example

function f interpolating parabola and a bracketing triplet of abscissas (a b c)

a<b<c & f(a) >f(b) <f(c)

(1 2 3) (1 2 4) (1 2 5)

One-Dimensional Search

one dimensional search – what’s the problem? determine the minimizer of a function f : R -

>R typically over a closed interval [a0,b0] assumption: function is unimodal (at least

over this interval)typical approaches:1 bisection method 2 golden section search 3 Newton’s method 4 secant method

golden section search

basic ideastart with interval [a0,b0] (1st derivative

not available) split the interval into three subintervals: pick two new values a1 and b1 and a0 <

a1 < b1 < b0

there are two possible outcomes: f(a0) >f(a1) > f(b1)< f(b0) f(a0) > f(a1) < f(b1) < f(b0)

First derivative method

F(x)=3x^4−4x^3−12x^2+3 We cannot find regions of which f is

increasing or decreasing, relative maxima or minima.

Graphing by hand is tedious and imprecise. Even the use of a graphing program will only give us an approximation for the locations and values of maxima and minima. We can use the first derivative of f, however, to find all these things quickly and easily.

example

F(x)=3x^4−4x^3−12x^2+3 has first derivative

F’(x) = 12x^3−12x^2−24x = 12x(x^2−x−2) 

=12x(x+1)(x−2)   f(x) is increasing on (−1,0) ∪(2 ， ∞ ) and

decreasing on(− ∞ ，− 1) ∪(0 ， 2).

use derivative information only as follows The sign of the derivative at the central point of

the bracketing triplet （ a b c ） the next test point should be taken in the

interval [a b] or [a c] The value of this derivative and of the derivative

at the second-best-so-far point are extrapolated to zero by the secant method

We impose the same sort of restrictions on this new trial point as in Brent’s method.

If the trial point must be rejected, we bisect the interval under scrutiny.