coulomb excitation data analysis codes; gosia 2007 - Physics and ...

Figure 8: Simple steepest descent minimization of the function f(x, y) =x 2 +(x − y) 2fïx 0 − X iα i ¯Pi!=min (4.41)with respect to the coefficients α i .Merging4.40 and 4.41 we get:f(¯x 0 ) − X α i ( ¯5 0 · P i )+ 1 2iwhich can be written in the vector form as:Xα i α j ¯Pi ¯Jj P j =min (4.42)f(¯x 0 ) − ᾱ · ¯β 1+ =min (4.43)2ᾱRᾱwith a set of coefficients α i treated as a vector ᾱ and with:β i = ¯5 0 · ¯P i (4.44)ijR ij = ¯P i J ¯P jThe matrix R is symmetric following the symmetry of J, thus the solution for the vector ᾱ is given by:ᾱ = R −1¯β (4.45)The gradient+derivative minimization algorithm uses two directions - the gradient, defining a directionof steepest descent, and a derivative of the gradient with respect to the displacement along its direction:¯D = limh→0¯5(¯x0 + h ¯5 0 ) − ¯5 0h(4.46)48