a Provide equations for position and velocity update of particles of the canonical particle swarm optimization algorithm.

b Find the maximum of the function $f(x)=-x^2+5x+20,$ where $-10x10$ using the canonical PSO algorithm. Let the initial positions of the particles at $t=0$ be:

$x_1=-9\mathrm{.}6,x_2=-6,x_3=-2\mathrm{.}6,x_4=-1\mathrm{.}1,x_5=0\mathrm{.}6,x_6=2\mathrm{.}3,x_7=2\mathrm{.}8,$ and $x_9=10.$

Set all initial velocities $v_1=v_2=v_3=v_4=v_5=v_6=v_7=v_8=v_9=0$ and let $c_1=c_2=1,r_1=0\mathrm{.}213,r_2=0\mathrm{.}876$ at $t=1,r_1=0\mathrm{.}113,r_2=0\mathrm{.}706$ at $t=2.$ Calculate the objective function value $f(x_i)$ at $t=0$ and find the personal best for each particle at iteration $1$ and $2.$ Also calculate the global best at the end of each iteration. Show the detailed computations for iterations $1$ and $2.$