gnucap:user:newtonnumericexample1 [*]

Newton Numeric Example 1

Let us consider simple 1-node circuit

       --------+-------------       
      |        |             |      
    I=0.001   R=1000     i=V^3/1000  
      |        |             |      
     ---      ---           ---

Which is equivalent to equation:

F(V) = I + V/R + N(V) = V/1000 + 1/1000*V^3 - 0.001

J(V) = 1/1000 + 3/1000 * V^2

F'(V)= 3/1000*V^2*V + 0.001 - 1/1000*V^3

model code and circuit file for this example are here

Iteration sequence for this equation will be:

Iteration	Newton classic (2,3)	Gnucap's Newton Modification (5,6,7)	Damped
0	V0=0	V0=0
	J=0.001	J=0.001
	F=-0.001	F' = 0.001
	S=F/J=-1	R =F'/J=0.001/0.001=1;
	V1= V0-S = 1	V1 = R = 1
1	V1=1	V1 =1
	J=0.004	J = 0.004
	F=0.001	F' =0.003
	S=F/J=0.25	R=F'/J=0.75
	V2=V1-S=0.75	V2=R=0.75
2	V2=0.75	V2=0.75
	J=0.0026875	J=0.0026875
	F=0.000171875	F'=0.00184375
	S=F/J=0.063953	R=F'/J=0.686047
	V3=V2-S=0.0686047	V3=R=0.686047
3	V3=0.0686047	V3=0.0686047	…
	J=0.0024119815	J=0.0024119815
	F=0.00000894221	F'=0.00164579043
	S=F/J=0.003707	R=F'/J=0.68234
	V4=V3-S=068234	V4=R=0.68234
…	…	…