So I am given three non linear equations:
and i need to solve using newtons method using 100 randomly selected initial guesses for x1^(0), x2^(0) and x3^(0).Code:x1(x2-2)=0, x2(x3-3)=0, x3(x1-1)=0
So I am given three non linear equations:
and i need to solve using newtons method using 100 randomly selected initial guesses for x1^(0), x2^(0) and x3^(0).Code:x1(x2-2)=0, x2(x3-3)=0, x3(x1-1)=0
First find the Jacobian:
$\displaystyle
J(x_1,x_2,x_3)=\left[ \begin{array}{ccc}x_2-2&x_1&0\\
0&x_3-3&x_2\\
x_3&0&x_1-1 \end{array}\right]
$
Now Newton's method proceeds from a initial approximation of guess: $\displaystyle P_{(0)}$ to the next approximation using by solving:
$\displaystyle J(P_{(0)}) \delta P = F(P_{(0)})$
for $\displaystyle \delta P$, then the next estimate is:
$\displaystyle P_{(1)}=P_{(0)}+ \delta P$.
Then the general itteration step is:
solve:
$\displaystyle J(P_{(n)}) \delta P = F(P_{(n)})$
for $\displaystyle \delta P$, then the next estimate is:
$\displaystyle P_{(n+1)}=P_{(n)}+ \delta P$.
In Matlab the equation J dP= P is solved using a statement of the form:
dP=J\P
if I recall correctly
RonL