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Jacobian

Defining the Jacobian matrix ˆJ we have

ˆJ=(f1/x1f1/x2f2/x1f2/x2),

we can rephrase Newton's method as

(xn+11xn+12)=(xn1xn2)+(hn1hn2),

where we have defined

(hn1hn2)=ˆJ1(f1(xn1,xn2)f2(xn1,xn2)).