方程组的牛顿法

Linearize the system and solve

A worked pair of iterations

The recipe as geometry

Using Newton's method well

For the system $F(x, y) = x^2 + y^2 - 4 = 0$ and $G(x, y) = x - y = 0$, what is the Jacobian matrix $J$?

Starting Newton's method for the system $x^2 + y^2 = 4$, $x = y$ from the point $(1, 1)$, what is the next iterate?

Newton's method for a system is started from an arbitrary point far from any solution. What is the honest statement about its behavior?

For the system $x^2 + y^2 = 4$, $x = y$, compute the value of $F(x, y) = x^2 + y^2 - 4$ at the point $(1.5, 1.5)$, as a decimal like 0.5.

The system $x^2 + y^2 = 4$, $x = y$ has its solution in the first quadrant at $x = y = \sqrt{a}$ for some integer $a$. Give the value of $a$, as a plain number.