Multivariable Calculus · free preview

Newton's Method for Systems

Solving systems of nonlinear equations by linearization, the two-variable Newton update built from the Jacobian matrix, and the behavior that convergence is local.

1. Linearize the system and solve

Linearize the system and solve

2. A worked pair of iterations

A worked pair of iterations

3. The recipe as geometry

The recipe as geometry

4. Using Newton's method well

Using Newton's method well

5. Assembling the Jacobian

For the system F(x,y)=x2+y2−4=0F(x, y) = x^2 + y^2 - 4 = 0 and G(x,y)=x−y=0G(x, y) = x - y = 0, what is the Jacobian matrix JJ?

6. One Newton step from a guess

Starting Newton's method for the system x2+y2=4x^2 + y^2 = 4, x=yx = y from the point (1,1)(1, 1), what is the next iterate?

7. Reading convergence honestly

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

8. Residual at a candidate

For the system x2+y2=4x^2 + y^2 = 4, x=yx = y, compute the value of F(x,y)=x2+y2−4F(x, y) = x^2 + y^2 - 4 at the point (1.5,1.5)(1.5, 1.5), as a decimal like 0.5.

9. The exact crossing

The system x2+y2=4x^2 + y^2 = 4, x=yx = y has its solution in the first quadrant at x=y=ax = y = \sqrt{a} for some integer aa. Give the value of aa, as a plain number.

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