Topic 10.7: Newton's Method in Higher Dimensions (Questions)

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Question 1

Perform three iterations to approximate a root of the system described by

f(x) = (x2 + y2 - 3, -2 x2 - ½ y2 + 2)T

starting with the point x0 = (1, 1)T.

Answer: x1 = (0.6666667, 1.833333)T, x2 = (0.5833333, 1.643939)T and x3 = (0.5773810, 1.633030)T. Incidentally, this is closest to the root (0.5773502693, 1.632993162)T (there are four altogether).

Question 2

What happens if you start with x0 = (0, 0)T in Question 1?

Question 3

Perform three iterations to approximate a root of the system described by

f(x) = (x2 - xy + y2 - 3, x + y - xy)T

starting with the point x0 = (-1.5, 0.5)T.

Answer: x1 = (-1.375, 0.575)T, x2 = (-1.36921, 0.577912)T and x3 = (-1.36921, 0.577918)T. Incidentally, this is closest to the root (-1.369205407, 0.577917560)T (there are four altogether).

Question 4

From the symmetry in the function given in Question 3, can you guess another root?

Answer: By symmetry, (0.577918, -1.36921)T would approximate another root because if you swap all xs and ys in the function you get the same expression.

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.