Numerical Methods

Question

f(x,y)= (x-2)2 + 2(y-2)2 - 100 g(x,y) = -34+x2+24x-9y

(x0,y0 = (4,9) is an initial estimate of the solution of f(x, y) = 0 and g(x,y) = 0, then using Newton's method find the next approximate solution. 

Do all calculations to at least eight decimal digit accuracy. Your answers can be rounded to five decimal digit accuracy when entered. Or they can be entered as an exact rational expression

 

X1 and Y1

Answer 1

Here’s one way to apply Newton’s Method.

g(x,y)=0⇒y=(x²+24x-34)/9,

y-2=(x²+24x-34)/9-2=(x²+24x-52)/9=(x-2)(x+26)/9.

Substitute for y in f(x,y)=0:

(x-2)²+2(x-2)²(x+26)²/81-100=0,

(x-2)²+2(x-2)²(x+26)²/81-100=0,

(x-2)²(1+2(x+26)²/81)-100=0,

(x-2)²(2x²+104x+1433)/81-100=0.

Differentiate:

2(x-2)(2x²+104x+1433)/81+(x-2)²(4x+104)/81.

x₀=4,

x₁=x₀-[(x₀-2)²(2x₀²+104x₀+1433)/81-100]/[2(x₀-2)(2x₀²+104x₀+1433)/81+(x₀-2)²(4x₀+104)/81]

More to follow...

Comments

Continued from above solution:

After substitution: x₁=4.071964018, y₁=(x₁²+24x₁-34)/9=8.923114155.

Rounding: (4.07196,8.92311).

A more accurate solution after a couple more iterations:

x₃=4.070487838, y₃=(x₃²+24x₃-34)/9=8.91784215.

Rounding to 5 decimal places (4.07049,8.91784).

Note that there is another intersection point (see graph).