Part 3
2nd Iteration
For x1 = [x1,y1,z1] = [0.72254, 0.86392, 0.02569]


1.467279449

1.709282902

0.6242208

Df(x0)

=

0.8639224733

0.72254268

0.0513797



0.4855161696

2.372448332

1

f1 = 1.131742062
f2 =  0.7148808356
f3 = 2.473654395
the system of equations, in matrix format, is Df(x0)∆x = −f(x0).
1.467279449

1.709282902

0.6242208

∆x

=

1.131742062

0.8639224733

0.72254268

0.0513797

∆y

=

0.7148808356

0.4855161696

2.372448332

1

∆z

=

2.473654395

or,
1.467279449∆x + 1.709282902∆y – 0.6242208∆z = 1.131742062
0.8639224733∆x – 0.72254268∆y – 0.0513797∆z = 0.7148808356
0.4855161696∆x + 2.372448332∆y + ∆z = 2.473654395
Solving this system of equations gives us,
∆x = 0.4174047065, ∆y = 0.360889953, ∆z = 1.820118362
From which,
x2 = x1 + ∆x, y2 = y1 + ∆y, z2 = z1 + ∆z,
x2 = 0.72254 + 0.4174047, y1 = 0.86392  0.36089, z2 = 0.02569  1.82012,
x2 = 0.305135, y2 = 0.50303, z2 = 1.79443,
x2 = [0.305135, 0.50303, 1.79443]
3rd Iteration
For x2 = [x2,y2,z2] = [0.305135, y2 = 0.50303, z2 = 1.79443]


0.292379716

1.553613689

0.1534944

Df(x0)

=

0.5030325661

0.30513813

3.58885701



0.7370215630

1.653728716

1

f1 = 0.9046327596
f2 = 3.463468076
f3 = 0. 186321774
the system of equations, in matrix format, is Df(x0)∆x = −f(x0).
0.292379716

1.553613689

0.1534944

∆x

=

0.9046327596

0.5030325661

0.30513813

3.58885701

∆y

=

3.463468076

0.7370215630

1.653728716

1

∆z

=

0.186321774

or,
0.292379716∆x + 1.553613689∆y – 0.1534944∆z = 0.9046327596
0.5030325661∆x – 0.30513813∆y + 3.58885701∆z = 3.463468076
0.737021563∆x + 1.653728716∆y + ∆z = 0.186321774
Solving this system of equations gives us,
∆x = 2.592141381, ∆y = 0.2276073932, ∆z = 1.347741436
From which,
x3 = x2 + ∆x, y3 = y2 + ∆y, z3 = z2 + ∆z,
x3 = 0.305135  2.592141381, y3 = 0.50303 + 0.2276073932, z3 = 1.79443 + 1.347741436,
x3 = 2.897279, y3 = 0.730637, z3 = 0.446688,
x3 = [2.897279, 0.730637, 0.446688]
This 3d iteration is far from accurate, although it is converging. 10 iterations will give 4 d.p. accuracy with the solution:
X10 = [0.3287, 1.3027. 0.5816]