Numerical Methods and Optimization

Solving Equations 107

where α =0. Then, obviously



f

1

(x, y)=x

f

2

(x, y)=y

⇔



αF

1

(x, y)= 0

αF

2

(x, y)=0.

To find a root (x

∗

,y

∗

) ∈ (a, b) × (c, d), we take some [x

0

,y

0

]

T

∈ (a, b) × (c, d)

and then recursively calculate (x

k+1

,y

k+1

) by the formula



x

k+1

= f

1

(x

k

,y

k

)

y

k+1

= f

2

(x

k

,y

k

)

(4.23)

for k ≥ 0. In the above, we assume that [f

1

(x

k

,y

k

),f

2

(x

k

,y

k

)]

T

∈ (a, b)×(c, d)

for all [x

k

,y

k

]

T

, so the next vector [x

k+1

,y

k+1

]

T

in the sequence is in the

domain of f

i

(x, y),i =1, 2.

Similar to the one-dimensional case, we call [x

∗

,y

∗

]

T

a fixed point of a

function f(x, y)=



f

1

(x, y)

f

2

(x, y)



if f(x

∗

,y

∗

)=



x

∗

y

∗



,thatis,f

1

(x

∗

,y

∗

)=x

∗

and f

2

(x

∗

,y

∗

)=y

∗

. Obviously, if [x

∗

,y

∗

]

T

is a fixed point of ...