Apply multivariate Newton Raphson method for the following system o f equations with initial values x 1 2 , y 1 2 t o carry out one iteration ( e x c l u d i n g relative errors ) Use Cramer's rule for solving the Jacobian equation x 2 x 0 7 5 y y 5 x y x 2 General formula for Jacobian o f a system o f n equations J d e l f 1 , i d e l x 1 d e l f 1 , i d e l x 2 c d o t s d e l f 1 , i d e l x n d e l f 2 , i d e l x 1 d e l f 2 , i d e l x 2 c d o t s d e l f 2 , i d e l x n v d o t s v d o t s v d o t s d e l f n , i d e l x 1 d e l f n , i d e l x 2 c d o t s d e l f n , i d e l x n x i 1 x i J 1 f Cramer's rule for 2 b y 2 system ( a s a reminder ) a 1 b 1 a 2 b 2 x y c 1 c 2 Assume a 1 b 2 b 1 a 2 nonzero Then, with help o f determinants, x and y can b e found with Cramer's rule a s x c 1 , b 1 , c 2 , b 2 a 1 , b 1 , a 2 , b 2 c 1 b 2 b 1 c 2 a 1 b 2 b 1 a 2 , y a 1 , c 1 , a 2 , c 2 a 1 , b 1 , a 2 , b 2 a 1 c 2 c 1 a 2 a 1 b 2 b 1 a 2

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Question: Apply multivariate Newton - Raphson method for the following system o f equations with initial values x = 1 . 2 , y = 1

Apply multivariate Newton

-

Raphson method for the following system

o f

equations with initial

values

x = 1.2, y = 1.2 t o

carry out one iteration

(e x c l u d i n g

relative errors

) .

Use Cramer's rule for solving the Jacobian equation.

- x^{2} + x + 0.75 = y

y + 5 x y = x^{2}

General formula for Jacobian

o f

a system

o f n

equations:

[J] = [\begin{matrix} \frac{d e l f_{1, i}}{d e l x_{1}} & \frac{d e l f_{1, i}}{d e l x_{2}} & c d o t s & \frac{d e l f_{1, i}}{d e l x_{n}} \\ \frac{d e l f_{2, i}}{d e l x_{1}} & \frac{d e l f_{2, i}}{d e l x_{2}} & c d o t s & \frac{d e l f_{2, i}}{d e l x_{n}} \\ v d o t s & v d o t s & v d o t s & ? \\ \frac{d e l f_{n, i}}{d e l x_{1}} & \frac{d e l f_{n, i}}{d e l x_{2}} & c d o t s & \frac{d e l f_{n, i}}{d e l x_{n}} \end{matrix}]

{x_{i + 1}} = x_{i} - [J]^{- 1} {f}

Cramer's rule for

2 b y 2

system

(a s

a reminder

)

[\begin{matrix} a_{1} & b_{1} \\ a_{2} & b_{2} \end{matrix}] [\begin{matrix} x \\ y \end{matrix}] = [\begin{matrix} c_{1} \\ c_{2} \end{matrix}] .

Assume

a_{1} b_{2} - b_{1} a_{2}

nonzero. Then, with help

o f

determinants,

x

and

y

can

b e

found with Cramer's rule

a s

x = \frac{| [c_{1}, b_{1}], [c_{2}, b_{2}] |}{| [a_{1}, b_{1}], [a_{2}, b_{2}] |} = \frac{c_{1} b_{2} - b_{1} c_{2}}{a_{1} b_{2} - b_{1} a_{2}}, y = \frac{| [a_{1}, c_{1}], [a_{2}, c_{2}] |}{| [a_{1}, b_{1}], [a_{2}, b_{2}] |} = \frac{a_{1} c_{2} - c_{1} a_{2}}{a_{1} b_{2} - b_{1} a_{2}} .

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