using this code TODO 0 uncomment r , Earth's radius in km TODO 0 uncomment c , the speed of light in km sec r 6 3 6 7 4 4 4 7 Earth's radius ( km ) c 2 9 9 7 9 2 4 5 8 speed of light in the void emptiness ( km sec ) Below are the known simultaneous satellite positions ( km ) and measured time intervals ( s ) A 1 1 5 6 0 0 B 1 7 5 4 0 C 1 2 0 1 4 0 t 1 0 0 7 0 7 4 A 2 1 8 7 6 0 B 2 2 7 5 0 C 2 1 8 6 1 0 t 2 0 0 7 2 2 0 A 3 1 7 6 1 0 B 3 1 4 6 3 0 C 3 1 3 4 8 0 t 3 0 0 7 6 9 0 A 4 1 9 1 7 0 B 4 6 1 0 C 4 1 8 3 9 0 t 4 0 0 7 2 4 2 Solve the system with Multivariate Newton's Method TODO 1 uncomment F ( x ) F lambda x np array ( sqrt ( ( x 0 , 0 A 1 ) 2 ( x 1 , 0 B 1 ) 2 ( x 2 , 0 C 1 ) 2 ) c ( t 1 x 3 , 0 ) , sqrt ( ( x 0 , 0 A 2 ) 2 ( x 1 , 0 B 2 ) 2 ( x 2 , 0 C 2 ) 2 ) c ( t 2 x 3 , 0 ) , sqrt ( ( x 0 , 0 A 3 ) 2 ( x 1 , 0 B 3 ) 2 ( x 2 , 0 C 3 ) 2 ) c ( t 3 x 3 , 0 ) , sqrt ( ( x 0 , 0 A 4 ) 2 ( x 1 , 0 B 4 ) 2 ( x 2 , 0 C 4 ) 2 ) c ( t 4 x 3 , 0 ) ) TODO 2 uncomment J ( x ) J lambda x np array ( ( x 0 , 0 A 1 ) sqrt ( ( x 0 , 0 A 1 ) 2 ( x 1 , 0 B 1 ) 2 ( x 2 , 0 C 1 ) 2 ) , ( x 1 , 0 B 1 ) sqrt ( ( x 0 , 0 A 1 ) 2 ( x 1 , 0 B 1 ) 2 ( x 2 , 0 C 1 ) 2 ) , ( x 2 , 0 C 1 ) sqrt ( ( x 0 , 0 A 1 ) 2 ( x 1 , 0 B 1 ) 2 ( x 2 , 0 C 1 ) 2 ) , c , ( x 0 , 0 A 2 ) sqrt ( ( x 0 , 0 A 2 ) 2 ( x 1 , 0 B 2 ) 2 ( x 2 , 0 C 2 ) 2 ) , ( x 1 , 0 B 2 ) sqrt ( ( x 0 , 0 A 2 ) 2 ( x 1 , 0 B 2 ) 2 ( x 2 , 0 C 2 ) 2 ) , ( x 2 , 0 C 2 ) sqrt ( ( x 0 , 0 A 2 ) 2 ( x 1 , 0 B 2 ) 2 ( x 2 , 0 C 2 ) 2 ) , c , ( x 0 , 0 A 3 ) sqrt ( ( x 0 , 0 A 3 ) 2 ( x 1 , 0 B 3 ) 2 ( x 2 , 0 C 3 ) 2 ) , ( x 1 , 0 B 3 ) sqrt ( ( x 0 , 0 A 3 ) 2 ( x 1 , 0 B 3 ) 2 ( x 2 , 0 C 3 ) 2 ) , ( x 2 , 0 C 3 ) sqrt ( ( x 0 , 0 A 3 ) 2 ( x 1 , 0 B 3 ) 2 ( x 2 , 0 C 3 ) 2 ) , c , ( x 0 , 0 A 4 ) sqrt ( ( x 0 , 0 A 4 ) 2 ( x 1 , 0 B 4 ) 2 ( x 2 , 0 C 4 ) 2 ) , ( x 1 , 0 B 4 ) sqrt ( ( x 0 , 0 A 4 ) 2 ( x 1 , 0 B 4 ) 2 ( x 2 , 0 C 4 ) 2 ) , ( x 2 , 0 C 4 ) sqrt ( ( x 0 , 0 A 4 ) 2 ( x 1 , 0 B 4 ) 2 ( x 2 , 0 C 4 ) 2 ) , c ) x 0 0 y 0 0 z 0 6 3 7 0 d 0 0 x k

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Question: using this code: ##### TODO # 0 : uncomment r , Earth's radius in km ##### TODO # 0 : uncomment c , the speed

using this code:

##### TODO #

0

: uncomment r

,

Earth's radius in km

##### TODO #

0

: uncomment c

,

the speed of light in km

/

sec

= 6367.4447

# Earth's radius

(

)

= 299792.458

# speed of light in the void

/

emptiness

(

/

sec

)

# Below are the known simultaneous satellite positions

(

)

and measured

# time intervals

(

) .

1 = 15600

; B

1 = 7540

; C

1 = 20140

; t

1 = 0.07074

;

2 = 18760

; B

2 = 2750

; C

2 = 18610

; t

2 = 0.07220

;

3 = 17610

; B

3 = 14630

; C

3 = 13480

; t

3 = 0.07690

;

4 = 19170

; B

4 = 610

; C

4 = 18390

; t

4 = 0.07242

;

# Solve the system with Multivariate Newton's Method

##### TODO #

1

: uncomment F

(

)

=

lambda x: np

.

array

(

[[

sqrt

((

[0, 0] -

1) * * 2 + (

[1, 0] -

1) * * 2 + (

[2, 0] -

1) * * 2) -

* (

1 -

[3, 0])],

[

sqrt

((

[0, 0] -

2) * * 2 + (

[1, 0] -

2) * * 2 + (

[2, 0] -

2) * * 2) -

* (

2 -

[3, 0])],

[

sqrt

((

[0, 0] -

3) * * 2 + (

[1, 0] -

3) * * 2 + (

[2, 0] -

3) * * 2) -

* (

3 -

[3, 0])],

[

sqrt

((

[0, 0] -

4) * * 2 + (

[1, 0] -

4) * * 2 + (

[2, 0] -

4) * * 2) -

* (

4 -

[3, 0])]])

##### TODO #

2

: uncomment J

(

)

=

lambda x: np

.

array

(

[[(

[0, 0] -

1) /

sqrt

((

[0, 0] -

1) * * 2 + (

[1, 0] -

1) * * 2 + (

[2, 0] -

1) * * 2),

(

[1, 0] -

1) /

sqrt

((

[0, 0] -

1) * * 2 + (

[1, 0] -

1) * * 2 + (

[2, 0] -

1) * * 2),

(

[2, 0] -

1) /

sqrt

((

[0, 0] -

1) * * 2 + (

[1, 0] -

1) * * 2 + (

[2, 0] -

1) * * 2),

],

[(

[0, 0] -

2) /

sqrt

((

[0, 0] -

2) * * 2 + (

[1, 0] -

2) * * 2 + (

[2, 0] -

2) * * 2),

(

[1, 0] -

2) /

sqrt

((

[0, 0] -

2) * * 2 + (

[1, 0] -

2) * * 2 + (

[2, 0] -

2) * * 2),

(

[2, 0] -

2) /

sqrt

((

[0, 0] -

2) * * 2 + (

[1, 0] -

2) * * 2 + (

[2, 0] -

2) * * 2),

],

[(

[0, 0] -

3) /

sqrt

((

[0, 0] -

3) * * 2 + (

[1, 0] -

3) * * 2 + (

[2, 0] -

3) * * 2),

(

[1, 0] -

3) /

sqrt

((

[0, 0] -

3) * * 2 + (

[1, 0] -

3) * * 2 + (

[2, 0] -

3) * * 2),

(

[2, 0] -

3) /

sqrt

((

[0, 0] -

3) * * 2 + (

[1, 0] -

3) * * 2 + (

[2, 0] -

3) * * 2),

],

[(

[0, 0] -

4) /

sqrt

((

[0, 0] -

4) * * 2 + (

[1, 0] -

4) * * 2 + (

[2, 0] -

4) * * 2),

(

[1, 0] -

4) /

sqrt

((

[0, 0] -

4) * * 2 + (

[1, 0] -

4) * * 2 + (

[2, 0] -

4) * * 2),

(

[2, 0] -

4) /

sqrt

((

[0, 0] -

4) * * 2 + (

[1, 0] -

4) * * 2 + (

[2, 0] -

4) * * 2),

]])

_0 = 0

; y

_0 = 0

; z

_0 = 6370

; d

_0 = 0

;

_

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