Introduction to Computational Linear Algebra

104 Introduction to Computational Linear Algebra

where

• a =

1

2

(< a

1

, a

1

>, ··· , < a

N

, a

N

>)

T

;

• A =











α

1

β

1

γ

1

−ρ

1

α

2

β

2

γ

2

−ρ

2

.

.

.

.

.

.

.

.

.

.

.

.

α

N

β

N

γ

N

−ρ

N











;

• λ =

1

2

< x, x > and e = (1, ··· , 1)

T

∈ R

N×4

.

From (4.47), the solution x is chosen to satisfy x = A

+

(a+λe) where A

+

is the

pseudo-inverse of A. Then, prove that the solution is given by the quadratic

equation

λ

2

< A

+

e, A

+

e > + 2λ (< A

+

a, A

+

e > −1) + < A

+

a, A

+

a >= 0. (4.48)

How does one select the correct solution?

3. Write the MATLAB function:

function [r, Deltat] = GPS(S,Te,Tr)

% Input

% S : (N x 3)-matrix : the i-th row contains the coordinates

% of Satellite Si.

% Te : N-vector : the i-th entry is the emitting ...