Numerical Methods and Optimization

60 Numerical Methods and Optimization: An Introduction

Denote the resulting matrix by M

(2)

:

M

(2)

=

⎡

⎢

⎢

⎢

⎢

⎢

⎣

a

(1)

11

a

(1)

12

··· a

(1)

1n

a

(1)

1(n+1)

0 a

(2)

22

··· a

(2)

2n

a

(2)

2(n+1)

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0 a

(2)

n2

··· a

(2)

nn

a

(2)

n(n+1)

⎤

⎥

⎥

⎥

⎥

⎥

⎦

.

Next, assuming that a

(2)

22

= 0, we use the second row to obtain zeros in rows

3,...,n of the second column of matrix M

(2)

:

a

(3)

ij

= a

(2)

ij

−

a

(2)

i2

a

(2)

22

a

(2)

2j

,i=3,...,n, j =2,...,n,n+1.

We denote the matrix obtained after this transformation by M

(3)

.

Continuing this way, at the k

th

step, k ≤ n −1, we have matrix M

(k)

,and

assuming that a

(k)

kk

= 0, eliminate the elements in rows k +1,...,n of the k

th

column:

a

(k+1)

ij

= a

(k)

ij

−

a

(k)

ik

a

(k)

kk

a

(k)

kj ...