Introduction and theoretical background - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 76

The element of the coe cient matrix with largest magnitude is a 32 = 9. Selecting

this element as the pivot, the array takes the form
⎝

⎠

10 0
01 0
001/9

C = P 1 C

⎝ ⎠

2 − 74 9100

= D = 38 5 6010

1/91

− . (1.534)

− 2/31/9001/9

Multiplying the last row by 7 and adding it to the first, then multiplying the last
row by 8 and subtracting it from the second, yields the array
⎝ ⎠

10 7
01

−

8
00 1

D = Q 1 D

⎝ ⎠

25/90 − 2/388/910 7/9

= E = − 35/9031/346/901 − 8/9 . (1.535)

1/91 − 2/3 /900 1/9

Elements on the row and column of the initial pivot are no longer available as
pivots. Of the available elements, a 23 31/3 has the largest magnitude. Selecting

this element as the next pivot, and using the resulting second row to reduce the
other elements in the third column to zero, gives the array
⎝

=

⎠ ⎝ ⎠

12/30
010
02/31

100
03/31 0
001

E = Q 2 P 2 E

⎝ ⎠

235/93 0 0 940/9312/31 67/93

= F = − 35/9301 46/9303/31 − 8/93 . (1.536)

− 13/9310 41/9302/31 5/93

The only available pivot is now a 11 235/93. Accordingly, the resulting first row

is used to reduce the remaining elements in the first column to zero, producing the
array

=

⎝ ⎠ ⎝ ⎠

100
35/9310
13/9301

93/235 0 0
0 1 0 F = Q 3 P 3 F

0 0 1

⎝ ⎠

100493/235 6/235 67/235
0012 7/47 5/47 1/47
010113/235 16/235 22/235

= G = . (1.537)