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10 Some Fundamental Numerical Methods
First normalize the first row with respect to "pivot coefficient" all:
[3
: ~.5 - ~.5 ~.5]. (2.8)
1 - 1 2
This is done to avoid potential numerical overflow or underflow in the next
multiplication. In order to make the a 21 coefficient zero, form a new row 2 by
multiplying row 1 by a21 and subtracting this from row' 2. Also, form a new
row 3 by multiplying row I by a 31
and subtracting this from row 3. The result
IS
0.5 - 0.5 0.5 j
0.5 1.5 5.5 .
(2.9)
-0.5 0.5 0.5
The next cycle is to normalize the coefficients of row 2 with respect to the
new pivot coefficient a 22 :
0.5
I
-0.5
-0.5
3
0.5
0.5 ]
II .
0.5
Note that after normalization the new coefficient is always
(2.10)
(2.11 )
where K is the pivot row, I is the new row being formed, and J is the
coefficient (column) being formed. Continue by forming new rows I and 3 in
(2.10):
- 5 ]
~I .
(2.12)
In the final cycle, normalize the coefficients of row 3 with respect to
coefficient a JJ
:
[
6 ~ -; -;1]. (2.13)
o 0 1 3
Finally, form new rows I and 2 using (2.11):
[ 6 ~ g i]. (2.14)
o 0 I 3
Because (2.14) now represents the system in (2.5), the solution is in column 4:
Xl = 1, X2 = 2, and x J = 3.
The BASIC language Program B2-1 in Appendix B implements this algorithm.
Note that new coefficients are generated according to (2.1 I) in program
line number 9330. Lines 9140-9240 implement a feature not yet mentioned. If
any pivot coefficient is too small (taken to be less than or equal to IE - 6 in
line 9020), then the rows are interchanged. The reader is encouraged to first