Engineering Optimization: Theory and Practice, Fourth Edition

3.7 Pivotal Reduction of a General System of Equations 135

0 x 1 + 0 x 2 + 1 x 3 + · · · + 0 xn=b′′ 3 (3.16)

0 x 1 + 0 x 2 + 0 x 3 + · · · + 1 xn=b′′n

This system of Eqs. (3.16) is said to be in canonical form and has been obtained after
carrying outnpivot operations. From the canonical form, the solution vector can be
directly obtained as

xi=b′′i, i= 1 , 2 ,... , n (3.17)

Since the set of Eqs. (3.16) has been obtained from Eqs. (3.14) only through elementary
operations, the system of Eqs. (3.16) is equivalent to the system of Eqs. (3.14). Thus
the solution given by Eqs. (3.17) is the desired solution of Eqs. (3.14).

3.7 Pivotal Reduction of a General System of Equations

Instead of a square system, let us consider a system ofmequations innvariables with
n≥m. This system of equations is assumed to be consistent so that it will have at
least one solution:

a 11 x 1 +a 12 x 2 + · · · +a 1 nxn=b 1

a 21 x 1 +a 22 x 2 + · · · +a 2 nxn=b 2

..

.

am 1 x 1 +am 2 x 2 + · · · +amnxn=bm

(3.18)

The solution vector(s)Xthat satisfy Eqs. (3.18) are not evident from the equations.
However, it is possible to reduce this system to an equivalent canonical system from
which at least one solution can readily be deduced. If pivotal operations with respect
to any set ofmvariables, say,x 1 , x 2 ,... , xm, are carried, the resulting set of equations
can be written as follows:

Canonical system with pivotal variablesx 1 , x 2 ,... , xm

1 x 1 + 0 x 2 + · · · + 0 xm+a′′ 1 ,m+ 1 xm+ 1 + · · · +a 1 ′′nxn=b′′ 1

0 x 1 + 1 x 2 + · · · + 0 xm+a′′ 2 ,m+ 1 xm+ 1 + · · · +a 2 ′′nxn=b′′ 2 ( 3. 19 )

..

.

0 x 1 + 0 x 2 + · · · + 1 xm+a′′m,m+ 1 xm+ 1 + · · · +a′′mnxn=b′′m

Pivotal Nonpivotal or Constants

variables independent

variables