Download Numerical Solutions of a System of Equations

Numerical Solutions of a System of Equations
One way to solve the linear system of two equations
x + y = 7,
−x + 3y = 1,
(1)
is to add the two equations, obtaining
4y = 8,
and hence y = 2. Substituting y = 2 into the first equation (1) yields x = 5.
Our objective in these chapters is to illustrate numerical methods for solving a linear
system of equations.
Linear System of Equations
a11 x1 + a12 x2 + · · · + a1n xn = b1
a21 x1 + a22 x2 + · · · + a2n xn = b2
..
..
.
.
.
= ..
..
..
.
.
.
= ..
an1 x1 + an2 x2 + · · · + ann xn = bn ,
where aij , bi , i, j = 1..n are real constants and the a0 s are the coefficients of the
variables xi . The above system can be written as AX = b, where





x
a
a
. . . a1n

 1 

 11 12






 x2 
 a21 a22 . . . a2n 
, b = 
, x = 
A=

 .. 
 ..
..
.. 

 . 
 .
. ...
. 





xn
an1 an2 . . . ann
Inhomogeneous o
AX = b
Linear system
b1
b2
..
.
bn




.



/ Homogeneous
Methods used to solve
1-Elementary row operations
AX = 0
2-Cramer’s Rule
Always has at least the trivial solution
3-Gaussian elimination
1
3 If
|A| =
6 0
• Inhomogeneous
/
Unique solution X = A−1 b
Infinitly
many solutions
3
+
Unknowns > Equations
+
No solution
Exercise 9 page (356)
Given the linear system
2x1 − 6αx2 = 3,
3αx1 − x2 = 3/2.
1. Find values of α for which the system has no solution?
2. Find values of α for which the system has infinite number of solutions, if they
exist?
3. Find values of α for which the system has unique solution if it exists?.
Solution
2 −6α = 18α2 − 2 = 0 =⇒ α = ±1/3.
|A| = 3α −1 • For α 6= ±1/3, the system has unique solution.
• For α = −1/3, the system becomes
x1 + x2 = 3/2,
x1 + x2 = −3/2,
and therefore it has no solution.
• For α = 1/3, the system becomes
x1 − x2 = 3/2,
x1 − x2 = 3/2,
and therefore it has infinite number of solutions.
2