Ill-Conditioned Systems

• A system whose coefficient matrix is singular has no unique solution. What if the matrix is almost singular?
  $$A=\left[ \begin{array}{rrr} 3.02 & -1.05 & 2.53 \\ 4.33 & 0.56 &-1.78 \\ -0.83 &-0.54 & 1.47\\ \end{array} \right]$$
  The LU equivalent has a very small element in position (3, 3), and the inverse has elements very large in comparison to $A$:
  $$LU=\left[ \begin{array}{rrr} 4.33 & 0.56 &-1.78 \\ 0.6975... ...3.7715 \\ -0.1917 & 0.3003 & -0.0039\\ \end{array} \right],$$
  $$inv(A)=\left[ \begin{array}{rrr} 5.6611 & -7.2732 &-18.5503... ...43 \\ 76.8511 & -102.6500 & -255.8846\\ \end{array} \right]$$

• Matrix is nonsingular but is almost singular
• Suppose we solve the system $Ax=b$, with $b$ equal to $[-1.61, 7.23, -3.38]^T$. The solution is $x = [1.0000, 2.0000, -1.0000]^T$.
• Now suppose that we make a small change in just the first element of the $b$-vector : $[-1.60, 7.23, -3.38]^T$. We get $x =[ 1.0566, 4.0051, -0.2315]^T$
• $b=[-1.61, 7.22, -3.38]^T$. The solution now is $x = [1.07271, 4.6826, 0.0265]^T$ which also differs.
• A system whose coefficient matrix is nearly singular is called
  ill-conditioned. When a system is ill-conditioned, the solution is very sensitive to changes in the right-hand vector. It is also sensitive to small changes in the coefficients.
• $A(1 , 1)$ is changed from 3.02 to 3.00, original b-vector, a large change in the solution $x =[1.1277, 6.5221, 0.7333]^T]$. This means that it is also very sensitive to round-off error.