Even Small Systems of Equations
Can be Hard!
1. A
typical problem without difficulties
1*x + 1*y = 2
1*x - 1*y = 0
The answer is x=1, y=1.
2. Missing Variable.
1*x + 0*y = 1
2*x + 0*y = 2
Our programs return x=1, y=0. This is the minimum norm solution.
3. Effectively missing equation.
1*x + 1*y = 2
0*x + 0*y = 0
Our programs return x=1, y=1. This is the minimum norm solution.
4. Dependent but consistent equations.
1*x + 1*y = 2
2*x + 2*y = 4
Our programs return x=1, y=1. This is the minimum norm solution.
5. Dependent and inconsistent equations.
1*x + 1*y = 2
2*x + 2*y = 6
Our programs recognize the problem as ill-conditioned, apply an automatic regularization method and give x=1.2625, y=1.2625. This is reasonable behavior.
6. Independent, consistent, but unreasonable equations.
1*x + 1*y = 2
1*x + 1.01*y = 3
A simple solution returns x=-98, y=101, which is probably NOT what the user expected.
Our programs recognize the problem as ill-conditioned, apply an automatic regularization method and return x=1.1165, y=1.1334. This is the sort of behavior which our programs have been specifically designed to do.
7. An under-determined system.
1x + 2y = 2
Our programs return the minimum-norm solution, x=0.4, y=0.8. This is actually an easy problem.
8. An over-determined system
1*x + 2 *y = 15.1
2*x + 2*y = 15.9
-1*x + 1*y = 6.5
Our programs return x=0.7276. y=7.2069. When you substitute these values into the equations, the resulting right side values are: 15.141, 15.869, and 6.479, which is excellent!
These are all illustrative, or “toy” problems. Our programs usually are used to solve much more involved problems!