**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!