Even Small Systems of Equations Can be Hard!

1.       A typical problem without difficulties

1*x + 1*y = 2

1*x -  1*y  = 0

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!