**What Do the Solve*
Programs Give for Solutions to these problems?**

We will give each problem to SolveS and SolveSA and see what solutions we get. We will assume that there are no error estimates available.

1. Missing Variable.

1*x + 0*y = 1

2*x + 0*y = 2

The input file for this problem would be

2,2

=,1,0,1,0

=,2,0,2,0

Both SolveS and SolveSA return x=1, y=0. This is the minimum norm solution.

2. Effectively missing equation.

1*x + 1*y = 2

0*x + 0*y = 0

Both SolveS and SolveSA return x=1, y=1. This is the minimum norm solution.

3. Dependent but consistent equations.

1*x + 1*y = 2

2*x + 2*y = 4

Both SolveS and SolveSA return x=1, y=1. This is the minimum norm solution.

4. Dependent and inconsistent equations.

1*x + 1*y = 2

2*x + 2*y = 6

SolveS treats this as a simple least squares problem and returns x=1.4, y=1.4.

SolveSA recognizes the problem as ill-conditioned, applies an automatic regularization method and gives x=1.2625, y=1.2625.

This is reasonable behavior. See details about this elsewhere in these tutorials.

5. Independent, consistent, but unreasonable equations.

1*x + 1*y = 2

1*x + 1.01*y = 3

SolveS treats this as a simple least squares problem and returns x=-98, y=101, which is probably NOT what the user expected.

SolveSA recognizes the problem as ill-conditioned, applies an automatic regularization method and gives x=1.1165, y=1.1334. This is the sort of behavior which the Solve*A programs have been specifically designed to do.

6. An under-determined system.

1x + 2y = 2

Both SolveS and SolveSA return the expected minimum-norm solution, x=0.4, y=0.8. This is actually an easy problem.

7. An over-determined system

1*x + 2 *y = 15.1

2*x + 2*y = 15.9

-1*x + 1*y = 6.5

Both SolveS and SolveSA 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!