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!