Interactive Educational Modules in
Scientific Computing

Conditioning of Solutions

This module graphically illustrates the uncertainty in the solution of a system of linear equations due to uncertainty in the input data. The relative error in the input data is amplified by the condition number of the matrix, so the relative error in the solution can be much greater than that in the input if the matrix is ill-conditioned.

The user first selects the precision to be used in the calculation, from one to ten decimal digits. The user then explicitly enters or randomly chooses a two by two matrix A and corresponding right-hand-side vector b. The solution set of each of the two equations in the linear system is drawn as a straight line in the plane. The width of the lines reflects the uncertainty in the data within the specified precision. The resulting uncertainty in the intersection (i.e., the solution) depends on the condition number of the matrix, which is also printed. The user can zoom in or out on the plot in order to see more or less detail. After clicking to focus on the plot, the arrow keys can be used to move around within the plot.

Reference: Michael T. Heath, Scientific Computing, An Introductory Survey, 2nd edition, McGraw-Hill, New York, 2002. See Section 2.3, especially Example 2.2 on page 52 and Figure 2.4 on page 61.

Developers: Nicholas Exner and Michael Heath