Condition of problems
Other well-known problems of rounding errors are described in the foundations of computational mathematics. In the following, we consider another rounding problem that is related to the numerical precision of matrix operations. For the condition of a problem, in most cases, the reciprocal condition number is estimated in computer programs. The smaller the reciprocal condition number (or the higher the condition number), the worse is the condition of the problem.
The 2-norm condition number (in R it is kappa()) represents the ratio of the largest to the smallest non-zero singular value of a matrix, while rcond() computes an approximation of the reciprocal condition number; take a look at the details.
A poorly conditioned problem is, for example, as follows:
library("Matrix") ## reciprocal approximate condition number rcond(Hilbert(9)) ## worse ## [1] 9.0938e-13 ## reciprocal condition number x1 <- cbind(1, 1:10) head(x1, 3) ## [,1] [,2] ## [1,] 1 ...
