Extremely ill-conditioned matrix, numerical examples, multiplicative correction, Accurate summation, Roundoff error, Error-free transformations, Numerical computation of matrix norms, conditioning, scaling, error-free transformations, Multiplicative correction, Direct numerical methods for linear systems and matrix inversion, Conditioning of matrices, Accurate dot product, approximate inverse, accurate dot product, extremely ill-conditioned matrix, rounding error, floating-point computations, Condition number, condition number, accurate summation