Fig. 2.

Benchmarks of randomized SVD. a Elapsed time of different SVD algorithms. The blue, orange, and green points indicate the elapsed time of the full, truncated, and randomized SVD, respectively. b Relative errors of the randomized SVD. The error bars denote the standard deviation of ten trials. The relative error of the ith largest singular value σ_i is defined as $\left|1-\frac{{\widehat{\sigma }}_i}{{\sigma }_i}\right|$, where ${\widehat{\sigma }}_i$ is an approximated value of σ_i. The error bars denote the standard deviation of ten trials. The approximation error for a real matrix A with a low-rank matrix is bounded by a singular value as illustrated in the following formula: minrank(X)≤j||A−X||=σ_j+1, where ||·|| denotes the operator norm of a matrix