Before revision:  With TLSE problems solved by the classical QR-SVD or randomized algorithms , numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, the estimates based on mixed and componentwise condition numbers are much tighter.
After revision:  With TLSE problems solved by the classical QR-SVD or randomized algorithms , numerical experiments illustrate that normwise condition number-based estimate is sharp to evaluate the forward error of the solution, while for sparse and badly scaled matrices, mixed and componentwise condition numbers are much tighter.