R: Estimated Matrix Norm

normest {pracma} | R Documentation

## Estimated Matrix Norm

### Description

Estimate the 2-norm of a real (or complex-valued) matrix. 2-norm is also the maximum absolute eigenvalue of M, computed here using the power method.

### Usage

    normest(M, maxiter = 100, tol = .Machine$double.eps^(1/2))

### Arguments

`M` | Numeric matrix; vectors will be considered as column vectors.
`maxiter` | Maximum number of iterations allowed; default: 100.
`tol` | Tolerance used for stopping the iteration.

### Details

Estimate the 2-norm of the matrix `M`, typically used for large or sparse matrices, where the cost of calculating the `norm (A)` is prohibitive and an approximation to the 2-norm is acceptable.

Theoretically, the 2-norm of a matrix M is defined as

||M||_2 = max \frac{||M*x||_2}{||x||_2} for all x \neq 0

where ||.||_2 is the Euclidean/Frobenius norm.

### Value

2-norm of the matrix as a positive real number.

### Note

If feasible, an accurate value of the 2-norm would simply be calculated as the maximum of the singular values (which are all positive):

`max(svd(M)\$d)`

### References

Trefethen, L. N., and D. Bau III. (1997). Numerical Linear Algebra. SIAM, Philadelphia.

### See Also

`cond`, `svd`

### Examples

    normest(magic(5)) == max(svd(magic(5))$d)  # TRUE
    normest(magic(100))                        # 500050

[Package pracma version 2.0.4 Index]