Computes the condition number of a matrix with respect to a matrix norm.

Letting be or , the condition number of a matrix is defined as

Usage

linalg_cond(A, p = NULL)

Arguments

A

(Tensor): tensor of shape (*, m, n) where * is zero or more batch dimensions for p in (2, -2), and of shape (*, n, n) where every matrix is invertible for p in ('fro', 'nuc', inf, -inf, 1, -1).

p

(int, inf, -inf, 'fro', 'nuc', optional): the type of the matrix norm to use in the computations (see above). Default: NULL

Value

A real-valued tensor, even when A is complex.

Details

$$\kappa(A) = \|A\|_p\|A^{-1}\|_p$$

The condition number of A measures the numerical stability of the linear system AX = B with respect to a matrix norm.

Supports input of float, double, cfloat and cdouble dtypes. Also supports batches of matrices, and if A is a batch of matrices then the output has the same batch dimensions.

p defines the matrix norm that is computed. See the table in 'Details' to find the supported norms.

For p is one of ('fro', 'nuc', inf, -inf, 1, -1), this function uses linalg_norm() and linalg_inv().

As such, in this case, the matrix (or every matrix in the batch) A has to be square and invertible.

For p in (2, -2), this function can be computed in terms of the singular values

$$\kappa_2(A) = \frac{\sigma_1}{\sigma_n}\qquad \kappa_{-2}(A) = \frac{\sigma_n}{\sigma_1}$$

In these cases, it is computed using linalg_svd(). For these norms, the matrix (or every matrix in the batch) A may have any shape.

p	matrix norm
NULL	2-norm (largest singular value)
'fro'	Frobenius norm
'nuc'	nuclear norm
Inf	max(sum(abs(x), dim=2))
-Inf	min(sum(abs(x), dim=2))
1	max(sum(abs(x), dim=1))
-1	min(sum(abs(x), dim=1))
2	largest singular value
-2	smallest singular value

Note

When inputs are on a CUDA device, this function synchronizes that device with the CPU if if p is one of ('fro', 'nuc', inf, -inf, 1, -1).

Examples

if (torch_is_installed()) {
a <- torch_tensor(rbind(c(1., 0, -1), c(0, 1, 0), c(1, 0, 1)))
linalg_cond(a)
linalg_cond(a, "fro")
}
#> torch_tensor
#> 3.16228
#> [ CPUFloatType{} ]