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


linalg_cond(A, p = NULL)



(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).


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


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


κ(A)=ApA1p\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

κ2(A)=σ1σnκ2(A)=σnσ1\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.

pmatrix norm
NULL2-norm (largest singular value)
'fro'Frobenius norm
'nuc'nuclear norm
Infmax(sum(abs(x), dim=2))
-Infmin(sum(abs(x), dim=2))
1max(sum(abs(x), dim=1))
-1min(sum(abs(x), dim=1))
2largest singular value
-2smallest singular value


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).


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