solve

→Tensor

solve(A: Tensor, b: Tensor)

source

Solve a square linear system $AX = B$ .

Returns the unique solution $X$ of the system

A X = B

where $A$ is a non-singular square matrix. The system is solved by LU decomposition with partial pivoting, which is both faster and more accurate than forming $A^{-1}$ explicitly.

Parameters

ATensor

Square coefficient matrix of shape (*, n, n).

bTensor

Right-hand side of shape (*, n, k) (multiple RHS columns) or (*, n) (single RHS vector).

Returns

Tensor

Solution $X$ with the same shape as b.

Notes

Algorithm: factor $PA = LU$ , then perform forward-substitution $L Y = P B$ followed by back-substitution $U X = Y$ . Total cost is $O(n^3 + k n^2)$ .

Examples

>>> import lucid
>>> from lucid.linalg import solve
>>> A = lucid.tensor([[3.0, 1.0], [1.0, 2.0]])
>>> b = lucid.tensor([9.0, 8.0])
>>> solve(A, b)
Tensor([2.0000, 3.0000])