solve_triangular

→Tensor

solve_triangular(A: Tensor, B: Tensor, upper: bool = True, left: bool = True, unitriangular: bool = False)

source

Solve a triangular linear system by back/forward substitution.

Solves the system $AX = B$ (or $XA = B$ ) in which the coefficient matrix $A$ is triangular. For upper-triangular $A$ the system is solved by back-substitution starting from the last row; for lower-triangular $A$ by forward substitution from the first row. Either direction runs in $O(n^2 k)$ time and is numerically stable when the diagonal of $A$ is well-conditioned.

Parameters

ATensor

Triangular coefficient matrix of shape (*, n, n). Only the relevant triangle is read; the other half is ignored.

BTensor

Right-hand side of shape (*, n, k) (or (*, n)).

upper(bool, keyword - only)= True

If True (default)

A

is upper-triangular; if False lower-triangular.

left(bool, keyword - only)= True

If True (default) solves

AX = B

; if False solves

XA = B

via transposition.

unitriangular(bool, keyword - only)= False

If True the diagonal of

A

is treated as all-ones regardless of its stored values (LAPACK's "unit-diagonal" mode).

Returns

Tensor

Solution $X$ shaped like $B$ .

Notes

Backed by LAPACK trsm. Triangular solves are the workhorse used inside Cholesky, LU, and QR back-substitution paths.

Examples

>>> import lucid
>>> from lucid.linalg import solve_triangular
>>> A = lucid.tensor([[2.0, 1.0], [0.0, 3.0]])  # upper
>>> b = lucid.tensor([[5.0], [9.0]])
>>> solve_triangular(A, b, upper=True)
Tensor([[1.0000],
        [3.0000]])