solve_triangular_op

→TensorImplPtr

int solve_triangular_op(const int & a, const int & b, bool upper, bool unitriangular)

source edit

Python wrappers

pylucid.linalg.solve_triangularfunction

Solve $A X = B$ for $X$ where $A$ is triangular.

Performs a single substitution sweep through $A$ ; no factorisation is required. $A$ and $B$ must share the same dtype and device. Batched inputs are supported on both arguments: the trailing two dimensions are treated as the matrix axes and the broadcast batch loop runs LAPACK once per slice.

Math

Solves

A X = B \quad\Longleftrightarrow\quad X = A^{-1} B,

without ever forming $A^{-1}$ . When unitriangular is set, $A$ is regarded as $\widetilde{A}$ with $\widetilde{A}_{ii} = 1$ for all $i$ . The corresponding reverse-mode rule is

\frac{\partial B}{\partial L} = A^{-\top} \frac{\partial X}{\partial L}, \qquad \frac{\partial A}{\partial L} = -\,\frac{\partial B}{\partial L}\,X^\top,

implemented in the Python wrapper.

Shape

a : (..., N, N).
b : (..., N, K) or (..., N).
return: same shape as b.

See Also

solve_op — full LU-based dense solve; use when $A$ is not known to be triangular.
cholesky_op / ldl_factor_op — produce the triangular factors this op back-substitutes through.

Parameters

aTensorImplPtr

Triangular coefficient matrix of shape (..., N, N) with dtype F32 or F64. Only the half indicated by upper is read.

bTensorImplPtr

Right-hand side of shape (..., N, K) (or (..., N) for a single RHS), same dtype and device as a.

upperbool= None

If true (default), a is interpreted as upper-triangular and back-substitution is used. If false, a is lower-triangular and forward-substitution is used.

unitriangularbool= None

If true, the diagonal of a is treated as all-ones regardless of its stored values. Useful for the unit-lower factor produced by LDL

^\top

and for Householder products. Default is false.

Returns

TensorImplPtr

Solution $X$ with the same shape and dtype as b.

Raises

LucidError

If a is not square, if a and b have mismatched dtype or device, or if either tensor has a non-float dtype.

LucidError

If LAPACK reports a singular triangular system (zero on the diagonal when unitriangular is false).

Notes

No autograd node is wired; the output TensorImpl is a leaf.