ldl_solve

→Tensor

ldl_solve(LD: Tensor, pivots: Tensor, B: Tensor)

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Solve a symmetric linear system using an LDL factorization.

Given $A = L\,D\,L^\top$ produced by ldl_factor, solves

A\,X \,=\, B

by chaining three substitutions:

L\,Y = P B, \quad D\,Z = Y, \quad L^\top X = Z,

followed by an inverse permutation to undo the Bunch-Kaufman row swaps.

Parameters

LDTensor

Packed LDL factor from ldl_factor, shape (n, n).

pivotsTensor

Pivot indices from ldl_factor. This implementation only supports 1×1 (simple) pivots — every entry must be strictly positive. Mixed 2×2 block pivots raise NotImplementedError.

BTensor

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

Returns

Tensor

Solution $X$ , same shape as B.

Notes

Supports indefinite symmetric $A$ (unlike Cholesky), so it is appropriate for KKT / saddle-point systems where Cholesky would fail. Cost per solve is $O(n^2 k)$ once the LDL factor is in hand.

Examples

>>> import lucid
>>> from lucid.linalg import ldl_factor, ldl_solve
>>> A = lucid.tensor([[4.0, 1.0], [1.0, 3.0]])
>>> LD, piv = ldl_factor(A)
>>> b = lucid.tensor([[5.0], [4.0]])
>>> ldl_solve(LD, piv, b)

>>> import lucid >>> from lucid.linalg import ldl_factor, ldl_solve >>> A = lucid.tensor([[4.0, 1.0], [1.0, 3.0]]) >>> LD, piv = ldl_factor(A) >>> b = lucid.tensor([[5.0], [4.0]]) >>> ldl_solve(LD, piv, b)