class

SolveBackward

SolveBackward

source edit

Python wrappers

pylucid.linalg.solvefunction

Autograd node for the general linear solve $AX = B$ .

The forward saves $A$ as saved_inputs_[0] (required to build $A^\top$ for the adjoint solve) and the solution $X$ as saved_output_ (required for the outer-product step that yields $\partial L / \partial A$ ). Although $B$ is technically saved by NaryKernel when save_inputs=true, the backward never reads it.

Math

Given upstream gradient $G = \partial L / \partial X$ :

\frac{\partial L}{\partial B} = \text{solve}(A^\top, G), \qquad \frac{\partial L}{\partial A} = -\,\frac{\partial L}{\partial B}\, X^\top

Both expressions compose existing differentiable ops ([[solve_op]] / [[matmul_op]] / [[neg_op]] / [[mT_op]]) so that second-order gradients flow automatically.

References

Giles, "Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation" (Oxford, 2008), §2.4.

Attributes

schema_v1OpSchema

"solve", one saved-input slot, AmpPolicy::KeepInput — prevents a lossy autocast promotion before the LAPACK call.

Instance methods

apply

→std::vector<Storage>

int apply(int grad_out)

source edit

Compute input gradients for the forward solve $AX = B$ .

Parameters

grad_outStorage

Upstream gradient

G = \partial L / \partial X

, shaped like the forward output

X

Returns

std::vector<Storage>

Two-element vector {dA, dB} matching the input ordering [A=0, B=1] that NaryKernel expects, where dB = solve(Aᵀ, G) and dA = -dB @ Xᵀ.

Notes

The transpose solve solve(Aᵀ, G) performs a fresh LU factorisation of $A^\top$ — the forward factors are not cached.

class

SolveBackward

SolveBackward

source edit

Python wrappers

pylucid.linalg.solvefunction

Autograd node for the general linear solve $AX = B$ .

Math

Given upstream gradient $G = \partial L / \partial X$ :

\frac{\partial L}{\partial B} = \text{solve}(A^\top, G), \qquad \frac{\partial L}{\partial A} = -\,\frac{\partial L}{\partial B}\, X^\top

Both expressions compose existing differentiable ops ([[solve_op]] / [[matmul_op]] / [[neg_op]] / [[mT_op]]) so that second-order gradients flow automatically.

References

Giles, "Collected Matrix Derivative Results for Forward and Reverse Mode Algorithmic Differentiation" (Oxford, 2008), §2.4.

Attributes

schema_v1OpSchema

"solve", one saved-input slot, AmpPolicy::KeepInput — prevents a lossy autocast promotion before the LAPACK call.

Instance methods

apply

→std::vector<Storage>

int apply(int grad_out)

source edit

Compute input gradients for the forward solve $AX = B$ .

Parameters

grad_outStorage

Upstream gradient

G = \partial L / \partial X

, shaped like the forward output

X

Returns

std::vector<Storage>

Two-element vector {dA, dB} matching the input ordering [A=0, B=1] that NaryKernel expects, where dB = solve(Aᵀ, G) and dA = -dB @ Xᵀ.

Notes

The transpose solve solve(Aᵀ, G) performs a fresh LU factorisation of $A^\top$ — the forward factors are not cached.