householder_product_op

→TensorImplPtr

int householder_product_op(const int & H, const int & tau)

source edit

Python wrappers

pylucid.linalg.householder_productfunction

Materialise the orthogonal $Q$ from packed Householder reflectors.

Math

Q = H_1 H_2 \cdots H_k, \qquad k = \min(m, n)

Shape

Input H : (m, n)
Input tau : (k,) with $k = \min(m, n)$
Output Q : (m, k)

References

LAPACK *orgqr; Golub & Van Loan §5.1.2.

See Also

[[eig_op]] : Produces packed reflectors for general eigenvector reconstruction. [[eigh_op]] : Symmetric eigendecomp; reflectors come from a tridiagonal reduction step.

Parameters

Hconst TensorImplPtr&

Packed reflector matrix of shape (m, n) produced by a QR-type factorisation. Column

i

stores reflector

v_i

in the strict lower triangle below the diagonal (the diagonal entry of

v_i

is implicitly

1

and the slot is reused for

R

). Must be float-typed and non-null.

tauconst TensorImplPtr&

1-D scaling factors of length

k = \min(m, n)

. tau[i] is the

\tau_i

scalar of the

i

-th reflector

H_i = I - \tau_i v_i v_i^\top

. Must be float-typed and non-null.

Returns

TensorImplPtr

Explicit orthogonal matrix $Q$ of shape (m, k) with the same dtype as H. Columns are orthonormal ( $Q^\top Q = I_k$ ).

Raises

LinAlgError

When H or tau is null / not float-typed.

Notes

Output is a leaf node — autograd is not wired. Gradients flow through the calling QR / eig op rather than through this unpack step.