direct_sum

std.linalg.direct_sum · Level L4

Whether Rⁿ is the internal direct sum of the subspaces U and W, each given by a basis (the rows of U and of W), by exact ranks: dim(U ∩ W) = dim U + dim W − dim(U + W); U ∩ W = {0} gives uniqueness of the decomposition inside U + W, and U + W = Rⁿ is a separate condition; the direct sum needs both. For three or more subspaces, pairwise trivial intersections are not enough.

Rⁿ = U ⊕ W ⇔ U ∩ W = {0} ∧ U + W = Rⁿ

Signature

direct_sum(U: f64[a, n], W: f64[b, n]) → f64[], f64[], f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference exact ranks of U, W and U + W in exact rational arithmetic, on all 40 test cases.
All 160 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 39 of them exact, with zero error. Uncertified, too ill-conditioned at float32 for a finite bound: 1.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

rank_exact

Called by

—

sha256:37a2db18246c74cd2d7da10194a4865685a2fcee3611186cc08d632a60d579c6

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.linalg.direct_sum · Ω:direct_sum
Pins: sha256:461f709e7ee6911b72a9794cf653dbe7681bfaaf11f1a1960736d7023d25bc63this graph and the 2 it reaches through calls
Evidence: sha256:36be4c22e4c922042c0b50c7493e2bcd6496dbd07e1292cb7e09ce8e06ab1017the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.