dual_basis

std.linalg.dual_basis · Level L3

The dual basis of a basis b₁ … bₙ of Rⁿ (the rows of B): the covectors φ₁ … φₙ (the rows of Φ) with φᵢ(bⱼ) = δᵢⱼ, which is Φ = (B⁻¹)ᵀ. A covector is a linear functional, not a vector of the space; without an inner product V and V* are not identified.

φᵢ(bⱼ) = δᵢⱼ, Φ = (B⁻¹)ᵀ

Signature

dual_basis(B: f64[n, n]) → f64[n, n]

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 the inverse over the rationals, transposed in exact rational arithmetic, on all 40 test cases.
All 749 float64 results inside the running error bound; the closest uses 1% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 37 of them inside the running error bound at u = 2⁻²⁴; the closest uses <1% of it. Uncertified, too ill-conditioned at float32 for a finite bound: 3.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 42%
bit-equal to the NumPy formula in float64: 42%
largest error, in units in the last place: 1.6e+5

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls

inverse

Called by

—

sha256:7baf259f34b60c44617bca8a8c1494b014084c82072853390e6ceb11ec23dd68

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.dual_basis · Ω:dual_basis
Pins: sha256:405a0384f22a04f412460936d6a95831262454c9928b27cac18ea44056372b76this graph and the 1 it reaches through calls
Evidence: sha256:88980ac11ac2f821ecc935cc54e280e4bd75521abc645144650cb6d3cd7dd111the 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.