pinv

std.linalg.pinv · Level L4

The Moore–Penrose pseudo-inverse of a tall A of full column rank, from its SVD: V·diag(1/σ)·Uᵀ. It maps b to the least-squares solution.

A⁺ = V·Σ⁻¹·Uᵀ

Signature

pinv(A: f64[n+k, n]) → f64[n, n+k]

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.

Af64[n+k, n]SVDLeftVectorsUSVDValuessSVDRightVectorsVTransposeUtDivideVsMatMulApinvApinvf64[n, n+k]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference np.linalg.pinv(A) to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 1,512 float64 results inside the running error bound; the closest uses 4% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
6%
bit-equal to the NumPy formula in float64
57%
largest error, in units in the last place
3.1e+3

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.

Note

A singular pair is defined only up to a joint sign, so NOVA fixes one: the right vector's largest component is positive. The error bound grows as neighbouring singular values approach each other or zero, and a case where they are too close, or where the sign itself could flip, is refused.

Identity

Calls
—
Called by
—
sha256:96d4d8b41715df18119d0391143c5b61c3166bb03b093dae82e9040c3f167395

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.pinv · Ω:pinv
Pins
sha256:d4225ba8731ba5088309cc5749ca7cbbdfdd1880366d5dcc6dacd7fdeaa6af0dthis graph alone
Evidence
sha256:53466a284e6401a50b20084e13e71eead192e45b187cd3393f9f71ee752184fathe 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.