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 zero or 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.

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
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Called by
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sha256:96d4d8b41715df18119d0391143c5b61c3166bb03b093dae82e9040c3f167395

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