projector

std.linalg.projector · Level L4

The orthogonal projection onto A's column space, Q·Qᵀ from the QR factorization.

P = Q·Qᵀ = A·(AᵀA)⁻¹·Aᵀ

Signature

projector(A: f64[n+k, n]) → f64[n+k, 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]QROrthogonalQTransposeQtMatMulPPf64[n+k, n+k]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference A @ np.linalg.solve(A.T @ A, A.T) to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 3,083 float64 results inside the running error bound; the closest uses 2% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
17%
bit-equal to the NumPy formula in float64
14%
largest error, in units in the last place
7.2e+4

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.

Identity

Calls
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Called by
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sha256:c92f29b2cd5883187213609e99ba1c55fd7852ae5566e24e6eddca8f986ff6c2

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