qr_q

std.linalg.qr_q · Level L4

The orthonormal factor Q of A = Q·R, for a tall matrix of full column rank: its columns are an orthonormal basis of A's column space.

A = Q·R, QᵀQ = I

Signature

qr_q(A: f64[n+k, n]) → f64[n+k, 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.

Af64[n+k, n]QROrthogonalQQf64[n+k, n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference np.linalg.qr(A)[0], signs fixed to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 1,254 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)
35%
bit-equal to the NumPy formula in float64
100%
largest error, in units in the last place
60

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

R's diagonal is made positive, which makes the factorization unique. The reference is Householder QR; the exact check reaches the same factors through the Cholesky factor of AᵀA, so two different algorithms have to agree.

Identity

Calls
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Called by
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sha256:62bdb99d3ea3c7b0ee0f491b599d6315b2ff135777f990c27d9ebc6e08f718dc

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