polar

std.linalg.polar · Level L4

The orthogonal factor of the polar decomposition A = W·H: the orthogonal matrix nearest to A, U·Vᵀ from the SVD.

W = U·Vᵀ = A·(AᵀA)^(−1/2)

Signature

polar(A: f64[k, k]) → f64[k, 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[k, k]SVDLeftVectorsUSVDRightVectorsVTransposeVtMatMulWWf64[k, k]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference A @ inv(sqrtm(A.T @ A)) to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 696 float64 results inside the running error bound; the closest uses 5% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
10%
bit-equal to the NumPy formula in float64
7%
largest error, in units in the last place
1.1e+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.

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:e12f93a76d0211736950a05298a2da0bbad54933b5978952785ddde64b99221a

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