rank1_approx

std.linalg.rank1_approx · Level L4

The best rank-1 approximation of A (Eckart–Young): σ₁·u₁·v₁ᵀ, from the leading singular triplet.

A₁ = σ₁·u₁·v₁ᵀ

Signature

rank1_approx(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]SVDLeftVectorsUSVDValuessSVDRightVectorsVSliceu1Slices1Slicev1MultiplyusTransposev1tMatMulA1A1f64[n+k, n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference A @ v1 @ v1.T, v1 the top eigenvector of A.T @ A to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 1,321 float64 results inside the running error bound; the closest uses 3% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
7%
bit-equal to the NumPy formula in float64
6%
largest error, in units in the last place
4.0e+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:c5936d8939955d2fbe80598e3bfbe8b62859d8e26cc315d5b4a853e8b46c64f9

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