lstsq

std.linalg.lstsq · Level L4

Least squares: the x minimizing ‖A·x − b‖ for a tall A of full column rank, through QR: R·x = Qᵀ·b. Unlike the normal equations, it does not square A's condition number.

R·x = Qᵀ·b, A = Q·R

Signature

lstsq(A: f64[n+k, n], b: f64[n+k]) → f64[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]bf64[n+k]QROrthogonalQReshapebcQRTriangularRTransposeQtMatMulQtb_cReshapeQtbTriangularSolvexxf64[n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference np.linalg.lstsq(A, b)[0] to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 127 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)
23%
bit-equal to the NumPy formula in float64
8%
largest error, in units in the last place
55

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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sha256:0790d3d1f811a7d49d059f077730517035362bbc5cbf00c73586ca6d6f9c1629

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