jacobi_step

std.linalg.jacobi_step · Level L0

One Jacobi step for A·x = b, with A split into its diagonal d and the rest R: solve each row for its own unknown, using the others' current values. The body that jacobi_solve runs.

x′ = (b − R·x) ⊘ d

Signature

jacobi_step(x: f64[k], R: f64[k, k], b: f64[k], d: f64[k]) → f64[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.

xf64[k]Rf64[k, k]bf64[k]df64[k]ReshapexcMatMulRxcReshapeRxSubtractrDividex2x2f64[k]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Equal to the reference (b - R @ x) / d in exact rational arithmetic, on all 40 test cases.
  • All 180 float64 results inside the running error bound; the closest uses 51% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
65%
bit-equal to the NumPy formula in float64
100%
largest error, in units in the last place
20

Large ulp counts appear where a result is 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. Results within their own error of zero are not counted.

Note

Written this way, a step contracts errors visibly: each new entry depends on the others through R/d, whose rows sum to at most ½ in magnitude for the test matrices, so the running error bound halves each step instead of growing.

Identity

Calls
—
Called by
jacobi_solve
sha256:f769b9be4fcadbeb52565d4f39513c65d1889bd64de3fb8281de820d75303756

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

Control handle

Symbol
Ω:std.linalg.jacobi_step · Ω:jacobi_step
Pins
sha256:40d0dce1538f9f15629bb5d3eb7b8ef016973f631451f253957ae4972c2bbdcdthis graph alone
Evidence
sha256:f410a3fb82156cb7c595c4afefd303e6806ee75f370659a1cb141305ed851856the hash of its verification record
Needs
no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.