lp_standard_form

std.optim.lp_standard_form · Level L2

The standard form of the constraints A·x ≤ b, x ≥ 0: the matrix [A | I], so that A·x + s = b with slacks s ≥ 0. Feasibility and the objective cᵀx carry over both ways between x and (x, s). The other conversions are done before it: a ≥ row is multiplied by −1, an equality becomes two ≤ rows, a free variable is x⁺ − x⁻, and minimizing cᵀx is maximizing −cᵀx.

A·x ≤ b ⇔ [A | I]·(x, s) = b, s ≥ 0

Signature

lp_standard_form(A: f64[m, n], b: f64[m]) → f64[m, n+m]

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.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference np.hstack([A, np.eye(m)]) in exact rational arithmetic, on all 40 test cases.
All 1,314 float64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them exact, with zero error.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 100%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 0

Identity

Calls

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Called by

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sha256:b93720282ddf46e747e63a701607423b628c391f3c33d9f9cbea86e3266cbc4d

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

Control handle

Symbol: Ω:std.optim.lp_standard_form · Ω:lp_standard_form
Pins: sha256:26747546bcf27393372a533672612151a24c688a64bf5b7ebceaf4b49ade6e8dthis graph alone
Evidence: sha256:0e2f3628f878d9dd653bf61f661ab7e00950085f92a160c0bb39d2bc2c9b5a4fthe 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.