simplex_step

std.optim.simplex_step · Level L4

One pivot of the fraction-free simplex method on an integer tableau, by Bland's rule: the entering column is the first with a negative reduced cost; the leaving row has the smallest ratio rhs/column over positive entries, compared by cross-multiplication, ties to the smallest basic variable. Every row but the pivot row becomes (p·row − T[i, k]·row r)/D, an exact integer division, and D becomes the pivot p. With no entering column the status becomes 1 (optimal); with no leaving row, 0 (unbounded). The body simplex_max runs.

Tᵢ ← (p·Tᵢ − Tᵢₖ·Tᵣ)/D (i ≠ r), D ← p

Signature

simplex_step(T: i64[m+1, n+m+1], basis: i64[m], D: i64[], status: i64[]) → i64[m+1, n+m+1], i64[m], i64[], i64[]

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 Bland's entering column and leaving row; (p·row − T[i, k]·row r) // D in exact rational arithmetic, on all 40 test cases.
All 1,196 int64 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 int64 nearest the exact value): 100%
bit-equal to the NumPy formula in int64: 100%
largest error, in units in the last place: 0

Identity

Calls

—

Called by

simplex_max

sha256:cfc09f3e102967cbf330d2511a0790df39a39edb3dde43691b5bc3f6b19e1fc0

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.simplex_step · Ω:simplex_step
Pins: sha256:6cd233d9a9df664dc367aaac612069144007c796d4a407bb9d0d84e2c270ffebthis graph alone
Evidence: sha256:5bcca69136af887d3dfe52542fd1fc389580fade8e7ab2089a3763b0579c9c00the 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.