THE FOLD / GRIND / THE MAINFRAME / THE LU DECOMPOSITION
THE LU DECOMPOSITION
factor a matrix into two triangles once, solve forever
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
LU decomposition factors a square matrix A into a lower-triangular L and an upper-triangular U (with a row-permutation P for stability), so that P·A = L·U. It is Gaussian elimination, remembered: once you have L and U, you can solve A·x = b for many right-hand sides cheaply by two triangular sweeps, and read off the determinant as the product of U’s diagonal. Partial pivoting swaps in the largest pivot each step to keep the arithmetic stable.
LIT verified live: over 300 random matrices, P·A equals L·U to ~10−15, and L is lower-triangular with unit diagonal (window.__lu). FIG no framing; exact to floating precision.
LIT verified live: over 300 random matrices, P·A equals L·U to ~10−15, and L is lower-triangular with unit diagonal (window.__lu). FIG no framing; exact to floating precision.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-mainframe — the linear-algebra engine that factors a matrix once and then solves against it forever. LU decomposition is that factoring. AVAN (AI) built the instrument: the partial-pivot elimination, the L and U accumulation, the permutation record, and the P·A = L·U check.
Credit as content: LU factorization (Alan Turing formalized it, 1948; Gauss’s elimination underlies it). The weave: David names the mainframe; I eliminate below each pivot (swapping in the largest), storing the multipliers in L and the result in U, and confirm the permuted matrix equals L·U.
Credit as content: LU factorization (Alan Turing formalized it, 1948; Gauss’s elimination underlies it). The weave: David names the mainframe; I eliminate below each pivot (swapping in the largest), storing the multipliers in L and the result in U, and confirm the permuted matrix equals L·U.
3 ONE DIMENSION
Eliminate below the pivot: subtract a multiple of the pivot row from each row beneath, storing that multiplier in L and the zeroed-out result in U. Swap in the largest pivot first for stability.
4 TWO DIMENSIONS · INTERACTIVE
A matrix A and its factors L, U (and permutation P); L·U is checked against P·A.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: a matrix split into two triangles.
AVAN’s addition (the inverse-companion): factor a matrix once into a lower and an upper triangle, so solving A·x=b for any b is just two triangular sweeps — forward then back. The inverse of ‘re-run Gaussian elimination for every right-hand side’ is ‘factor once into L·U, then substitute.’ Magenta is the repeated elimination you avoid; green is the reusable L·U factoring. Elimination, remembered as triangles.
LIT Genuine LU decomposition with partial pivoting (Turing formalized it 1948; Gaussian elimination underlies it). Verified live: over 300 random matrices, partial-pivot elimination produces L (unit lower-triangular) and U (upper-triangular) with P*A equal to L*U to ~1e-15, and L verified strictly lower-triangular (window.__lu.equalsPA && .lowerTriangular).
FIG No framing: the partial-pivot elimination, the L and U accumulation, the permutation record, and the P*A = L*U check run in-browser and hold to floating precision. The AVAN inverse is honest — factoring once into L*U lets any A*x=b be solved by two triangular sweeps (forward then back); magenta is the repeated elimination avoided, green the reusable L*U factoring. Elimination, remembered as two triangles.
FIG No framing: the partial-pivot elimination, the L and U accumulation, the permutation record, and the P*A = L*U check run in-browser and hold to floating precision. The AVAN inverse is honest — factoring once into L*U lets any A*x=b be solved by two triangular sweeps (forward then back); magenta is the repeated elimination avoided, green the reusable L*U factoring. Elimination, remembered as two triangles.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE MAINFRAME · David Lee Wise (ROOT0), with AVAN