Numerical Linear Algebra
where the algebra stops being the arithmetic
Ladder

Admissibility — the ladder

3 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. 545545545455554545545545545454555555454545545545545455554545545545rows and columns, in the order the points arrivedark: kept dense · light: two thin factors, rank printedthe strong partitionblocks112kept dense46largest rank5numbers stored2.7·10⁴‖A − A_H‖ ⁄ ‖A‖3.4·10⁻¹⁰the picture is decidedbefore a number is read

    Which pairs are allowed to be small

    A hierarchical representation is a partition of the matrix into blocks, and the rule that produces it reads four numbers per pair of index clusters and not one entry of the matrix. On a 256-square it yields 112 blocks, 66 of them stored as two thin factors, none of rank above five.

    rung 1 · hierarchy
  2. 567891003691215log₂ nlargest rank in the partitionweak: touching pairs compressedstrong: touching pairs refusedthe test bounds a rank and costs storagestrong, blocks250weak, blocks94strong, numbers6.8·10⁴weak, numbers6.1·10⁴strong ⁄ weak1.1the better partitionis the more expensive one

    The test that costs what it saves

    The partition that refuses to compress a touching pair keeps every rank at five while the other lets them climb from nine to thirteen. It also stores more numbers at every size measured — 67,968 against 61,440 at n = 512 — and which of those two facts matters is a question about how large the problem is going to get.

    rung 2 · hierarchy
  3. numbers stored, 256 × 256clustered, strong27,008clustered, weak24,064the dense matrix65,536shuffled, strong65,536shuffled, weak118,208the same matrix, twiceκ, clustered24κ, shuffled24‖A‖_F, clustered6140‖A‖_F, shuffled6140shuffled weak ⁄ dense1.8the compressibility is in the numberingand the numbering is not in the matrix

    The same matrix, numbered twice

    One symmetric permutation. The condition number is 24.3948 either way to eight digits and the Frobenius norm is 6.13996414·10³ either way to twelve. The partition that stored 27,008 numbers now finds no admissible pair anywhere and stores all 65,536, and the format that compresses regardless stores 118,208.

    rung 3 · hierarchy

All ladders