Add functions for computation of krylov basis and krylov kernel basis by Biffo89 · Pull Request #40508 · sagemath/sage

Biffo89 · 2025-07-29T16:59:11Z

This PR implements the following functions, adds aliases row_rank_profile for pivot_rows and column_rank_profile for pivots, and caches the row rank profile that is calculated as part of linbox_echelonize in matrix_modn_dense_template.pxi.

Given a $\mathbb{K}[x]$-module $V$ with multiplication matrix $J \in \mathbb{K}^{\sigma\times\sigma}$, a set of $m$ elements of $V$ expressed in matrix form $E \in \mathbb{K}^{m\times\sigma}$, a priority shift $s = (s_1,\dots,s_m) \in \mathbb{Z}^m$, and an upper bound $d$ on the degree of the minimal polynomial of $J$, krylov_rank_profile finds the lexicographical first maximal linearly independent set of rows from the matrix, with rows ordered according to the priority $P(E_{c,*} J^d) = s_c + d$:

[   E  ]
[  EJ  ]
[  ... ]
[ EJ^δ ]

It returns the positions of these rows in the permuted matrix, the submatrix formed by these rows, and the lexicographical first maximal linearly independent set of columns of the submatrix.

Given the same arguments and a variable name, linear_interpolation_basis calculates an $s$-minimal interpolation basis for $(E,J)$, i.e. an $s$-reduced basis for the space of polynomials satisfying $p_1 E_{1,*}+\dots+p_mE_{m,*}=0$.

📝 Checklist

The title is concise and informative.
The description explains in detail what this PR is about.
I have linked a relevant issue or discussion.
I have created tests covering the changes.
I have updated the documentation and checked the documentation preview.

…lambda

…dn_dense, add aliases for pivots and pivot_rows, optimisation for krylov_rank_profile by minimising index conversion and full permutation calculation

…ial creation in basis calculation

Biffo89 · 2025-07-29T17:11:50Z

Some benchmarks are provided here (done in a branch with #40423 and #40435 merged):

GF(2)

matrix multiplication
+------+---------+
|    n | time    |
+======+=========+
|  128 | 31.4 μs |
+------+---------+
|  256 | 40 μs   |
+------+---------+
|  512 | 181 μs  |
+------+---------+
| 1024 | 980 μs  |
+------+---------+
| 2048 | 10.8 ms |
+------+---------+
| 4096 | 73.5 ms |
+------+---------+

uniform profile
+-------------+--------+--------+---------+--------+--------+
|   *   sigma | 256    | 512    | 1024    | 2048   | 4096   |
|       *     |        |        |         |        |        |
|   m       * |        |        |         |        |        |
+=============+========+========+=========+========+========+
|           1 |        |        | 121 ms  | 209 ms | 1.36 s |
+-------------+--------+--------+---------+--------+--------+
|           4 |        |        | 92.2 ms | 478 ms | 1.27 s |
+-------------+--------+--------+---------+--------+--------+
|          16 |        |        | 113 ms  | 243 ms | 1.67 s |
+-------------+--------+--------+---------+--------+--------+
|         256 | 115 ms | 236 ms | 822 ms  |        |        |
+-------------+--------+--------+---------+--------+--------+
|         512 | 207 ms | 459 ms | 1.18 s  |        |        |
+-------------+--------+--------+---------+--------+--------+
|        1024 | 404 ms | 1.13 s | 2.75 s  |        |        |
+-------------+--------+--------+---------+--------+--------+

hermite profile
+-------------+--------+--------+---------+--------+--------+
|   *   sigma | 256    | 512    | 1024    | 2048   | 4096   |
|       *     |        |        |         |        |        |
|   m       * |        |        |         |        |        |
+=============+========+========+=========+========+========+
|           1 |        |        | 53.7 ms | 451 ms | 1.28 s |
+-------------+--------+--------+---------+--------+--------+
|           4 |        |        | 92.2 ms | 364 ms | 2.27 s |
+-------------+--------+--------+---------+--------+--------+
|          16 |        |        | 335 ms  | 488 ms | 3.25 s |
+-------------+--------+--------+---------+--------+--------+
|         256 | 309 ms | 464 ms | 789 ms  |        |        |
+-------------+--------+--------+---------+--------+--------+
|         512 | 400 ms | 898 ms | 1.78 s  |        |        |
+-------------+--------+--------+---------+--------+--------+
|        1024 | 515 ms | 1.41 s | 2.76 s  |        |        |
+-------------+--------+--------+---------+--------+--------+

uniform basis
+-------------+--------+--------+--------+--------+--------+--------+
|   *   sigma | 128    | 256    | 512    | 1024   | 2048   | 4096   |
|       *     |        |        |        |        |        |        |
|   m       * |        |        |        |        |        |        |
+=============+========+========+========+========+========+========+
|           1 |        |        |        | 146 ms | 249 ms | 1.7 s  |
+-------------+--------+--------+--------+--------+--------+--------+
|           4 |        |        |        | 158 ms | 263 ms | 1.65 s |
+-------------+--------+--------+--------+--------+--------+--------+
|          16 |        |        |        | 253 ms | 628 ms | 2 s    |
+-------------+--------+--------+--------+--------+--------+--------+
|         128 | 177 ms | 225 ms | 779 ms |        |        |        |
+-------------+--------+--------+--------+--------+--------+--------+
|         256 | 461 ms | 686 ms | 1.23 s |        |        |        |
+-------------+--------+--------+--------+--------+--------+--------+
|         512 | 1.91 s | 2.88 s | 4.22 s |        |        |        |
+-------------+--------+--------+--------+--------+--------+--------+

hermite basis
+-------------+--------+--------+--------+--------+--------+--------+
|   *   sigma | 128    | 256    | 512    | 1024   | 2048   | 4096   |
|       *     |        |        |        |        |        |        |
|   m       * |        |        |        |        |        |        |
+=============+========+========+========+========+========+========+
|           1 |        |        |        | 143 ms | 247 ms | 1.7 s  |
+-------------+--------+--------+--------+--------+--------+--------+
|           4 |        |        |        | 264 ms | 380 ms | 2.57 s |
+-------------+--------+--------+--------+--------+--------+--------+
|          16 |        |        |        | 180 ms | 614 ms | 3.38 s |
+-------------+--------+--------+--------+--------+--------+--------+
|         128 | 259 ms | 185 ms | 745 ms |        |        |        |
+-------------+--------+--------+--------+--------+--------+--------+
|         256 | 332 ms | 700 ms | 762 ms |        |        |        |
+-------------+--------+--------+--------+--------+--------+--------+
|         512 | 1.68 s | 2.05 s | 3.26 s |        |        |        |
+-------------+--------+--------+--------+--------+--------+--------+

GF(243)

matrix multiplication
+-----+---------+
|   n | time    |
+=====+=========+
|  16 | 1.77 ms |
+-----+---------+
|  32 | 14 ms   |
+-----+---------+
|  64 | 111 ms  |
+-----+---------+

uniform profile
+-------------+---------+---------+--------+
|   *   sigma | 16      | 32      | 64     |
|       *     |         |         |        |
|   m       * |         |         |        |
+=============+=========+=========+========+
|           1 | 17.1 ms | 119 ms  | 485 ms |
+-------------+---------+---------+--------+
|           4 | 11.9 ms | 86.7 ms | 387 ms |
+-------------+---------+---------+--------+
|          16 | 6.35 ms | 26.5 ms | 450 ms |
+-------------+---------+---------+--------+
|          32 | 3.83 ms | 16.4 ms | 395 ms |
+-------------+---------+---------+--------+
|          64 | 10.6 ms | 40.8 ms | 226 ms |
+-------------+---------+---------+--------+

hermite profile
+-------------+---------+---------+--------+
|   *   sigma | 16      | 32      | 64     |
|       *     |         |         |        |
|   m       * |         |         |        |
+=============+=========+=========+========+
|           1 | 16.9 ms | 118 ms  | 999 ms |
+-------------+---------+---------+--------+
|           4 | 23 ms   | 162 ms  | 777 ms |
+-------------+---------+---------+--------+
|          16 | 28.5 ms | 97.2 ms | 1.65 s |
+-------------+---------+---------+--------+
|          32 | 14.6 ms | 220 ms  | 1.14 s |
+-------------+---------+---------+--------+
|          64 | 33.4 ms | 226 ms  | 1.87 s |
+-------------+---------+---------+--------+

uniform basis
+-------------+---------+--------+--------+
|   *   sigma | 16      | 32     | 64     |
|       *     |         |        |        |
|   m       * |         |        |        |
+=============+=========+========+========+
|           1 | 21.5 ms | 111 ms | 975 ms |
+-------------+---------+--------+--------+
|           4 | 25.6 ms | 117 ms | 883 ms |
+-------------+---------+--------+--------+
|          16 | 36 ms   | 110 ms | 701 ms |
+-------------+---------+--------+--------+
|          32 | 50.8 ms | 110 ms | 318 ms |
+-------------+---------+--------+--------+
|          64 | 66.3 ms | 139 ms | 554 ms |
+-------------+---------+--------+--------+

hermite basis
+-------------+---------+---------+--------+
|   *   sigma | 16      | 32      | 64     |
|       *     |         |         |        |
|   m       * |         |         |        |
+=============+=========+=========+========+
|           1 | 20.4 ms | 58.1 ms | 1.08 s |
+-------------+---------+---------+--------+
|           4 | 36.3 ms | 191 ms  | 1.3 s  |
+-------------+---------+---------+--------+
|          16 | 59.3 ms | 256 ms  | 1.01 s |
+-------------+---------+---------+--------+
|          32 | 72.2 ms | 287 ms  | 1.98 s |
+-------------+---------+---------+--------+
|          64 | 90.3 ms | 154 ms  | 1.44 s |
+-------------+---------+---------+--------+

GF(256)

matrix multiplication
+------+---------+
|    n | time    |
+======+=========+
|   64 | 558 μs  |
+------+---------+
|  128 | 1.52 ms |
+------+---------+
|  256 | 5.45 ms |
+------+---------+
|  512 | 16.2 ms |
+------+---------+
| 1024 | 36.9 ms |
+------+---------+

uniform profile
+-------------+---------+---------+--------+--------+
|   *   sigma | 128     | 256     | 512    | 1024   |
|       *     |         |         |        |        |
|   m       * |         |         |        |        |
+=============+=========+=========+========+========+
|           1 |         | 61.9 ms | 515 ms | 1.99 s |
+-------------+---------+---------+--------+--------+
|           4 |         | 56.7 ms | 471 ms | 2.09 s |
+-------------+---------+---------+--------+--------+
|          16 |         | 104 ms  | 421 ms | 1.84 s |
+-------------+---------+---------+--------+--------+
|         128 | 77.6 ms | 179 ms  | 544 ms |        |
+-------------+---------+---------+--------+--------+
|         256 | 126 ms  | 172 ms  | 751 ms |        |
+-------------+---------+---------+--------+--------+
|         512 | 236 ms  | 499 ms  | 1.07 s |        |
+-------------+---------+---------+--------+--------+

hermite profile
+-------------+--------+---------+--------+--------+
|   *   sigma | 128    | 256     | 512    | 1024   |
|       *     |        |         |        |        |
|   m       * |        |         |        |        |
+=============+========+=========+========+========+
|           1 |        | 59.9 ms | 519 ms | 2.37 s |
+-------------+--------+---------+--------+--------+
|           4 |        | 83.2 ms | 797 ms | 3.62 s |
+-------------+--------+---------+--------+--------+
|          16 |        | 220 ms  | 1.09 s | 3.87 s |
+-------------+--------+---------+--------+--------+
|         128 | 145 ms | 390 ms  | 1.72 s |        |
+-------------+--------+---------+--------+--------+
|         256 | 133 ms | 537 ms  | 2.18 s |        |
+-------------+--------+---------+--------+--------+
|         512 | 343 ms | 396 ms  | 2.76 s |        |
+-------------+--------+---------+--------+--------+

uniform basis
+-------------+---------+---------+--------+--------+
|   *   sigma | 64      | 128     | 256    | 512    |
|       *     |         |         |        |        |
|   m       * |         |         |        |        |
+=============+=========+=========+========+========+
|           1 |         | 34.9 ms | 208 ms | 750 ms |
+-------------+---------+---------+--------+--------+
|           4 |         | 98.3 ms | 218 ms | 720 ms |
+-------------+---------+---------+--------+--------+
|          16 |         | 111 ms  | 215 ms | 319 ms |
+-------------+---------+---------+--------+--------+
|          64 | 93.4 ms | 71.5 ms | 293 ms |        |
+-------------+---------+---------+--------+--------+
|         128 | 134 ms  | 101 ms  | 457 ms |        |
+-------------+---------+---------+--------+--------+
|         256 | 273 ms  | 408 ms  | 635 ms |        |
+-------------+---------+---------+--------+--------+

hermite basis
+-------------+--------+---------+--------+--------+
|   *   sigma | 64     | 128     | 256    | 512    |
|       *     |        |         |        |        |
|   m       * |        |         |        |        |
+=============+========+=========+========+========+
|           1 |        | 72.8 ms | 227 ms | 360 ms |
+-------------+--------+---------+--------+--------+
|           4 |        | 130 ms  | 335 ms | 1.08 s |
+-------------+--------+---------+--------+--------+
|          16 |        | 200 ms  | 585 ms | 2.47 s |
+-------------+--------+---------+--------+--------+
|          64 | 167 ms | 184 ms  | 805 ms |        |
+-------------+--------+---------+--------+--------+
|         128 | 121 ms | 686 ms  | 2.85 s |        |
+-------------+--------+---------+--------+--------+
|         256 | 529 ms | 1.28 s  | 3.95 s |        |
+-------------+--------+---------+--------+--------+

GF(257)

matrix multiplication
+------+---------+
|    n | time    |
+======+=========+
|  128 | 532 μs  |
+------+---------+
|  256 | 2.39 ms |
+------+---------+
|  512 | 12.4 ms |
+------+---------+
| 1024 | 73.4 ms |
+------+---------+

uniform profile
+-------------+---------+---------+--------+--------+
|   *   sigma | 128     | 256     | 512    | 1024   |
|       *     |         |         |        |        |
|   m       * |         |         |        |        |
+=============+=========+=========+========+========+
|           1 |         | 70 ms   | 308 ms | 1.62 s |
+-------------+---------+---------+--------+--------+
|           4 |         | 72.8 ms | 296 ms | 1.55 s |
+-------------+---------+---------+--------+--------+
|          16 |         | 78 ms   | 286 ms | 1.44 s |
+-------------+---------+---------+--------+--------+
|         128 | 71.3 ms | 169 ms  | 459 ms |        |
+-------------+---------+---------+--------+--------+
|         256 | 127 ms  | 288 ms  | 684 ms |        |
+-------------+---------+---------+--------+--------+
|         512 | 232 ms  | 506 ms  | 857 ms |        |
+-------------+---------+---------+--------+--------+

hermite profile
+-------------+--------+---------+--------+--------+
|   *   sigma | 128    | 256     | 512    | 1024   |
|       *     |        |         |        |        |
|   m       * |        |         |        |        |
+=============+========+=========+========+========+
|           1 |        | 70.6 ms | 302 ms | 1.73 s |
+-------------+--------+---------+--------+--------+
|           4 |        | 114 ms  | 477 ms | 2.48 s |
+-------------+--------+---------+--------+--------+
|          16 |        | 160 ms  | 641 ms | 3.31 s |
+-------------+--------+---------+--------+--------+
|         128 | 119 ms | 325 ms  | 1.09 s |        |
+-------------+--------+---------+--------+--------+
|         256 | 188 ms | 472 ms  | 1.01 s |        |
+-------------+--------+---------+--------+--------+
|         512 | 322 ms | 757 ms  | 2.01 s |        |
+-------------+--------+---------+--------+--------+

uniform basis
+-------------+--------+---------+--------+--------+
|   *   sigma | 128    | 256     | 512    | 1024   |
|       *     |        |         |        |        |
|   m       * |        |         |        |        |
+=============+========+=========+========+========+
|           1 |        | 97.1 ms | 200 ms | 2.19 s |
+-------------+--------+---------+--------+--------+
|           4 |        | 94 ms   | 390 ms | 2.08 s |
+-------------+--------+---------+--------+--------+
|          16 |        | 108 ms  | 399 ms | 2.04 s |
+-------------+--------+---------+--------+--------+
|         128 | 337 ms | 480 ms  | 874 ms |        |
+-------------+--------+---------+--------+--------+
|         256 | 343 ms | 1.32 s  | 2.11 s |        |
+-------------+--------+---------+--------+--------+
|         512 | 1.98 s | 2.24 s  | 4.72 s |        |
+-------------+--------+---------+--------+--------+

hermite basis
+-------------+--------+--------+--------+--------+
|   *   sigma | 128    | 256    | 512    | 1024   |
|       *     |        |        |        |        |
|   m       * |        |        |        |        |
+=============+========+========+========+========+
|           1 |        | 94 ms  | 206 ms | 2.18 s |
+-------------+--------+--------+--------+--------+
|           4 |        | 135 ms | 557 ms | 3.04 s |
+-------------+--------+--------+--------+--------+
|          16 |        | 195 ms | 739 ms | 3.88 s |
+-------------+--------+--------+--------+--------+
|         128 | 191 ms | 432 ms | 1.24 s |        |
+-------------+--------+--------+--------+--------+
|         256 | 414 ms | 408 ms | 1.85 s |        |
+-------------+--------+--------+--------+--------+
|         512 | 1.2 s  | 1.71 s | 3.29 s |        |
+-------------+--------+--------+--------+--------+

GF(59049)

matrix multiplication
+-----+---------+
|   n | time    |
+=====+=========+
|  16 | 3.37 ms |
+-----+---------+
|  32 | 34.2 ms |
+-----+---------+
|  64 | 283 ms  |
+-----+---------+

uniform profile
+-------------+---------+---------+--------+
|   *   sigma | 16      | 32      | 64     |
|       *     |         |         |        |
|   m       * |         |         |        |
+=============+=========+=========+========+
|           1 | 37.5 ms | 286 ms  | 2.82 s |
+-------------+---------+---------+--------+
|           4 | 34 ms   | 239 ms  | 2.55 s |
+-------------+---------+---------+--------+
|          16 | 8.33 ms | 119 ms  | 1.03 s |
+-------------+---------+---------+--------+
|          32 | 12.9 ms | 87.3 ms | 757 ms |
+-------------+---------+---------+--------+
|          64 | 21.5 ms | 94.8 ms | 542 ms |
+-------------+---------+---------+--------+

hermite profile
+-------------+---------+--------+--------+
|   *   sigma | 16      | 32     | 64     |
|       *     |         |        |        |
|   m       * |         |        |        |
+=============+=========+========+========+
|           1 | 49.8 ms | 290 ms | 3.23 s |
+-------------+---------+--------+--------+
|           4 | 67.9 ms | 495 ms | 2.74 s |
+-------------+---------+--------+--------+
|          16 | 43 ms   | 406 ms | 3.38 s |
+-------------+---------+--------+--------+
|          32 | 48.9 ms | 421 ms | 3.49 s |
+-------------+---------+--------+--------+
|          64 | 45.5 ms | 431 ms | 4.1 s  |
+-------------+---------+--------+--------+

uniform basis
+-------------+---------+--------+--------+
|   *   sigma | 16      | 32     | 64     |
|       *     |         |        |        |
|   m       * |         |        |        |
+=============+=========+========+========+
|           1 | 27.2 ms | 241 ms | 2.23 s |
+-------------+---------+--------+--------+
|           4 | 52.6 ms | 295 ms | 2.24 s |
+-------------+---------+--------+--------+
|          16 | 101 ms  | 357 ms | 2.24 s |
+-------------+---------+--------+--------+
|          32 | 180 ms  | 488 ms | 2.37 s |
+-------------+---------+--------+--------+
|          64 | 336 ms  | 809 ms | 2.77 s |
+-------------+---------+--------+--------+

hermite basis
+-------------+---------+----------+--------+
|   *   sigma | 16      | 32       | 64     |
|       *     |         |          |        |
|   m       * |         |          |        |
+=============+=========+==========+========+
|           1 | 26.8 ms | 268 ms   | 2.07 s |
+-------------+---------+----------+--------+
|           4 | 78.6 ms | 477 ms   | 4 s    |
+-------------+---------+----------+--------+
|          16 | 146 ms  | 775 ms   | 5.56 s |
+-------------+---------+----------+--------+
|          32 | 224 ms  | 1e+03 ms | 6.65 s |
+-------------+---------+----------+--------+
|          64 | 434 ms  | 1.32 s   | 6.91 s |
+-------------+---------+----------+--------+

GF(65536)

matrix multiplication
+-----+---------+
|   n | time    |
+=====+=========+
|  16 | 13.4 ms |
+-----+---------+
|  32 | 37.1 ms |
+-----+---------+
|  64 | 129 ms  |
+-----+---------+

uniform profile
+-------------+---------+--------+--------+
|   *   sigma | 16      | 32     | 64     |
|       *     |         |        |        |
|   m       * |         |        |        |
+=============+=========+========+========+
|           1 | 140 ms  | 426 ms | 1.91 s |
+-------------+---------+--------+--------+
|           4 | 93.1 ms | 355 ms | 1.45 s |
+-------------+---------+--------+--------+
|          16 | 51.5 ms | 222 ms | 930 ms |
+-------------+---------+--------+--------+
|          32 | 57.9 ms | 153 ms | 700 ms |
+-------------+---------+--------+--------+
|          64 | 70.3 ms | 163 ms | 471 ms |
+-------------+---------+--------+--------+

hermite profile
+-------------+--------+--------+--------+
|   *   sigma | 16     | 32     | 64     |
|       *     |        |        |        |
|   m       * |        |        |        |
+=============+========+========+========+
|           1 | 114 ms | 455 ms | 1.89 s |
+-------------+--------+--------+--------+
|           4 | 187 ms | 642 ms | 2.25 s |
+-------------+--------+--------+--------+
|          16 | 213 ms | 689 ms | 2.44 s |
+-------------+--------+--------+--------+
|          32 | 230 ms | 738 ms | 2.64 s |
+-------------+--------+--------+--------+
|          64 | 233 ms | 743 ms | 2.79 s |
+-------------+--------+--------+--------+

uniform basis
+-------------+--------+--------+--------+
|   *   sigma | 16     | 32     | 64     |
|       *     |        |        |        |
|   m       * |        |        |        |
+=============+========+========+========+
|           1 | 147 ms | 462 ms | 1.62 s |
+-------------+--------+--------+--------+
|           4 | 141 ms | 408 ms | 1.51 s |
+-------------+--------+--------+--------+
|          16 | 194 ms | 548 ms | 1.77 s |
+-------------+--------+--------+--------+
|          32 | 311 ms | 652 ms | 1.96 s |
+-------------+--------+--------+--------+
|          64 | 499 ms | 1.16 s | 2.84 s |
+-------------+--------+--------+--------+

hermite basis
+-------------+--------+--------+--------+
|   *   sigma | 16     | 32     | 64     |
|       *     |        |        |        |
|   m       * |        |        |        |
+=============+========+========+========+
|           1 | 147 ms | 490 ms | 1.62 s |
+-------------+--------+--------+--------+
|           4 | 216 ms | 637 ms | 2.26 s |
+-------------+--------+--------+--------+
|          16 | 340 ms | 973 ms | 3.19 s |
+-------------+--------+--------+--------+
|          32 | 442 ms | 1.19 s | 3.84 s |
+-------------+--------+--------+--------+
|          64 | 801 ms | 1.76 s | 5.51 s |
+-------------+--------+--------+--------+

GF(65537)

matrix multiplication
+------+---------+
|    n | time    |
+======+=========+
|   64 | 203 μs  |
+------+---------+
|  128 | 689 μs  |
+------+---------+
|  256 | 3.48 ms |
+------+---------+
|  512 | 19.6 ms |
+------+---------+
| 1024 | 123 ms  |
+------+---------+

uniform profile
+-------------+---------+---------+--------+--------+
|   *   sigma | 128     | 256     | 512    | 1024   |
|       *     |         |         |        |        |
|   m       * |         |         |        |        |
+=============+=========+=========+========+========+
|           1 |         | 86.1 ms | 547 ms | 2.65 s |
+-------------+---------+---------+--------+--------+
|           4 |         | 104 ms  | 408 ms | 2.36 s |
+-------------+---------+---------+--------+--------+
|          16 |         | 92.4 ms | 383 ms | 2.06 s |
+-------------+---------+---------+--------+--------+
|         128 | 84.6 ms | 191 ms  | 532 ms |        |
+-------------+---------+---------+--------+--------+
|         256 | 136 ms  | 315 ms  | 777 ms |        |
+-------------+---------+---------+--------+--------+
|         512 | 248 ms  | 556 ms  | 1.26 s |        |
+-------------+---------+---------+--------+--------+

hermite profile
+-------------+--------+---------+--------+--------+
|   *   sigma | 128    | 256     | 512    | 1024   |
|       *     |        |         |        |        |
|   m       * |        |         |        |        |
+=============+========+=========+========+========+
|           1 |        | 98.6 ms | 478 ms | 2.64 s |
+-------------+--------+---------+--------+--------+
|           4 |        | 147 ms  | 646 ms | 3.51 s |
+-------------+--------+---------+--------+--------+
|          16 |        | 193 ms  | 810 ms | 4.57 s |
+-------------+--------+---------+--------+--------+
|         128 | 135 ms | 378 ms  | 1.34 s |        |
+-------------+--------+---------+--------+--------+
|         256 | 209 ms | 571 ms  | 1.7 s  |        |
+-------------+--------+---------+--------+--------+
|         512 | 355 ms | 847 ms  | 2.33 s |        |
+-------------+--------+---------+--------+--------+

uniform basis
+-------------+---------+--------+--------+--------+--------+
|   *   sigma | 64      | 128    | 256    | 512    | 1024   |
|       *     |         |        |        |        |        |
|   m       * |         |        |        |        |        |
+=============+=========+========+========+========+========+
|           1 |         |        | 150 ms | 672 ms | 3.63 s |
+-------------+---------+--------+--------+--------+--------+
|           4 |         |        | 149 ms | 673 ms | 3.35 s |
+-------------+---------+--------+--------+--------+--------+
|          16 |         |        | 150 ms | 612 ms | 2.91 s |
+-------------+---------+--------+--------+--------+--------+
|          64 | 96.4 ms | 153 ms | 282 ms |        |        |
+-------------+---------+--------+--------+--------+--------+
|         128 | 207 ms  | 370 ms | 561 ms |        |        |
+-------------+---------+--------+--------+--------+--------+
|         256 | 496 ms  | 801 ms | 1.46 s |        |        |
+-------------+---------+--------+--------+--------+--------+

hermite basis
+-------------+---------+--------+--------+--------+--------+
|   *   sigma | 64      | 128    | 256    | 512    | 1024   |
|       *     |         |        |        |        |        |
|   m       * |         |        |        |        |        |
+=============+=========+========+========+========+========+
|           1 |         |        | 151 ms | 667 ms | 3.61 s |
+-------------+---------+--------+--------+--------+--------+
|           4 |         |        | 198 ms | 855 ms | 4.25 s |
+-------------+---------+--------+--------+--------+--------+
|          16 |         |        | 238 ms | 982 ms | 5.22 s |
+-------------+---------+--------+--------+--------+--------+
|          64 | 64.3 ms | 132 ms | 364 ms |        |        |
+-------------+---------+--------+--------+--------+--------+
|         128 | 128 ms  | 224 ms | 520 ms |        |        |
+-------------+---------+--------+--------+--------+--------+
|         256 | 332 ms  | 468 ms | 878 ms |        |        |
+-------------+---------+--------+--------+--------+--------+

GF(67108879)

matrix multiplication
+-----+---------+
|   n | time    |
+=====+=========+
|  64 | 447 μs  |
+-----+---------+
| 128 | 3.1 ms  |
+-----+---------+
| 256 | 27.4 ms |
+-----+---------+

uniform profile
+-------------+---------+---------+--------+
|   *   sigma | 64      | 128     | 256    |
|       *     |         |         |        |
|   m       * |         |         |        |
+=============+=========+=========+========+
|           1 | 12.5 ms | 50.2 ms | 384 ms |
+-------------+---------+---------+--------+
|           4 | 11.7 ms | 45 ms   | 333 ms |
+-------------+---------+---------+--------+
|          16 | 12.5 ms | 42.8 ms | 287 ms |
+-------------+---------+---------+--------+
|          64 | 20.5 ms | 55.2 ms | 275 ms |
+-------------+---------+---------+--------+
|         128 | 37 ms   | 78.3 ms | 316 ms |
+-------------+---------+---------+--------+
|         256 | 61.2 ms | 135 ms  | 394 ms |
+-------------+---------+---------+--------+

hermite profile
+-------------+---------+---------+--------+
|   *   sigma | 64      | 128     | 256    |
|       *     |         |         |        |
|   m       * |         |         |        |
+=============+=========+=========+========+
|           1 | 12.4 ms | 50.3 ms | 383 ms |
+-------------+---------+---------+--------+
|           4 | 18.4 ms | 73.9 ms | 632 ms |
+-------------+---------+---------+--------+
|          16 | 26.2 ms | 102 ms  | 812 ms |
+-------------+---------+---------+--------+
|          64 | 42 ms   | 145 ms  | 1.05 s |
+-------------+---------+---------+--------+
|         128 | 59.5 ms | 193 ms  | 1.2 s  |
+-------------+---------+---------+--------+
|         256 | 92.2 ms | 263 ms  | 1.44 s |
+-------------+---------+---------+--------+

uniform basis
+-------------+---------+---------+--------+
|   *   sigma | 64      | 128     | 256    |
|       *     |         |         |        |
|   m       * |         |         |        |
+=============+=========+=========+========+
|           1 | 16.8 ms | 66.5 ms | 500 ms |
+-------------+---------+---------+--------+
|           4 | 16.9 ms | 63 ms   | 440 ms |
+-------------+---------+---------+--------+
|          16 | 23.6 ms | 68.8 ms | 410 ms |
+-------------+---------+---------+--------+
|          64 | 96.6 ms | 161 ms  | 510 ms |
+-------------+---------+---------+--------+
|         128 | 209 ms  | 383 ms  | 792 ms |
+-------------+---------+---------+--------+
|         256 | 496 ms  | 823 ms  | 1 s    |
+-------------+---------+---------+--------+

hermite basis
+-------------+---------+---------+--------+
|   *   sigma | 64      | 128     | 256    |
|       *     |         |         |        |
|   m       * |         |         |        |
+=============+=========+=========+========+
|           1 | 16.8 ms | 66.3 ms | 502 ms |
+-------------+---------+---------+--------+
|           4 | 23.3 ms | 91.1 ms | 686 ms |
+-------------+---------+---------+--------+
|          16 | 34.4 ms | 124 ms  | 898 ms |
+-------------+---------+---------+--------+
|          64 | 70.5 ms | 197 ms  | 1.26 s |
+-------------+---------+---------+--------+
|         128 | 134 ms  | 296 ms  | 1.5 s  |
+-------------+---------+---------+--------+
|         256 | 329 ms  | 575 ms  | 1.93 s |
+-------------+---------+---------+--------+

xcaruso

Just a few comments on the doctests to start with.

src/sage/matrix/matrix2.pyx

Krylov kernel basis documentation and examples + handling var argument

vneiger · 2025-08-31T07:31:15Z

The latest changes handle the input var being either a generator or a string, based on a similar mechanism as cyclic_subspace. The documentation has to be update to reflect this, and also the examples for krylov_kernel_basis are now more detailed and explained. In my opinion this PR is ready for a final round of proofreading.

vneiger

All my comments have been handled properly and I see no other immediate enhancement. Looks good to me!

vneiger · 2025-09-01T15:24:46Z

@xcaruso I have not yet put the "positive review" label, in case you still have some suggestions to add.

src/sage/matrix/matrix2.pyx

…ecessary imports

xcaruso

A last round of comments. After this, I give a positive review, I promise :-)

src/sage/matrix/matrix2.pyx

xcaruso · 2025-09-05T10:58:01Z

Thanks for all!

sagemathgh-40508: Add functions for computation of krylov basis and krylov kernel basis This PR implements the following functions, adds aliases `row_rank_profile` for `pivot_rows` and `column_rank_profile` for `pivots`, and caches the row rank profile that is calculated as part of `linbox_echelonize` in `matrix_modn_dense_template.pxi`. Given a $\mathbb{K}[x]$-module $V$ with multiplication matrix $J \in \mathbb{K}^{\sigma\times\sigma}$, a set of $m$ elements of $V$ expressed in matrix form $E \in \mathbb{K}^{m\times\sigma}$, a priority shift $s = (s_1,\dots,s_m) \in \mathbb{Z}^m$, and an upper bound $d$ on the degree of the minimal polynomial of $J$, **`krylov_rank_profile`** finds the lexicographical first maximal linearly independent set of rows from the matrix, with rows ordered according to the priority $P(E_{c,*} J^d) = s_c + d$: ``` [ E ] [ EJ ] [ ... ] [ EJ^δ ] ``` It returns the positions of these rows in the permuted matrix, the submatrix formed by these rows, and the lexicographical first maximal linearly independent set of columns of the submatrix. Given the same arguments and a variable name, **`linear_interpolation_basis`** calculates an $s$-minimal interpolation basis for $(E,J)$, i.e. an $s$-reduced basis for the space of polynomials satisfying $p_1 E_{1,*}+\dots+p_mE_{m,\*}=0$. ### 📝 Checklist  - [X] The title is concise and informative. - [X] The description explains in detail what this PR is about. - [ ] I have linked a relevant issue or discussion. - [X] I have created tests covering the changes. - [X] I have updated the documentation and checked the documentation preview. URL: sagemath#40508 Reported by: Biffo89 Reviewer(s): Biffo89, Vincent Neiger, Xavier Caruso

Biffo89 and others added 23 commits July 3, 2025 15:44

implementation

e13a26b

compression for linear interpolation basis

f69c402

fast krylov almost working, problem with indices being too large for …

a415dcc

…lambda

krylov_rank_profile working now

59663bc

optimised target calculation

21882b5

combine row and column submatrix operations in basis calc

ce717fe

small bug fixes and cases with 0 rows

c4ae3ea

performance fix for lambda - calculate inverse once

308333f

remove debug code, tidy up variable handling for polynomial ring

4126fec

add caching for row_pivots in calculation of col_pivots for matrix_mo…

346aaab

…dn_dense, add aliases for pivots and pivot_rows, optimisation for krylov_rank_profile by minimising index conversion and full permutation calculation

make optimisation fast perm profile main implementation

a41e6ec

optimised version of basis calculation without permutation

3cc05bd

make optimisation final

59dea57

optimise matrix creation for basis, fix echelon_form bug

ae04c03

use matrix_from_rows in krylov_rank_profile, optimisation for polynom…

796155e

…ial creation in basis calculation

add early exit filtering for krylov rank profile

a6de952

mitigate performance loss on hermite shift

a010078

Merge branch 'sagemath:develop' into gsoc-2025-rebase

4659e9e

make early exit permanent

b6df7d3

fix boundary conditions for polynomial conversion

661db96

tidy up code

01f9301

update doc, add tests, remove whitespace

9264205

remove testing functions from development

fa0bf8c

github-actions bot added the s: needs review label Jul 29, 2025

Merge branch 'sagemath:develop' into gsoc-2025-rebase-2

f9a82ad

xcaruso self-requested a review July 29, 2025 19:27

xcaruso reviewed Jul 29, 2025

View reviewed changes

xcaruso added the gsoc: 2025 Tag for GSoC2025 issues/PRs label Jul 29, 2025

xcaruso requested a review from vneiger July 31, 2025 09:48

add support for generator var; improve documentation and examples

eb87928

vneiger mentioned this pull request Aug 30, 2025

Matrices: make cyclic_subspace faster #40735

Open

Merge pull request #4 from vneiger/doc_enhancements

72b6344

Krylov kernel basis documentation and examples + handling var argument

vneiger added t: feature c: linear algebra t: performance labels Aug 31, 2025

Biffo89 marked this pull request as ready for review August 31, 2025 22:18

github-actions bot added the s: needs review label Aug 31, 2025

vneiger self-requested a review September 1, 2025 15:22

vneiger approved these changes Sep 1, 2025

View reviewed changes

xcaruso reviewed Sep 2, 2025

View reviewed changes

Biffo89 added 4 commits September 4, 2025 12:08

add commas after spaces, add test cases for type checking, remove unn…

3cd8b9d

…ecessary imports

add spaces after commas

5f8d45f

fix doc tests

d540cc3

review changes

6582ade

xcaruso reviewed Sep 4, 2025

View reviewed changes

src/sage/matrix/matrix2.pyx Outdated Show resolved Hide resolved

Biffo89 added 5 commits September 5, 2025 10:04

review change

7277781

add spaces

94fc36a

limit to 80 characters

fae74c3

fix short line, bad indentation

8367783

fix doc tests

4ad72a6

xcaruso added s: positive review and removed s: needs review labels Sep 5, 2025

vbraun merged commit d33e0c1 into sagemath:develop Sep 14, 2025
23 of 24 checks passed

github-actions bot removed the s: positive review label Sep 14, 2025

Uh oh!

Conversation

Biffo89 commented Jul 29, 2025 • edited Loading Uh oh! There was an error while loading. Please reload this page.

Uh oh!

📝 Checklist

Uh oh!

Biffo89 commented Jul 29, 2025 • edited Loading Uh oh! There was an error while loading. Please reload this page.

Uh oh!

Uh oh!

xcaruso left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

vneiger commented Aug 31, 2025 • edited Loading Uh oh! There was an error while loading. Please reload this page.

Uh oh!

Uh oh!

vneiger left a comment

Choose a reason for hiding this comment

Uh oh!

vneiger commented Sep 1, 2025

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

xcaruso left a comment

Choose a reason for hiding this comment

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

Uh oh!

xcaruso commented Sep 5, 2025

Uh oh!

Uh oh!

Reviewers

Assignees

Labels

Projects

Milestone

Development

Uh oh!

4 participants

Biffo89 commented Jul 29, 2025 •

edited

Loading

Biffo89 commented Jul 29, 2025 •

edited

Loading

vneiger commented Aug 31, 2025 •

edited

Loading