Add nmod_vec_invert: invert an array of nmod coefficients

[vneiger]

This inverts each entry in an nmod_vec, doing mostly multiplications instead of inversions.

See the table below for efficiency comparisons.

In short, this is in most cases significantly faster than the naive approach which uses nmod_inv repeatedly. When the vector length grows, the speed-up factor becomes close to the ratio between the time spent in a modular inversion vs. the time spent in a modular multiplication. The table below includes cases with factors beyond 20.

The only exceptions for "faster" are for very small moduli (like bitsize 2 or 3) since inversion is then quite fast; and for very small vector lengths. But already for bitsize 3 and length 5, this starts to be beneficial.

And actually, for not-too-small moduli (say bitsize 16 or more), this has an interesting speed-up already for very small lengths (more than 1.5x for length 2, more than 2x for length 3)... if this operation of inverting 2 or 3 (or a small number) of elements is not rare (I'm not sure about this), it could make sense to add some function for this in nmod? Something like:

void nmod_inv2(ulong * inv_a1, ulong * inv_a2, ulong a1, ulong a2, nmod_t mod)
void nmod_inv3(ulong * inv_a1, ulong * inv_a2, ulong * inv_a3, ulong a1, ulong a2, ulong a3, nmod_t mod)

Any insight about this is welcome before this PR gets finalized.

╰─ ./p-invert
unit: all measurements in c/l (up to constant multiplicative factor)
profiled: naive | precomp shoup | generic
bit/len 1               2               3               4               5               6               7               8               9               10              11              12              13              14              15              16              1024            65536
2       0.26|0.56|0.45  0.22|0.36|0.33  0.22|0.30|0.30  0.20|0.28|0.29  0.20|0.25|0.30  0.19|0.24|0.30  0.19|0.23|0.29  0.18|0.23|0.29  0.18|0.21|0.28  0.18|0.21|0.29  0.18|0.22|0.30  0.19|0.21|0.28  0.18|0.21|0.29  0.18|0.20|0.28  0.17|0.20|0.28  0.19|0.21|0.29  0.17|0.18|0.28  0.17|0.18|0.28
3       0.33|0.72|0.53  0.32|0.45|0.40  0.31|0.36|0.34  0.29|0.32|0.34  0.28|0.28|0.33  0.27|0.27|0.34  0.27|0.25|0.31  0.27|0.25|0.31  0.28|0.22|0.30  0.27|0.22|0.31  0.27|0.23|0.31  0.27|0.22|0.31  0.27|0.22|0.30  0.27|0.21|0.29  0.27|0.21|0.31  0.27|0.21|0.30  0.25|0.18|0.28  0.24|0.18|0.28
4       0.38|0.80|0.60  0.37|0.49|0.43  0.36|0.37|0.37  0.36|0.33|0.35  0.34|0.29|0.34  0.33|0.27|0.35  0.33|0.26|0.32  0.34|0.25|0.33  0.33|0.23|0.33  0.33|0.23|0.31  0.34|0.23|0.31  0.33|0.24|0.31  0.32|0.23|0.30  0.32|0.22|0.30  0.32|0.22|0.30  0.32|0.22|0.30  0.30|0.18|0.28  0.30|0.18|0.28
5       0.48|0.89|0.68  0.48|0.55|0.48  0.45|0.40|0.41  0.44|0.35|0.37  0.43|0.31|0.36  0.42|0.29|0.38  0.42|0.27|0.34  0.43|0.26|0.34  0.43|0.25|0.33  0.42|0.24|0.33  0.42|0.25|0.32  0.42|0.24|0.32  0.41|0.23|0.31  0.41|0.23|0.31  0.42|0.22|0.31  0.43|0.22|0.31  0.39|0.18|0.29  0.39|0.18|0.28
6       0.50|0.91|0.70  0.48|0.56|0.49  0.47|0.41|0.40  0.46|0.36|0.38  0.47|0.31|0.37  0.44|0.29|0.37  0.44|0.29|0.35  0.46|0.26|0.34  0.44|0.25|0.33  0.44|0.24|0.34  0.44|0.24|0.32  0.45|0.24|0.32  0.44|0.24|0.32  0.43|0.23|0.31  0.43|0.23|0.31  0.43|0.23|0.32  0.41|0.18|0.28  0.41|0.18|0.28
7       0.61|1.02|0.79  0.59|0.59|0.53  0.57|0.45|0.44  0.57|0.38|0.41  0.55|0.33|0.39  0.54|0.30|0.38  0.54|0.29|0.35  0.56|0.28|0.35  0.55|0.26|0.35  0.54|0.25|0.33  0.54|0.25|0.33  0.54|0.25|0.34  0.56|0.24|0.32  0.53|0.23|0.32  0.53|0.24|0.32  0.53|0.24|0.32  0.51|0.19|0.28  0.52|0.18|0.28
8       0.67|1.06|0.85  0.65|0.61|0.56  0.63|0.46|0.45  0.64|0.40|0.42  0.63|0.34|0.40  0.61|0.32|0.40  0.61|0.29|0.36  0.61|0.29|0.36  0.64|0.27|0.36  0.61|0.26|0.34  0.60|0.26|0.34  0.60|0.25|0.34  0.60|0.25|0.32  0.61|0.24|0.32  0.60|0.24|0.33  0.59|0.24|0.32  0.57|0.18|0.28  0.57|0.19|0.29
9       0.79|1.19|0.97  0.76|0.67|0.61  0.76|0.50|0.49  0.73|0.42|0.45  0.73|0.37|0.42  0.73|0.33|0.41  0.71|0.31|0.37  0.72|0.30|0.37  0.73|0.28|0.35  0.71|0.28|0.36  0.72|0.27|0.36  0.72|0.26|0.34  0.71|0.25|0.34  0.71|0.24|0.34  0.72|0.24|0.33  0.71|0.24|0.33  0.69|0.18|0.29  0.68|0.18|0.28
10      0.85|1.25|1.09  0.83|0.73|0.65  0.83|0.52|0.51  0.82|0.44|0.46  0.81|0.39|0.43  0.80|0.33|0.44  0.81|0.32|0.39  0.79|0.31|0.38  0.78|0.29|0.36  0.78|0.28|0.36  0.79|0.28|0.35  0.79|0.27|0.35  0.78|0.26|0.34  0.78|0.26|0.34  0.78|0.25|0.33  0.77|0.25|0.34  0.78|0.18|0.28  0.75|0.18|0.28
11      0.93|1.32|1.11  0.92|0.73|0.69  0.92|0.55|0.54  0.96|0.47|0.48  0.90|0.39|0.45  0.89|0.35|0.44  0.94|0.33|0.40  0.91|0.32|0.39  0.91|0.31|0.37  0.94|0.30|0.37  0.90|0.28|0.36  0.90|0.28|0.36  0.93|0.28|0.36  0.89|0.26|0.34  0.89|0.25|0.34  0.91|0.25|0.34  0.91|0.19|0.29  0.88|0.18|0.28
12      0.92|1.35|1.18  0.99|0.77|0.72  0.99|0.57|0.57  0.99|0.48|0.51  0.94|0.40|0.45  0.96|0.36|0.44  0.92|0.33|0.42  0.96|0.32|0.39  0.92|0.31|0.38  0.93|0.29|0.38  0.92|0.28|0.36  0.92|0.27|0.36  0.92|0.27|0.36  0.93|0.26|0.36  0.94|0.25|0.34  0.92|0.25|0.35  0.89|0.18|0.28  0.89|0.18|0.28
13      1.09|1.59|1.35  1.14|0.83|0.78  1.13|0.61|0.60  1.12|0.52|0.53  1.10|0.43|0.48  1.09|0.38|0.46  1.10|0.37|0.42  1.10|0.34|0.42  1.09|0.32|0.40  1.09|0.30|0.38  1.13|0.32|0.39  1.10|0.29|0.38  1.09|0.28|0.36  1.10|0.27|0.35  1.08|0.26|0.35  1.07|0.26|0.37  1.09|0.18|0.28  1.06|0.18|0.28
14      1.30|1.73|1.58  1.32|0.94|0.89  1.29|0.70|0.68  1.29|0.55|0.59  1.29|0.48|0.53  1.25|0.42|0.50  1.25|0.39|0.46  1.30|0.38|0.46  1.32|0.35|0.42  1.25|0.33|0.41  1.29|0.32|0.40  1.25|0.31|0.39  1.24|0.31|0.38  1.30|0.28|0.37  1.26|0.28|0.37  1.26|0.28|0.36  1.27|0.18|0.28  1.23|0.19|0.29
15      1.36|1.79|1.58  1.37|0.97|0.92  1.34|0.69|0.69  1.33|0.58|0.60  1.35|0.52|0.54  1.32|0.42|0.51  1.30|0.40|0.46  1.36|0.38|0.45  1.30|0.36|0.43  1.30|0.35|0.42  1.36|0.32|0.40  1.30|0.32|0.39  1.29|0.30|0.38  1.33|0.29|0.38  1.29|0.28|0.39  1.32|0.28|0.37  1.31|0.18|0.28  1.27|0.18|0.28
16      1.49|1.93|1.72  1.50|1.09|1.00  1.47|0.74|0.74  1.45|0.60|0.64  1.47|0.52|0.57  1.43|0.45|0.54  1.42|0.42|0.49  1.51|0.39|0.47  1.43|0.37|0.45  1.42|0.35|0.42  1.42|0.34|0.41  1.42|0.32|0.42  1.45|0.31|0.40  1.42|0.31|0.39  1.41|0.29|0.38  1.41|0.29|0.38  1.39|0.18|0.29  1.41|0.18|0.28
17      1.53|1.97|1.76  1.51|1.05|1.01  1.49|0.76|0.75  1.48|0.64|0.67  1.53|0.53|0.58  1.47|0.46|0.54  1.46|0.42|0.50  1.47|0.39|0.47  1.47|0.38|0.47  1.53|0.36|0.43  1.46|0.34|0.42  1.46|0.33|0.41  1.53|0.32|0.40  1.46|0.30|0.39  1.45|0.30|0.38  1.51|0.29|0.38  1.43|0.18|0.28  1.42|0.18|0.28
18      1.64|2.07|1.81  1.57|1.08|1.04  1.53|0.77|0.77  1.57|0.63|0.65  1.50|0.54|0.57  1.48|0.46|0.57  1.56|0.43|0.48  1.51|0.40|0.47  1.49|0.38|0.45  1.49|0.36|0.43  1.48|0.34|0.42  1.57|0.34|0.41  1.49|0.32|0.40  1.48|0.30|0.39  1.48|0.30|0.38  1.47|0.30|0.39  1.48|0.18|0.28  1.46|0.18|0.28
19      1.70|2.10|1.87  1.62|1.13|1.08  1.62|0.80|0.86  1.69|0.67|0.68  1.60|0.55|0.60  1.59|0.48|0.57  1.58|0.45|0.51  1.59|0.42|0.49  1.63|0.39|0.47  1.59|0.38|0.45  1.58|0.35|0.43  1.58|0.34|0.42  1.58|0.33|0.41  1.59|0.31|0.40  1.56|0.30|0.39  1.56|0.30|0.39  1.54|0.18|0.28  1.54|0.18|0.28
20      1.76|2.21|1.94  1.68|1.15|1.12  1.69|0.83|0.83  1.67|0.67|0.69  1.66|0.56|0.61  1.68|0.50|0.56  1.65|0.46|0.52  1.66|0.42|0.49  1.65|0.39|0.47  1.64|0.38|0.46  1.70|0.36|0.43  1.66|0.35|0.43  1.65|0.34|0.41  1.66|0.32|0.40  1.64|0.31|0.39  1.68|0.31|0.38  1.62|0.18|0.28  1.62|0.18|0.28
21      1.85|2.25|2.03  1.77|1.19|1.16  1.86|0.86|0.86  1.75|0.68|0.71  1.73|0.58|0.62  1.72|0.50|0.58  1.72|0.47|0.53  1.73|0.45|0.52  1.75|0.39|0.47  1.72|0.39|0.47  1.72|0.36|0.44  1.71|0.35|0.43  1.71|0.34|0.42  1.72|0.33|0.42  1.72|0.32|0.40  1.71|0.32|0.39  1.70|0.18|0.28  1.68|0.20|0.28
22      1.98|2.56|2.16  1.91|1.27|1.22  1.88|0.90|0.89  1.87|0.75|0.75  1.86|0.63|0.66  1.86|0.52|0.61  1.85|0.48|0.57  1.90|0.45|0.52  1.85|0.41|0.49  1.85|0.39|0.49  1.86|0.37|0.45  1.83|0.36|0.44  1.82|0.34|0.44  1.85|0.33|0.42  1.81|0.32|0.42  1.92|0.31|0.39  1.79|0.18|0.28  1.80|0.18|0.28
23      2.03|2.48|2.27  1.96|1.27|1.27  1.95|0.91|0.92  1.96|0.73|0.76  1.90|0.61|0.66  1.90|0.54|0.61  1.89|0.49|0.57  1.96|0.46|0.52  1.91|0.41|0.49  1.89|0.39|0.48  1.95|0.37|0.45  1.88|0.37|0.45  1.93|0.34|0.43  1.95|0.34|0.42  1.88|0.33|0.40  1.87|0.32|0.40  1.92|0.18|0.30  1.90|0.18|0.28
24      2.12|2.53|2.29  2.09|1.31|1.28  2.00|0.93|0.93  1.98|0.74|0.78  2.11|0.65|0.68  1.97|0.54|0.62  1.97|0.50|0.57  1.98|0.46|0.53  1.97|0.42|0.50  2.10|0.40|0.49  1.97|0.38|0.46  1.97|0.37|0.46  1.96|0.35|0.44  1.98|0.34|0.43  1.98|0.33|0.41  1.96|0.33|0.41  1.94|0.18|0.28  1.93|0.18|0.28
25      2.15|2.55|2.31  2.05|1.36|1.31  2.05|0.95|0.95  2.02|0.77|0.78  2.01|0.63|0.68  2.00|0.56|0.63  2.00|0.52|0.58  2.05|0.47|0.54  2.01|0.43|0.51  2.01|0.41|0.49  2.01|0.38|0.46  1.99|0.38|0.48  2.11|0.36|0.45  1.99|0.35|0.43  1.99|0.33|0.41  1.99|0.33|0.43  2.02|0.18|0.30  2.01|0.18|0.28
26      2.27|2.66|2.49  2.18|1.38|1.35  2.13|0.98|0.97  2.12|0.78|0.81  2.18|0.66|0.71  2.11|0.58|0.65  2.11|0.52|0.59  2.18|0.49|0.54  2.11|0.44|0.52  2.24|0.42|0.50  2.21|0.39|0.47  2.11|0.39|0.46  2.11|0.36|0.44  2.21|0.35|0.43  2.09|0.34|0.43  2.11|0.33|0.41  2.07|0.18|0.28  2.07|0.18|0.28
27      2.34|2.73|2.48  2.34|1.48|1.42  2.24|1.00|1.00  2.20|0.79|0.82  2.19|0.68|0.72  2.16|0.58|0.66  2.17|0.52|0.60  2.20|0.51|0.54  2.18|0.45|0.53  2.18|0.43|0.50  2.17|0.40|0.49  2.18|0.39|0.48  2.20|0.37|0.45  2.17|0.36|0.45  2.18|0.34|0.43  2.16|0.33|0.42  2.14|0.18|0.29  2.17|0.18|0.28
28      2.41|2.79|2.55  2.28|1.50|1.43  2.27|1.01|1.01  2.25|0.82|0.89  2.29|0.68|0.73  2.24|0.60|0.67  2.23|0.54|0.60  2.24|0.50|0.56  2.24|0.46|0.53  2.38|0.45|0.52  2.26|0.41|0.49  2.26|0.40|0.47  2.26|0.38|0.46  2.25|0.37|0.45  2.30|0.35|0.44  2.26|0.34|0.43  2.23|0.18|0.28  2.22|0.18|0.28
29      2.49|2.85|2.64  2.41|1.50|1.46  2.36|1.07|1.05  2.45|0.85|0.86  2.33|0.69|0.75  2.32|0.61|0.69  2.46|0.57|0.62  2.33|0.52|0.58  2.33|0.48|0.54  2.33|0.46|0.53  2.32|0.41|0.49  2.32|0.41|0.48  2.33|0.38|0.48  2.32|0.37|0.45  2.30|0.35|0.44  2.31|0.35|0.44  2.30|0.19|0.30  2.32|0.18|0.28
30      2.59|2.99|2.77  2.47|1.54|1.50  2.44|1.10|1.08  2.42|0.86|0.93  2.48|0.71|0.77  2.42|0.62|0.71  2.42|0.57|0.63  2.48|0.51|0.59  2.41|0.48|0.58  2.45|0.44|0.53  2.49|0.41|0.50  2.49|0.41|0.49  2.46|0.40|0.48  2.51|0.38|0.46  2.45|0.36|0.45  2.41|0.36|0.44  2.49|0.19|0.28  2.36|0.18|0.28
31      2.62|2.99|2.87  2.60|1.59|1.53  2.48|1.08|1.09  2.45|0.87|0.90  2.45|0.73|0.77  2.43|0.63|0.69  2.55|0.59|0.64  2.46|0.54|0.60  2.44|0.49|0.56  2.44|0.45|0.53  2.44|0.43|0.51  2.55|0.41|0.50  2.44|0.39|0.47  2.43|0.39|0.47  2.44|0.36|0.45  2.42|0.36|0.44  2.40|0.19|0.29  2.38|0.18|0.28
32      2.72|3.10|2.86  2.60|1.68|1.60  2.58|1.15|1.13  2.57|0.93|0.93  2.59|0.74|0.80  2.56|0.65|0.73  2.55|0.59|0.65  2.56|0.55|0.61  2.56|0.51|0.58  2.60|0.47|0.55  2.55|0.43|0.52  2.55|0.42|0.49  2.55|0.40|0.48  2.54|0.39|0.49  2.67|0.38|0.46  2.54|0.36|0.45  2.52|0.18|0.29  2.58|0.18|0.29
33      2.79|3.17|2.94  2.68|1.64|1.59  2.63|1.17|1.14  2.61|0.93|0.94  2.59|0.75|0.81  2.58|0.65|0.74  2.68|0.60|0.66  2.59|0.54|0.61  2.59|0.51|0.57  2.59|0.46|0.56  2.59|0.43|0.52  2.74|0.43|0.51  2.60|0.40|0.50  2.60|0.39|0.47  2.57|0.37|0.46  2.57|0.36|0.46  2.60|0.18|0.28  2.55|0.18|0.28
34      2.87|3.29|3.03  2.73|1.68|1.64  2.71|1.20|1.17  2.70|0.96|1.00  2.74|0.77|0.82  2.67|0.67|0.75  2.67|0.61|0.67  2.75|0.54|0.63  2.69|0.54|0.61  2.82|0.48|0.56  2.79|0.44|0.53  2.70|0.43|0.51  2.70|0.41|0.49  2.79|0.40|0.49  2.72|0.38|0.47  2.67|0.37|0.46  2.75|0.18|0.28  2.63|0.18|0.28
35      2.91|3.41|3.09  2.85|1.68|1.66  2.74|1.18|1.18  2.73|0.93|0.96  2.83|0.77|0.82  2.71|0.68|0.76  2.76|0.61|0.67  2.86|0.57|0.63  2.72|0.51|0.59  2.71|0.48|0.56  2.71|0.46|0.56  2.87|0.44|0.51  2.72|0.41|0.49  2.73|0.41|0.49  2.74|0.38|0.46  2.73|0.37|0.46  2.76|0.19|0.28  2.71|0.18|0.28
36      2.98|3.33|3.08  2.82|1.74|1.68  2.79|1.21|1.26  2.83|0.96|0.99  2.82|0.79|0.85  2.80|0.70|0.77  2.79|0.63|0.68  2.80|0.57|0.64  2.95|0.53|0.59  2.78|0.48|0.56  2.76|0.45|0.53  2.77|0.43|0.51  2.77|0.41|0.49  2.76|0.41|0.49  2.77|0.38|0.47  2.76|0.37|0.46  2.73|0.18|0.28  2.73|0.18|0.28
37      3.10|3.51|3.25  2.92|1.76|1.72  2.90|1.25|1.22  2.87|0.98|1.00  2.86|0.80|0.85  2.85|0.72|0.79  2.89|0.64|0.69  2.86|0.57|0.63  2.86|0.53|0.60  2.85|0.48|0.59  2.88|0.46|0.55  2.90|0.44|0.53  2.87|0.42|0.53  2.91|0.41|0.49  2.84|0.39|0.47  2.84|0.38|0.49  2.90|0.18|0.28  2.82|0.18|0.28
38      3.14|3.57|3.34  3.03|1.82|1.77  3.00|1.30|1.27  3.00|0.99|1.06  3.05|0.81|0.87  2.93|0.71|0.78  2.93|0.63|0.70  3.08|0.59|0.65  3.14|0.56|0.63  2.97|0.50|0.59  2.97|0.48|0.56  2.96|0.45|0.54  2.96|0.43|0.51  2.96|0.43|0.51  2.96|0.40|0.48  2.96|0.39|0.47  2.94|0.19|0.29  2.93|0.18|0.28
39      3.27|3.63|3.38  3.22|1.85|1.82  3.08|1.33|1.29  3.07|1.02|1.05  3.06|0.86|0.90  3.04|0.73|0.82  3.06|0.66|0.73  3.06|0.60|0.67  3.05|0.55|0.62  3.04|0.52|0.58  3.04|0.48|0.59  3.14|0.46|0.54  3.04|0.44|0.52  3.04|0.43|0.52  3.08|0.40|0.49  3.03|0.40|0.50  3.13|0.18|0.28  2.98|0.18|0.28
40      3.29|3.66|3.39  3.12|1.88|1.88  3.14|1.32|1.34  3.16|1.03|1.06  3.10|0.86|0.91  3.09|0.74|0.81  3.09|0.66|0.72  3.09|0.60|0.68  3.17|0.56|0.63  3.10|0.51|0.60  3.09|0.48|0.56  3.08|0.45|0.54  3.09|0.44|0.52  3.27|0.44|0.50  3.08|0.40|0.49  3.07|0.40|0.48  3.05|0.18|0.28  3.05|0.18|0.28
41      3.41|3.73|3.48  3.21|1.92|1.88  3.20|1.35|1.33  3.18|1.05|1.07  3.18|0.86|0.92  3.16|0.78|0.86  3.22|0.70|0.73  3.17|0.62|0.69  3.22|0.56|0.64  3.18|0.53|0.63  3.21|0.49|0.58  3.21|0.46|0.55  3.16|0.44|0.55  3.25|0.43|0.51  3.15|0.41|0.49  3.15|0.40|0.51  3.31|0.19|0.28  3.13|0.18|0.28
42      3.45|3.80|3.69  3.35|1.94|1.90  3.25|1.35|1.42  3.31|1.06|1.09  3.40|0.88|0.93  3.22|0.76|0.84  3.22|0.67|0.73  3.23|0.64|0.68  3.49|0.58|0.66  3.31|0.53|0.61  3.29|0.51|0.59  3.28|0.47|0.56  3.29|0.45|0.54  3.30|0.46|0.54  3.32|0.42|0.50  3.28|0.40|0.49  3.27|0.19|0.29  3.26|0.18|0.28
43      3.53|3.79|3.58  3.30|2.01|1.91  3.26|1.38|1.35  3.25|1.05|1.09  3.25|0.88|0.96  3.29|0.79|0.85  3.29|0.69|0.75  3.26|0.62|0.73  3.33|0.58|0.65  3.31|0.53|0.61  3.30|0.50|0.61  3.42|0.47|0.56  3.29|0.46|0.54  3.26|0.44|0.52  3.37|0.41|0.50  3.26|0.41|0.52  3.27|0.19|0.28  3.29|0.18|0.28
44      3.61|3.98|3.70  3.44|2.03|1.99  3.53|1.39|1.49  3.60|1.12|1.14  3.41|0.92|0.97  3.41|0.79|0.87  3.39|0.71|0.77  3.41|0.65|0.72  3.63|0.59|0.66  3.39|0.54|0.62  3.35|0.51|0.59  3.40|0.49|0.57  3.39|0.46|0.55  3.39|0.45|0.54  3.41|0.43|0.51  3.39|0.41|0.50  3.37|0.19|0.28  3.35|0.18|0.28
45      3.73|4.07|3.82  3.54|2.08|2.04  3.50|1.46|1.43  3.49|1.14|1.15  3.47|0.93|0.98  3.46|0.84|0.88  3.52|0.75|0.76  3.47|0.65|0.73  3.48|0.60|0.67  3.46|0.55|0.65  3.48|0.52|0.63  3.65|0.50|0.58  3.45|0.47|0.57  3.51|0.45|0.53  3.44|0.43|0.52  3.45|0.42|0.50  3.53|0.19|0.28  3.43|0.18|0.28
46      3.73|4.15|3.91  3.57|2.09|2.05  3.54|1.43|1.45  3.58|1.13|1.18  3.62|0.93|0.99  3.51|0.81|0.88  3.50|0.72|0.79  3.67|0.67|0.73  3.49|0.61|0.69  3.52|0.55|0.64  3.50|0.54|0.61  3.47|0.49|0.57  3.46|0.46|0.55  3.45|0.47|0.54  3.48|0.43|0.51  3.45|0.41|0.50  3.43|0.18|0.29  3.43|0.18|0.28
47      3.76|4.27|4.04  3.71|2.18|2.06  3.61|1.46|1.45  3.54|1.13|1.17  3.53|0.97|1.01  3.54|0.80|0.90  3.56|0.73|0.79  3.56|0.66|0.76  3.59|0.60|0.68  3.52|0.55|0.64  3.52|0.53|0.64  3.63|0.50|0.58  3.52|0.47|0.55  3.52|0.46|0.54  3.61|0.43|0.52  3.52|0.42|0.51  3.62|0.19|0.28  3.50|0.18|0.28
48      3.90|4.24|3.96  3.69|2.17|2.24  3.85|1.52|1.66  3.82|1.18|1.20  3.68|0.96|1.02  3.65|0.83|0.91  3.65|0.74|0.80  3.65|0.67|0.74  3.91|0.64|0.71  3.69|0.56|0.65  3.65|0.54|0.61  3.64|0.52|0.59  3.64|0.48|0.56  3.64|0.48|0.55  3.69|0.44|0.53  3.65|0.43|0.51  3.62|0.18|0.28  3.62|0.18|0.28
49      4.02|4.29|4.03  3.76|2.18|2.13  3.71|1.51|1.51  3.70|1.20|1.20  3.68|0.97|1.02  3.68|0.87|0.92  3.71|0.78|0.80  3.67|0.68|0.75  3.72|0.61|0.70  3.67|0.58|0.68  3.72|0.54|0.65  3.75|0.51|0.60  3.68|0.48|0.59  3.75|0.47|0.54  3.66|0.44|0.53  3.66|0.43|0.53  3.88|0.19|0.28  3.68|0.18|0.28
50      4.04|4.42|4.22  3.88|2.23|2.20  3.84|1.55|1.55  3.88|1.21|1.29  3.79|0.99|1.05  3.80|0.86|0.94  3.80|0.77|0.83  3.97|0.69|0.76  3.98|0.63|0.72  3.82|0.58|0.67  3.83|0.56|0.64  3.80|0.53|0.61  3.80|0.49|0.57  3.79|0.50|0.57  3.85|0.45|0.54  3.80|0.43|0.53  3.76|0.19|0.29  3.74|0.18|0.28
51      4.17|4.51|4.23  3.96|2.25|2.24  3.91|1.59|1.57  3.90|1.22|1.26  3.89|1.03|1.07  3.86|0.90|0.99  3.98|0.78|0.84  3.88|0.70|0.79  3.89|0.63|0.71  3.86|0.59|0.68  3.86|0.56|0.67  4.07|0.55|0.62  3.85|0.50|0.58  3.86|0.48|0.59  3.95|0.45|0.54  3.85|0.44|0.53  3.98|0.19|0.29  3.84|0.18|0.28
52      4.22|4.57|4.28  4.03|2.31|2.34  4.05|1.60|1.69  4.07|1.26|1.29  3.98|1.03|1.09  3.97|0.89|0.96  3.98|0.79|0.86  3.97|0.72|0.78  3.98|0.65|0.74  3.99|0.60|0.69  3.97|0.57|0.65  3.96|0.54|0.62  3.96|0.51|0.59  3.96|0.51|0.60  4.08|0.46|0.55  3.97|0.45|0.54  3.93|0.19|0.28  3.93|0.18|0.28
53      4.32|4.56|4.38  4.09|2.36|2.31  4.05|1.62|1.62  4.03|1.28|1.30  4.00|1.03|1.09  4.00|0.90|0.98  4.06|0.80|0.85  4.02|0.72|0.79  4.00|0.65|0.73  4.01|0.61|0.72  4.05|0.57|0.68  4.08|0.54|0.63  4.01|0.51|0.59  4.14|0.49|0.57  4.00|0.47|0.55  4.00|0.46|0.54  4.18|0.19|0.29  3.97|0.18|0.28
54      4.34|4.70|4.52  4.18|2.37|2.33  4.11|1.63|1.64  4.32|1.30|1.32  4.21|1.05|1.10  4.08|0.91|0.96  4.07|0.80|0.87  4.27|0.75|0.80  4.05|0.67|0.75  4.10|0.62|0.70  4.08|0.59|0.66  4.06|0.54|0.63  4.06|0.51|0.60  4.07|0.52|0.59  4.10|0.47|0.56  4.07|0.46|0.54  4.04|0.19|0.29  4.03|0.18|0.28
55      4.51|4.78|4.47  4.22|2.39|2.35  4.17|1.65|1.66  4.19|1.28|1.32  4.17|1.06|1.14  4.19|0.90|1.04  4.19|0.82|0.89  4.19|0.75|0.85  4.27|0.66|0.74  4.16|0.63|0.71  4.18|0.58|0.67  4.39|0.58|0.65  4.17|0.52|0.61  4.17|0.51|0.58  4.35|0.48|0.56  4.15|0.47|0.55  4.26|0.19|0.29  4.15|0.18|0.29
56      4.52|4.84|4.55  4.29|2.46|2.44  4.36|1.68|1.69  4.34|1.32|1.36  4.27|1.09|1.14  4.25|0.93|1.01  4.25|0.83|0.89  4.24|0.75|0.82  4.24|0.68|0.75  4.22|0.63|0.71  4.20|0.58|0.67  4.20|0.55|0.63  4.18|0.52|0.60  4.19|0.49|0.59  4.24|0.48|0.56  4.19|0.46|0.55  4.17|0.18|0.28  4.17|0.18|0.28
57      4.57|5.03|4.77  4.39|2.46|2.43  4.30|1.71|1.69  4.30|1.35|1.36  4.28|1.09|1.14  4.26|0.93|1.03  4.31|0.84|0.91  4.32|0.77|0.83  4.32|0.69|0.77  4.30|0.64|0.75  4.43|0.58|0.67  4.39|0.56|0.64  4.27|0.53|0.61  4.41|0.50|0.58  4.24|0.49|0.58  4.32|0.46|0.56  4.51|0.19|0.29  4.29|0.18|0.28
58      4.64|5.01|4.79  4.46|2.52|2.48  4.40|1.75|1.74  4.55|1.34|1.40  4.36|1.11|1.16  4.38|0.95|1.03  4.38|0.85|0.91  4.59|0.78|0.84  4.36|0.73|0.81  4.56|0.65|0.74  4.44|0.62|0.71  4.43|0.57|0.66  4.46|0.55|0.62  4.44|0.51|0.62  4.48|0.50|0.59  4.42|0.47|0.57  4.39|0.19|0.29  4.37|0.18|0.28
59      4.69|5.14|4.79  4.51|2.61|2.48  4.44|1.77|1.76  4.44|1.35|1.40  4.42|1.12|1.19  4.42|0.96|1.08  4.49|0.86|0.92  4.43|0.78|0.87  4.48|0.69|0.77  4.40|0.64|0.73  4.41|0.60|0.69  4.51|0.57|0.65  4.42|0.54|0.62  4.42|0.51|0.60  4.55|0.49|0.58  4.50|0.49|0.57  4.40|0.19|0.30  4.37|0.18|0.28
60      4.82|5.14|4.83  4.57|2.60|2.65  4.62|1.78|1.79  4.65|1.39|1.44  4.56|1.15|1.19  4.53|0.98|1.06  4.53|0.86|0.92  4.53|0.79|0.86  4.55|0.71|0.79  4.58|0.67|0.75  4.54|0.61|0.70  4.54|0.58|0.66  4.53|0.55|0.63  4.52|0.54|0.62  4.56|0.51|0.59  4.53|0.49|0.57  4.51|0.19|0.28  4.49|0.18|0.28
61      4.90|5.15|4.98  4.68|2.65|2.61  4.64|1.83|1.82  4.63|1.44|1.45  4.59|1.15|1.21  4.60|1.03|1.12  4.76|0.88|0.92  4.60|0.80|0.87  4.61|0.71|0.79  4.59|0.67|0.78  4.69|0.62|0.74  4.67|0.59|0.67  4.60|0.56|0.64  4.78|0.54|0.61  4.58|0.51|0.59  4.58|0.48|0.58  4.78|0.19|0.29  4.58|0.18|0.28
62      4.93|5.31|5.12  4.75|2.65|2.62  4.70|1.82|1.91  4.72|1.42|1.47  4.67|1.17|1.22  4.67|1.00|1.08  4.66|0.88|0.95  4.88|0.82|0.87  4.68|0.73|0.81  4.68|0.68|0.76  4.67|0.64|0.72  4.64|0.59|0.68  4.65|0.56|0.64  4.65|0.55|0.65  4.79|0.51|0.60  4.68|0.50|0.58  4.62|0.19|0.29  4.61|0.18|0.28
63      5.02|5.25|5.11  4.85|2.69|2.67  4.79|1.85|1.87  4.78|1.43|1.48  4.77|1.23|1.26  4.74|1.05|1.11  4.83|0.89|0.97  4.78|0.82|0.91  4.82|0.73|0.81  4.76|0.69|0.77  4.75|0.63|0.76  4.93|0.60|0.68  4.78|0.57|0.65  4.75|0.54|0.62  4.91|0.52|0.60  4.73|0.49|0.59  4.69|0.19|0.28  4.68|0.18|0.28
64      5.10| na |5.10  4.81| na |2.79  4.90| na |1.96  5.06| na |1.50  4.84| na |1.25  4.82| na |1.11  4.80| na |0.96  4.80| na |0.89  5.06| na |0.82  4.83| na |0.78  4.83| na |0.72  4.80| na |0.69  4.79| na |0.66  4.79| na |0.64  4.83| na |0.61  4.80| na |0.59  4.76| na |0.29  4.76| na |0.28

[vneiger]

On my list to finalize this:

• add documentation
• see if memory consumption of the Shoup variant can be reduced to 2*len instead of 4*len
• add case for doing the naive approach when len == 1
• check performance a bit more carefully on a few machines, and add some logic in the main function to refine the choice of variant depending on length/bitsize (notably for very small lengths and very small bitsize)

[vneiger]

This PR is ready for review. Performance on three machines attached. On recent-ish machines, the non naive approach is basically always interesting.

On older machines with a slower integer division, the gain is less significant, and the naive approach would actually be faster for moduli of very small bitsizes (up to bitsize 5 or so). This was not enough to convince me that having thresholds for this could be useful, but this can be discussed.

profile.txt

[albinahlback]

On older machines with a slower integer division, the gain is less significant, and the naive approach would actually be faster for moduli of very small bitsizes (up to bitsize 5 or so). This was not enough to convince me that having thresholds for this could be useful, but this can be discussed.

Cool! I think it is okay, Cascade Lake is 6 years old anyway.

[fredrik-johansson]

This is a nice speedup. Do you have an application?

I know some functions where we need to construct [1, 1/2, 1/3, 1/4, ...], but this is a special case where one should be able to do a bit better than a general algorithm (there is a slightly-less than naive _gr_nmod_vec_reciprocals which isn't really optimized).

[vneiger]

This is a nice speedup. Do you have an application?

This appears in functions for Cauchy / Cauchy-like matrices, but I'm not sure about any plans for such structured linear algebra being in FLINT in the near future.

Recently I needed this when writing draft code for rational reconstruction, which I would like to add to FLINT in the somehow-near future. More precisely this was the Cauchy interpolation case of rational reconstruction (as in [von zur Gathen and Gerhard, Section 5.8]), where I had to invert a bunch of evaluations of a polynomial. This might also be useful more generally for related algorithms that compute with 2 x 2 univariate polynomial matrices like the half-gcd or Padé approximation, when multiplications are done through FFT evaluation-interpolation.

[edgarcosta]

This is also useful for BSGS algorithms. Many times one needs to patch the output of the main loop with an inverse coming from a known sequence apriori.

[vneiger]

This is also useful for BSGS algorithms. Many times one needs to patch the output of the main loop with an inverse coming from a known sequence apriori.

Thanks, that's good to know.

I don't have more to add to this PR, and regarding the initial PR message, I am not sure about adding a function like

void nmod_inv2(ulong * inv_a1, ulong * inv_a2, ulong a1, ulong a2, nmod_t mod)

because I don't have a use case in mind. Unless someone has suggestions, for me this is ready for merge.

[fredrik-johansson]

Thanks!