SLIDE 1
Ore Polynomials in Sage
Manuel Kauers
joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU.
1
Ore Polynomials in Sage
Manuel Kauers
joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU.
1
Ore Polynomials in Sage Manuel Kauers joint work with Maximilian - - PowerPoint PPT Presentation
Ore Polynomials in Sage Manuel Kauers joint work with Maximilian - - PowerPoint PPT Presentation
Ore Polynomials in Sage Manuel Kauers joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU. 1 Ore Polynomials in Sage Manuel Kauers joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU. 1 Sage Mathematica
SLIDE 2
SLIDE 3
Sage Mathematica computation speed programming speed
2
SLIDE 4
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 10 20 30 40 time
3
SLIDE 5
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 10 20 30 40 time
3
SLIDE 6
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 7
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 8
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 9
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 10
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 11
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 12
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 13
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 14
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 15
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 35 time
3
SLIDE 16
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 time
3
SLIDE 17
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 time
3
SLIDE 18
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 time
3
SLIDE 19
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 time
3
SLIDE 20
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 time
3
SLIDE 21
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 30 time
3
SLIDE 22
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 time
3
SLIDE 23
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 time
3
SLIDE 24
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 time
3
SLIDE 25
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 time
3
SLIDE 26
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 25 time
3
SLIDE 27
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 time
3
SLIDE 28
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 time
3
SLIDE 29
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 time
3
SLIDE 30
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 time
3
SLIDE 31
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 20 time
3
SLIDE 32
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 time
3
SLIDE 33
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 time
3
SLIDE 34
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 time
3
SLIDE 35
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 5 10 15 time
3
SLIDE 36
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 10 12 14 time
3
SLIDE 37
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 10 12 14 time
3
SLIDE 38
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 10 12 14 time
3
SLIDE 39
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 10 12 time
3
SLIDE 40
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 10 12 time
3
SLIDE 41
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 10 12 time
3
SLIDE 42
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 10 time
3
SLIDE 43
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 10 time
3
SLIDE 44
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 time
3
SLIDE 45
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 time
3
SLIDE 46
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 time
3
SLIDE 47
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 2 4 6 8 time
3
SLIDE 48
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 1 2 3 4 5 6 7 time
3
SLIDE 49
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 1 2 3 4 5 6 7 time
3
SLIDE 50
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 1 2 3 4 5 6 time
3
SLIDE 51
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 1 2 3 4 5 6 time
3
SLIDE 52
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 1 2 3 4 5 time
3
SLIDE 53
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 1 2 3 4 5 time
3
SLIDE 54
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 1 2 3 4 time
3
SLIDE 55
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 1 2 3 4 time
3
SLIDE 56
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 1 2 3 4 time
3
SLIDE 57
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.5 1.0 1.5 2.0 2.5 3.0 3.5 time
3
SLIDE 58
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.5 1.0 1.5 2.0 2.5 3.0 time
3
SLIDE 59
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.5 1.0 1.5 2.0 2.5 3.0 time
3
SLIDE 60
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.5 1.0 1.5 2.0 2.5 time
3
SLIDE 61
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.5 1.0 1.5 2.0 2.5 time
3
SLIDE 62
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.5 1.0 1.5 2.0 time
3
SLIDE 63
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.5 1.0 1.5 2.0 time
3
SLIDE 64
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.5 1.0 1.5 time
3
SLIDE 65
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.5 1.0 1.5 time
3
SLIDE 66
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.2 0.4 0.6 0.8 1.0 1.2 1.4 time
3
SLIDE 67
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.2 0.4 0.6 0.8 1.0 1.2 time
3
SLIDE 68
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.2 0.4 0.6 0.8 1.0 time
3
SLIDE 69
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.2 0.4 0.6 0.8 1.0 time
3
SLIDE 70
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.2 0.4 0.6 0.8 time
3
SLIDE 71
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.1 0.2 0.3 0.4 0.5 0.6 0.7 time
3
SLIDE 72
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.1 0.2 0.3 0.4 0.5 0.6 time
3
SLIDE 73
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.1 0.2 0.3 0.4 0.5 time
3
SLIDE 74
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.1 0.2 0.3 0.4 0.5 time
3
SLIDE 75
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.1 0.2 0.3 0.4 time
3
SLIDE 76
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.05 0.10 0.15 0.20 0.25 0.30 0.35 time
3
SLIDE 77
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.05 0.10 0.15 0.20 0.25 0.30 time
3
SLIDE 78
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.05 0.10 0.15 0.20 0.25 time
3
SLIDE 79
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.05 0.10 0.15 0.20 time
3
SLIDE 80
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.05 0.10 0.15 time
3
SLIDE 81
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.02 0.04 0.06 0.08 0.10 0.12 0.14 time
3
SLIDE 82
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.02 0.04 0.06 0.08 0.10 0.12 time
3
SLIDE 83
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.02 0.04 0.06 0.08 0.10 time
3
SLIDE 84
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.02 0.04 0.06 0.08 time
3
SLIDE 85
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.01 0.02 0.03 0.04 0.05 0.06 time
3
SLIDE 86
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.01 0.02 0.03 0.04 0.05 time
3
SLIDE 87
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.005 0.010 0.015 0.020 0.025 0.030 0.035 time
3
SLIDE 88
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.005 0.010 0.015 0.020 0.025 time
3
SLIDE 89
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.005 0.010 0.015 0.020 time
3
SLIDE 90
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.005 0.010 0.015 time
3
SLIDE 91
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.002 0.004 0.006 0.008 0.010 time
3
SLIDE 92
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.002 0.004 0.006 0.008 time
3
SLIDE 93
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.001 0.002 0.003 0.004 0.005 time
3
SLIDE 94
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 0.0035 time
3
SLIDE 95
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.0005 0.0010 0.0015 0.0020 time
3
SLIDE 96
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.0002 0.0004 0.0006 0.0008 0.0010 0.0012 0.0014 time
3
SLIDE 97
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.0002 0.0004 0.0006 0.0008 time
3
SLIDE 98
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.0001 0.0002 0.0003 0.0004 time
3
SLIDE 99
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.00005 0.00010 0.00015 0.00020 0.00025 time
3
SLIDE 100
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.00005 0.00010 0.00015 time
3
SLIDE 101
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.00002 0.00004 0.00006 0.00008 0.00010 0.00012 time
3
SLIDE 102
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.00002 0.00004 0.00006 0.00008 0.0001 time
3
SLIDE 103
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.00002 0.00004 0.00006 0.00008 0.0001 time
3
SLIDE 104
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.00002 0.00004 0.00006 0.00008 0.0001 time
3
SLIDE 105
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.00002 0.00004 0.00006 0.00008 0.0001 time
3
SLIDE 106
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.00002 0.00004 0.00006 0.00008 0.0001 time 670000x
3
SLIDE 107
A Computation Speed Comparison
Multiplication time for dense polynomials with integer coefficients in Mathematica and Sage (Flint)
2000 4000 6000 8000 10 000 deg 0.00002 0.00004 0.00006 0.00008 0.0001 time 74000x 670000x
3
SLIDE 108
A Programming Speed Comparison
4
SLIDE 109
A ///////////////// Code Length Programming Speed Comparison
4
SLIDE 110
A ///////////////// Code Length Programming Speed Comparison
Find a polynomial solution of prescribed degree of a given recurrence.
4
SLIDE 111
A ///////////////// Code Length Programming Speed Comparison
Find a polynomial solution of prescribed degree of a given recurrence.
4
SLIDE 112
A ///////////////// Code Length Programming Speed Comparison
Find a polynomial solution of prescribed degree of a given recurrence.
4
SLIDE 113
Ore Polynomials in Sage
Manuel Kauers
joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU.
5
SLIDE 114
Ore Polynomials in Sage
Manuel Kauers
joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU.
5
SLIDE 115
◮ Idea: Represent a “function” or a “sequence” f by an
equation of which it is a solution.
6
SLIDE 116
◮ Idea: Represent a “function” or a “sequence” f by an
equation of which it is a solution.
◮ Represent an “equation” for f by an “operator” which maps
this function to zero.
6
SLIDE 117
◮ Idea: Represent a “function” or a “sequence” f by an
equation of which it is a solution.
◮ Represent an “equation” for f by an “operator” which maps
this function to zero.
◮ Examples:
◮ e2x is killed by L = D − 2, where D =
d dx.
6
SLIDE 118
◮ Idea: Represent a “function” or a “sequence” f by an
equation of which it is a solution.
◮ Represent an “equation” for f by an “operator” which maps
this function to zero.
◮ Examples:
◮ e2x is killed by L = D − 2, where D =
d dx.
◮ 2n is killed by L = E − 2, where E ≡ n n + 1 6
SLIDE 119
◮ Idea: Represent a “function” or a “sequence” f by an
equation of which it is a solution.
◮ Represent an “equation” for f by an “operator” which maps
this function to zero.
◮ Examples:
◮ log(1 − √x) is killed by
L = 2x(x − 1)D3 + (7x − 3)D2 + 3D.
6
SLIDE 120
◮ Idea: Represent a “function” or a “sequence” f by an
equation of which it is a solution.
◮ Represent an “equation” for f by an “operator” which maps
this function to zero.
◮ Examples:
◮ log(1 − √x) is killed by
L = 2x(x − 1)D3 + (7x − 3)D2 + 3D.
◮
n
- k=1
k
- i=1
1 i + k is killed by L = (2n + 7)(n + 4)E3 − (6n2 − 41n − 71)E2 + (6n2 + 37n + 58)E − (n + 3)(2n + 5)
6
SLIDE 121
◮ These operators are called Ore Polynomials.
7
SLIDE 122
◮ These operators are called Ore Polynomials. ◮ They live in an Ore Algebra.
7
SLIDE 123
◮ These operators are called Ore Polynomials. ◮ They live in an Ore Algebra. ◮ They act on a “Function Space.”
7
SLIDE 124
Definition (Ore Algebra)
8
SLIDE 125
Definition (Ore Algebra)
◮ Let R be a ring
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SLIDE 126
Definition (Ore Algebra)
◮ Let R be a ring ◮ Let σ: R → R be an endomorphism, i.e.,
σ(a + b) = σ(a) + σ(b) and σ(ab) = σ(a)σ(b)
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SLIDE 127
Definition (Ore Algebra)
◮ Let R be a ring ◮ Let σ: R → R be an endomorphism ◮ Let δ: R → R be a “σ-derivation”, i.e.,
δ(a + b) = δ(a) + δ(b) and δ(ab) = δ(a)b + σ(a)δ(b)
8
SLIDE 128
Definition (Ore Algebra)
◮ Let R be a ring ◮ Let σ: R → R be an endomorphism ◮ Let δ: R → R be a “σ-derivation” ◮ Let A = R[∂] be the set of all univariate polynomials in ∂
with coefficients in R.
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SLIDE 129
Definition (Ore Algebra)
◮ Let R be a ring ◮ Let σ: R → R be an endomorphism ◮ Let δ: R → R be a “σ-derivation” ◮ Let A = R[∂] be the set of all univariate polynomials in ∂
with coefficients in R.
◮ Let + be the usual polynomial addition.
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SLIDE 130
Definition (Ore Algebra)
◮ Let R be a ring ◮ Let σ: R → R be an endomorphism ◮ Let δ: R → R be a “σ-derivation” ◮ Let A = R[∂] be the set of all univariate polynomials in ∂
with coefficients in R.
◮ Let + be the usual polynomial addition. ◮ Let · be the unique (noncommutative) multiplication in A
which extends the multiplication in R and satisfies ∂a = σ(a)∂ + δ(a) for all a ∈ R.
8
SLIDE 131
Definition (Ore Algebra)
◮ Let R be a ring ◮ Let σ: R → R be an endomorphism ◮ Let δ: R → R be a “σ-derivation” ◮ Let A = R[∂] be the set of all univariate polynomials in ∂
with coefficients in R.
◮ Let + be the usual polynomial addition. ◮ Let · be the unique (noncommutative) multiplication in A
which extends the multiplication in R and satisfies ∂a = σ(a)∂ + δ(a) for all a ∈ R.
◮ Then A together with this + and · is called an Ore Algebra.
8
SLIDE 132
Examples:
9
SLIDE 133
Examples:
◮ For R = Q[x], σ = id, δ = d dx, we have that
A = R[∂] = Q[x][∂] is the ring of linear differential operators with polynomial coefficients.
9
SLIDE 134
Examples:
◮ For R = Q[x], σ = id, δ = d dx, we have that
A = R[∂] = Q[x][∂] is the ring of linear differential operators with polynomial coefficients.
◮ For R = Q[n], σ: R → R defined by σ(c) = c for all c ∈ Q
and σ(n) = n + 1, and δ = 0, we have that A = R[∂] = Q[n][∂] is the ring of linear recurrence operators with polynomial coefficients.
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Examples:
◮ For R = Q[x], σ = id, δ = d dx, we have that
A = R[∂] = Q[x][∂] is the ring of linear differential operators with polynomial coefficients.
◮ For R = Q[n], σ: R → R defined by σ(c) = c for all c ∈ Q
and σ(n) = n + 1, and δ = 0, we have that A = R[∂] = Q[n][∂] is the ring of linear recurrence operators with polynomial coefficients.
◮ There are other examples. . .
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Ore algebras A = R[∂] can act on an R-module F via a suitable “interpretation” of the algebra’s generator ∂.
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Ore algebras A = R[∂] can act on an R-module F via a suitable “interpretation” of the algebra’s generator ∂. We want the action A × F → F, (a, f) → a · f
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Ore algebras A = R[∂] can act on an R-module F via a suitable “interpretation” of the algebra’s generator ∂. We want the action A × F → F, (a, f) → a · f to be such that (a + b) · f = a · f + b · f (ab) · f = a · (b · f) a · (f + g) = a · f + a · g for all a, b ∈ A, f, g ∈ F.
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Ore algebras A = R[∂] can act on an R-module F via a suitable “interpretation” of the algebra’s generator ∂. Examples:
◮ The Ore algebra A = Q[x][Dx] acts on C∞(C, C) via
(a0(x) + a1(x)Dx + · · · + ar(x)Dr
x) · f(z)
= a0(z)f(z) + a1(z)f′(z) + · · · + ar(z)f(r)(z).
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Ore algebras A = R[∂] can act on an R-module F via a suitable “interpretation” of the algebra’s generator ∂. Examples:
◮ The Ore algebra A = Q[x][Dx] acts on C∞(C, C) via
(a0(x) + a1(x)Dx + · · · + ar(x)Dr
x) · f(z)
= a0(z)f(z) + a1(z)f′(z) + · · · + ar(z)f(r)(z).
◮ The Ore algebra A = Q[n][En] acts on the space CN via
(a0(n) + a1(n)En + · · · + ar(n)Er
n) · f(n)
= a0(n)f(n) + a1(n)f(n + 1) + · · · + ar(n)f(n + r).
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◮ The annihilator of f ∈ F is defined as
ann(f) :=
- a ∈ R[∂] : a · f = 0
- .
It is a subset of R[∂]. Its elements are called annihilating
- perators for f.
◮ The solution space of a ∈ R[∂] is defined as
V (a) :=
- f ∈ F : a · f = 0
- .
It is a subset of F. Its elements are called solutions of a.
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Want: Obtain information about f by doing computations in R[∂].
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Want: Obtain information about f by doing computations in R[∂]. Don’t want: do these computations by hand.
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Want: Obtain information about f by doing computations in R[∂]. Don’t want: do these computations by hand. Instead: have them done by a computer algebra package.
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Ore Polynomials in Sage
Manuel Kauers
joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU.
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Ore Polynomials /////// elsewhere in Sage
Manuel Kauers
joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU.
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◮ For Mathematica:
◮ univariate: Mallinger’s package ◮ multivariate: Koutschan’s package.
◮ For Maple:
◮ univariate: gfun by Salvy/Zimmermann
- r OreTools by Abramov et al.
◮ multivariate: mgfun by Chyzak 14
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Ore Polynomials /////// elsewhere in Sage
Manuel Kauers
joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU.
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Ore Polynomials in Sage
Manuel Kauers
joint work with Maximilian Jaroschek and Fredrik Johansson RISC, JKU.
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Key Features:
◮ Construction of Ore algebras and Ore polynomials ◮ GCRD, Closure properties, Desingularization ◮ Various types of solutions ◮ Guessing
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Key Features:
◮ Construction of Ore algebras and Ore polynomials ◮ GCRD, Closure properties, Desingularization ◮ Various types of solutions ◮ Guessing
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Algebra RingElement OreAlgebra OreOperator UnivariateOreOperator MultivariateOreOperator UnivariateOreOperatorOverUnivariateRing UnivariateDifferentialOperatorOverUnivariateRing . . . . . . . . . . . . 17
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Key Features:
◮ Construction of Ore algebras and Ore polynomials ◮ GCRD, Closure properties, Desingularization ◮ Various types of solutions ◮ Guessing
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Key Features:
◮ Construction of Ore algebras and Ore polynomials ◮ GCRD, Closure properties, Desingularization ◮ Various types of solutions ◮ Guessing
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Runtime for computing the least common left multiple of two random operators from Z[x][Dx] of order n and degree 2n
◮ in Mathematica (i.e., Koutschan’s code) ◮ in Sage (i.e., our code)
5 10 15 20 n 100 200 300 400 500 600 700 time
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Key Features:
◮ Construction of Ore algebras and Ore polynomials ◮ GCRD, Closure properties, Desingularization ◮ Various types of solutions ◮ Guessing
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Key Features:
◮ Construction of Ore algebras and Ore polynomials ◮ GCRD, Closure properties, Desingularization ◮ Various types of solutions ◮ Guessing
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Key Features:
◮ Construction of Ore algebras and Ore polynomials ◮ GCRD, Closure properties, Desingularization ◮ Various types of solutions ◮ Guessing
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Key Features:
◮ Construction of Ore algebras and Ore polynomials ◮ GCRD, Closure properties, Desingularization ◮ Various types of solutions ◮ Guessing ◮ Built-in code for polynomial matrices
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A typical linear algebra problem arising in this context: compute a nullspace vector for the following matrix over Z[x]: Degrees:
200 400 600 800 1000
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A typical linear algebra problem arising in this context: compute a nullspace vector for the following matrix over Z[x]: Total matrix size (number of monomials) during the elimination:
100 200 300 400 500 600 5.0 106 1.0 107 1.5 107 2.0 107 2.5 107 3.0 107 3.5 107
green: naive code, blue: our code, red: Axel Riese’s Mathematica code.
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Key Features:
◮ Construction of Ore algebras and Ore polynomials ◮ GCRD, Closure properties, Desingularization ◮ Various types of solutions ◮ Guessing ◮ Built-in code for polynomial matrices
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To do:
◮ operator factorization and fast arithmetic ◮ arbitrary precision evaluation of analytic D-finite functions ◮ construction of an annihilator from an expression ◮ the multivariate case, incl. Gr¨
- bner bases and creative
telescoping.
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