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update info for real_roots in the documentation
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Fredrik Johansson
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1 changed files with 64 additions and 30 deletions
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@ -130,6 +130,10 @@ The following functions (specified by an integer code) are implemented:
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* 1 - `\sin(x)`
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* 2 - `\sin(x^2)`
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* 3 - `\sin(1/x)`
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* 4 - `\operatorname{Ai}(x)` (Airy function)
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* 5 - `\operatorname{Ai}'(x)` (Airy function)
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* 6 - `\operatorname{Bi}(x)` (Airy function)
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* 7 - `\operatorname{Bi}'(x)` (Airy function)
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The following options are available:
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@ -153,78 +157,108 @@ on the critical line, and guarantees that no roots are missed
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(there are more efficient ways to do this, but it is a nice example)::
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> build/examples/real_roots 0 0.0 50.0 -verbose
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interval: 25 +/- 25
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interval: [0, 50]
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maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
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found isolated root in: 14.12353515625 +/- 0.012207
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found isolated root in: 21.0205078125 +/- 0.024414
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found isolated root in: 25.0244140625 +/- 0.024414
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found isolated root in: 30.43212890625 +/- 0.012207
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found isolated root in: 32.9345703125 +/- 0.024414
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found isolated root in: 37.5732421875 +/- 0.024414
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found isolated root in: 40.9423828125 +/- 0.024414
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found isolated root in: 43.32275390625 +/- 0.012207
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found isolated root in: 48.01025390625 +/- 0.012207
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found isolated root in: 49.76806640625 +/- 0.012207
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found isolated root in: [14.111328125, 14.16015625]
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found isolated root in: [20.99609375, 21.044921875]
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found isolated root in: [25, 25.048828125]
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found isolated root in: [30.419921875, 30.4443359375]
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found isolated root in: [32.91015625, 32.958984375]
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found isolated root in: [37.548828125, 37.59765625]
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found isolated root in: [40.91796875, 40.966796875]
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found isolated root in: [43.310546875, 43.3349609375]
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found isolated root in: [47.998046875, 48.0224609375]
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found isolated root in: [49.755859375, 49.7802734375]
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---------------------------------------------------------------
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Found roots: 10
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Subintervals possibly containing undetected roots: 0
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Function evaluations: 3425
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cpu/wall(s): 1.22 1.229
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virt/peak/res/peak(MB): 20.63 20.66 2.23 2.23
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Function evaluations: 3058
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cpu/wall(s): 0.202 0.202
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virt/peak/res/peak(MB): 26.12 26.14 2.76 2.76
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Find just one root and refine it to approximately 75 digits::
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> build/examples/real_roots 0 0.0 50.0 -maxfound 1 -refine 75
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interval: 25 +/- 25
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interval: [0, 50]
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maxdepth = 30, maxeval = 100000, maxfound = 1, low_prec = 30
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refined root:
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14.134725141734693790457251983562470270784257115699243175685567460149963429809 +/- 8.4532e-81
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refined root (0/8):
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[14.134725141734693790457251983562470270784257115699243175685567460149963429809 +/- 2.57e-76]
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---------------------------------------------------------------
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Found roots: 1
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Subintervals possibly containing undetected roots: 8
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Function evaluations: 992
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cpu/wall(s): 0.41 0.415
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virt/peak/res/peak(MB): 20.76 20.76 2.23 2.23
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Subintervals possibly containing undetected roots: 7
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Function evaluations: 761
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cpu/wall(s): 0.055 0.056
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virt/peak/res/peak(MB): 26.12 26.14 2.75 2.75
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Find the first few roots of an Airy function and refine them to 50 digits each::
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> build/examples/real_roots 4 -10 0 -refine 50
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interval: [-10, 0]
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maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
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refined root (0/6):
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[-9.022650853340980380158190839880089256524677535156083 +/- 4.85e-52]
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refined root (1/6):
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[-7.944133587120853123138280555798268532140674396972215 +/- 1.92e-52]
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refined root (2/6):
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[-6.786708090071758998780246384496176966053882477393494 +/- 3.84e-52]
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refined root (3/6):
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[-5.520559828095551059129855512931293573797214280617525 +/- 1.05e-52]
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refined root (4/6):
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[-4.087949444130970616636988701457391060224764699108530 +/- 2.46e-52]
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refined root (5/6):
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[-2.338107410459767038489197252446735440638540145672388 +/- 1.48e-52]
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---------------------------------------------------------------
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Found roots: 6
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Subintervals possibly containing undetected roots: 0
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Function evaluations: 200
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cpu/wall(s): 0.003 0.003
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virt/peak/res/peak(MB): 26.12 26.14 2.24 2.24
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Find roots of `\sin(x^2)` on `(0,100)`. The algorithm cannot isolate
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the root at `x = 0` (it is at the endpoint of the interval, and in any
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case a root of multiplicity higher than one). The failure is reported::
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> build/examples/real_roots 2 0 100
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interval: 50 +/- 50
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interval: [0, 100]
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maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
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---------------------------------------------------------------
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Found roots: 3183
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Subintervals possibly containing undetected roots: 1
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Function evaluations: 34058
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cpu/wall(s): 0.26 0.263
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virt/peak/res/peak(MB): 20.73 20.76 1.72 1.72
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cpu/wall(s): 0.032 0.032
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virt/peak/res/peak(MB): 26.32 26.37 2.04 2.04
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This does not miss any roots::
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> build/examples/real_roots 2 1 100
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interval: 50.5 +/- 49.5
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interval: [1, 100]
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maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
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---------------------------------------------------------------
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Found roots: 3183
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Subintervals possibly containing undetected roots: 0
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Function evaluations: 34039
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cpu/wall(s): 0.26 0.266
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virt/peak/res/peak(MB): 20.73 20.76 1.70 1.70
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cpu/wall(s): 0.023 0.023
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virt/peak/res/peak(MB): 26.32 26.37 2.01 2.01
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Looking for roots of `\sin(1/x)` on `(0,1)`, the algorithm finds many roots,
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but will never find all of them since there are infinitely many::
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> build/examples/real_roots 3 0.0 1.0
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interval: 0.5 +/- 0.5
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interval: [0, 1]
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maxdepth = 30, maxeval = 100000, maxfound = 100000, low_prec = 30
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---------------------------------------------------------------
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Found roots: 10198
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Subintervals possibly containing undetected roots: 24695
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Function evaluations: 202587
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cpu/wall(s): 1.73 1.731
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virt/peak/res/peak(MB): 21.84 22.89 2.76 2.76
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cpu/wall(s): 0.171 0.171
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virt/peak/res/peak(MB): 28.39 30.38 4.05 4.05
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Remark: the program always computes rigorous containing intervals
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for the roots, but the accuracy after refinement could be less than *d* digits.
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