Iterations of the Quadratic Map

This website (link to bookmark: http://www.pawelpilarczyk.com/quadr/) contains software and data relatd to two papers:

[1] Rigorous numerics for the quadratic family by A. Golmakani, C. E. Koudjinan, S. Luzzatto, and P. Pilarczyk
[2] Rigorous computation of escape times for parameter intervals in the quadratic map by P. Pilarczyk and S. Luzzatto

Files for download related to the paper [1]:

progcode1.zip – the source code of the program, published here under the terms of the GNU General Public License, version 3
phpcode.zip – the source code of the website (programmed in PHP), published here under the terms of the GNU Affero General Public License, version 3
computation_log.zip (91 MB) – the original computation log produced by the program, which is a basis for the results discussed in the paper and for the figures contained therein (227MB after unzipping)

Files for download related to the paper [2] (see note below on the data contained in the archive files):

progcode2.zip – the source code of the program, published here under the terms of the GNU General Public License, version 3
noChoppingTest.zip (39 MB) – computations used in Section 2 to analyze the iterated intervals at first encounter with Δ
w10data.zip (31 MB) – data from computations for six intervals: [1.4,1.5], …, [1.9,2.0] (used for Figure 6)
w4data.zip (0.7 MB) – complete computation with w=10⁻⁴ (for Figures in Section 3)
w5data.zip (5 MB) – complete computation with w=10⁻⁵ (for Figures in Section 3)
w6data.zip (41 MB) – complete computation with w=10⁻⁶ (for Figures in Section 3)
w7data.zip (273 MB) – complete computation with w=10⁻⁷ (for Figures in Section 3)
w8data_15-30.zip (453 MB) – complete computation with w=10⁻⁸ and N₀=15,…,30 (for Figures in Section 3)
w8data_31-35.zip (588 MB) – complete computation with w=10⁻⁸ and N₀=31,…,35 (for Figures in Section 3)
w8data_36-40.zip (620 MB) – complete computation with w=10⁻⁸ and N₀=36,…,40 (for Figures in Section 3)
bisectionTest.zip (0.1 MB) – computations used in Section 4 to check the effect of the number of bisection steps on the results

In most cases, instead of providing the complete ".log" files produced by the program, ".csv" files containing only the data used for the graphs are provided, extracted from the ".log" files, and the original ".log" files without the raw data are provided as ".txt" files. Additionally, ".res" files are available that contain the information on the resources used in each case, measured by David MacKenzie's time program (a timing program available in most Linux systems) on a Debian Linux 10.8 system with the Intel® Core™2 Duo Processor E8400 running at 3 GHz.

Description of the CSV files. Each CSV file name contains a capital letter indicating the type of data:

G (files "*G.csv") – "good" intervals, that is, intervals satisfying the escape condition (case P1a in [1], pg. 12),
A (files "*A.csv") – intervals abandoned due to reaching the limit of iterations (case P3 in [1], pg. 13),
C and F (files "*C.csv" and "*F.csv") – intervals with a mismatch in the sign of the derivative of c_n or fⁿ, respectively (case P2a in [1], pg. 13),
D (files "*D.csv") – cases of intervals hitting Δ but at an iterate N<N₀ or not satisfying the escape condition (case P1b in [1], pg. 12).

Each line of each CSV file contains numbers regarding specific computed intervals:

the G files: the actual number N of iterates, a lower bound on the width of the iterated interval ω_N, a lower bound on the left endpoint of the interval ω, an upper bound on the right endpoint of the interval ω, a lower bound on the width of the interval ω; for example:
22 2.20444158646 1.88489967547 1.88498738876 8.77132775368e-5
the A files: the actual number N of the iterate at which an interval ω was abandoned, a lower bound on the left endpoint of the interval ω, an upper bound on the width of the interval ω; for example:
100 1.75700000000 1.00000000001e-3
the C and F files: the actual number N of the iterate at which the problem with an interval ω was encountered, the left endpoint of the interval ω rounded to the nearest, for example:
24 1.48300000000
the D files: the actual number N of the iterate at which the interval ω_N hit Δ, the left endpoint of the interval ω rounded to the nearest, a lower bound on the width of the interval ω_N; for example:
39 1.41500000000 3.99738138690e-1

Computations

An interface is provided to running the program introduced in the paper [1]. Since some computations can be very lengthy and memory-consuming, there are some limits set up on what one can do using this interface. Please, download and compile the program to run it at your computer in order to use full capabilities of the software.

Please, fill out the form with desired values of parameters, and click the RUN button to see the result under the form. Then you are welcome to adjust some parameters, and run the computation again.
Please, note that in the short version of the form, there are only a few selected parameters shown below (thus the numbering is not contiguous). Click the button below to toggle between the short list and the long one.

1. The left endpoint of the parameter interval (obligatory argument)	`‑l`	(min 1) (max 2)
2. The right endpoint of the parameter interval (obligatory argument)	`‑r`	(min 1) (max 2)
3. The number of intervals into which the above interval is uniformly split (1)	`‑u`	(min 1)
4. The radius of the critical neighborhood, a.k.a. delta (0.001)	`‑d`
5. The binary precision of high-prec computations (300)	`‑p`	(max 10000)
6. The number of bisection steps when excluding (40)	`‑b`	(max 100)
7. The maximal allowed number of iterations at each step (200)	`‑n`	(max 10000)
8. The minimal interval fraction required to continue iterating (0)	`‑f`
9. The minimal width required for an interval, relative to the original one (10^-10)	`‑w`
10. The number of contractions encountered to stop iterating (0=off)	`‑c`
11. The maximal number of intervals to process (0 = no limit)	`‑i`	(max 100000)
12. The maximal allowed queue size (0 = no limit)	`‑q`
13. The maximal number of messages to display (0 = no limit)	`‑m`	(max 1000)
14. The maximal running time after which the program is stopped, in seconds	`‑t`	(min 1) (max 20)
15. The minimal number of iterates to reach by all to stop (0=unused)	`‑N`
16. The required minimal number of iterates to count as an 'escape'	`‑rN`
17. The required minimal image width to count as an 'escape'	`‑rW`
18. The number of decimal digits in the output of numbers	`‑o`	(max 1000)
19. The verbosity of program's output (0 = summary only, 100 = everything)	`‑v`	(max 100)

[After you hit the RUN button to run the program, the results of the computation will be displayed here after the computation has been completed.]