Skip to content

Commit 6937a1a

Browse files
ratwolfzeroratwolfzero
committed
modified: README.md
1 parent e347b4c commit 6937a1a

File tree

1 file changed

+30
-30
lines changed

1 file changed

+30
-30
lines changed

‎README.md‎

Lines changed: 30 additions & 30 deletions
Original file line numberDiff line numberDiff line change
@@ -1,6 +1,6 @@
11
<!--markdownlint-disable MD033 MD041-->
22

3-
# Hopalong Attractor: Pixel-Based Density Approximation
3+
# Pixel-Based Density Approximation: Hopalong Attractor (Python)
44

55
![Header Image](./Documentation/Figure_0.png)
66

@@ -108,7 +108,7 @@ Just-in-time (JIT) compilation is applied and supported by a low-complexity code
108108

109109
For details regarding two-pass strategy, see [Two-Pass Approach](#two-pass-approach)
110110

111-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
111+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
112112

113113
---
114114

@@ -162,7 +162,7 @@ Otherwise, please comment out the corresponding code fragments in the import sec
162162
#calculate_and_display_resource_usage(start_time, end_time)
163163
...
164164

165-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
165+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
166166

167167
---
168168

@@ -202,7 +202,7 @@ The programs generate a visual representation of the attractor trajectory as a d
202202
**Extended Version**
203203
![Extended Version](./Documentation/Figure_4.png)
204204

205-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
205+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
206206

207207
---
208208

@@ -218,7 +218,7 @@ The programs generate a visual representation of the attractor trajectory as a d
218218

219219
Examples of outputs can be found in the "Usage" section above.
220220

221-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
221+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
222222

223223
---
224224

@@ -296,30 +296,30 @@ The images were rendered as follows:
296296

297297
**Matrix Similarity with Increasing Iterations**
298298

299-
| Parameters | Iterations | PCC | CS | SSIM |
300-
|---------------------------|------------------|----------|----------|---------|
301-
| a = -2, b = -0.33, c = 0.01 | n = 10,000 | 0.6217 | 0.6246 | 0.9462 |
302-
| a = -2, b = -0.33, c = 0.01 | n = 100,000 | 0.8453 | 0.8540 | 0.9015 |
303-
| a = -2, b = -0.33, c = 0.01 | n = 1,000,000 | 0.9204 | 0.9342 | 0.8853 |
304-
| a = -2, b = -0.33, c = 0.01 | n = 200,000,000 | 0.9712 | 0.9766 | 0.9520 |
299+
| Parameters | Iterations | PCC | CS | SSIM |
300+
| --------------------------- | --------------- | ------ | ------ | ------ |
301+
| a = -2, b = -0.33, c = 0.01 | n = 10,000 | 0.6217 | 0.6246 | 0.9462 |
302+
| a = -2, b = -0.33, c = 0.01 | n = 100,000 | 0.8453 | 0.8540 | 0.9015 |
303+
| a = -2, b = -0.33, c = 0.01 | n = 1,000,000 | 0.9204 | 0.9342 | 0.8853 |
304+
| a = -2, b = -0.33, c = 0.01 | n = 200,000,000 | 0.9712 | 0.9766 | 0.9520 |
305305

306306
**Matrix Similarity Across Parameter Sets**
307307

308-
| Parameters | Iterations | PCC | CS | SSIM |
309-
|---------------------------|------------------|----------|----------|---------|
310-
| a = -2, b = -0.33, c = 0.01 | n = 200,000,000 | 0.9712 | 0.9766 | 0.9520 |
311-
| a = -1.7, b = -0.3, c = 0.7 | n = 200,000,000 | 0.9576 | 0.9717 | 0.9293 |
312-
| a = 0.6, b = 0.5, c = 0 | n = 200,000,000 | 0.9924 | 0.9935 | 0.9789 |
313-
| a = 0.7, b = 0.4, c = 0 | n = 200,000,000 | 0.9914 | 0.9944 | 0.9280 |
314-
| a = -55, b = -1, c = -42 | n = 200,000,000 | 0.9780 | 0.9809 | 0.9877 |
308+
| Parameters | Iterations | PCC | CS | SSIM |
309+
| --------------------------- | --------------- | ------ | ------ | ------ |
310+
| a = -2, b = -0.33, c = 0.01 | n = 200,000,000 | 0.9712 | 0.9766 | 0.9520 |
311+
| a = -1.7, b = -0.3, c = 0.7 | n = 200,000,000 | 0.9576 | 0.9717 | 0.9293 |
312+
| a = 0.6, b = 0.5, c = 0 | n = 200,000,000 | 0.9924 | 0.9935 | 0.9789 |
313+
| a = 0.7, b = 0.4, c = 0 | n = 200,000,000 | 0.9914 | 0.9944 | 0.9280 |
314+
| a = -55, b = -1, c = -42 | n = 200,000,000 | 0.9780 | 0.9809 | 0.9877 |
315315

316316
**Matrix Similarity with Periodic Orbit: A 3-Cycle Example** --> [Special Constellations and Edge Cases](#special-constellations-and-edge-cases-of-the-attractor)
317317

318-
| Parameters | Iterations | PCC | CS | SSIM |
319-
|---------------------------|------------------|----------|----------|---------|
320-
| a = 5, b = 5, c = 0 | n = 1,000,000 | 1.0000 | 1.0000 | 1.0000 |
318+
| Parameters | Iterations | PCC | CS | SSIM |
319+
| ------------------- | ------------- | ------ | ------ | ------ |
320+
| a = 5, b = 5, c = 0 | n = 1,000,000 | 1.0000 | 1.0000 | 1.0000 |
321321

322-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
322+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
323323

324324
---
325325

@@ -406,7 +406,7 @@ For example:
406406

407407
This scenario is an ideal use case for the features of the extended program variant, such as pixel hit count statistics, to analyze high-density cycle behavior.
408408

409-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
409+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
410410

411411
---
412412

@@ -420,7 +420,7 @@ Note: Since user interactions with the plot window, such as zooming, panning, or
420420
plt.pause(1)
421421
plt.close(fig)
422422

423-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
423+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
424424

425425
---
426426

@@ -438,7 +438,7 @@ Dummy calls are preliminary invocations of JIT-compiled functions that prompt th
438438

439439
The parallel loop function `prange` from the Numba library is not suitable for cross-iteration dependencies, such as those encountered when iterating recursive functions. While it is possible to restructure the second pass to use prange for populating the image array, this could introduce race conditions—situations where multiple threads access and modify shared data simultaneously, leading to inconsistent or unpredictable results. Therefore, this approach was not implemented.
440440

441-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
441+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
442442

443443
### Two-Pass Approach
444444

@@ -515,7 +515,7 @@ Trade-Off: The trajectory points are computed twice — once for extent calculat
515515
# Dummy call to ensure the function is pre-compiled by the JIT compiler before it's called by the interpreter.
516516
_ = compute_trajectory_and_image(1.0, 1.0, 1.0, 2, (-1, 0, 0, 1), (2, 2))
517517

518-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
518+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
519519

520520
---
521521

@@ -551,13 +551,13 @@ No other feasible alternative one-pass methods offering substantial advantages h
551551

552552
Overall, the two-pass approach offers an excellent balance of speed, efficiency, and simplicity, making it ideal for attractor calculations that involve large number of iterations. While the trajectory points must be computed during both passes, this method successfully avoids the drawbacks associated with alternative solutions.
553553

554-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
554+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
555555

556556
---
557557

558558
## Recent Code Changes
559559

560-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
560+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
561561

562562
---
563563

@@ -613,7 +613,7 @@ ISBN-10: 3922508502, ISBN-13: 978-3922508502.
613613
6. [Python Time Module](https://docs.python.org/3/library/time.html#module-time): Time access and conversions.
614614
7. [Python Resource Module](https://docs.python.org/3/library/resource.html): Interface for getting and setting resource limits.
615615

616-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
616+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))
617617

618618
## Mathematical Appendix
619619

@@ -704,4 +704,4 @@ $$
704704

705705
[Back to Pixel-Based Density Approximation](#pixel-based-density-approximation)
706706

707-
[Back to Table of Contents](#calculate--visualize-the-hopalong-attractor-with-python)
707+
[Back to Table of Contents](Pixel-Based Density Approximation: Hopalong Attractor (Python))

0 commit comments

Comments
 (0)