Revisiting the subject of approximating roots of polynomials.

In an earlier posting, I had written about an approach, for how to find approximations of the roots of polynomials, of an arbitrary degree, but in such away, also to find all the complex roots. (:1)

But with such strategies, there are issues. One concept was, that a home-grown search algorithm would get close to the actual root. Next, polishing would make the result more accurate. And then, an augmented division would be computed, which is also referred to as “Deflation”, resulting in a deflated polynomial, as many times as the original polynomial’s degree, minus one.

Pondering this issue today, I realized that there was still a conceptual weakness in that approach, that being, the fact that some small amount of error is tolerated in the first root found, so that each successive deflated polynomial contains progressively greater degrees of error. What effectively happens next is, that accurate roots are found, of increasingly inaccurate polynomials, and, that there appeared to be few ways to detect and correct the resulting errors, in roots found afterwards. Theoretically, this problem could progress to the point, where doubt is evoked, in whether or not roots found later, were even roots of the original polynomial, since by that time, the object which the roots are being found of, is no longer that original polynomial.

(Update 6/08/2020, 18h35… )

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Why some Linux devs still use spinlocks.

There is An interesting article, on another BB, which explains what some of the effects are, of spinlocks under Linux. But, given that at least under Linux, spinlocks are regarded as ‘bad code’ and constitute ‘a busy-wait loop’, I think that that posting does not explain the subject very well of why, in certain cases, they are still used.

A similar approach to acquiring an exclusive lock – that can exist as an abstract object, which does not really need to be linked to the resource that it’s designed to protect – could be programmed by a novice programmer, if there were not already a standard implementation somewhere. That other approach could read:

 

from time import sleep
import inspect
import thread

def acquire( obj ):
    assert inspect.isclass(obj)
    myID = thread.get_ident()
    if obj.owner is None:
        # Replace .owner with a unique attribute name.
        # In C, .owner needs to be declared volatile.
        # Compiler: Don't optimize out any reads.
        obj.owner = myID
    while obj.owner != myID:
        if obj.owner == 0:
            obj.owner = myID
        sleep(0.02)

def release( obj ):
    assert inspect.isclass(obj)
    if obj.owner is None:
        # In C, .owner needs to be declared volatile.
        obj.owner = 0
        return
    if obj.owner == thread.get_ident():
        obj.owner = 0

 

(Code updated 2/26/2020, 8h50…

I should add that, in the example above, Lines 8-12 will just happen to work under Python, because multi-threading under Python is in fact single-threaded, and governed by a Global Interpreter Lock. The semantics that take place between Lines 12 and 13 would break in the intended case, where this would just be pseudo-code, and where several clock cycles elapse, so that the ‘Truth’ which Line 13 tests may not last, just because another thread would have added a corresponding attribute on Line 12. OTOH, adding another sleep() statement is unnecessary, as those semantics are not available, outside Python.

In the same vein, if the above code is ported to C, then what matters is the fact that in the current thread, several clock-cycles elapse between Lines 14 and 15. Within those clock cycles, other threads could also read that obj.owner == 0, and assign themselves. Therefore, the only sleep() instruction is dual-purpose. Its duration might exceed the amount of time the cache was slowed down to, to execute multiple assignments to the same memory location. After that, one out of possibly several threads would have been the last, to assign themselves. And then, that thread would also be the one that breaks out of the loop.

However, there is more that could happen between Lines 14 and 15 above, than cache-inefficiency. An O/S scheduler could cause a context-switch, and the current thread could be out of action for some time. If that amount of time exceeds 20 milliseconds, then the current thread would assign itself afterwards, even though another thread has already passed the retest of Line 13, and assumes that it owns the Mutex. Therefore, better suggested pseudo-code is offered at the end of this posting…

)


 

This pseudo-code has another weakness. It assumes that every time the resource is not free, the program can afford to wait for more than 20 milliseconds, before re-attempting to acquire it. The problem can crop up, that the current process or thread must acquire the resource, within microseconds, or even within nanoseconds, after it has become free. And for such a short period of time, there is no way that the O/S can reschedule the current CPU core, to perform any useful amount of work on another process or thread. Therefore, in such cases, a busy-wait loop becomes The Alternative.

I suppose that another reason, for which some people have used spinlocks, is just due to bad code design.


 

Note: The subject has long been debated, of what the timer interrupt frequency should be. According to kernel v2.4 or earlier, it was 100Hz. According to kernel v2.6 and later, it has been 1000Hz. Therefore, in principle, an interval of 2 milliseconds could be inserted above (in case the resource had not become free). However, I don’t really think that doing so would change the nature of the problem.

Explanation: One of the (higher-priority, hardware) interrupt requests consists of nothing but a steady pulse-train, from a programmable oscillator. Even though the kernel can set its frequency over a wide range, this frequency is known not to be terribly accurate. Therefore, assuming that the machine has the software installed, that provides ‘strict kernel marshalling’, every time this interrupt fires, the system time is advanced by a floating-point number, that is also adjusted over a period of hours and days, so that the system time keeps in-sync with an external time reference. Under Debian Linux, the package which installs that is named ‘ntp’.

 

There exist a few other tricks to understand, about how, in practice, to force an Operating System which is not an RTOS, to behave like a Real-Time O/S. I’ve known a person who understood computers on a theoretical basis, and who had studied Real-Time Operating Systems. That’s a complicated subject. But, given the reality that his operating system was not a Real-Time O/S, he was somewhat stumped by why then, it was able to play a video-clip at 24 FPS…

(Updated on 3/12/2020, 13h20 …)

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Inkscape Extension ‘svg2tikz’ revisited.

In this earlier posting, I had written about the low-performing 3rd-party Inkscape Extension known as ‘svg2tikz’. Nevertheless, this extension may prove useful to some users, who wish to import an arbitrary document-type into Inkscape – preferably vector-based – and who wish to convert that into LaTeX in some way. And it seems that, even though this project was abandoned some time ago, work has slowly begun to resume on its source-code. And so, I should also fine-tune some of the earlier commentary I had made about this extension.

First off, there is an important detail about how to compile and install this extension, which its devs fail to point out anywhere. It needs to be built and installed, using Python 3, while many Linux computers still default to Python 2.7. Therefore, the commands to build and install it are:

 


$ python3 setup.py build
$ su
(...)
# python3 setup.py install

 

If one neglects this detail, then Unicode support is left out, and usually, SVG Files etc., will contain some Unicode characters. Further, as the Github comment states, while the importing of raster-based images is now supported, their import as Base-64 encoded, inline data is not. Therefore, within Inkscape, for example if a PDF File is being imported, the option needs to be unchecked, to ‘Embed’ graphics. And when Saving a Copy to TiKz Format, the option should also be unchecked, to ‘Indent Groups’.

But this last detail leads me to an important, additional observation. I have always known that the export of Text with the Figure has been dodgy. But lately, either because I’ve become more observant, or, because the behaviour of the latest version of the extension has improved, I’ve noticed what, exactly, goes wrong with Exporting Text along with the Figure.

(Updated 2/11/2020, 1h05 … )

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Dealing with a picture frame that freezes.

I recently bought myself a (1920×1080 pixel) digital picture frame, that had rave reviews among other customers, but that began the habit of freezing after about 12 hours of continuous operation, with my JPEG Images on its SD Card.

This could signal that there is some sort of hardware error, including in the internal logic, or of the SD Card itself. And one of the steps which I took to troubleshoot this problem was, to try saving the ‘.jpg’ Files to different SD Cards, and once even, to save those pictures to a USB Key, since the picture frame in question accepts a USB Memory Stick. All these efforts resulted in the same behaviour. This brought me back to the problem, that there could be some sort of data-error, i.e., of the JPEG Files in question already being corrupted, as they were stored on my hard drives. I had known of this possibility, and so I already tried the following:

 


find . -type f -name '*.jpg' | jpeginfo -c -f - | grep -v 'OK'

 

Note: To run this command requires that the Debian package ‘jpeginfo’ be installed, which was not installed out-of-the-box on my computer.

This is the Linux way to find JPEG Files that Linux deems to be corrupted. But, aside from some trivial issues which this command found, and which I was easily able to correct, Linux deemed all the relevant JPEG Files to be clean.

And this is where my thinking became more difficult. I was not looking for a quick reimbursement for the picture frame, and continued to operate on the assumption that mine was working as well, as the frames that other users had given such good reviews for. And so, another type of problem came to my attention, which I had run in to previously, in a way that I could be sure of. Sometimes Linux will find media files to be ‘OK’, that non-Linux software (or embedded firmware) deems to be unacceptable. And with my collection of 253 photos, all it would take is one such photo, which, as soon as the frame selected it to be viewed, could still have caused the frame to crash.

(Updated 1/16/2020, 17h15 … )

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