Matthew van Eerde's web log

  • Matthew van Eerde's web log

    Using the Speech API to convert speech to text

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    Some time ago I created a "listen.exe" tool which used SAPI's ISpRecoContext to listen to the microphone and dump any recognized text to the console.

    Today I had to debug an issue with SAPI reading from a .wav file, so I updated it to accept a listen.exe --file foo.wav argument; this consumes the audio in the .wav file instead of listening to the microphone.

    Pseudocode for the difference:

    CoCreate(ISpRecognizer);
    CoCreate(ISpStream);
    pSpStream->BindToFile(file);
    pSpRecognizer->SetInput(pSpStream);

    Also, we have to tell the ISpRecoContext that we're interested in SPEI_END_SR_STREAM events as well as SPEI_RECOGNITION events.

    Full source and binaries attached.

    A gotcha: the .wav file has to have a WAVEFORMATEX.wFormatTag = WAVE_FORMAT_PCM. If it's anything else, ISpRecoGrammar::SetDictationState fails with SPERR_UNSUPPORTED_FORMAT. Neither WAVE_FORMAT_IEEE_FLOAT nor (WAVE_FORMAT_EXTENSIBLE with SubFormat = KSDATAFORMAT_SUBTYPE_PCM) work.

  • Matthew van Eerde's web log

    More on audio buffer alignment requirements

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    I chatted in the past about how audio device alignment requirements impact the buffer size and the WASAPI alignment dance.

    There are three alignment requirements on audio buffers:

    1. The buffer size must be a multiple of WAVEFORMATEX.nBlockAlign. This allows individual audio frames to be copied around without worrying about them being cut in half and then having to glue them together at the end.
    2. KSPROPERTY_RTAUDIO_BUFFER must be a multiple of a page - that is, 4096 bytes. This allows multiply mapping the buffer into consecutive pages, which in turn simplifies memory copies where the buffer is the source or the destination. KSPROPERTY_RTAUDIO_BUFFER is for timer-driven streaming; there is an event-driven analog, KSPROPERTY_RTAUDIO_BUFFER_WITH_NOTIFICATION, which has no corresponding alignment requirement.
    3. HD Audio buffer allocations must be a multiple of 256 bytes. For timer-driven buffers, this applies to the whole buffer. For event-driven buffers, this applies to the sum of the "ping" and "pong" buffers, so the individual "ping" or "pong" buffer must be a multiple of 128 bytes.

    Consider a 5.1 16-bit 48 kHz stream playing to HD Audio hardware via KSPROPERTY_RTAUDIO_BUFFER. Where multiple alignment requirements apply, the effective alignment requirement is the least common multiple of all the applicable requirements.

    From the nBlockAlign requirement, the buffer must be a multiple of (6 * 16) / 8 bytes = 12 bytes.

    From the KSPROPERTY_RTAUDIO_BUFFER requirement, the buffer must be a multiple of PAGE_SIZE = 4096 bytes.

    From the HD Audio requirement, the buffer must be a multiple of 256 bytes (this is timer-driven, so we do not divide by 2.)

    In all, then, the buffer must be a multiple of LCM(12, 4096, 256) = 12288 bytes.

    Since WAVEFORMATEX.nAvgBytesPerSec = ((6 * 16) / 8) * 48000 = 576000 byte/sec, this corresponds to 12288 / 576000 * 1000 = 21.333 milliseconds.

  • Matthew van Eerde's web log

    Linearity of Windows volume APIs - render session and stream volumes

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    We have talked about some of the volume APIs Windows exposes. We have also talked about what it means for a volume control to be linear in magnitude, linear in power, or linear in dB. We have also talked about how to read IAudioMeterInformation and how the limiter can attenuate full-scale signals.

    The last post had a volume-linearity.exe which, when called with --signal, showed that IAudioMeterInformation is linear in amplitude.

    Today we'll look at the --stream, --channel, and --session arguments, which explore the linearity of IAudioStreamVolume, IChannelAudioVolume, and ISimpleAudioVolume respectively. Each of these modes plays a half-scale square wave, then set the volume API to various levels, and reads the resulting IAudioMeterInformation. We use a half-scale square wave to avoid running afoul of the limiter; we expect a meter reading of 0.5 when the volume is set to 1.  The graphs below have their meter readings doubled to account for the fact that we're using a half-scale square wave rather than a full-scale.

    Here's what we get for IAudioStreamVolume, graph-inated for your convenience:

    And IChannelAudioVolume:

    And ISimpleAudioVolume:

    We already know that IAudioMeterInformation is linear in amplitude. We now know that IAudioStreamVolume, IChannelAudioVolume, and ISimpleAudioVolume have a linear effect (with slope 1 and intercept 0) on IAudioMeterInformation. We infer that IAudioStreamVolume, IChannelAudioVolume, and ISimpleAudioVolume are linear in amplitude.

  • Matthew van Eerde's web log

    Nitpicking Sam Loyd - a wheel within a wheel

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    In August 1878 Sam Loyd published this mate in two and dedicated it to a friend of his named Wheeler:


    Mate in two; Black to move and mate in two; Selfmate in two; Black to move and selfmate in two

    While the mates appear to stand up, the problem position is not legal. White has three a-pawns; this implies at least three Black pieces were captured by a White pawn. But Black has fifteen pieces on the board; only one is missing!

    Looking at Black pawn captures - the b2-, c-, and d- pawns together account for three pawn captures. This seems OK at first glance since White has three pieces missing. But all the missing White pieces are pawns, and they are from the right half of the board... so they must have promoted. This implies more pawn captures to either get the Black pawns out of the way or to get the White pawns around them. (The promoted pieces could have been captured by the Black pawns, or the original pieces could have been captured in which case the promoted pieces are on the board now.)

    Finally, the h-pawns on h5 and h6 could not have got into their present position without at least one pawn capture by White, or at least two pawn captures by Black.

  • Matthew van Eerde's web log

    Mark your variadic logging function with __format_string to have PREfast catch format specifier errors

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    There are a handful of Problems (with a capital P) which occur over and over again in programming. One of them is Logging.

    It is incredibly convenient to use the variadic printf function to log strings with values of common types embedded in them:

    // spot the bug
    LOG(L"Measurement shows %lg% deviation", 100.0 * abs(expected - actual) / expected);

    However, printf is very error prone. It is very easy to use the wrong format specifier like %d instead of %Id, or to forget to escape a special character like % or \.
    In particular, the above line contains a bug.

    Static code analysis tools like PREfast are quite good at catching these kinds of errors. If my LOG macro was something like this, PREfast would catch the bug:

    #define LOG(fmt, ...) wprintf(fmt L"\n", __VA_ARGS__)

    This works because PREfast knows that the first argument to wprintf is a format string, and can match up the format specifiers with the trailing arguments and verify that they match.

    If you implement your own variadic logger function, though, PREfast doesn't necessarily know that the last explicit argument is a format specifier - you have to tell it. For example, PREfast will NOT catch format specifier issues if the LOG macro is defined like this:

    // PREfast doesn't know Format is a format string
    interface IMyLogger { virtual void Log(LPCWSTR Format, ...) = 0; };
    extern IMyLogger *g_Logger;
    #define LOG(fmt, ...) g_Logger->Log(fmt, __VA_ARGS__)

    How do you tell it? Well, let's look at the declaration of wprintf. It's in (SDK)\inc\crt\stdio.h:

    _CRTIMP __checkReturn_opt int __cdecl wprintf(__in_z __format_string const wchar_t * _Format, ...);

    The relevant part here is __format_string. So the fixed IMyLogger declaration looks like this:

    // Now PREfast can catch format specifier issues
    interface IMyLogger { virtual void Log(__format_string LPCWSTR Format, ...) = 0; };
    extern IMyLogger *g_Logger;
    #define LOG(fmt, ...) g_Logger->Log(fmt, __VA_ARGS__)

  • Matthew van Eerde's web log

    Beep sample

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    A question came in today about the Beep(...) API1 not being able to set the frequency of the beep that was generated. In order to confirm that it worked I whipped up a quick sample which would take the frequency (and duration) on the command line. Source and binaries attached.

    For fun I added the ability to pass in the frequency using Scientific pitch notation. Note that A4 is about 431 Hz using this scale, rather than the more standard 440 Hz2.

    for (int i = 1; i + 1 < argc; i += 2) {

         ULONG frequency;
         HRESULT hr = HertzFromScientificPitchNotation(argv[i], &frequency);
         if (FAILED(hr)) { return -__LINE__; }

         ULONG duration;
         hr = UlongFromString(argv[i + 1], &duration);
         if (FAILED(hr)) { return -__LINE__; }

         if (!Beep(frequency, duration)) {
             LOG(L"Beep(%u, %u) failed: GetLastError() = %u", frequency, duration, GetLastError());
             return -__LINE__;
         }
    }

    So, for example, you can play a certain well-known tune via Beep() using this command:

    >beep.exe C3 2000 G3 2000 C4 4000 E4 500 Eb4 3500 C3 500 G2 500 C3 500 G2 500 C3 500 G2 500 C3 2000

    1 More on the Beep(...) API:

    The official Beep(...) documentation

    A couple of blog posts from Larry Osterman:
    Beep Beep
    What’s up with the Beep driver in Windows 7?

    2 If you want the more standard pitch, change this line:

    double freq = 256.0;

    To this:

    double freq = 440.0 * pow(semitoneRatio, -9.0); // C4 is 9 semitones below A4

  • Matthew van Eerde's web log

    Command-line app to set the desktop wallpaper

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    Working on Windows, I find myself installing Windows a lot.

    I find that I like to change a lot of the settings that Windows offers to non-default values.  (That is, I'm picky.)

    I have a script which automates some of these things, which I add to now and again.  Some of the bits of the script are straightforward, but once in a while the tweak itself is of interest.

    One of the things I love about my work setup is the many large monitors.  So, one of the things I like to change is the desktop wallpaper image.

    Changing the desktop wallpaper required some code, which makes it "of interest."  Here's the code.

    // main.cpp

    #include <windows.h>
    #include <winuser.h>
    #include <stdio.h>

    int _cdecl wmain(int argc, LPCWSTR argv[]) {
        if (1 != argc - 1) {
            wprintf(L"expected a single argument, not %d\n", argc);
            return -__LINE__;
        }
       
        if (!SystemParametersInfo(
            SPI_SETDESKWALLPAPER,
            0,
            const_cast<LPWSTR>(argv[1]),
            SPIF_SENDCHANGE
        )) {
            DWORD dwErr = GetLastError();
            wprintf(L"SystemParametersInfo(...) failed with error %d\n", dwErr);
            return -__LINE__;
        }
       
        wprintf(L"Setting the desktop wallpaper to %s succeeded.\n", argv[1]);   
        return 0;
    }

     

    Binaries attached.

    Warning: if you pass a relative path to this tool, it won't qualify it for you, and the SystemParametersInfo call won't either - so the wallpaper you want won't be set, though all the calls will succeed.  Make sure to specify a fully-qualified path.

     

  • Matthew van Eerde's web log

    How to create a shortcut from the command line

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    Working on Windows, I install Windows a lot.  This means a lot of my customizations have to be re-applied every time I install.  To save myself some time I created a script which applies some of them. Last time I showed how to set the desktop wallpaper from a command-line app.

    This time, a script to create a shortcut.  The example usage creates a shortcut to Notepad and puts that in the "SendTo" folder.  I find this very useful because I often need to edit text files that have non-".txt" assocations.  (There are also other shortcuts I create with it.)

     

    Here's the script:

    >create-shortcut.vbs

    If WScript.Arguments.Count < 2 Or WScript.Arguments.Count > 3 Then
        WScript.Echo "Expected two or three arguments; got " & WScript.Arguments.Count
        WScript.Echo "First argument is the file to create"
        WScript.Echo "Second is the command to link to"
        WScript.Echo "Third, if present, is the arguments to pass"
        WScript.Quit
    End If

    Set shell = WScript.CreateObject("WScript.Shell")

    Set link = shell.CreateShortcut(WScript.Arguments(0))
    link.TargetPath = WScript.Arguments(1)

    If WScript.Arguments.Count = 3 Then
        link.Arguments = WScript.Arguments(2)
    End If

    link.Save

    >cscript create-shortcut.vbs "%appdata%\Microsoft\Windows\SendTo\Notepad.lnk" notepad.exe

  • Matthew van Eerde's web log

    Generating primes using the Sieve of Eratosthenes plus a few optimizations

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    When solving Project Euler problems I frequently need to iterate over prime numbers less than a given n. A Sieve of Eratosthenes method quickly and easily finds the small prime numbers; there are more complicated methods that find larger prime numbers, but with a couple of tweaks the Sieve of Eratosthenes can get quite high.

    A naive implementation for finding the set of primes below n will:

    1. Allocate an array of n booleans, initialized to false.
    2. Allocate an empty list
    3. For each i in the range 2 to n:
      1. If the boolean value at this index in the array is true, i is composite. Skip to the next value and check that.
      2. If the boolean value at this index in the array is false, i is prime!
      3. Add i to the list of primes
      4. For each multiple of i in the range 2i to n, set the boolean value at that index in the array to true

    There are a handful of simple optimizations that can be made to this naive implementation:

    1. Step 3d) will have no effect until the multiple of i reaches i2, so the range can be changed to "i2 to n"
    2. As a direct consequence of this, step 3d) can be skipped entirely once i2 passes n.
    3. Instead of allocating an array of n booleans, an array of nbits will suffice.
    4. All the even-indexed bits are set to true on the first pass. Manually recognize that 2 is prime, and only allocate bits for odd-numbered values. Change the outer loop in 3) to "in the range 3 to n", incrementing by two each time. Change the loop 3d) to increment by 2i each time.
    5. Storing the list of primes takes a lot of memory - more than the sieve. Don't bother creating a list of primes, just write an enumerator that travels the sieve directly.

    With these optimizations I can enumerate primes from 2 up to 5 billion (5 * 109) in about seven minutes.  Source and binaries attached.

    >primes 5000000000
    Will enumerate primes <= 5000000000 = 5e+009
    Memory for sieve: 298.023 MB
    Initialization complete: 983 milliseconds since start
    Sieving to 70711
    Sieving complete: 4.70292 minutes since start
    Picking up the rest to 5000000000
    Pickup complete: 6.12252 minutes since start
    Primes: 234954223
    1: 2
    23495423: 442876981
    46990845: 920233121
    70486267: 1410555607
    93981689: 1909272503
    117477111: 2414236949
    140972533: 2924158169
    164467955: 3438252577
    187963377: 3955819157
    211458799: 4476550979
    234954221: 4999999883
    Enumerating complete: 7.43683 minutes since start
    Freeing CPrimes object

    There are more complicated sieves like the Sieve of Atkin which perform better but at the cost of being much more complex. So far I haven't had to resort to any of those.

  • Matthew van Eerde's web log

    Programmatically grabbing a screenshot of the primary display

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    It's sometimes difficult to explain to people what my job actually is. "I test Windows sound." "Cool.  How does that work?"

    A product like Windows has a lot of components that interact with each other.  If everything works, the user doesn't know that most of these components even exist; everything is invisible and seamless.

    Most testing involves the connection ("interface") between two components.  "I test APIs."  To the uninitiated, this is just a word.  It sounds like "I test wakalixes."  "You test what, now?"

    There are two interfaces which are easier to explain.  There's the software-to-hardware interface, where the driver talks to the hardware.  "I test the HD Audio, USB Audio, and Bluetooth audio class drivers."  "Huh?" "They make the speakers and the microphone work."  "Oh, cool.  So you sit around and use Skype all day?"

    But the easiest of all to explain is the user interface.  "I make sure that the Sound Recorder app, the volume slider, and the Sound control panel work." "Oh, that!  I had this annoying problem once where..."

    What does the test result for an invisible interface look like? A lot of logging.  "I expected this call to succeed; it returned this HRESULT."  "I poked the hardware like this and got a bluescreen."  "There seems to be an infinite loop here."  Lots of text. 

    Boring.

    UI testing has logging too.  But with UI testing you can also... TAKE PICTURES!  A UI bug is a lot easier to understand (triage, and fix) if there's a screenshot attached (preferably with a big red rectangle highlighting the problem.)

    It is therefore valuable to have an automatable utility that can take a screenshot and dump it to a file.  Here's one I cribbed together from the "Capturing an Image" sample code on http://msdn.microsoft.com/en-us/library/dd183402(v=VS.85).aspx.  Source and binaries attached.

    This version only captures the main display, and not secondary monitors (if any.)

    Pseudocode:

    screen_dc = GetDC(nullptr);

    memory_dc = CreateCompatibleDC(screen);

    rect = GetClientRect(GetDesktopWindow());

    hbmp = CreateCompatibleBitmap(screen_dc, rect);

    SelectObject(memory_dc, hbmp);

    BitBlt(memory_dc, rect, screen_dc);

    bmp = GetObject(hbmp);

    bytes = allocate enough memory

    bytes = GetDIBits(screen_dc, bmp, hbmp)

    file = CreateFile();

    WriteFile(bitmap header);

    WriteFile(bytes);

  • Matthew van Eerde's web log

    Excruciating rhymes

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    I was watching How the Grinch Stole Christmas and I was struck by this rhyme (from the song  "You're a Mean One, Mr. Grinch:")

    You're a nauseous super naus!
    You're a dirty crooked jockey, and you drive a crooked hoss
       
    -- Dr. Seuss, How the Grinch Stole Christmas

    I tried to see what other particularly excruciating rhymes I could remember.  I came up with two:

    You know, that little guy, he's got me feeling all contempt-ey:
    He takes his boat out loaded up and brings it back in empty.
       
    -- Phil Vischer, Lyle the Kindly Viking

    And then of course:

    In short, when I've a smattering of elemental strategy,
    You'll say a better Major-General had never sat a-gee!
       
    -- W. S. Gilbert, The Pirates of Penzance

  • Matthew van Eerde's web log

    Teaching someone to fish and the AKS primality test

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    This morning my wife (whom I love and adore) woke me up at 3:00 AM with an urgent question.

    "Hey!" she said, shaking me awake.

    "Is 19 prime?"

    ...

    Like a fool, I answered her.  "Yes.  Yes it is."  Off she went.

    This is a true response, but not a correct response.  I realized shortly afterwards that a correct response would look more like:

    I'm glad you asked me that. dear.  Eratosthenes, the Greek mathematician, discovered a very efficient way to list primes in about 200 BC that is still in use today.  You start by writing out all the numbers from 1 to 19: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19.  1 is a very special number (it's the multiplicative identity, or what algebraists would call a unit) so we put a square around it.  The first number we didn't consider was 2, so we circle it - that means it's prime - and then cross out every multiple of 2 after that.  Going back, the first number we didn't consider was 3... and so on until we get 1 2 3 X 5 X 7 X X X 11 X 13 X X X 17 X 19.  A common optimization is to realize that after circling a prime p, the first number you cross out (that wasn't crossed out before) is always p2, which means that after circling a number you can immediately jump to its square, and also means you can stop crossing out altogether once you hit p> N...

    This would allow her to fish rather than waking me up when she wanted a fish.

    An even better response would have been:

    It's funny you've asked me that.  Number theorists and cryptanalysts have considered this question for thousands of years.  Eratosthenes' method (see above) is a very simple way to find all the primes below a given number, but an efficient way to determine whether a given number is prime was found only very recently.

    In practice, the test that is usually used is the randomized version of the Miller-Rabin test.  Although this is nondeterministic, it is very fast indeed, and will tell you to a very high degree of certainty whether the given number is prime.  This usually suffices.

    There is a deterministic version of the Miller-Rabin test too, which is guaranteed to tell you with perfect certainty whether the given number is prime.  But it only works if you believe in the generalized Riemann hypothesis.  Most mathematicians nowadays believe the hypothesis, but no-one has (yet) been able to prove it.

    Amazingly, in 2002 three mathematicians named Manindra Agrawal, Neeraj Kayal, and Nitin Saxena came up with a deterministic, proven, polynomial-time (specifically, polynomial in the number of digits in the input) method for telling whether a given number is prime.   This is known as the AKS primality test.  The most striking thing about this test is its simplicity - if something this straightforward can be found after thousands of years of looking, what else out there remains to be found?

    Such a response would probably prevent her from waking me again for any mathematical problem at all.  Boo-ya.

    Here's Agrawal, Kayal, and Saxena's "PRIMES is in P" paper.

    Here's Yves Gallot's C++ implementation of AKS.

    My own Perl implementation follows:

    use strict;
    # use bignum;

    print <<USAGE and exit 0 unless @ARGV;
    $0 [-v] n
        Use the AKS primality test to check whether n is prime
        -v adds verbose log spew
    USAGE

    sub is_power($$);
    sub ceil_log2($);
    sub first_r($$);
    sub check_gcds($$);
    sub check_polynomials($$$);
    sub gcd($$);
    sub totient($);
    sub polypow($$\@);
    sub polymult($$\@\@);
    sub polyeq(\@\@);

    my $verbose = $ARGV[0] eq "-v";
    shift @ARGV if $verbose;

    die "Expected only one argument" unless 1 == @ARGV;
    my $n = shift;

    # step 0: restrict to integers >= 2
    print "$n is not an integer and so is NEITHER PRIME NOR COMPOSITE\n" and exit 0 unless int($n) == $n;
    print "$n < 2 and so is NEITHER PRIME OR COMPOSITE\n" and exit 0 unless $n >= 2;

    # step 1: check if the number is a power of some lower number.
    # this can be done quickly by iterating over the exponent (2, 3, ...)
    # and doing a binary search on the base.
    # we start at the top and work down for performance reasons;
    # several subroutines need to know ceil(log2(n)) so we calculate it once and pass it around.
    my $log2_n = ceil_log2($n);
    is_power($n, $log2_n) and exit 0;
    print "Not a power.\n";

    # step 2: find the smallest r such that o_r(n) > (log2 n)^2
    # where o_r(n) is the multiplicative order of n mod r
    # that is, the smallest k such that n^k == 1 mod r
    my $r = first_r($n, $log2_n);
    print "r = $r\n";

    # step 3: for all a between 2 and r inclusive, check whether gcd(a, n) > 1
    check_gcds($n, $r) or exit 0;

    # step 4: if r >= n, we're done
    if ($r >= $n) {
        print "$r >= $n so $n is PRIME\n";
        exit 0;
    }

    # step 5: for all a between 1 and floor( sqrt(phi(r)) log2(n) )
    # check whether (x + a)^n = x^n + a mod x^r - 1, n
    check_polynomials($n, $r, $log2_n) or exit 0;

    # step 6: if we got this far, n is prime
    print "$n is PRIME\n";

    sub is_power($$) {
        my $n = shift;
        my $log2_n = shift; # actually ceil(log2(n))

        print "Checking for power-ness...\n";

        # we consider numbers of the form b^i
        # we iterate over the exponent i
        # starting at i = ceil(log2(n)) and working down to i = 2
        #
        # for each exponent we do a binary search on the base
        # the lowest the base can be is 2
        # and the highest the base can be (initially) is 2
        #
        # we set up bounds on the base that are guaranteed to
        # surround the actual base
        my $b_low = 1; # 1 ^ ceil(log2(n)) = 1 < n
        my $b_high = 3; # 3 ^ ceil(log2(n)) > 2 ^ log2(n) = n

        for (my $i = $log2_n; $i >= 2; $i--) {
            print "\tb^$i\n" if $verbose;

            # let's check that the bounds are really correct
            die "$b_low ^ $i is not < $n" unless $b_low ** $i < $n;
            die "$b_high ^ $i is not > $n" unless $b_high ** $i > $n;

            # do a binary search to find b such that b ^ i = n
            while ($b_high - $b_low > 1) {
                print "\t\tb^$i: b is between $b_low and $b_high\n" if $verbose;
                my $b = int(($b_low + $b_high)/2);

                my $t = $b ** $i;
                if ($t == $n) {
                    print "$n = $b^$i; $n is COMPOSITE\n";
                    return 1;
                }

                ($t > $n ? $b_high : $b_low) = $b;
            }

            # as we pass from the exponent (say, 5)
            # to the exponent below (say, 4)
            # we need to reconsider our bounds
            #
            # b_low can remain the same because b ^ (i - 1) is even less than b ^ i
            # OPEN ISSUE: can we even raise b_low?
            #
            # but we need to raise b_high since b ^ i > n does NOT imply b ^ (i - 1) > n
            #
            # we'll square b_high; b ^ i > n => (b ^ 2) ^ (i - 1) = b ^ (2 i - 2) > n
            # since i >= 2
            #
            # OPEN ISSUE: is there a better way to raise this higher bound? Does this help much?
            $b_high *= $b_high;
        }

        # nope, not a power
        return 0;
    }

    sub ceil_log2($) {
        my $n = shift;

        my $i = 0;
        my $t = 1;

        until ($t >= $n) {
            $i++;
            $t *= 2;
        }

        return $i;
    }

    sub first_r($$) {
        my $n = shift;
        my $log2_n = shift; # actually ceil(log2(n))

        my $s = $log2_n ** 2;
        print "Looking for the first r where o_r($n) > $s...\n";

        # for each r we want to find the smallest k such that
        # n^k == 1 mod r

        my $r;
        for ($r = 2; ; $r++) {
            # print "\tTrying $r...\n";

            # find the multiplicative order of n mod r
            my $k = 1;
            my $t = $n % $r;

            until (1 == $t or $k > $s) {
                $t = ($t * $n) % $r;
                $k++;
            }

            if ($k > $s) {
                # print "\to_$r($n) is at least $k\n";
                last;
            } else {
                # print "\to_$r($n) = $k\n";
            }
        }

        return $r;
    }

    sub check_gcds($$) {
        my ($n, $r) = @_;

        print "Checking GCD($n, a) for a = 2 to $r...\n";

        for (my $a = 2; $a <= $r; $a++) {
            my $g = gcd($n, $a);

            next if ($g == $n); # this is OK

            if (1 != $g) {
                print "gcd($n, $a) = $g; $n is COMPOSITE\n";
                return 0;
            }
        }

        print "All GCDs are 1 or $n\n";

        return 1;
    }

    sub gcd($$) {
        my ($x, $y) = @_;

        ($x, $y) = ($y, $x) unless $x > $y;

        while ($y) {
            ($x, $y) = ($y, $x % $y);
        }

        return $x;
    }

    sub check_polynomials($$$) {
        my $n = shift;
        my $r = shift;
        my $log2_n = shift; # actually ceil(log2(n))

        # iterate over a from 1 to floor( sqrt(phi(r)) log2(n) )
        # for each a, check whether the polynomial equality holds:
        # (x + a)^n = x^n + a mod (x^r - 1, n)
        # if it fails to hold, the number is composite
        #
        # first we need to evaluate phi(r) so we can determine the upper bound
        # OPEN ISSUE: this seems to be a potential weakness in the algorithm
        # because the usual way to evaluate phi(r) is to find the prime factorization of r
        # and then form the product r*PI(1 - 1/p) where the product ranges over all primes
        # which divide r

        my $phi = totient($r);

        # a < sqrt(phi(r)) * log2(n) => a^2 < phi(r) * (log2(n))^2
        my $a2_max = $phi * $log2_n * $log2_n;
        print "Checking polynomials up to roughly ", int sqrt($a2_max), "...\n";

        for (my $a = 1; $a * $a <= $a2_max; $a++) {
            print "\ta = $a...\n" if $verbose;

            # polynomials are of the form (c0, c1, c2, ..., ci, ...)
            # which corresponds to c0 + c1 x + c2 x^2 + ... + ci x^i + ...)
            my @x = (0, 1);
            my @x_plus_a = ($a % $n, 1);

            my @lhs = polypow($n, $r, @x_plus_a);

            # POTENTIAL OPTIMIZATION:
            # x^n + a mod (x^r - 1) is just x^(n % r) + a
            # and we know n % r != 0
            my @rhs = polypow($n, $r, @x); # x^n
            $rhs[0] = ($rhs[0] + $a) % $n; # + a

            next if polyeq(@lhs, @rhs);

            print "(x + $a)^$n is not equal to x^$n + $a mod(x^$r - 1, $n)\n";
            print "So $n is COMPOSITE\n";
            return 0;
        }

        return 1;
    }

    sub totient($) {
        my $r = shift;

        print "Finding the Euler totient of $r\n";

        # we'll do a trial division to find the totient
        # there are faster ways that use a sieve
        # but we don't know how big r is
        my $t = $r;

        # by construction p will always be prime when it is used
        # OPEN ISSUE: this might be slow
        for (my $p = 2; $r > 1; $p++) {
            next if $r % $p;

            print "\t$p is a factor\n" if $verbose;
            # decrease the totient
            $t /= $p;
            $t *= $p - 1;

            # decrease r
            $r /= $p; # we know there's at least one factor of p
            $r /= $p until $r % $p; # there might be more
        }

        print "Totient is $t\n";

        return $t;
    }

    sub polypow($$\@) {
        my $n = shift; # this is both the mod and the exponent
        my $r = shift;
        my @base = @{ +shift };

        my $exp = $n;
        my @result = (1); # 1

        # print "\t(", join(" ", @base), ")^$exp mod (x^$r - 1, $n)\n" if $verbose;

        # basic modpow routine, but with polynomials
        while ($exp) {
            if ($exp % 2) {
                @result = polymult($n, $r, @result, @base);
            }

            $exp = int ($exp / 2);
            @base = polymult($n, $r, @base, @base);
        }

        # print "\t= (", join(" ", @result), ")\n" if $verbose;
        return @result;
    }

    sub polymult($$\@\@) {
        my $n = shift;
        my $r = shift;
        my @first = @{ +shift };
        my @second = @{ +shift };

        # print "\t\t(", join(" ", @first), ") * (", join(" ", @second), ") mod (x^$r - 1, $n)\n" if $verbose;

        my @result = ();

        # first do a straight multiplication first * second
        my $s = @second - 1;
        for (my $i = @first - 1; $i >= 0; $i--) {
            for (my $j = $s; $j >= 0; $j--) {
                my $k = $i + $j;
                $result[$k] += $first[$i] * $second[$j];
                $result[$k] %= $n;
            }
        }

        # then do a straight mod x^r - 1
        # consider a polynomial
        # c0 + ... + ck x^k
        # with k >= r
        # we can subtract ck (x^r - 1)
        # without changing the mod value
        # the net effect is to eliminate the x^k term
        # and add ck to the x^(k - r) term

        for (my $i = @result - 1; $i >= $r; $i--) {
            my $j = $i - $r;
            $result[$j] += $result[$i];
            $result[$j] %= $n;

            pop @result;
        }

        # eliminate any leading zero terms
        for (my $i = @result - 1; 0 == $result[$i]; $i--) {
            pop @result;
        }

        # print "\t\t= (", join(" ", @result), ")\n" if $verbose;
        return @result;
    }

    sub polyeq(\@\@) {
        my @lhs = @{ +shift };
        my @rhs = @{ +shift };

        # print "(", join(" ", @lhs), ") = (", join(" ", @rhs), ")?\n" if $verbose;

        return 0 unless @lhs == @rhs;

        for (my $i = @lhs - 1; $i >= 0; $i--) {
            return 0 unless $lhs[$i] == $rhs[$i];
        }

        return 1;
    }

    Here's the output when I run it on 19:

    >perl -w aks.pl 19
    Checking for power-ness...
    Not a power.
    Looking for the first r where o_r(19) > 25...
    r = 19
    Checking GCD(19, a) for a = 2 to 19...
    All GCDs are 1 or 19
    19 >= 19 so 19 is PRIME

    And here's the output with a bigger input:

    >perl -w aks.pl 99
    Checking for power-ness...
    Not a power.
    Looking for the first r where o_r(997) > 100...
    r = 103
    Checking GCD(997, a) for a = 2 to 103...
    All GCDs are 1 or 997
    Finding the Euler totient of 103
    Totient is 102
    Checking polynomials up to roughly 100...
    997 is PRIME
  • Matthew van Eerde's web log

    Enumerating MIDI devices

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    In addition to audio playback and recording, Windows Multimedia (WinMM) provides a Musical Instrument Digital Interface (MIDI) API.

    Here's how to make a list of all the MIDI devices on the system, their capabilities, and the hardware device interface associated with each of them.

    Source and binaries attached.

    Pseudocode:

    midiInGetNumDevs or midiOutGetNumDevs
    for each device
        midiInGetDevCaps or midiOutGetDevCaps
        log device capabilities
        midiInMessage or midiOutMessage
            with DRV_QUERYDEVICEINTERFACESIZE
            and DRV_QUERYDEVICEINTERFACE
        log the device interface

    Output:

    >midienum.exe
    midiIn devices: 1
    -- 0: USB2.0 MIDI Device --
        Device ID: 0
        Manufacturer identifier: 65535
        Product identifier: 65535
        Driver version: 1.6
        Product name: USB2.0 MIDI Device
        Support: 0x0
        Device interface: "\\?\usb#vid_xxxx&pid_yyyy&..."
    midiOut devices: 2
    -- 0: Microsoft GS Wavetable Synth --
        Device ID: 0
        Manufacturer identifier: 1
        Product identifier: 27
        Driver version: 1.0
        Product name: Microsoft GS Wavetable Synth
        Technology: 7 (MOD_SWSYNTH)
        Voices: 32
        Notes: 32
        Channel mask: 0xffff
        Support: 0x1
            MIDICAPS_VOLUME
        Device interface: ""
    -- 1: USB2.0 MIDI Device --
        Device ID: 1
        Manufacturer identifier: 65535
        Product identifier: 65535
        Driver version: 1.6
        Product name: USB2.0 MIDI Device
        Technology: 1 (MOD_MIDIPORT)
        Voices: 0
        Notes: 0
        Channel mask: 0xffff
        Support: 0x0
        Device interface: "\\?\usb#vid_xxxx&pid_yyyy&..."

    (Actual device interface string suppressed.)

    Note the Microsoft GS Wavetable Synth device, which is always present.

    Why would you want to know the device interface? In our case, because we want to test all the audio-related interfaces of a particular device on the system.

  • Matthew van Eerde's web log

    Enumerating mixer devices, mixer lines, and mixer controls

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    The WinMM multimedia APIs include an API for enumerating and controlling all the paths through the audio device; things like bass boost, treble control, pass-through audio from your CD player to your headphones, etc.  This is called the "mixer" API and is the forerunner of the IDeviceTopology API.

    I wrote a quick app to enumerate all the mixer devices on the system; for each mixer device, enumerate each mixer line (that is, each source and destination); for each mixer line, enumerate all the controls (volume, mute, equalization, etc.); and for each control, query the associated text (if any) and the current value.

    Source and binaries attached.

    Pseudocode:

    mixerGetNumDevs()
    for each mixer device
        mixerGetDevCaps(dev)
        for each destination (line) on the device
            mixerGetLineInfo(dest)
            mixerGetLineControls(dest)
            for each control on the line
                if the control supports per-item description
                    mixerGetControlDetails(control, MIXER_GETCONTROLDETAILSF_LISTTEXT)
                    log the per-item description
                mixerGetControlDetails(control, MIXER_GETCONTROLDETAILSF_VALUE)
                log the value(s)

    Usage:

    >mixerenum.exe
    Mixer devices: 5
    -- 0: Contoso headset --
        Device ID: 0
        Manufacturer identifier: 1
        Product identifier: 104
        Driver version: 6.2
        Product name: Contoso headset
        Support: 0x0
        Destinations: 1
            -- Destination 0: Master Volume --
                Destination: 0
                Source: -1
                Line ID: 0xffff0000
                Status: MIXERLINE_LINEF_ACTIVE (1)
                User: 0x00000000
                Component Type: MIXERLINE_COMPONENTTYPE_DST_HEADPHONES (5)
                Channels: 1
                Connections: 2
                Controls: 2
                Short name: Volume
                Long name: Master Volume
                -- Target:  --
                    Type: MIXERLINE_TARGETTYPE_UNDEFINED (0)
                    Device ID: 0
                    Manufacturer identifier: 65535
                    Product identifier: 65535
                    Driver version: 0.0
                    Product name:
                -- Control 1: Mute --
                    Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Mute
                    Long name: Mute
                    -- Values --
                        FALSE
                -- Control 2: Volume --
                    Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Volume
                    Long name: Volume
                    -- Values --
                        0xffff on a scale of 0x0 to 0xffff
    -- 1: HDMI Audio (Contoso --
        Device ID: 1
        Manufacturer identifier: 1
        Product identifier: 104
        Driver version: 6.2
        Product name: HDMI Audio (Contoso
        Support: 0x0
        Destinations: 1
            -- Destination 0: Master Volume --
                Destination: 0
                Source: -1
                Line ID: 0xffff0000
                Status: MIXERLINE_LINEF_ACTIVE (1)
                User: 0x00000000
                Component Type: MIXERLINE_COMPONENTTYPE_DST_DIGITAL (1)
                Channels: 1
                Connections: 2
                Controls: 2
                Short name: Volume
                Long name: Master Volume
                -- Target:  --
                    Type: MIXERLINE_TARGETTYPE_UNDEFINED (0)
                    Device ID: 0
                    Manufacturer identifier: 65535
                    Product identifier: 65535
                    Driver version: 0.0
                    Product name:
                -- Control 1: Mute --
                    Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Mute
                    Long name: Mute
                    -- Values --
                        FALSE
                -- Control 2: Volume --
                    Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Volume
                    Long name: Volume
                    -- Values --
                        0xffff on a scale of 0x0 to 0xffff
    -- 2: Speakers (Contoso --
        Device ID: 2
        Manufacturer identifier: 1
        Product identifier: 104
        Driver version: 6.2
        Product name: Speakers (Contoso
        Support: 0x0
        Destinations: 1
            -- Destination 0: Master Volume --
                Destination: 0
                Source: -1
                Line ID: 0xffff0000
                Status: MIXERLINE_LINEF_ACTIVE (1)
                User: 0x00000000
                Component Type: MIXERLINE_COMPONENTTYPE_DST_SPEAKERS (4)
                Channels: 1
                Connections: 2
                Controls: 2
                Short name: Volume
                Long name: Master Volume
                -- Target:  --
                    Type: MIXERLINE_TARGETTYPE_UNDEFINED (0)
                    Device ID: 0
                    Manufacturer identifier: 65535
                    Product identifier: 65535
                    Driver version: 0.0
                    Product name:
                -- Control 1: Mute --
                    Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Mute
                    Long name: Mute
                    -- Values --
                        FALSE
                -- Control 2: Volume --
                    Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Volume
                    Long name: Volume
                    -- Values --
                        0xffff on a scale of 0x0 to 0xffff
    -- 3: Contoso headset --
        Device ID: 3
        Manufacturer identifier: 1
        Product identifier: 104
        Driver version: 6.2
        Product name: Contoso headset
        Support: 0x0
        Destinations: 1
            -- Destination 0: Master Volume --
                Destination: 0
                Source: -1
                Line ID: 0xffff0000
                Status: MIXERLINE_LINEF_ACTIVE (1)
                User: 0x00000000
                Component Type: MIXERLINE_COMPONENTTYPE_DST_WAVEIN (7)
                Channels: 1
                Connections: 1
                Controls: 2
                Short name: Volume
                Long name: Master Volume
                -- Target: Contoso headset --
                    Type: MIXERLINE_TARGETTYPE_WAVEIN (2)
                    Device ID: 0
                    Manufacturer identifier: 1
                    Product identifier: 101
                    Driver version: 6.2
                    Product name: Contoso headset
                -- Control 1: Mute --
                    Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Mute
                    Long name: Mute
                    -- Values --
                        FALSE
                -- Control 2: Volume --
                    Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Volume
                    Long name: Volume
                    -- Values --
                        0xf332 on a scale of 0x0 to 0xffff
    -- 4: Microphone (Contoso --
        Device ID: 4
        Manufacturer identifier: 1
        Product identifier: 104
        Driver version: 6.2
        Product name: Microphone (Contoso
        Support: 0x0
        Destinations: 1
            -- Destination 0: Master Volume --
                Destination: 0
                Source: -1
                Line ID: 0xffff0000
                Status: MIXERLINE_LINEF_ACTIVE (1)
                User: 0x00000000
                Component Type: MIXERLINE_COMPONENTTYPE_DST_WAVEIN (7)
                Channels: 1
                Connections: 1
                Controls: 2
                Short name: Volume
                Long name: Master Volume
                -- Target: Microphone (Contoso --
                    Type: MIXERLINE_TARGETTYPE_WAVEIN (2)
                    Device ID: 1
                    Manufacturer identifier: 1
                    Product identifier: 101
                    Driver version: 6.2
                    Product name: Microphone (Contoso
                -- Control 1: Mute --
                    Type: MIXERCONTROL_CONTROLTYPE_MUTE (0x20010002)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Mute
                    Long name: Mute
                    -- Values --
                        FALSE
                -- Control 2: Volume --
                    Type: MIXERCONTROL_CONTROLTYPE_VOLUME (0x50030001)
                    Status: MIXERCONTROL_CONTROLF_UNIFORM (0x1)
                    Item count: 0
                    Short name: Volume
                    Long name: Volume
                    -- Values --
                        0xf332 on a scale of 0x0 to 0xffff


  • Matthew van Eerde's web log

    Muting all audio outputs with IAudioEndpointVolume

    • 0 Comments

    I have a selfhost tool that I use to mute all audio outputs programmatically.

    Pseudocode:

    IMMDeviceEnumerator::EnumAudioEndpoints
    for each device:
        IMMDevice::Activate(IAudioEndpointVolume)
        IAudioEndpointVolume::SetMute(TRUE)

    Source and binaries attached.

  • Matthew van Eerde's web log

    Implementing a "say" command using ISpVoice from the Microsoft Speech API

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    I've known for a while that Microsoft Windows comes with text-to-speech and speech-to-text APIs, which power the Narrator and Speech Recognition features respectively.

    This forum post prompted me to mess around with them a little.

    I came up with this implementation of a say.exe command which takes a single argument as text, and then uses the ISpVoice text-to-speech API to have the computer speak it aloud.

    Source and binaries attached.

    Pseudocode:

    CoInitialize(nullptr);
    CoCreateInstance(ISpVoice)
    pSpVoice->Speak(text);

    Usage:

    >say.exe "Daisy, Daisy; give me your answer, do."

    More information on the Speech APIs available here: http://msdn.microsoft.com/en-us/library/ms723627(v=vs.85).aspx

  • Matthew van Eerde's web log

    unattend.xml: turning on Remote Desktop automatically

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    Here are the portions of my unattend.xml file which are needed to turn on Remote Desktop automatically.

    This piece flips the "no remote desktop" kill switch to "allow."

        <settings pass="specialize">
            ...
            <component name="Microsoft-Windows-TerminalServices-LocalSessionManager" ...>
                <fDenyTSConnections>false</fDenyTSConnections>

    That's not enough; it is also necessary to poke a hole in the firewall to allow inbound connections.  I use an indirect string for the Group name, to allow for installing localized builds.  This points to the "Remote Desktop" feature group.

        <settings pass="specialize">
            ...
            <component name="Networking-MPSSVC-Svc" ...>
                <FirewallGroups>
                    <FirewallGroup wcm:action="add" wcm:keyValue="RemoteDesktop">
                        <Active>true</Active>
                        <Profile>all</Profile>
                        <Group>@FirewallAPI.dll,-28752</Group>

    If your user account is a member of the "Administrators" group, you're done:

        <settings pass="oobeSystem">
            <component name="Microsoft-Windows-Shell-Setup" ...>
                <UserAccounts>
                    <LocalAccounts>
                        <LocalAccount wcm:action="add">
                            <Password>
                                <Value>#PASSWORD_ADMIN#</Value>
                                <PlainText>true</PlainText>
                            </Password>
                            <Name>Admin</Name>
                            <Group>Administrators</Group>

    But if you're like me and you don't want to live in the Administrators group, you need to join the Remote Desktop Users group to be able to log in remotely:

        <settings pass="oobeSystem">
            <component name="Microsoft-Windows-Shell-Setup" ...>
                <UserAccounts>
                    <DomainAccounts>
                        <DomainAccountList wcm:action="add">
                            <Domain>redmond.corp.microsoft.com</Domain>
                            <DomainAccount wcm:action="add">
                                <Group>RemoteDesktopUsers</Group>
                                <Name>MatEer</Name>

  • Matthew van Eerde's web log

    Weighing the Sun and the Moon

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    In an earlier post I mentioned how the Cavendish experiment allowed us to weigh the Earth - to determine the mass of the Earth mE.  Newton knew the acceleration due to gravity on the surface of the Earth and was able to use that to find the product G mE; Cavendish determined G directly, and was thus able to solve for mE.  He would also have been able to find the mass of the sun as follows:

    mE aE = G mE mS / rE2

    G, rE, and aE = vE2 / rE are known, so we can solve for mS.

    But calculating the mass of the moon is trickier.

    Once we were able to put a satellite around the moon we could measure its orbital radius and speed, deduce the acceleration, and use that plus the known G to calculate the mass of the moon.  But prior to that we were limited to techniques like:

    The moon does not exactly orbit the Earth, but instead orbits the center of mass of the moon/Earth system.  By careful observation we can determine where this center of mass is.  We can then measure the distance between the center of mass and the Earth's center.  This plus the known mass of the Earth and the distance of the Earth from the Moon allows us to determine the mass of the Moon.

    If we're lucky enough to see a foreign object come close to the moon, we can determine how much it is accelerated by the Moon.  This will allow us to determine the mass of the Moon using the technique above.  (We won't be able to determine the mass of the foreign object, but we don't need it.)

    When the USSR launched Sputnik, American scientists really wanted to know what its mass was.  But because none of the techniques above were useful, they were unable to determine it.

  • Matthew van Eerde's web log

    Programmatically rearranging displays

    • 0 Comments

    Most of my test machines and my laptop have a single display; but I have two dev machines which are each connected to two displays.

    When I clean install Windows, I sometimes need to rearrange the displays:

    Since I clean install Windows frequently, I wrote myself a little C++ app which does this programmatically using EnumDisplayDevices / EnumDisplaySettings / ChangeDisplaySettingsEx.

    Source and binaries attached.

    Pseudocode:

    for (each device returned by EnumDisplayDevices) {

       grab the position and the height/width using EnumDisplaySettings

    }

    calculate the desired position of the secondary monitor

    Set it using ChangeDisplaySettingsEx with DM_POSITION

     

    Call as:

    >swapmonitors
    Moved secondary monitor to (1920, 0)

    EDIT: Oops: 0 should be CDS_GLOBAL | CDS_UPDATEREGISTRY, to make the settings apply to all users, and to persist across display resets / reboots

    LONG status = ChangeDisplaySettingsEx(
        nameSecondary,
        &mode,
        nullptr, // reserved
        CDS_GLOBAL | CDS_UPDATEREGISTRY, // was 0
        nullptr // no video parameter
    );

  • Matthew van Eerde's web log

    Windows Sound test team rowing morale event

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    Last Friday the Windows Sound test team went kayaking.  We went to the Agua Verde paddle club and kayaked around Union Bay for a while.

    Here's the route we took:

     

    More detail:

    http://connect.garmin.com/activity/179545084

     

  • Matthew van Eerde's web log

    Why square waves have ears: Gibbs' phenomenon (Wilbraham's phenomenon)

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    In a recent post I sung the praises of square waves as a way to get a heckuva lot of power (3 dB more power than a sine wave) into a sample-range-limited signal.  It's time to take them down a notch now.

    A problem with square waves is they're impossible to generate in the analog domain.  In fact, you can't even get close.

    Signals generated in the analog domain are subject to physical laws regarding continuity (no teleportation) and whatnot. A common way to model these is to express (periodic) analog signals using a basis of sine waves with integral periods.  Suppose I want to generate the square wave:

    f(x) = 1, -π < x < 0
    -1, 0 < x < π

    Graphed below are some approximations of this function using sine waves as a basis.  Note that only odd values of n are used in the sums of sin(nx).


     

    The sums converge to +/-1 quite well, but there are definite "ears" at x near 0 where there's overshoot.  This doesn't appear to die down.  A closeup of one of the "ears":


     

    If anything, rather than dying down, the "ears" converge to max(fn) → about 1.18, or about 9% of the "jump" from -1 to 1.  (Dym and McKean, in their 1972 book Fourier Series and Integrals, get the 9% right but incorrectly assert that the convergence is to 1.09.)

    This mathematical phenomenon - Gibbs' phenomenon - is a good illustration of the difference between convergence of a series of functions and uniform convergence.

    In this case, the series of partial sums pointwise converge to the square wave... for any given point x > 0 and ε > 0, the ear will eventually move to the left, and I can choose an N such that fn(x) is within of ε of 1 for all n > N...

    ... but the series does not uniformly converge to the square wave.  The following assertion is false: "for any given ε > 0, I can pick an N such that fn(x) is within of ε of 1 for all n > N and all x > 0."  This can be demonstrated by picking ε = 0.17, say.  For any n, even, say, n = 10100, there is an x close to 0 where f1e100(x) > 1.17.

  • Matthew van Eerde's web log

    Minimal unsatisfiable regular expression, XPath query

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    Regular expressions are a tool for matching generic text.  XPath queries are a tool for matching chunks of XML.  Both are search technologies.

    When using search technologies it is occasionally quite useful to have a query that will never match anything - for SQL, this would be something like "SELECT 1 WHERE 1 = 0".

    My candidates for minimal unsatisfiable regular expression:

    /a\bc/
    \b is a zero-width assertion that matches a boundary at the beginning or end of a word - specifically, it is true between a word-ish character (\w) and a non-wordish character (\W).  Since literal "a" and "c" are both wordish characters, this will never match.

    Or, if you allow Perl extensions:

    /(?!)/
    This is a negative lookahead for an empty string.  Since the empty string always matches everywhere, this will never match.

    My candidate for minimal unsatisfiable XPath query:

    /parent::*
    This matches everything at or under the parent of the root element.  Since, by definition, the root element has no parent, this will never match.

  • Matthew van Eerde's web log

    Second cousins, cousins once removed; relationships by generations to common ancestor

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    Raymond Chen explains some common terms for blood relatives of varying distance across cultures in his blog post "What kind of uncle am I?"

    He links to a diagram on genealogy.com that I felt was lacking something... so here's my version, with consanguinary colors.

    Red means "marriage is almost certainly legally prohibited."

    Yellow means "marriage may be legally prohibited - check your region's laws."

    Green means "marriage is amost certainly legal."


    Relationship by generations to common ancestor
    # 0 1 2 3 4 ... m
    0 self father/mother grand
    (father/mother)
    great‑
    grand
    (father/mother)
    (great‑)2
    grand
    (father/mother)

    (great-)m ‑ 2
    grand
    (father/mother)
    1 son/daughter brother/sister aunt/uncle grand
    (aunt/uncle)
    great‑
    grand
    (aunt/uncle)

    (great‑)m ‑ 3
    grand
    (aunt/uncle)
    2 grand
    (son/daughter)
    niece/nephew (first) cousin first cousin,
    once removed
    first cousin,
    twice removed

    first cousin,
    (m ‑ 2) times removed
    3 great‑
    grand
    (son/daughter)
    grand
    (niece/nephew)
    first cousin,
    once removed
    second cousin second cousin,
    once removed

    second cousin,
    (m ‑ 3) times removed
    4 (great‑)2
    grand
    (son/daughter)
    great-
    grand
    (niece/nephew)
    first cousin,
    twice removed
    second cousin,
    once removed
    third cousin
    third cousin,
    (m ‑ 4) times removed
    ...
    n (great‑)n ‑ 2
    grand
    (son/daughter)
    (great‑)n ‑ 3
    grand
    (niece/nephew)
    first cousin,
    (n ‑ 2) times removed
    second cousin,
    (n ‑ 3) times removed
    third cousin,
    (n ‑ 4) times removed

    (n == m) ?
    ((n ‑ 1)th cousin) :
    ((min(n, m) ‑ 1)th cousin, |n ‑ m| times removed)
  • Matthew van Eerde's web log

    Bad Perl: locker problem

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    Bad Perl solution to the "print the open lockers" problem:

    perl -e"print join', ',map{$_*$_}1..sqrt pop" 100

    54 characters.  I prefer this to the 53-character solution obtained by omitting the space after the first comma.

    EDIT: 49 characters:

    perl -e"print map{$_*$_,' '}1..sqrt pop" 100

    EDIT: 48:

    perl -e"print map{$_*$_.$/}1..sqrt pop" 100

    EDIT: 47:

    perl -e"map{print$/.$_*$_}1..sqrt pop" 100

    I still think "say" is cheating but it does afford this very short solution:

    perl -E"map{say$_*$_}1..sqrt pop" 100

    EDIT: Apparently I need to learn how to count. Counts above are off. Anyway, 41:

    perl -e"print$_*$_,$/for 1..sqrt pop" 100

  • Matthew van Eerde's web log

    Good Perl, Bad Perl

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    One of my favorite languages is Perl.  Perl has an ambivalent reputation; some people take to it, some accuse it of being a syntax-complete language.  (There's some truth to this.)

    My view is that Perl gives you a very direct link into the mind of the programmer - much more so than other languages.  Perl is designed very much like a spoken language, perhaps because Larry Wall's background is linguistics.

    There was a little girl
    Who had a little curl
    Right in the middle of her forehead.
    And when she was good,
    She was very, very, good;
    But when she was bad
    She was horrid.
       -- Henry Wadsworth Longfellow

    (In an English accent, "forehead" and "horrid" actually rhyme.)

    Two examples of my own Perl to illustrate my point.  This is in my email signature:

    perl -e "print join er,reverse',','l hack',' P','Just anoth'"

    And this little seasonal gem:

    use strict;
    use warnings;

    sub receive($);

    my @ordinals = qw(
    zeroth
    first second third fourth fifth sixth
    seventh eighth ninth tenth eleventh twelfth
    );

    my @gifts = reverse split /\n/, <<END_OF_LIST;
    Twelve drummers drumming;
    Eleven pipers piping;
    Ten lords a-leaping;
    Nine ladies dancing;
    Eight maids a-milking;
    Seven swans a-swimming;
    Six geese a-laying;
    Five golden ringeds;
    Four colly birds;
    Three French hens;
    Two turtle doves;
    A partridge in a pear tree.
    END_OF_LIST

    for (my $day = 1; $day <= 12; $day++) {
    receive($day);
    }

    sub receive($) {
    my $day = shift;

    print("On the ", $ordinals[$day], " day of Christmas, my true love sent to me:\n");

    for (my $i = $day; $i > 0; $i--) {
    my $gift = $gifts[$i - 1];

    if ($i == 1 && $day != 1) {
    $gift =~ s/^(\s*)A/$1And a/;
    }

    print $gift, "\n";
    }

    if ($day != 12) {
    print "\n";
    }
    }

    The latter kind of Perl I like to call "good Perl".  It's easy to read, I think.  There are a couple of idioms that take getting used to, just like with any new language, but well-written Perl is (I think) easier to read than any other language.

    But flexibility has its dark sides as well.  Black Perl is the canonical example, but there are others such as Perl golf.  This kind of thing (the first sample above is an example) is responsible for at least part of Perl's reputation for opacity; its compatibility with shell scripting, and most particularly its embedded regular expression support, is responsible for much of the rest.

    Exercise: duplicate the output of the second sample above using as short a Perl program as possible.

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