I am trying to compare two vectors.
v1 = {0.520974 , 0.438171 , 0.559061}
v2 = [0.520974 , 0.438171 , 0.559061}
I write v1 to a file, read and that's v2. For some reason when I compare the two vectors, I am getting false!
When I do: v1[0]-v2[0] I get 4.3123e-8
Thanks,
Double values, unlike integers, are fragile against write and read. That means, the information that represents them in a string is not necessarily complete.
The leading reason of that is rounding: it's like if you had 1/7 and wanted to write it on a paper in the same format as in your question, you'd get:
0.142857
That's exact to 6 decimal places, but no more than that, and the difference shows up. The only difference in the computer is that it counts in binary (and rounds in binary, too), and is further complicated by the fact that at output (or input) you coerce that into decimal (or back, respectively) and round again at each step. All of those are sources of little errors.
If you want to be able to save and reload your doubles exactly (on the same machine), do it in their native binary representation using a write and read. If you want them to be human-readable, you need to sacrifice the exact reconstruction. You'd then need to compare them up to a little allowed deviation.
Related
I want to write a program to convert hexadecimal numbers into their decimal forms without using a variable of fixed length to store the result because that would restrict the range of inputs that my program can work with.
Let's say I were to use a variable of type long long int to calculate, store and print the result. Doing so would limit the range of hexadecimal numbers that my program can handle to between 8000000000000001 and 7FFFFFFFFFFFFFFF. Anything outside this range would cause the variable to overflow.
I did write a program that calculates and stores the decimal result in a dynamically allocated string by performing carry and borrow operations but it runs much slower, even for numbers that are as big as 7FFFFFFFF!
Then I stumbled onto this site which could take numbers that are way outside the range of a 64 bit variable. I tried their converter with numbers much larger than 16^65 - 1 and still couldn't get it to overflow. It just kept on going and printing the result.
I figured that they must be using a much better algorithm for hex to decimal conversion, one that isn't limited to 64 bit values.
So far, Google's search results have only led me to algorithms that use some fixed-length variable for storing the result.
That's why I am here. I wanna know if such an algorithm exists and if it does, what is it?
Well, it sounds like you already did it when you wrote "a program that calculates and stores the decimal result in a dynamically allocated string by performing carry and borrow operations".
Converting from base 16 (hexadecimal) to base 10 means implementing multiplication and addition of numbers in a base 10x representation. Then for each hex digit d, you calculate result = result*16 + d. When you're done you have the same number in a 10-based representation that is easy to write out as a decimal string.
There could be any number of reasons why your string-based method was slow. If you provide it, I'm sure someone could comment.
The most important trick for making it reasonably fast, though, is to pick the right base to convert to and from. I would probably do the multiplication and addition in base 109, so that each digit will be as large as possible while still fitting into a 32-bit integer, and process 7 hex digits at a time, which is as many as I can while only multiplying by single digits.
For every 7 hex digts, I'd convert them to a number d, and then do result = result * (16^7) + d.
Then I can get the 9 decimal digits for each resulting digit in base 109.
This process is pretty easy, since you only have to multiply by single digits. I'm sure there are faster, more complicated ways that recursively break the number into equal-sized pieces.
I'm looking for something like a hash function but for which it's output is closer the closer two different inputs are?
Something like:
f(1010101) = 0 #original hash
f(1010111) = 1 #very close to the original hash as they differ by one bit
f(0101010) = 9999 #not very close to the original hash they all bits are different
(example outputs for demonstration purposes only)
All of the input data will be of the same length.
I want to make comparisons between a file a lots of other files and be able to determine which other file has the fewest differences from it.
You may try this algorithm.
http://en.wikipedia.org/wiki/Levenshtein_distance
Since this is string only.
You may convert all your binary to string
for example:
0 -> "00000000"
1 -> "00000001"
You might be interested in either simhashing or shingling.
If you are only trying to detect similarity between documents, there are other techniques that may suit you better (like TF-IDF.) The second link is part of a good book whose other chapters delve into general information retrieval topics, including these other techniques.
You should not use a hash for this.
You must compute signatures containing several characteristic values like :
file name
file size
Is binary / Is ascii only
date (if needed)
some other more complex like :
variance of the values of bytes
average value of bytes
average length of same value bits sequence (in compressed files there are no long identical bit sequences)
...
Then you can compare signatures.
But the most important is to know what kind of data is in these files. If it is images, the size and main color are more important. If it is sound, you could analyse only some frequencies...
You might want to look at the source code to unix utilities like cmp or the FileCmp stuff in Python and use that to try to determine a reasonable algorithm.
In my uninformed opinion, calculating a hash is not likely to work well. First, it can be expensive to calculate a hash. Second, what you're trying to do sounds more like a job for encoding than a hash; once you start thinking of it that way, it's not clear that it's even worth transforming the file that way.
If you have some constraints, specifying them might be useful. For example, if all the files are the exact same length, that may simplify things. Or if you are only interested in differences between bits in the same position and not interested in things that are similar only if you compare bits in different positions (e.g., two files are identical, except that one has everything shifted three bits--should those be considered similar or not similar?).
You could calculate the population count of the XOR of the two files, which is exactly the number of bits that are not the same between the two files. So it just does precisely what you asked for, no approximations.
You can represent your data as a binary vector of features and then use dimensionality reduction either with SVD or with random indexing.
What you're looking for is a file fingerprint of sorts. For plain text, something like Nilsimsa (http://ixazon.dynip.com/~cmeclax/nilsimsa.html) works reasonably well.
There are a variety of different names for this type of technique. Fuzzy Hashing/Locality Sensitive Hashing/Distance Based Hashing/Dimensional reduction and a few others. Tools can generate a fixed length output or variable length output, but the outputs are generally comparable (eg by levenshtein distance) and similar inputs yield similar outputs.
The link above for nilsimsa gives two similar spam messages and here are the example outputs:
773e2df0a02a319ec34a0b71d54029111da90838cbc20ecd3d2d4e18c25a3025 spam1
47182cf0802a11dec24a3b75d5042d310ca90838c9d20ecc3d610e98560a3645 spam2
* * ** *** * ** ** ** ** * ******* **** ** * * *
Spamsum and sdhash are more useful for arbitrary binary data. There are also algorithms specifically for images that will work regardless of whether it's a jpg or a png. Identical images in different formats wouldn't be noticed by eg spamsum.
I am writing a utility class which converts strings from one alphabet to another, this is useful in situations where you have a target alphabet you wish to use, with a restriction on the number of characters available. For example, if you can use lower case letters and numbers, but only 12 characters its possible to compress a timestamp from the alphabet 01234567989 -: into abcdefghijklmnopqrstuvwxyz01234567989 so 2010-10-29 13:14:00 might become 5hhyo9v8mk6avy (19 charaters reduced to 16).
The class is designed to convert back and forth between alphabets, and also calculate the longest source string that can safely be stored in a target alphabet given a particular number of characters.
Was thinking of publishing this through Google code, however I'd obviously like other people to find it and use it - hence the question on what this is called. I've had to use this approach in two separate projects, with Bloomberg and a proprietary system, when you need to generate unique file names of a certain length, but want to keep some plaintext, so GUIDs aren't appropriate.
Your examples bear some similarity to a Dictionary coder with a fixed target and source dictionaries. Also worthwhile to look at is Fibonacci coding, which has a fixed target dictionary (of variable-length bits), which is variably targeted.
I think it also depends whether it is very important that your target alphabet has fixed width entries - if you allow for a fixed alphabet with variable length codes, your compression ratio will approach your entropy that much more optimally! If the source alphabet distribution is known in advance, a static Huffman tree could easily be generated.
Here is a simple algorithm:
Consider that you don't have to transmit the alphabet used for encoding. Also, you don't use (and transmit) the probabilities of the input symbols, as in standard compressions, so we just re-encode somehow the data.
In this case we can consider that the input data are in number represented with base equal to the cardinality of the input alphabet. We just have to change its representation to another base, that is a simple task.
EDITED example:
input alpabet: ABC, output alphabet: 0123456789
message ABAC will translate to 0102 in base 3, that is 11 (9 + 2) in base 10.
11 to base 10: 11
We could have a problem decoding it, because we don't know how many 0-es to use at the begining of the decoded result, so we have to use one of the modifications:
1) encode somehow in the stream the size of compressed data.
2) use a dummy 1 at the start of the stream: in this way our example will become:
10102 (base 3) = 81 + 9 + 2 = 92 (base 10).
Now after decoding we just have to ignore the first 1 (this also provides a basic error detection).
The main problem of this approach is that in most cases (GCD == 1) each new encoded character will completely change the output. This will be very inneficient and difficult to implement. We end up with arithmetic coding as the best solution (actually a simplified version of it).
You probably know about Base64 which does the same thing just usually the other way around. Too bad there are way too many Google results on BaseX or BaseN...
I am a beginner trying to do some engineering experiments using fortran 77. I am using Force 2.0 compiler and editor. I have the following queries:
How can I generate a random number between a specified range, e.g. if I need to generate a single random number between 3.0 and 10.0, how can I do that?
How can I use the data from a text file to be called in calculations in my program. e.g I have temperature, pressure and humidity values (hourly values for a day, so total 24 values in each text file).
Do I also need to define in the program how many values are there in the text file?
Knuth has released into the public domain sources in both C and FORTRAN for the pseudo-random number generator described in section 3.6 of The Art of Computer Programming.
2nd question:
If your file, for example, looks like:
hour temperature pressure humidity
00 15 101325 60
01 15 101325 60
... 24 of them, for each hour one
this simple program will read it:
implicit none
integer hour, temp, hum
real p
character(80) junkline
open(unit=1, file='name_of_file.dat', status='old')
rewind(1)
read(1,*)junkline
do 10 i=1,24
read(1,*)hour,temp,p,hum
C do something here ...
10 end
close(1)
end
(the indent is a little screwed up, but I don't know how to set it right in this weird environment)
My advice: read up on data types (INTEGER, REAL, CHARACTER), arrays (DIMENSION), input/output (READ, WRITE, OPEN, CLOSE, REWIND), and loops (DO, FOR), and you'll be doing useful stuff in no time.
I never did anything with random numbers, so I cannot help you there, but I think there are some intrinsic functions in fortran for that. I'll check it out, and report tomorrow. As for the 3rd question, I'm not sure what you ment (you don't know how many lines of data you'll be having in a file ? or ?)
You'll want to check your compiler manual for the specific random number generator function, but chances are it generates random numbers between 0 and 1. This is easy to handle - you just scale the interval to be the proper width, then shift it to match the proper starting point: i.e. to map r in [0, 1] to s in [a, b], use s = r*(b-a) + a, where r is the value you got from your random number generator and s is a random value in the range you want.
Idigas's answer covers your second question well - read in data using formatted input, then use them as you would any other variable.
For your third question, you will need to define how many lines there are in the text file only if you want to do something with all of them - if you're looking at reading the line, processing it, then moving on, you can get by without knowing the number of lines ahead of time. However, if you are looking to store all the values in the file (e.g. having arrays of temperature, humidity, and pressure so you can compute vapor pressure statistics), you'll need to set up storage somehow. Typically in FORTRAN 77, this is done by pre-allocating an array of a size larger than you think you'll need, but this can quickly become problematic. Is there any chance of switching to Fortran 90? The updated version has much better facilities for dealing with standardized dynamic memory allocation, not to mention many other advantages. I would strongly recommend using F90 if at all possible - you will make your life much easier.
Another option, depending on the type of processing you're doing, would be to investigate algorithms that use only single passes through data, so you won't need to store everything to compute things like means and standard deviations, for example.
This subroutine generate a random number in fortran 77 between 0 and ifin
where i is the seed; some great number such as 746397923
subroutine rnd001(xi,i,ifin)
integer*4 i,ifin
real*8 xi
i=i*54891
xi=i*2.328306e-10+0.5D00
xi=xi*ifin
return
end
You may modifies in order to take a certain range.
I have an array of numbers that potentially have up to 8 decimal places and I need to find the smallest common number I can multiply them by so that they are all whole numbers. I need this so all the original numbers can all be multiplied out to the same scale and be processed by a sealed system that will only deal with whole numbers, then I can retrieve the results and divide them by the common multiplier to get my relative results.
Currently we do a few checks on the numbers and multiply by 100 or 1,000,000, but the processing done by the *sealed system can get quite expensive when dealing with large numbers so multiplying everything by a million just for the sake of it isn’t really a great option. As an approximation lets say that the sealed algorithm gets 10 times more expensive every time you multiply by a factor of 10.
What is the most efficient algorithm, that will also give the best possible result, to accomplish what I need and is there a mathematical name and/or formula for what I’m need?
*The sealed system isn’t really sealed. I own/maintain the source code for it but its 100,000 odd lines of proprietary magic and it has been thoroughly bug and performance tested, altering it to deal with floats is not an option for many reasons. It is a system that creates a grid of X by Y cells, then rects that are X by Y are dropped into the grid, “proprietary magic” occurs and results are spat out – obviously this is an extremely simplified version of reality, but it’s a good enough approximation.
So far there are quiet a few good answers and I wondered how I should go about choosing the ‘correct’ one. To begin with I figured the only fair way was to create each solution and performance test it, but I later realised that pure speed wasn’t the only relevant factor – an more accurate solution is also very relevant. I wrote the performance tests anyway, but currently the I’m choosing the correct answer based on speed as well accuracy using a ‘gut feel’ formula.
My performance tests process 1000 different sets of 100 randomly generated numbers.
Each algorithm is tested using the same set of random numbers.
Algorithms are written in .Net 3.5 (although thus far would be 2.0 compatible)
I tried pretty hard to make the tests as fair as possible.
Greg – Multiply by large number
and then divide by GCD – 63
milliseconds
Andy – String Parsing
– 199 milliseconds
Eric – Decimal.GetBits – 160 milliseconds
Eric – Binary search – 32
milliseconds
Ima – sorry I couldn’t
figure out a how to implement your
solution easily in .Net (I didn’t
want to spend too long on it)
Bill – I figure your answer was pretty
close to Greg’s so didn’t implement
it. I’m sure it’d be a smidge faster
but potentially less accurate.
So Greg’s Multiply by large number and then divide by GCD” solution was the second fastest algorithm and it gave the most accurate results so for now I’m calling it correct.
I really wanted the Decimal.GetBits solution to be the fastest, but it was very slow, I’m unsure if this is due to the conversion of a Double to a Decimal or the Bit masking and shifting. There should be a
similar usable solution for a straight Double using the BitConverter.GetBytes and some knowledge contained here: http://blogs.msdn.com/bclteam/archive/2007/05/29/bcl-refresher-floating-point-types-the-good-the-bad-and-the-ugly-inbar-gazit-matthew-greig.aspx but my eyes just kept glazing over every time I read that article and I eventually ran out of time to try to implement a solution.
I’m always open to other solutions if anyone can think of something better.
I'd multiply by something sufficiently large (100,000,000 for 8 decimal places), then divide by the GCD of the resulting numbers. You'll end up with a pile of smallest integers that you can feed to the other algorithm. After getting the result, reverse the process to recover your original range.
Multiple all the numbers by 10
until you have integers.
Divide
by 2,3,5,7 while you still have all
integers.
I think that covers all cases.
2.1 * 10/7 -> 3
0.008 * 10^3/2^3 -> 1
That's assuming your multiplier can be a rational fraction.
If you want to find some integer N so that N*x is also an exact integer for a set of floats x in a given set are all integers, then you have a basically unsolvable problem. Suppose x = the smallest positive float your type can represent, say it's 10^-30. If you multiply all your numbers by 10^30, and then try to represent them in binary (otherwise, why are you even trying so hard to make them ints?), then you'll lose basically all the information of the other numbers due to overflow.
So here are two suggestions:
If you have control over all the related code, find another
approach. For example, if you have some function that takes only
int's, but you have floats, and you want to stuff your floats into
the function, just re-write or overload this function to accept
floats as well.
If you don't have control over the part of your system that requires
int's, then choose a precision to which you care about, accept that
you will simply have to lose some information sometimes (but it will
always be "small" in some sense), and then just multiply all your
float's by that constant, and round to the nearest integer.
By the way, if you're dealing with fractions, rather than float's, then it's a different game. If you have a bunch of fractions a/b, c/d, e/f; and you want a least common multiplier N such that N*(each fraction) = an integer, then N = abc / gcd(a,b,c); and gcd(a,b,c) = gcd(a, gcd(b, c)). You can use Euclid's algorithm to find the gcd of any two numbers.
Greg: Nice solution but won't calculating a GCD that's common in an array of 100+ numbers get a bit expensive? And how would you go about that? Its easy to do GCD for two numbers but for 100 it becomes more complex (I think).
Evil Andy: I'm programing in .Net and the solution you pose is pretty much a match for what we do now. I didn't want to include it in my original question cause I was hoping for some outside the box (or my box anyway) thinking and I didn't want to taint peoples answers with a potential solution. While I don't have any solid performance statistics (because I haven't had any other method to compare it against) I know the string parsing would be relatively expensive and I figured a purely mathematical solution could potentially be more efficient.
To be fair the current string parsing solution is in production and there have been no complaints about its performance yet (its even in production in a separate system in a VB6 format and no complaints there either). It's just that it doesn't feel right, I guess it offends my programing sensibilities - but it may well be the best solution.
That said I'm still open to any other solutions, purely mathematical or otherwise.
What language are you programming in? Something like
myNumber.ToString().Substring(myNumber.ToString().IndexOf(".")+1).Length
would give you the number of decimal places for a double in C#. You could run each number through that and find the largest number of decimal places(x), then multiply each number by 10 to the power of x.
Edit: Out of curiosity, what is this sealed system which you can pass only integers to?
In a loop get mantissa and exponent of each number as integers. You can use frexp for exponent, but I think bit mask will be required for mantissa. Find minimal exponent. Find most significant digits in mantissa (loop through bits looking for last "1") - or simply use predefined number of significant digits.
Your multiple is then something like 2^(numberOfDigits-minMantissa). "Something like" because I don't remember biases/offsets/ranges, but I think idea is clear enough.
So basically you want to determine the number of digits after the decimal point for each number.
This would be rather easier if you had the binary representation of the number. Are the numbers being converted from rationals or scientific notation earlier in your program? If so, you could skip the earlier conversion and have a much easier time. Otherwise you might want to pass each number to a function in an external DLL written in C, where you could work with the floating point representation directly. Or you could cast the numbers to decimal and do some work with Decimal.GetBits.
The fastest approach I can think of in-place and following your conditions would be to find the smallest necessary power-of-ten (or 2, or whatever) as suggested before. But instead of doing it in a loop, save some computation by doing binary search on the possible powers. Assuming a maximum of 8, something like:
int NumDecimals( double d )
{
// make d positive for clarity; it won't change the result
if( d<0 ) d=-d;
// now do binary search on the possible numbers of post-decimal digits to
// determine the actual number as quickly as possible:
if( NeedsMore( d, 10e4 ) )
{
// more than 4 decimals
if( NeedsMore( d, 10e6 ) )
{
// > 6 decimal places
if( NeedsMore( d, 10e7 ) ) return 10e8;
return 10e7;
}
else
{
// <= 6 decimal places
if( NeedsMore( d, 10e5 ) ) return 10e6;
return 10e5;
}
}
else
{
// <= 4 decimal places
// etc...
}
}
bool NeedsMore( double d, double e )
{
// check whether the representation of D has more decimal points than the
// power of 10 represented in e.
return (d*e - Math.Floor( d*e )) > 0;
}
PS: you wouldn't be passing security prices to an option pricing engine would you? It has exactly the flavor...