I've been poring through .NET disassemblies and the GCC source code, but can't seem to find anywhere the actual implementation of sin() and other math functions... they always seem to be referencing something else.
Can anyone help me find them? I feel like it's unlikely that ALL hardware that C will run on supports trig functions in hardware, so there must be a software algorithm somewhere, right?
I'm aware of several ways that functions can be calculated, and have written my own routines to compute functions using taylor series for fun. I'm curious about how real, production languages do it, since all of my implementations are always several orders of magnitude slower, even though I think my algorithms are pretty clever (obviously they're not).
In GNU libm, the implementation of sin is system-dependent. Therefore you can find the implementation, for each platform, somewhere in the appropriate subdirectory of sysdeps.
One directory includes an implementation in C, contributed by IBM. Since October 2011, this is the code that actually runs when you call sin() on a typical x86-64 Linux system. It is apparently faster than the fsin assembly instruction. Source code: sysdeps/ieee754/dbl-64/s_sin.c, look for __sin (double x).
This code is very complex. No one software algorithm is as fast as possible and also accurate over the whole range of x values, so the library implements several different algorithms, and its first job is to look at x and decide which algorithm to use.
When x is very very close to 0, sin(x) == x is the right answer.
A bit further out, sin(x) uses the familiar Taylor series. However, this is only accurate near 0, so...
When the angle is more than about 7°, a different algorithm is used, computing Taylor-series approximations for both sin(x) and cos(x), then using values from a precomputed table to refine the approximation.
When |x| > 2, none of the above algorithms would work, so the code starts by computing some value closer to 0 that can be fed to sin or cos instead.
There's yet another branch to deal with x being a NaN or infinity.
This code uses some numerical hacks I've never seen before, though for all I know they might be well-known among floating-point experts. Sometimes a few lines of code would take several paragraphs to explain. For example, these two lines
double t = (x * hpinv + toint);
double xn = t - toint;
are used (sometimes) in reducing x to a value close to 0 that differs from x by a multiple of π/2, specifically xn × π/2. The way this is done without division or branching is rather clever. But there's no comment at all!
Older 32-bit versions of GCC/glibc used the fsin instruction, which is surprisingly inaccurate for some inputs. There's a fascinating blog post illustrating this with just 2 lines of code.
fdlibm's implementation of sin in pure C is much simpler than glibc's and is nicely commented. Source code: fdlibm/s_sin.c and fdlibm/k_sin.c
Functions like sine and cosine are implemented in microcode inside microprocessors. Intel chips, for example, have assembly instructions for these. A C compiler will generate code that calls these assembly instructions. (By contrast, a Java compiler will not. Java evaluates trig functions in software rather than hardware, and so it runs much slower.)
Chips do not use Taylor series to compute trig functions, at least not entirely. First of all they use CORDIC, but they may also use a short Taylor series to polish up the result of CORDIC or for special cases such as computing sine with high relative accuracy for very small angles. For more explanation, see this StackOverflow answer.
OK kiddies, time for the pros....
This is one of my biggest complaints with inexperienced software engineers. They come in calculating transcendental functions from scratch (using Taylor's series) as if nobody had ever done these calculations before in their lives. Not true. This is a well defined problem and has been approached thousands of times by very clever software and hardware engineers and has a well defined solution.
Basically, most of the transcendental functions use Chebyshev Polynomials to calculate them. As to which polynomials are used depends on the circumstances. First, the bible on this matter is a book called "Computer Approximations" by Hart and Cheney. In that book, you can decide if you have a hardware adder, multiplier, divider, etc, and decide which operations are fastest. e.g. If you had a really fast divider, the fastest way to calculate sine might be P1(x)/P2(x) where P1, P2 are Chebyshev polynomials. Without the fast divider, it might be just P(x), where P has much more terms than P1 or P2....so it'd be slower. So, first step is to determine your hardware and what it can do. Then you choose the appropriate combination of Chebyshev polynomials (is usually of the form cos(ax) = aP(x) for cosine for example, again where P is a Chebyshev polynomial). Then you decide what decimal precision you want. e.g. if you want 7 digits precision, you look that up in the appropriate table in the book I mentioned, and it will give you (for precision = 7.33) a number N = 4 and a polynomial number 3502. N is the order of the polynomial (so it's p4.x^4 + p3.x^3 + p2.x^2 + p1.x + p0), because N=4. Then you look up the actual value of the p4,p3,p2,p1,p0 values in the back of the book under 3502 (they'll be in floating point). Then you implement your algorithm in software in the form:
(((p4.x + p3).x + p2).x + p1).x + p0
....and this is how you'd calculate cosine to 7 decimal places on that hardware.
Note that most hardware implementations of transcendental operations in an FPU usually involve some microcode and operations like this (depends on the hardware).
Chebyshev polynomials are used for most transcendentals but not all. e.g. Square root is faster to use a double iteration of Newton raphson method using a lookup table first.
Again, that book "Computer Approximations" will tell you that.
If you plan on implmementing these functions, I'd recommend to anyone that they get a copy of that book. It really is the bible for these kinds of algorithms.
Note that there are bunches of alternative means for calculating these values like cordics, etc, but these tend to be best for specific algorithms where you only need low precision. To guarantee the precision every time, the chebyshev polynomials are the way to go. Like I said, well defined problem. Has been solved for 50 years now.....and thats how it's done.
Now, that being said, there are techniques whereby the Chebyshev polynomials can be used to get a single precision result with a low degree polynomial (like the example for cosine above). Then, there are other techniques to interpolate between values to increase the accuracy without having to go to a much larger polynomial, such as "Gal's Accurate Tables Method". This latter technique is what the post referring to the ACM literature is referring to. But ultimately, the Chebyshev Polynomials are what are used to get 90% of the way there.
Enjoy.
For sin specifically, using Taylor expansion would give you:
sin(x) := x - x^3/3! + x^5/5! - x^7/7! + ... (1)
you would keep adding terms until either the difference between them is lower than an accepted tolerance level or just for a finite amount of steps (faster, but less precise). An example would be something like:
float sin(float x)
{
float res=0, pow=x, fact=1;
for(int i=0; i<5; ++i)
{
res+=pow/fact;
pow*=-1*x*x;
fact*=(2*(i+1))*(2*(i+1)+1);
}
return res;
}
Note: (1) works because of the aproximation sin(x)=x for small angles. For bigger angles you need to calculate more and more terms to get acceptable results.
You can use a while argument and continue for a certain accuracy:
double sin (double x){
int i = 1;
double cur = x;
double acc = 1;
double fact= 1;
double pow = x;
while (fabs(acc) > .00000001 && i < 100){
fact *= ((2*i)*(2*i+1));
pow *= -1 * x*x;
acc = pow / fact;
cur += acc;
i++;
}
return cur;
}
Concerning trigonometric function like sin(), cos(),tan() there has been no mention, after 5 years, of an important aspect of high quality trig functions: Range reduction.
An early step in any of these functions is to reduce the angle, in radians, to a range of a 2*π interval. But π is irrational so simple reductions like x = remainder(x, 2*M_PI) introduce error as M_PI, or machine pi, is an approximation of π. So, how to do x = remainder(x, 2*π)?
Early libraries used extended precision or crafted programming to give quality results but still over a limited range of double. When a large value was requested like sin(pow(2,30)), the results were meaningless or 0.0 and maybe with an error flag set to something like TLOSS total loss of precision or PLOSS partial loss of precision.
Good range reduction of large values to an interval like -π to π is a challenging problem that rivals the challenges of the basic trig function, like sin(), itself.
A good report is Argument reduction for huge arguments: Good to the last bit (1992). It covers the issue well: discusses the need and how things were on various platforms (SPARC, PC, HP, 30+ other) and provides a solution algorithm the gives quality results for all double from -DBL_MAX to DBL_MAX.
If the original arguments are in degrees, yet may be of a large value, use fmod() first for improved precision. A good fmod() will introduce no error and so provide excellent range reduction.
// sin(degrees2radians(x))
sin(degrees2radians(fmod(x, 360.0))); // -360.0 < fmod(x,360) < +360.0
Various trig identities and remquo() offer even more improvement. Sample: sind()
Yes, there are software algorithms for calculating sin too. Basically, calculating these kind of stuff with a digital computer is usually done using numerical methods like approximating the Taylor series representing the function.
Numerical methods can approximate functions to an arbitrary amount of accuracy and since the amount of accuracy you have in a floating number is finite, they suit these tasks pretty well.
Use Taylor series and try to find relation between terms of the series so you don't calculate things again and again
Here is an example for cosinus:
double cosinus(double x, double prec)
{
double t, s ;
int p;
p = 0;
s = 1.0;
t = 1.0;
while(fabs(t/s) > prec)
{
p++;
t = (-t * x * x) / ((2 * p - 1) * (2 * p));
s += t;
}
return s;
}
using this we can get the new term of the sum using the already used one (we avoid the factorial and x2p)
It is a complex question. Intel-like CPU of the x86 family have a hardware implementation of the sin() function, but it is part of the x87 FPU and not used anymore in 64-bit mode (where SSE2 registers are used instead). In that mode, a software implementation is used.
There are several such implementations out there. One is in fdlibm and is used in Java. As far as I know, the glibc implementation contains parts of fdlibm, and other parts contributed by IBM.
Software implementations of transcendental functions such as sin() typically use approximations by polynomials, often obtained from Taylor series.
Chebyshev polynomials, as mentioned in another answer, are the polynomials where the largest difference between the function and the polynomial is as small as possible. That is an excellent start.
In some cases, the maximum error is not what you are interested in, but the maximum relative error. For example for the sine function, the error near x = 0 should be much smaller than for larger values; you want a small relative error. So you would calculate the Chebyshev polynomial for sin x / x, and multiply that polynomial by x.
Next you have to figure out how to evaluate the polynomial. You want to evaluate it in such a way that the intermediate values are small and therefore rounding errors are small. Otherwise the rounding errors might become a lot larger than errors in the polynomial. And with functions like the sine function, if you are careless then it may be possible that the result that you calculate for sin x is greater than the result for sin y even when x < y. So careful choice of the calculation order and calculation of upper bounds for the rounding error are needed.
For example, sin x = x - x^3/6 + x^5 / 120 - x^7 / 5040... If you calculate naively sin x = x * (1 - x^2/6 + x^4/120 - x^6/5040...), then that function in parentheses is decreasing, and it will happen that if y is the next larger number to x, then sometimes sin y will be smaller than sin x. Instead, calculate sin x = x - x^3 * (1/6 - x^2 / 120 + x^4/5040...) where this cannot happen.
When calculating Chebyshev polynomials, you usually need to round the coefficients to double precision, for example. But while a Chebyshev polynomial is optimal, the Chebyshev polynomial with coefficients rounded to double precision is not the optimal polynomial with double precision coefficients!
For example for sin (x), where you need coefficients for x, x^3, x^5, x^7 etc. you do the following: Calculate the best approximation of sin x with a polynomial (ax + bx^3 + cx^5 + dx^7) with higher than double precision, then round a to double precision, giving A. The difference between a and A would be quite large. Now calculate the best approximation of (sin x - Ax) with a polynomial (b x^3 + cx^5 + dx^7). You get different coefficients, because they adapt to the difference between a and A. Round b to double precision B. Then approximate (sin x - Ax - Bx^3) with a polynomial cx^5 + dx^7 and so on. You will get a polynomial that is almost as good as the original Chebyshev polynomial, but much better than Chebyshev rounded to double precision.
Next you should take into account the rounding errors in the choice of polynomial. You found a polynomial with minimum error in the polynomial ignoring rounding error, but you want to optimise polynomial plus rounding error. Once you have the Chebyshev polynomial, you can calculate bounds for the rounding error. Say f (x) is your function, P (x) is the polynomial, and E (x) is the rounding error. You don't want to optimise | f (x) - P (x) |, you want to optimise | f (x) - P (x) +/- E (x) |. You will get a slightly different polynomial that tries to keep the polynomial errors down where the rounding error is large, and relaxes the polynomial errors a bit where the rounding error is small.
All this will get you easily rounding errors of at most 0.55 times the last bit, where +,-,*,/ have rounding errors of at most 0.50 times the last bit.
The actual implementation of library functions is up to the specific compiler and/or library provider. Whether it's done in hardware or software, whether it's a Taylor expansion or not, etc., will vary.
I realize that's absolutely no help.
There's nothing like hitting the source and seeing how someone has actually done it in a library in common use; let's look at one C library implementation in particular. I chose uLibC.
Here's the sin function:
http://git.uclibc.org/uClibc/tree/libm/s_sin.c
which looks like it handles a few special cases, and then carries out some argument reduction to map the input to the range [-pi/4,pi/4], (splitting the argument into two parts, a big part and a tail) before calling
http://git.uclibc.org/uClibc/tree/libm/k_sin.c
which then operates on those two parts.
If there is no tail, an approximate answer is generated using a polynomial of degree 13.
If there is a tail, you get a small corrective addition based on the principle that sin(x+y) = sin(x) + sin'(x')y
They are typically implemented in software and will not use the corresponding hardware (that is, aseembly) calls in most cases. However, as Jason pointed out, these are implementation specific.
Note that these software routines are not part of the compiler sources, but will rather be found in the correspoding library such as the clib, or glibc for the GNU compiler. See http://www.gnu.org/software/libc/manual/html_mono/libc.html#Trig-Functions
If you want greater control, you should carefully evaluate what you need exactly. Some of the typical methods are interpolation of look-up tables, the assembly call (which is often slow), or other approximation schemes such as Newton-Raphson for square roots.
If you want an implementation in software, not hardware, the place to look for a definitive answer to this question is Chapter 5 of Numerical Recipes. My copy is in a box, so I can't give details, but the short version (if I remember this right) is that you take tan(theta/2) as your primitive operation and compute the others from there. The computation is done with a series approximation, but it's something that converges much more quickly than a Taylor series.
Sorry I can't rembember more without getting my hand on the book.
Whenever such a function is evaluated, then at some level there is most likely either:
A table of values which is interpolated (for fast, inaccurate applications - e.g. computer graphics)
The evaluation of a series that converges to the desired value --- probably not a taylor series, more likely something based on a fancy quadrature like Clenshaw-Curtis.
If there is no hardware support then the compiler probably uses the latter method, emitting only assembler code (with no debug symbols), rather than using a c library --- making it tricky for you to track the actual code down in your debugger.
If you want to look at the actual GNU implementation of those functions in C, check out the latest trunk of glibc. See the GNU C Library.
As many people pointed out, it is implementation dependent. But as far as I understand your question, you were interested in a real software implemetnation of math functions, but just didn't manage to find one. If this is the case then here you are:
Download glibc source code from http://ftp.gnu.org/gnu/glibc/
Look at file dosincos.c located in unpacked glibc root\sysdeps\ieee754\dbl-64 folder
Similarly you can find implementations of the rest of the math library, just look for the file with appropriate name
You may also have a look at the files with the .tbl extension, their contents is nothing more than huge tables of precomputed values of different functions in a binary form. That is why the implementation is so fast: instead of computing all the coefficients of whatever series they use they just do a quick lookup, which is much faster. BTW, they do use Tailor series to calculate sine and cosine.
I hope this helps.
I'll try to answer for the case of sin() in a C program, compiled with GCC's C compiler on a current x86 processor (let's say a Intel Core 2 Duo).
In the C language the Standard C Library includes common math functions, not included in the language itself (e.g. pow, sin and cos for power, sine, and cosine respectively). The headers of which are included in math.h.
Now on a GNU/Linux system, these libraries functions are provided by glibc (GNU libc or GNU C Library). But the GCC compiler wants you to link to the math library (libm.so) using the -lm compiler flag to enable usage of these math functions. I'm not sure why it isn't part of the standard C library. These would be a software version of the floating point functions, or "soft-float".
Aside: The reason for having the math functions separate is historic, and was merely intended to reduce the size of executable programs in very old Unix systems, possibly before shared libraries were available, as far as I know.
Now the compiler may optimize the standard C library function sin() (provided by libm.so) to be replaced with an call to a native instruction to your CPU/FPU's built-in sin() function, which exists as an FPU instruction (FSIN for x86/x87) on newer processors like the Core 2 series (this is correct pretty much as far back as the i486DX). This would depend on optimization flags passed to the gcc compiler. If the compiler was told to write code that would execute on any i386 or newer processor, it would not make such an optimization. The -mcpu=486 flag would inform the compiler that it was safe to make such an optimization.
Now if the program executed the software version of the sin() function, it would do so based on a CORDIC (COordinate Rotation DIgital Computer) or BKM algorithm, or more likely a table or power-series calculation which is commonly used now to calculate such transcendental functions. [Src: http://en.wikipedia.org/wiki/Cordic#Application]
Any recent (since 2.9x approx.) version of gcc also offers a built-in version of sin, __builtin_sin() that it will used to replace the standard call to the C library version, as an optimization.
I'm sure that is as clear as mud, but hopefully gives you more information than you were expecting, and lots of jumping off points to learn more yourself.
Don't use Taylor series. Chebyshev polynomials are both faster and more accurate, as pointed out by a couple of people above. Here is an implementation (originally from the ZX Spectrum ROM): https://albertveli.wordpress.com/2015/01/10/zx-sine/
Computing sine/cosine/tangent is actually very easy to do through code using the Taylor series. Writing one yourself takes like 5 seconds.
The whole process can be summed up with this equation here:
Here are some routines I wrote for C:
double _pow(double a, double b) {
double c = 1;
for (int i=0; i<b; i++)
c *= a;
return c;
}
double _fact(double x) {
double ret = 1;
for (int i=1; i<=x; i++)
ret *= i;
return ret;
}
double _sin(double x) {
double y = x;
double s = -1;
for (int i=3; i<=100; i+=2) {
y+=s*(_pow(x,i)/_fact(i));
s *= -1;
}
return y;
}
double _cos(double x) {
double y = 1;
double s = -1;
for (int i=2; i<=100; i+=2) {
y+=s*(_pow(x,i)/_fact(i));
s *= -1;
}
return y;
}
double _tan(double x) {
return (_sin(x)/_cos(x));
}
Improved version of code from Blindy's answer
#define EPSILON .0000000000001
// this is smallest effective threshold, at least on my OS (WSL ubuntu 18)
// possibly because factorial part turns 0 at some point
// and it happens faster then series element turns 0;
// validation was made against sin() from <math.h>
double ft_sin(double x)
{
int k = 2;
double r = x;
double acc = 1;
double den = 1;
double num = x;
// precision drops rapidly when x is not close to 0
// so move x to 0 as close as possible
while (x > PI)
x -= PI;
while (x < -PI)
x += PI;
if (x > PI / 2)
return (ft_sin(PI - x));
if (x < -PI / 2)
return (ft_sin(-PI - x));
// not using fabs for performance reasons
while (acc > EPSILON || acc < -EPSILON)
{
num *= -x * x;
den *= k * (k + 1);
acc = num / den;
r += acc;
k += 2;
}
return (r);
}
The essence of how it does this lies in this excerpt from Applied Numerical Analysis by Gerald Wheatley:
When your software program asks the computer to get a value of
or , have you wondered how it can get the
values if the most powerful functions it can compute are polynomials?
It doesnt look these up in tables and interpolate! Rather, the
computer approximates every function other than polynomials from some
polynomial that is tailored to give the values very accurately.
A few points to mention on the above is that some algorithms do infact interpolate from a table, albeit only for the first few iterations. Also note how it mentions that computers utilise approximating polynomials without specifying which type of approximating polynomial. As others in the thread have pointed out, Chebyshev polynomials are more efficient than Taylor polynomials in this case.
if you want sin then
__asm__ __volatile__("fsin" : "=t"(vsin) : "0"(xrads));
if you want cos then
__asm__ __volatile__("fcos" : "=t"(vcos) : "0"(xrads));
if you want sqrt then
__asm__ __volatile__("fsqrt" : "=t"(vsqrt) : "0"(value));
so why use inaccurate code when the machine instructions will do?
I'm in the process of converting a program to C++ from Scilab (similar to Matlab) and I'm required to maintain the same level of precision that is kept by the previous code.
Note: Although maintaining the same level of precision would be ideal. It's acceptable if there is some error with the finished result. The problem I'm facing (as I'll show below) is due to looping, so the calculation error compounds rather quickly. But if the final result is only a thousandth or so off (e.g. 1/1000 vs 1/1001) it won't be a problem.
I've briefly looked into a number of different ways to do this including:
GMP (A Multiple Precision
Arithmetic Library)
Using integers instead of floats (see example below)
Int vs Float Example: Instead of using the float 12.45, store it as an integer being 124,500. Then simply convert everything back when appropriate to do so. Note: I'm not exactly sure how this will work with the code I'm working with (more detail below).
An example of how my program is producing incorrect results:
for (int i = 0; i <= 1000; i++)
{
for (int j = 0; j <= 10000; j++)
{
// This calculation will be computed with less precision than in Scilab
float1 = (1.0 / 100000.0);
// The above error of float2 will become significant by the end of the loop
float2 = (float1 + float2);
}
}
My question is:
Is there a generally accepted way to go about retaining accuracy in floating point arithmetic OR will one of the above methods suffice?
Maintaining precision when porting code like this is very difficult to do. Not because the languages have implicitly different perspectives on what a float is, but because of what the different algorithms or assumptions of accuracy limits are. For example, when performing numerical integration in Scilab, it may use a Gaussian quadrature method. Whereas you might try using a trapezoidal method. The two may both be working on identical IEEE754 single-precision floating point numbers, but you will get different answers due to the convergence characteristics of the two algorithms. So how do you get around this?
Well, you can go through the Scilab source code and look at all of the algorithms it uses for each thing you need. You can then replicate these algorithms taking care of any pre- or post-conditioning of the data that Scilab implicitly does (if any at all). That's a lot of work. And, frankly, probably not the best way to spend your time. Rather, I would look into using the Interfacing with Other Languages section from the developer's documentation to see how you can call the Scilab functions directly from your C, C++, Java, or Fortran code.
Of course, with the second option, you have to consider how you are going to distribute your code (if you need to).Scilab has a GPL-compatible license, so you can just bundle it with your code. However, it is quite big (~180MB) and you may want to just bundle the pieces you need (e.g., you don't need the whole interpreter system). This is more work in a different way, but guarantees numerical-compatibility with your current Scilab solutions.
Is there a generally accepted way to go about retaining accuracy in floating
point arithmetic
"Generally accepted" is too broad, so no.
will one of the above methods suffice?
Yes. Particularly gmp seems to be a standard choice. I would also have a look at the Boost Multiprecision library.
A hand-coded integer approach can work as well, but is surely not the method of choice: it requires much more coding, and more severe a means to store and process aritrarily precise integers.
If your compiler supports it use BCD (Binary-coded decimal)
Sam
Well, another alternative if you use GCC compilers is to go with quadmath/__float128 types.
According to this link and this one, it is said that opencv is much faster than matlab. First link is written in March 2012, second one is a bit later than that.
In the first link, it says, "Programs written in OpenCV run much faster than similar programs written in Matlab." and rates Matlab: 2/10 and OpenCV: 9/10
Consider, I have two float Matrix whose size are 1024*1024(mat1 and mat2). I want to correlate this matrices.
In matlab,
corr2(mat1,mat2); //70-75 ms
In opencv, c++
Mat result(1,1,CV_32F);
matchTemplate(mat1,mat2,result, CV_TM_CCOEFF_NORMED); // 145-150 ms
As far as I know, c and c++ are in approximately same speed.
So, I wonder, why matlab is faster than opencv/c++ while doing cross correlation. Is it because I am comparing wrong things (even though the results are same) or is the cross correlation implementation of matlab is double faster than opencv implementation?
Note that, I'm using Matlab 2013a and Visual Studio 2010.
Thanks,
Matlab built in functions comes with mkl and opencv's dont. So if two exactly equivalent functions are present in both, matlab is likely to be faster(much) than opencv. I have tried to do pseudo inverse on a large matrix and matlab beat everything(openblas,Armadillo,self integrated mkl etc) by at least 2 times. Then I just stopped figuring out why and just load the data into matlab and let it do the thing. opencv is by far the slowest. Try matrix multiplication on a 10000x10000 matrix in opencv. it took 10 minutes on my laptop. Matlab took 1 minute.
Matlab is not as bad as you may think at doing matrix calculations. For many of the Basic Linear Algebra operation Matlab is calling rutines written in fortran and c++. So as long as you dont use loops and formulate it in matrix operations Matlab is actually very fast.
http://www.mathworks.se/company/newsletters/articles/matlab-incorporates-lapack.html
In your scenario, there is no reason to expect matlab to be slower. You are calling a single function, the overhead caused by the language interpreter and passing the data to a native function (mex function) has to be paid only once.
If you would call the same function 1024 times for a small 32*32 matrices, you will probably notice the overhead (unless the JIT-Compiler finds a neat trick to optimize the code).
Matlab can be fast if you vectorize everything and use native functions.
But if you would do some operations in a loop i.e.
A = zeros(100,100);
for m = 1:100
for n = 1:100
A(m, n) = 1/(m + n - 1);
end
end
vs.
Mat A(100, 100, CV_64F);
for (int r = 0; r < A.rows; r++)
for (int c = 0; c < A.cols; c++)
A.at<double>(r, c) = 1 / (r + c - 1);
you would notice the difference.
For correlation functions (and many more) matlab uses an advance libraries which uses an advanced instruction set.
However Matlab is smart than you think, Matlab Checks on runtime if the operation would execute faster on spatial domain or frequency domain, than execute fastest solution.
I couldn't find a mention for corr2, however I found for normxcorr2
Calculate cross-correlation in the spatial or the frequency domain, depending on size of images.
This question already has answers here:
Optimized matrix multiplication in C
(14 answers)
Closed 4 years ago.
I need to perform a lot of matrix operations in my application. The most time consuming is matrix multiplication. I implemented it this way
template<typename T>
Matrix<T> Matrix<T>::operator * (Matrix& matrix)
{
Matrix<T> multipliedMatrix = Matrix<T>(this->rows,matrix.GetColumns(),0);
for (int i=0;i<this->rows;i++)
{
for (int j=0;j<matrix.GetColumns();j++)
{
multipliedMatrix.datavector.at(i).at(j) = 0;
for (int k=0;k<this->columns ;k++)
{
multipliedMatrix.datavector.at(i).at(j) += datavector.at(i).at(k) * matrix.datavector.at(k).at(j);
}
//cout<<(*multipliedMatrix)[i][j]<<endl;
}
}
return multipliedMatrix;
}
Is there any way to write it in a better way?? So far matrix multiplication operations take most of time in my application. Maybe is there good/fast library for doing this kind of stuff ??
However I rather can't use libraries which uses graphic card for mathematical operations, because of the fact that I work on laptop with integrated graphic card.
Eigen is by far one of the fastest, if not the fastest, linear algebra libraries out there. It is well written and it is of high quality. Also, it uses expression template which makes writing code that is more readable. Version 3 just released uses OpenMP for data parallelism.
#include <iostream>
#include <Eigen/Dense>
using Eigen::MatrixXd;
int main()
{
MatrixXd m(2,2);
m(0,0) = 3;
m(1,0) = 2.5;
m(0,1) = -1;
m(1,1) = m(1,0) + m(0,1);
std::cout << m << std::endl;
}
Boost uBLAS I think is definitely the way to go with this sort of thing. Boost is well designed, well tested and used in a lot of applications.
Consider GNU Scientific Library, or MV++
If you're okay with C, BLAS is a low-level library that incorporates both C and C-wrapped FORTRAN instructions and is used a huge number of higher-level math libraries.
I don't know anything about this, but another option might be Meschach which seems to have decent performance.
Edit: With respect to your comment about not wanting to use libraries that use your graphics card, I'll point out that in many cases, the libraries that use your graphics card are specialized implementations of standard (non-GPU) libraries. For example, various implementations of BLAS are listed on it's Wikipedia page, only some are designed to leverage your GPU.
There is a book called Introduction to Algorithms. You may like to check the chapter of Dynamic Programming. It has an excellent matrix multiplication algo using dynamic programming. Its worth a read. Well, this info was in case you want to write your own logic instead of using a library.
There are plenty of algorithms for efficient matrix multiplication.
Algorithms for efficient matrix multiplication
Look at the algorithms, find an implementations.
You can also make a multi-threaded implementation for it.
In a C++ application I'm coding, I need to solve a system of non-linear equations (N equations, N unknowns).
The systems I'm solving will be rather small (up to 10 equations/unknowns), so performance is not going to be a real issue.
I've searched the web a bit for a non-linear solver library, and I couldn't get to something which looks easy to use (got to NOX and C/C++ Minpack, but both seem to be an overkill for my need).
Any thoughts and ideas of easy-to-use libraries for this purpose?
One thing should be clear: non-linear equation solution isn't easy. It's not the same as solving linear equations. You aren't always guaranteed to get a solution. And your choice of initial condition and incrementation strategy can have a profound effect on the solution you do get.
With that said, I can't recommend a particular library, but you should be on the lookout for a linear algebra package that includes Newton-Raphson iteration in its menu of choices.
There are two options for you, you can use the sundials packages which includes a nonlinear solver, written in C I think. The only problem I've found with it is that you need to give it good initial estimates. The second option is to use NLEQ or NLEQ2 which I think are superior (writtein in FORTRAN but easy to link to C like langages. However I have had some problems locating it just now. There is a good web site with a list of possible options at: http://plato.asu.edu/sub/zero.html
Numerical Recipes has a routine that will do the job for you.
It depends on how non-linear the equations are. If they possess some "nice" properties...most obvious being positive-semi-definite matrix or convexity, there may be specialized algorithms available. I use IBM/ILOG CPLEX for most of my linear programming needs. Libraries are provided that can be pulled into C++ applications. Although I have not used their quadratic programming module, it is really the state-of-the-art in high horse-power linear and (well-behaved) non-linear programming.
There is always GSL, but all the comments made in the other answers apply to this as well:
http://www.gnu.org/software/gsl/manual/html_node/Multidimensional-Root_002dFinding.html#index-nonlinear-systems-of-equations_002c-solution-of-2426
Have you looked at COIN-OR? It might help if you submit your question to the OR-Exchange.
It's not free by any means, but Solver would work here.
Microsoft Z3 https://github.com/Z3Prover/z3/blob/master/examples/c%2B%2B/example.cpp
also consider omnn::math:
https://github.com/ohhmm/openmind/blob/master/omnn/math/test/08_System.cpp
Lets say system of equations is like this:
(x-a1)^2 + (y-b1)^2 = c1
(x-a2)^2 + (y-b2)^2 = c2
Then you have couple options:
Valuable a1, a2, b1, b2; // init with values
System sys;
Variable x,y;
sys << (x-a1)^2 + (y-b1)^2 - c1; // addin an equation as an equality to 0
sys << (x-a2)^2 + (y-b2)^2 - c2;
for(auto& solution : sys.Solve(x))
std::cout << solution;
alternative way is to make single equation (see why):
((x-a1)^2 + (y-b1)^2 - c1)^2 + ((x-a2)^2 + (y-b2)^2 - c2)^2 = 0
Variable x,y;
Valuable a1, a2, b1, b2; // init with values
auto eq = ((x-a1)^2 + (y-b1)^2 - c1)^2 + ((x-a2)^2 + (y-b2)^2 - c2)^2;
eq.SetView(Valuable::View::Equation); // optional: equation optimizations
// get y function:
auto fn = eq(y);
// show
std::cout << fn << std::endl;
// evaluate
auto evaluate = fn;
evaluate.eval(x, 10);
evaluate.optimize(); // calculate
// show calculated value at x=10:
std::cout << evaluate << std::endl;