The GEM library is a variable precision library for MATLAB and GNU Octave. It provides an open source solution for basic high precision computations in standard numerical computing environments.
The library implements two data types:
- gem for high precision dense matrices
- sgem for high precision sparse matrices
and overloads a number of MATLAB/GNU Octave functions. The full library can be downloaded here.
The GEM library is coded in C++ and MATLAB/GNU Octave. It currently relies on GMP for high precision arithmetic (through MPFR and MPFR C++), and on Eigen and Spectra for matrix manipulations.
At the moment, priority is given to functionality over performance: the code comes with no garantee of optimality. Nevertheless, appreciable improvement is already available compared to MATLAB's builtin types performance. With 100 digits of precision, for instance, a 100x100 matrix is transfered from double to high precision 10x faster, from high precision to double format 250x faster, and its column-wise minimum is computed 25x faster. Multiplication of two 100x100 dense matrices with 100-digits precision is 10 times faster with gem objects as compared to MATLAB 2016a's vpa type. For a matrix of size 1000x1000 these ratios become respectively 14x, 1500~20000x, 500x, 10x (the GEM library's multithreading capabilities were disactivated when performing these comparisons).
The GEM library also implements sparse matrix multiplication, which is not available with MATLAB's vpa type.
Here are a few simple examples of GEM library usage.
toStrings(gem('pi',100))prints the 100 first decimals of pigem(2),gem(1.23),gem([1 2 3])create 50-digits precision representations of the numbers 2 and 1.23, and of the vector [1 2 3]. When translating a number from double form, exactly 15 digits are taken into account.gem('1.23456789123456789+2i')creates a 50-digits representation of the number provided in text form (all digits within the working precision are taken into account for numbers specified in string format)gemWorkingPrecision(100)changes the working precision to 100 digits
Once a high precision matrix is created, it can be manipulated by calling usual MATLAB/GNU Octave functions.
eig(gemRand(100,100)): computes the eigenvalues of a random 100x100 matrixeigs(gemRand(100,100),1): computes the largest eigenvalue of a random 100x100 matrixsum(gem([1:100000]).^8)-5e39-2e24gives 111111111177777777771111111111333333333330000 (all digits can be displayed with the help of the functiongemDisplayPrecision)notAnInteger = exp(sqrt(gem(163))*gem('pi')); display(notAnInteger, -1)gives 262537412640768743.9999999999992500725971981856889 (the number of digits displayed can also be specified on a case by case fashion as a parameter to the display fuction; a precision of -1 displays all available digits)sgem(speye(3))creates a high precision sparse representation of the 3x3 identity matrixa=1./gem([1:7]); save('filename','a'); load('filename');saves and loads a gem object- Watch how how 50 digits of the golden ratio are being calculated by the continued fraction formula with
x= gem(1); for i=1:118 x=1/(1+x); disp(num2str(1+x,50)); end; disp(num2str((1+sqrt(gem(5)))/2, 50))
Check out getting started with the GEM library for more usage examples.
The library comes pre-compiled for linux, macos and windows (64bits). It is therefore straightforward to use : after downloading the latest release, just add the gem folder into MATLAB/GNU Octave's path (this can be done by running the command path(path,'/path_to_the_gem_folder')), and it is ready to use.
If you experience trouble with the provided binaries, because for instance you are using a different platform (32 bits?), a significantly older version of the operating system than the one on which the provided binaries were compiled, a significantly older version of MATLAB or GNU Octave, you may use the provided script to compile it. For more details on this, please refer to the compilation instructions.
The GEM library is free and open source. It can be used for both open-source and proprietary application. Therefore, it is also free for academic use. Anyone can contribute on the github page. The source code is distributed under a MPL2 license. See COPYING.md for more details.