Computing a function when only its inverse is known, using Newson-Raphson method for 1D,2D,3D arrays in parallel. When user has parallelized version of f(x), it becomes faster. The overall speedup is about 7x for AVX and 15x for AVX512 with x-square function.
Requirements:
- C++14 compiler
- Auto-vectorization of compiler
- Auto-vectorization flags -std=c++14 -O3 -march=native -mavx2 -mprefer-vector-width=256 -ftree-vectorize -fno-math-errno
- Or AVX512 version of flags and 512 vector width
How it works:
- user has an f(x) function given to the constructor
- constructor also takes h(step length) argument for the discrete-derivative calculations such as "two point derivative"/"five point stencil"
- whenever inverse of f(x) at x=x0 is needed, user can call obj.computeInverse...() method
- the computeInverse..() method uses Newton-Raphson method and discrete-derivative to approach inverse f(x0)
Usage:
- parallelized (fast) inversion
InverseFX::ParallelInverse<float> invPar(
// user's parallel f(x) function
// maps one to one from inp to out, for n elements from first element
[](float * inp, float * out, int n){
// C++ compiler vectorizes simple loop easily and possibly inlines this lambda for efficient SIMD
for(int i=0;i<n;i++)
{
out[i]=inp[i]*inp[i]; // square of x
}
},
// h value that is used for computing two-point derivative inside the inversion logic
0.001f
);
float inp[n],outp[n];
// compute inverse of f(x) at all points (n elements), reading from inp and writing result to outp
invPar.computeInverseLowQuality(inp,outp,n); // square-root is found for n-elements- parallelized inversion using scalar f(x) function
InverseFX::ParallelInverse<float> invPar(
// f(x)=x*x, for answering question of "what is inverse of x*x?"
// not as efficient as the parallel f(x) but rest of the algorithm is still parallelized
[](float inp){ return inp*inp;},
0.001f /* h */
);
float inp[n],outp[n];
// compute inverse of f(x) at all points (n elements), reading from inp and writing result to outp
invPar.computeInverseLowQuality(inp,outp,n); - scalar inversion on scalar f(x) function
InverseFX::ScalarInverse<float> inv([](float inp){
// black-box function sample
// f(x)=x*x
return inp*inp;
},0.001f);
// inverse of x*x is sqrt(x)
float squareRootOfPI = inv.computeInverseLowQuality(3.1415f);