InverseFX

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);