Performance bottleneck because of high “erf” function use

问题

I'm using large matrices (100x100 to 3000x3000) to do some claculations (a lot of sums and matrix-vector multiplications), I'm using the Eigen Library for my vectors and matrices. My code is simple C-like code (only functions, no classes) and is going to be compiled as a DLL to be used on Excel.

I identified a bottleneck in the following code :

// Q(z) matrix function
Eigen::MatrixXd qzMatrix(const Eigen::MatrixXd& xjk, const float& riskFreeRate, 
                         const float& volatility, const float& rebalancingPeriod)
{
    int j, k, r = xjk.rows(), c = xjk.cols();
    Eigen::MatrixXd matrix(r, c);
    double mu = (riskFreeRate - volatility * volatility / 2) * rebalancingPeriod;
    double s = volatility * rebalancingPeriod;

    for (j = 0; j <= r - 1; ++j)
        for (k = 0; k <= c - 1; ++k)
            matrix(j, k) = (xjk(j, k) > 0) ? 0.5*(1 + erf(((log(xjk(j, k)) - mu) / s) * M_SQRT1_2)) : 0;

    return matrix;
}

There is also a second function, which is similar to this one, where the erf function takes a different argument (erf function is used here to calculate the standard normal cdf). The two functions take matrices (xjk in this case) of large sizes (usually around 1000x1000), and are looped at least 120 times (1000x1000x2x120 = 240 million calls of erf function). I already tried using the GSL normal CDF function, but is it slighly slower than the native C++ erf function.

Running the algorithm on a 1000x1000 matrix for 120 times takes about 45 seconds. I'm using mingw-w64 4.9.2, CodeBlocks, I have a Windows 7 x64, 4Go RAM, i5.

Is there a way I could speed it this algorithm ?

来源：https://stackoverflow.com/questions/28990706/performance-bottleneck-because-of-high-erf-function-use