## Archive for the ‘Math’ Category

### Round to midnight

A problem I’ve run into a few times is taking the current unix timestamp and rounding it to midnight, so that I can get the unix time for the start of the day. In PHP, I’ve commonly done the following:

`\$timestamp = strtotime('today midnight');`

It’s one of the solutions presented in this StackOverflow post.

The solution above works fine, but I began thinking about how to actually do the computation and bypass the string parsing done by strtotime(). The computation is actually pretty simple, as it’s in the same vein as snapping a point to a grid. The verbose code snippet below shows the step-by-step process in the computation.

`// Given the number of seconds in a day\$numSecondsInDay = 86400;// .. and the current unix time\$currentTime = time();// We can compute the number of days since the unix epoch (the decimal/fractional part is the portion of the current day that's elapsed)\$daysSinceEpoch = \$currentTime / \$numSecondsInDay;// We can throw away the fractional part by rounding down with the floor() function\$wholeDaysSinceEpoch = floor(\$daysSinceEpoch);// The number of whole days since the epoch x the number of seconds in a day will give the time for the current day at midnight\$midnightToday = \$wholeDaysSinceEpoch * \$numSecondsInDay; `

One interesting thing to notice: if you replace the floor() function with the ceil() function, rounding up the number of days since the epoch, you’ll get the start of the next day – midnight tomorrow.

### Monte Carlo integration

I was reading a bit about random numbers and remembered that I wrote a simple Monte Carlo integrator in C++ a few years ago. I took a few minutes to cleanup and comment the code, which is presented below.

Monte Carlo integration is simple, but surprisingly powerful:

$I=∫abf(x)dx$
$I≈∑i=1nf(xi)p(xi)$

xi is a random value within the range [a,b]

p(x) represents the distribution of random values, for a uniform distribution:

$p(x)=1b–a$

This presentation by Fabio Pellacini provides a lot more details.

The test code in the main() method computes the integral of sin2(x) in the interval [3,5].

`#include <iostream>#include <cmath>#include <ctime>using namespace std;// Functor base class for encapsulating 1-dimensional function to be integratedclass Function1d{    public:        virtual double operator()(double x)         {            return x;         }};// Functor for sine squared functionclass SineSquared : public Function1d{    public:        double operator()(double x)        {            return pow(sin(x), 2);        }};// Monte Carlo integrator class declarationclass MonteCarloIntegrator{    public:        double Run(int numSamples, Function1d& func, double intervalMin, double intervalMax);};// Monte Carlo integrator implementation// ::Run() method implementationdouble MonteCarloIntegrator::Run(int numSamples, Function1d& func, double intervalMin, double intervalMax){    double sum = 0;    double div = 1.0 / (double)RAND_MAX;    double intervalScale = 1.0 / (intervalMax-intervalMin);    for(int i=0; i<numSamples; i++)    {        double rnd1 = intervalMin + ( ((double)rand() * div) * (intervalMax-intervalMin) );         sum += (func)(rnd1) / intervalScale;    }    return (1.0/(double)numSamples) * sum;}// main() function with test codevoid main(){    MonteCarloIntegrator    integrator;        double output = integrator.Run(5000000, SineSquared(), 3, 5);    srand(time(NULL));    std::cout << output << endl;    getchar();}`