#include "storm/utility/numerical.h"
#include <cmath>
#include <boost/math/constants/constants.hpp>
#include "storm/utility/macros.h"
#include "storm/utility/constants.h"
#include "storm/exceptions/InvalidArgumentException.h"
#include "storm/exceptions/PrecisionExceededException.h"
namespace storm {
namespace utility {
namespace numerical {
template<typename ValueType>
FoxGlynnResult<ValueType>::FoxGlynnResult() : left(0), right(0), totalWeight(storm::utility::zero<ValueType>()) {
// Intentionally left empty.
}
/*!
* The following implementation of Fox and Glynn's algorithm is taken from David Jansen's patched version
* in MRMC, which is based on his paper:
*
* https://pms.cs.ru.nl/iris-diglib/src/getContent.php?id=2011-Jansen-UnderstandingFoxGlynn
*
* We have only adapted the code to match more of C++'s and our coding guidelines.
*/
template<typename ValueType>
FoxGlynnResult<ValueType> foxGlynnFinder(ValueType lambda, ValueType epsilon) {
ValueType tau = std::numeric_limits<ValueType>::min();
ValueType omega = std::numeric_limits<ValueType>::max();
ValueType const sqrt_2_pi = boost::math::constants::root_two_pi<ValueType>();
ValueType const log10_e = std::log10(boost::math::constants::e<ValueType>());
uint64_t m = static_cast<uint64_t>(lambda);
int64_t left = 0;
int64_t right = 0;
// tau is only used in underflow checks, which we are going to do in the logarithm domain.
tau = log(tau);
// In error bound comparisons, we always compare with epsilon*sqrt_2_pi.
epsilon *= sqrt_2_pi;
// Compute left truncation point.
if (m < 25) {
// For lambda below 25 the exponential can be smaller than tau. If that is the case we expect
// underflows and warn the user.
if (-lambda <= tau) {
STORM_LOG_WARN("Fox-Glynn: 0 < lambda < 25, underflow near Poi(" << lambda << ", 0) = " << std::exp(-lambda) << ". The results are unreliable.");
}
// Zero is used as left truncation point for lambda <= 25.
left = 0;
} else {
// Compute the left truncation point for lambda >= 25 (for lambda < 25 we use zero as left truncation point).
ValueType const bl = (1 + 1 / lambda) * std::exp((1/lambda) * 0.125);
ValueType const sqrt_lambda = std::sqrt(lambda);
int64_t k;
// Start looking for the left truncation point:
// * start search at k=4 (taken from original Fox-Glynn paper)
// * increase the left truncation point until we fulfil the error condition
for (k = 4;; ++k) {
ValueType max_err;
left = m - static_cast<int64_t>(std::ceil(k*sqrt_lambda + 0.5));
// For small lambda the above calculation can yield negative truncation points, crop them here.
if (left <= 0) {
left = 0;
break;
}
// Note that Propositions 2-4 in Fox--Glynn mix up notation: they write Phi where they mean
// 1 - Phi. (In Corollaries 1 and 2, phi is used correctly again.)
max_err = bl * exp(-0.5 * (k*k)) / k;
if (max_err * 2 <= epsilon) {
// If the error on the left hand side is smaller, we can be more lenient on the right hand
// side. To this end, we now set epsilon to the part of the error that has not yet been eaten
// up by the left-hand truncation.
epsilon -= max_err;
break;
}
}
// Finally the left truncation point is found.
}
// Compute right truncation point.
{
ValueType lambda_max;
int64_t m_max, k;
// According to Fox-Glynn, if lambda < 400 we should take lambda = 400, otherwise use the original
// value. This is for computing the right truncation point.
if (m < 400) {
lambda_max = 400;
m_max = 400;
epsilon *= 0.662608824988162441697980;
/* i.e. al = (1+1/400) * exp(1/16) * sqrt_2; epsilon /= al; */
} else {
lambda_max = lambda;
m_max = m;
epsilon *= (1 - 1 / (lambda + 1)) * 0.664265347050632847802225;
/* i.e. al = (1+1/lambda) * exp(1/16) * sqrt_2; epsilon /= al; */
}
// Find right truncation point.
// This loop is a modification to the original Fox-Glynn paper.
// The search for the right truncation point is only terminated by the error condition and not by
// the stop index from the FG paper. This can yield more accurate results if necessary.
for (k = 4;; ++k) {
// dkl_inv is between 1 - 1e-33 and 1 if lambda_max >= 400 and k >= 4; this will always be
// rounded to 1.0. We therefore leave the factor out.
// double dkl_inv=1 - exp(-266/401.0 * (k*sqrt(2*lambda_max) + 1.5));
// actually: "k * (dkl_inv*epsilon/al) >= exp(-0.5 * k^2)", but epsilon has been changed appropriately.
if (k * epsilon >= exp(-0.5*(k*k))) {
break;
}
}
right = m_max + static_cast<int64_t>(std::ceil(k * std::sqrt(2 * lambda_max) + 0.5));
if (right > m_max + static_cast<int64_t>(std::ceil((lambda_max + 1) * 0.5))) {
STORM_LOG_WARN("Fox-Glynn: right = " << right << " >> lambda = " << lambda_max << ", cannot bound the right tail. The results are unreliable.");
}
}
// Time to set the initial value for weights.
FoxGlynnResult<ValueType> fgresult;
fgresult.left = static_cast<uint64_t>(left);
fgresult.right = static_cast<uint64_t>(right);
fgresult.weights.resize(fgresult.right - fgresult.left + 1);
fgresult.weights[m - left] = omega / (1.0e+10 * (right - left));
if (m >= 25) {
// Perform underflow check.
ValueType result, log_c_m_inf;
int64_t i;
// we are going to compare with tau - log(w[m]).
tau -= std::log(fgresult.weights[m - left]);
// We take the c_m_inf = 0.14627 / sqrt( m ), as for lambda >= 25
// c_m = 1 / ( sqrt( 2.0 * pi * m ) ) * exp( m - lambda - 1 / ( 12.0 * m ) ) => c_m_inf.
// Note that m-lambda is in the interval (-1,0], and -1/(12*m) is in [-1/(12*25),0).
// So, exp(m-lambda - 1/(12*m)) is in (exp(-1-1/(12*25)),exp(0)).
// Therefore, we can improve the lower bound on c_m to exp(-1-1/(12*25)) / sqrt(2*pi) = ~0.14627.
// Its logarithm is -1 - 1/(12*25) - log(2*pi) * 0.5 = ~ -1.922272 (rounded towards -infinity).
log_c_m_inf = -1.922272 - log((double) m) * 0.5;
// We use FG's Proposition 6 directly (and not Corollary 4 i and ii), as k_prime may be too large
// if pFG->left == 0.
i = m - left;
// Equivalent to 2*i <= m, equivalent to i <= lambda/2.
if (i <= left) {
// Use Proposition 6 (i). Note that Fox--Glynn are off by one in the proof of this proposition;
// they sum up to i-1, but should have summed up to i. */
result = log_c_m_inf
- i * (i+1) * (0.5 + (2*i+1)/(6*lambda)) / lambda;
} else {
// Use Corollary 4 (iii). Note that k_prime <= sqrt(m+1)/m is a misprint for k_prime <= m/sqrt(m+1),
// which is equivalent to left >= 0, which holds trivially.
result = -lambda;
if (left != 0) {
// Also use Proposition 6 (ii).
double result_1 = log_c_m_inf + i * log(1 - i/(double) (m+1));
// Take the maximum.
if (result_1 > result) {
result = result_1;
}
}
}
if (result <= tau) {
int64_t const log10_result = static_cast<int64_t>(std::floor(result * log10_e));
STORM_LOG_WARN("Fox-Glynn: lambda >= 25, underflow near Poi(" << lambda << "," << left << ") <= " << std::exp(result - log10_result/log10_e) << log10_result << ". The results are unreliable.");
}
// We still have to perform an underflow check for the right truncation point when lambda >= 400.
if (m >= 400) {
// Use Proposition 5 of Fox--Glynn.
i = right - m;
result = log_c_m_inf - i * (i + 1) / (2 * lambda);
if (result <= tau) {
int64_t const log10_result = static_cast<int64_t>(std::floor(result * log10_e));
STORM_LOG_WARN("Fox-Glynn: lambda >= 25, underflow near Poi(" << lambda << "," << right << ") <= " << std::exp(result - log10_result/log10_e) << log10_result << ". The results are unreliable.");
}
}
}
return fgresult;
}
template<typename ValueType>
FoxGlynnResult<ValueType> foxGlynnWeighter(ValueType lambda, ValueType epsilon) {
ValueType tau = std::numeric_limits<ValueType>::min();
// The magic m point.
uint64_t m = static_cast<uint64_t>(lambda);
int64_t j, t;
FoxGlynnResult<ValueType> result = foxGlynnFinder(lambda, epsilon);
// Fill the left side of the array.
for (j = m - result.left; j > 0; --j) {
result.weights[j - 1] = (j + result.left) / lambda * result.weights[j];
}
t = result.right - result.left;
// Fill the right side of the array, have two cases lambda < 400 & lambda >= 400.
if (m < 400) {
// Perform the underflow check, according to Fox-Glynn.
STORM_LOG_ERROR_COND(result.right <= 600, "Fox-Glynn: " << result.right << " > 600, underflow is possible.");
// Compute weights.
for (j = m - result.left; j < t; ++j) {
ValueType q = lambda / (j + 1 + result.left);
if (result.weights[j] > tau / q) {
result.weights[j + 1] = q * result.weights[j];
} else {
t = j;
result.right = j + result.left;
result.weights.resize(result.right - result.left + 1);
// It's time to compute W.
break;
}
}
} else {
// Compute weights.
for (j = m - result.left; j < t; ++j) {
result.weights[j + 1] = lambda / (j + 1 + result.left) * result.weights[j];
}
}
// It is time to compute the normalization weight W.
result.totalWeight = storm::utility::zero<ValueType>();
j = 0;
// t was set above.
while(j < t) {
if (result.weights[j] <= result.weights[t]) {
result.totalWeight += result.weights[j];
j++;
} else {
result.totalWeight += result.weights[t];
t--;
}
}
result.totalWeight += result.weights[j];
STORM_LOG_TRACE("Fox-Glynn: ltp = " << result.left << ", rtp = " << result.right << ", w = " << result.totalWeight << ", " << result.weights.size() << " weights.");
return result;
}
template<typename ValueType>
FoxGlynnResult<ValueType> foxGlynn(ValueType lambda, ValueType epsilon) {
STORM_LOG_THROW(lambda > 0, storm::exceptions::InvalidArgumentException, "Fox-Glynn requires positive lambda.");
return foxGlynnWeighter(lambda, epsilon);
}
template FoxGlynnResult<double> foxGlynn(double lambda, double epsilon);
}
}
}