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1F1: begin to document error rates and testing methodology.

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jzmaddock
2019-03-14 12:04:42 +00:00
parent e07c435a0c
commit 092fcf278f
10 changed files with 23131 additions and 2 deletions
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10
doc/Jamfile.v2
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@@ -6,11 +6,18 @@
using quickbook ;
using auto-index ;
import modules ;
import regex ;
path-constant images_location : html ;
path-constant here : . ;
constant here-url : [ regex.replace $(here) "\\\\" "/" ] ;
xml math : math.qbk ;
xml math : math.qbk :
<quickbook-define>enable_index
<quickbook-define>__base_path__=$(here-url)
<format>pdf:<quickbook-define>__build_pdf
<format>html:<quickbook-define>__build_html
;
boostbook standalone
:
math
@@ -75,7 +82,6 @@ boostbook standalone
<auto-index-script>$(here)/index.idx
<auto-index-prefix>$(here)/../../..
#<auto-index-verbose>on
<quickbook-define>enable_index
<format>pdf:<xsl:param>index.on.type=1
;

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@@ -175,6 +175,42 @@ over it's entire domain are not yet available. We have worked extremely hard to
or where the series become divergent after a few terms limiting the precision that can be achieved.][[sub 1]F[sub 1](-5.9981750131794866e-15, 0.499999999999994, -240.42092034220695)]]
]
[h4 Testing]
There are 3 main groups of tests:
* Spot tests for special inputs with known values.
* Sanity checks which use integrals of hypergeometric functions of known value. These are particularly useful
since for fixed ['a] and ['b] they evaluate ['[sub 1]F[sub 1](a,b,z)] for all /z/.
* Testing against values precomputed using arbitrary precision arithmetic and the /pFq/ series.
We also have a [@../../tools/hypergeometric_1F1_error_plot.cpp small program] for testing random values over a user-specified domain of /a/, /b/ and /z/, this program
is also used for the error rate plots below and has been extremely useful in fine tuning the implementation.
It should be noted however, that there are some domains for large negative /a/ and /b/ that have poor test coverage as we were
simply unable to compute test values in reasonable time: it is not uncommon for the /pFq/ series to cancel many humdreds of digits
and sometimes into the thousands of digits.
[h4 Errors]
We divide the domain of [sub 1]F[sub 1] up into several domains to explain the error rates.
In the following graphs we ran 100000 random test cases over each domain, note that the scatter plots show only the very worst errors
as otherwise the graphs are both incomprehensible and virtually unplottable (as in sudden browser death).
For 0 < a,b,z the error rates are trivial small unless we are forced to take steps to avoid very-slow convergence and use a method other than direct evaluation of the series
for performance reasons. Even so the errors rates are broadly acceptible, with the scatter graph shows where the worst errors are located:
[$../graphs/hypergeometric_1f1/positive_abz_bins.svg]
[? __build_html '''<?dbhtml-include href="'''__base_path__'''/graphs/hypergeometric_1f1/positive_abz_include.html"?>''']
[?! __build_html [$../graphs/hypergeometric_1f1/positive_abz.png]]
For -1000 < a < 0 and 0 < b,z < 1000 the maximum error rates are higher, but most are still low, and the worst errors are fairly evenly distributed:
[$../graphs/hypergeometric_1f1/negative_a_bins.svg]
[? __build_html '''<?dbhtml-include href="'''__base_path__'''/graphs/hypergeometric_1f1/negative_a.html"?>''']
[?! __build_html [$../graphs/hypergeometric_1f1/negative_a.png]]
[h4 Implementation]
The implementation implementation for this function is really remarkably complex, and new methods for evaluating the function are still a matter of active research,

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#define BOOST_ENABLE_ASSERT_HANDLER
#define BOOST_MATH_MAX_SERIES_ITERATION_POLICY INT_MAX
// for consistent behaviour across compilers/platforms:
#define BOOST_MATH_PROMOTE_DOUBLE_POLICY false
// overflow to infinity is OK, we treat these as zero error as long as the sign is correct!
#define BOOST_MATH_OVERFLOW_ERROR_POLICY ignore_error
#include <iostream>
#include <ctime>
#include <boost/math/special_functions/hypergeometric_1F1.hpp>
#include <boost/math/special_functions/hypergeometric_pFq.hpp>
#include <boost/math/special_functions/relative_difference.hpp>
#include <boost/multiprecision/mpfr.hpp>
#include <boost/random.hpp>
#include <set>
#include <fstream>
#include <boost/iostreams/tee.hpp>
#include <boost/iostreams/stream.hpp>
typedef double test_type;
using boost::multiprecision::mpfr_float;
namespace boost {
//
// We convert assertions into exceptions, so we can log them and continue:
//
void assertion_failed(char const * expr, char const *, char const * file, long line)
{
std::ostringstream oss;
oss << file << ":" << line << " Assertion failed: " << expr;
throw std::runtime_error(oss.str());
}
}
void print_value(double x, std::ostream& os = std::cout)
{
int e;
double m = std::frexp(x, &e);
m = std::ldexp(m, 54);
e -= 54;
boost::int64_t val = (boost::int64_t)m;
BOOST_ASSERT(std::ldexp((double)val, e) == x);
os << "std::ldexp((double)" << val << ", " << e << ")";
}
void print_row(double a, double b, double z, mpfr_float result, std::ostream& os = std::cout)
{
os << " {{ ";
print_value(a, os);
os << ", ";
print_value(b, os);
os << ", ";
print_value(z, os);
os << ", SC_(" << std::setprecision(45) << result << ") }}" << std::endl;
}
struct error_data
{
error_data(double a, double b, double z, int e)
: a(a), b(b), z(z), error(e) {}
double a, b, z;
int error;
bool operator<(const error_data& other)const
{
return error < other.error;
}
};
int main()
{
try {
test_type max_a, max_b, max_z, min_a, min_b, min_z;
unsigned number_of_samples;
std::ofstream log_stream, incalculable_stream, unevaluated_stream, bins_stream;
std::string basename;
std::cout << "Enter range for a: ";
std::cin >> min_a >> max_a;
std::cout << "Enter range for b: ";
std::cin >> min_b >> max_b;
std::cout << "Enter range for z: ";
std::cin >> min_z >> max_z;
std::cout << "Enter number of samples: ";
std::cin >> number_of_samples;
std::cout << "Enter basename for log files: ";
std::cin >> basename;
typedef boost::iostreams::tee_device<std::ostream, std::ostream> tee_sink;
typedef boost::iostreams::stream<tee_sink> tee_stream;
log_stream.open((basename + ".log").c_str());
tee_stream tee_log(tee_sink(std::cout, log_stream));
incalculable_stream.open((basename + "_incalculable.log").c_str());
unevaluated_stream.open((basename + "_unevaluated.log").c_str());
bins_stream.open((basename + "_bins.csv").c_str());
tee_stream tee_bins(tee_sink(std::cout, bins_stream));
boost::random::mt19937 gen(std::time(0));
boost::random::uniform_real_distribution<test_type> a_dist(min_a, max_a);
boost::random::uniform_real_distribution<test_type> b_dist(min_b, max_b);
boost::random::uniform_real_distribution<test_type> z_dist(min_z, max_z);
std::multiset<error_data> errors;
std::map<std::pair<int, int>, int> bins;
unsigned incalculable = 0;
unsigned evaluation_errors = 0;
test_type max_error = 0;
do
{
test_type a = a_dist(gen);
test_type b = b_dist(gen);
test_type z = z_dist(gen);
test_type found, expected;
mpfr_float mp_expected;
try {
mp_expected = boost::math::hypergeometric_pFq_precision({ mpfr_float(a) }, { mpfr_float(b) }, mpfr_float(z), 25, 3.0);
expected = (test_type)mp_expected;
}
catch (const std::exception&)
{
// Unable to compute reference value:
++incalculable;
tee_log << "Unable to compute reference value in reasonable time: " << std::endl;
print_row(a, b, z, mpfr_float(0), tee_log);
incalculable_stream << std::setprecision(6) << std::scientific << a << "," << b << "," << z << "\n";
continue;
}
try
{
found = boost::math::hypergeometric_1F1(a, b, z);
}
catch (const std::exception&)
{
++evaluation_errors;
--number_of_samples;
tee_log << "Unexpected exception calculating value: " << std::endl;
print_row(a, b, z, mp_expected, tee_log);
unevaluated_stream << std::setprecision(6) << std::scientific << a << "," << b << "," << z << "\n";
continue;
}
test_type err = boost::math::epsilon_difference(found, expected);
if (err > max_error)
{
tee_log << "New maximum error is: " << err << std::endl;
print_row(a, b, z, mp_expected, tee_log);
max_error = err;
}
try {
errors.insert(error_data(a, b, z, boost::math::itrunc(err)));
}
catch (...)
{
errors.insert(error_data(a, b, z, INT_MAX));
}
--number_of_samples;
if (number_of_samples % 500 == 0)
std::cout << number_of_samples << " samples to go" << std::endl;
} while (number_of_samples);
tee_log << "Max error found was: " << max_error << std::endl;
unsigned current_bin = 0;
unsigned lim = 1;
unsigned old_lim = 0;
while (errors.size())
{
std::ofstream os((basename + "_errors_" + std::to_string(current_bin + 1) + ".csv").c_str());
os << "a,b,z,error\n";
old_lim = lim;
lim *= 2;
//std::cout << "Enter upper limit for bin " << current_bin << ": ";
//std::cin >> lim;
auto p = errors.upper_bound(error_data(0, 0, 0, lim));
int bin_count = std::distance(errors.begin(), p);
bins[std::make_pair(old_lim, lim)] = bin_count;
for (auto pos = errors.begin(); pos != p; ++pos)
{
os << pos->a << "," << pos->b << "," << pos->z << "," << pos->error << "\n";
}
errors.erase(errors.begin(), p);
++current_bin;
}
tee_bins << "Results:\n\n";
tee_bins << "#bin,Range,2^N,Count\n";
int hash = 0;
for (auto p = bins.begin(); p != bins.end(); ++p, ++hash)
{
tee_bins << hash << "," << p->first.first << "-" << p->first.second << "," << hash+1 << "," << p->second << std::endl;
}
if (evaluation_errors)
{
tee_bins << ",Failed,," << evaluation_errors << std::endl;
}
if (incalculable)
{
tee_bins << ",Incalculable,," << incalculable << std::endl;
}
}
catch (const std::exception& e)
{
std::cout << "Terminating with unhandled exception: " << e.what() << std::endl;
}
return 0;
}