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blatt05.cpp
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blatt05.cpp
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#include <iostream>
#include <iomanip>
#include <limits>
#include <stdexcept>
#include <vector>
namespace checked_int
{
template<typename T>
bool has_twos_complement = std::numeric_limits<T>::is_signed && std::numeric_limits<T>::min() + std::numeric_limits<T>::max() < T(0);
template<typename T>
class checked
{
T value;
public:
checked(T value = T()) : value(value) { }
operator T() const { return value; }
checked operator+() const { return *this; }
checked operator-() const
{
if (has_twos_complement<T> && value == std::numeric_limits<T>::min())
{
throw std::overflow_error("operator-: arithmetic overflow on negation");
}
return checked(-value);
}
template<typename S> checked& operator+=(S rhs) { value = *this + rhs; return *this; }
template<typename S> checked& operator-=(S rhs) { value = *this - rhs; return *this; }
template<typename S> checked& operator*=(S rhs) { value = *this * rhs; return *this; }
template<typename S> checked& operator/=(S rhs) { value = *this / rhs; return *this; }
template<typename S> checked& operator%=(S rhs) { value = *this % rhs; return *this; }
template<typename T1, typename T2> friend auto operator+(checked<T1> lhs, checked<T2> rhs);
template<typename T1, typename T2> friend auto operator-(checked<T1> lhs, checked<T2> rhs);
template<typename T1, typename T2> friend auto operator*(checked<T1> lhs, checked<T2> rhs);
template<typename T1, typename T2> friend auto operator/(checked<T1> lhs, checked<T2> rhs);
template<typename T1, typename T2> friend auto operator%(checked<T1> lhs, checked<T2> rhs);
friend std::ostream& operator<<(std::ostream& stream, checked c) { return stream << c.value; }
friend std::istream& operator>>(std::istream& stream, checked & c) { return stream >> c.value; }
};
template<typename T1, typename T2> bool operator==(checked<T1> lhs, checked<T2> rhs) { return static_cast<T1>(lhs) == static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator==(checked<T1> lhs, T2 rhs) { return static_cast<T1>(lhs) == rhs ; }
template<typename T1, typename T2> bool operator==( T1 lhs, checked<T2> rhs) { return lhs == static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator!=(checked<T1> lhs, checked<T2> rhs) { return static_cast<T1>(lhs) != static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator!=(checked<T1> lhs, T2 rhs) { return static_cast<T1>(lhs) != rhs ; }
template<typename T1, typename T2> bool operator!=( T1 lhs, checked<T2> rhs) { return lhs != static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator< (checked<T1> lhs, checked<T2> rhs) { return static_cast<T1>(lhs) < static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator< (checked<T1> lhs, T2 rhs) { return static_cast<T1>(lhs) < rhs ; }
template<typename T1, typename T2> bool operator< ( T1 lhs, checked<T2> rhs) { return lhs < static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator> (checked<T1> lhs, checked<T2> rhs) { return static_cast<T1>(lhs) > static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator> (checked<T1> lhs, T2 rhs) { return static_cast<T1>(lhs) > rhs ; }
template<typename T1, typename T2> bool operator> ( T1 lhs, checked<T2> rhs) { return lhs > static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator<=(checked<T1> lhs, checked<T2> rhs) { return static_cast<T1>(lhs) <= static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator<=(checked<T1> lhs, T2 rhs) { return static_cast<T1>(lhs) <= rhs ; }
template<typename T1, typename T2> bool operator<=( T1 lhs, checked<T2> rhs) { return lhs <= static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator>=(checked<T1> lhs, checked<T2> rhs) { return static_cast<T1>(lhs) >= static_cast<T2>(rhs); }
template<typename T1, typename T2> bool operator>=(checked<T1> lhs, T2 rhs) { return static_cast<T1>(lhs) >= rhs ; }
template<typename T1, typename T2> bool operator>=( T1 lhs, checked<T2> rhs) { return lhs >= static_cast<T2>(rhs); }
template<typename T1, typename T2> auto operator+( T1 lhs, checked<T2> rhs) { return checked<T1>(lhs) + rhs ; }
template<typename T1, typename T2> auto operator+(checked<T1> lhs, T2 rhs) { return lhs + checked<T2>(rhs); }
template<typename T1, typename T2> auto operator-( T1 lhs, checked<T2> rhs) { return checked<T1>(lhs) - rhs ; }
template<typename T1, typename T2> auto operator-(checked<T1> lhs, T2 rhs) { return lhs - checked<T2>(rhs); }
template<typename T1, typename T2> auto operator*( T1 lhs, checked<T2> rhs) { return checked<T1>(lhs) * rhs ; }
template<typename T1, typename T2> auto operator*(checked<T1> lhs, T2 rhs) { return lhs * checked<T2>(rhs); }
template<typename T1, typename T2> auto operator/( T1 lhs, checked<T2> rhs) { return checked<T1>(lhs) / rhs ; }
template<typename T1, typename T2> auto operator/(checked<T1> lhs, T2 rhs) { return lhs / checked<T2>(rhs); }
template<typename T1, typename T2> auto operator%( T1 lhs, checked<T2> rhs) { return checked<T1>(lhs) % rhs ; }
template<typename T1, typename T2> auto operator%(checked<T1> lhs, T2 rhs) { return lhs % checked<T2>(rhs); }
template<typename T1, typename T2>
auto operator+(checked<T1> lhs, checked<T2> rhs)
{
using Common = decltype(lhs.value + rhs.value);
if (rhs.value > T2(0) && lhs.value > std::numeric_limits<Common>::max() - rhs.value
|| rhs.value < T2(0) && lhs.value < std::numeric_limits<Common>::min() - rhs.value)
{
throw std::overflow_error("operator+: arithmetic overflow");
}
return checked<Common>(lhs.value + rhs.value);
}
template<typename T1, typename T2>
auto operator-(checked<T1> lhs, checked<T2> rhs)
{
using Common = decltype(lhs.value - rhs.value);
if (rhs.value < T2(0) && lhs.value > std::numeric_limits<Common>::max() + rhs.value
|| rhs.value > T2(0) && lhs.value < std::numeric_limits<Common>::min() + rhs.value)
{
throw std::overflow_error("operator-: arithmetic overflow");
}
return checked<Common>(lhs.value - rhs.value);
}
template<typename T1, typename T2>
auto operator*(checked<T1> lhs, checked<T2> rhs)
{
using Common = decltype(lhs.value * rhs.value);
if (has_twos_complement<Common>)
if (
lhs.value == T1(-1) && rhs.value == std::numeric_limits<Common>::min()
|| rhs.value == T2(-1) && lhs.value == std::numeric_limits<Common>::min())
{
throw std::overflow_error("operator*: arithmetic overflow on negation");
}
if (rhs < 0 && lhs < 0)
{
rhs.value = -rhs.value;
lhs.value = -lhs.value;
}
// one of rhs and lhs >= 0; for == 0, nothing to check; for < 0, the other must be >= 0, so if > 0, divide max by it.
if (rhs > 0 && (
lhs.value > std::numeric_limits<Common>::max() / rhs.value
|| lhs.value < std::numeric_limits<Common>::min() / rhs.value)
|| lhs > 0 && (
rhs.value > std::numeric_limits<Common>::max() / lhs.value
|| rhs.value < std::numeric_limits<Common>::min() / lhs.value))
{
throw std::overflow_error("operator*: arithmetic overflow");
}
return checked<Common>(lhs.value * rhs.value);
}
template<typename T1, typename T2>
auto operator/(checked<T1> lhs, checked<T2> rhs)
{
if (rhs == T2(0)) throw std::domain_error("operator/: division by zero");
using Common = decltype(lhs.value / rhs.value);
if (has_twos_complement<Common> && std::numeric_limits<T2>::is_signed
&& rhs == T2(-1) && lhs.value == std::numeric_limits<Common>::min())
{
throw std::overflow_error("operator/: arithmetic overflow on negation");
}
return checked<Common>(lhs.value / rhs.value);
}
template<typename T1, typename T2>
auto operator%(checked<T1> lhs, checked<T2> rhs)
{
if (rhs == T2(0)) throw std::domain_error("operator%: division by zero");
using Common = decltype(lhs.value % rhs.value);
return checked<Common>(lhs.value % rhs.value);
}
template<typename T>
checked<T> check(T value) { return checked<T>(value); }
} // namespace
/*
int main()
{
// to verify the calls compile.
checked_int::checked<long> a, b;
auto p = a + b;
auto s = a - b;
auto t = a * b;
auto d = a * b;
auto m = a % b;
}
*/
using namespace std;
using namespace checked_int;
// for debugging.
#define READ(x) do { std::cerr << #x << ": "; std::cin >> (x); } while (0)
#define INSPECT(x) do { std::cerr << #x << " = " << (x) << std::endl; } while (0)
#define EXPECT(x, r) do { std::cerr << "is " << #x << " = " << (x) << " equal to " << (r) << " = " << #r << "?" << std::endl; } while (0)
template <typename T> class rational;
#define INSPECT_RAT(x) do { std::cerr << #x << " = "; _inspect_rat(x); } while (0)
template<typename T> void _inspect_rat(rational<T> const & r) { std::cerr << r << " = " << static_cast<long double>(r) << std::endl; }
template <typename T>
class rational
{
private:
checked<T> p, q;
static T gcd(T a, T b)
{
if (a < 0) return gcd(-check(a), b);
if (b < 0) return gcd(a, -check(b));
if (b == 0) return a;
return gcd(b, a % b);
}
public:
rational(T z = 0, T n = 1)
: p(z), q(n)
{
if (q == 0) throw domain_error("denominator must not be zero");
if (q < 0)
{
p = -check(p);
q = -check(q);
}
T g = gcd(p, q);
p /= g;
q /= g;
}
explicit operator long double() const { return static_cast<long double>(p) / static_cast<long double>(q); }
friend rational operator+(rational const & r) { return r; }
friend rational operator-(rational r) { r.p = -r.p; return r; }
rational& operator+=(rational const & r) { return *this = *this + r; }
rational& operator-=(rational const & r) { return *this = *this - r; }
rational& operator*=(rational const & r) { return *this = *this * r; }
rational& operator/=(rational const & r) { return *this = *this / r; }
rational& operator%=(rational const & r) { return *this = *this % r; }
friend rational operator+(rational const & s, rational const & t)
{
T g = gcd(s.q, t.q);
T k = s.q / g * t.q;
T p = s.p * (t.q / g) + t.p * (s.q / g);
return rational(p, k);
}
friend rational operator-(rational const & s, rational const & t)
{
T g = gcd(s.q, t.q);
T k = (s.q / g) * t.q;
T p = s.p * (t.q / g) - t.p * (s.q / g);
return rational(p, k);
}
friend rational operator*(rational const & s, rational const & t)
{
// strategy: swap denominators to reduce the fractions:
// (26/35) * (15/22) = (26/22) * (15/35) = (13/11) * (3/7)
// the two resulting fractions not only have co-prime numerators and denominators,
// the numerator of one is co-prime to the denominator of the other,
// (13/11) * (3/7) = (13*3)/(11*7)
// therefore the numerator- and denominator-wise multiplication is optimal.
rational a(s.p, t.q), b(t.p, s.q), result;
result.p = a.p * b.p;
result.q = a.q * b.q;
return result;
}
friend rational operator/(rational const & s, rational const & t)
{
if (t == 0) throw invalid_argument("operator/: division by zero");
return s * rational(t.q, t.p);
}
friend T whole(rational const & r) { /* note: integer division is intentional! */ return r.p / r.q; }
friend T div(rational const & s, rational const & t) { return whole(s / t); }
friend rational operator%(rational const & s, rational const & t) { return s - div(s, t) * t; }
/*
// recursive version
friend rational pow(rational b, int n)
{
if (n == 0) return 1;
if (b == 0) return b;
if (n < 0) return pow(1 / b, -checked<int>(n));
if (n % 2 != 0)
{
return pow(b * b, n / 2) * b;
}
else
{
return pow(b * b, n / 2);
}
}
*/
friend rational pow(rational b, int n)
{
if (n == 0) return 1;
if (n < 0)
{
b = 1 / b;
n = -checked<int>(n);
}
if (b == 0 || n == 1) return b;
unsigned h = ~(~0u >> 1);
while (!(n & h)) h >>= 1;
rational result = b;
while (h >>= 1)
{
result *= result;
if (n & h) result *= b;
}
return result;
}
friend bool operator==(rational const & s, rational const & t) { return s.p == t.p && s.q == t.q; }
friend bool operator!=(rational const & s, rational const & t) { return !(s == t); }
friend bool operator> (rational const & s, rational const & t) { return t < s; }
friend bool operator<=(rational const & s, rational const & t) { return s == t || s < t; }
friend bool operator>=(rational const & s, rational const & t) { return s == t || s > t; }
friend bool operator< (rational const & s, rational const & t)
{
// trivial criteria
if (s.p <= 0 && 0 <= t.p) return s != t; // makes comparisons to 0 fast.
if (s.p <= t.p && s.q >= t.q) return s != t;
// We'll just use the fact that s.p/s.q < t.p/t.q iff s.p * t.q < t.p * s.q
// as denominators are always positive.
// For optimization, we use the fact that for (s.p/s.q) < (t.p/t.q), we can
// divide both sides by (gcd(s.p, t.p)/gcd(s.q, t.q)) easily:
// 15/22 < 35/26 using / (5/2)
// <=> 5/11 < 7/13 using cross multiply, i.e. * (11*13)
// <=> 5*13 < 7*11
// <=> 65 < 77
auto gp = gcd(s.p, t.p), gq = gcd(t.q, s.q);
return (s.p / gp) * (t.q / gq) < (t.p / gp) * (s.q / gq);
}
friend ostream& operator<<(ostream& stream, rational const & r)
{
if (r.q == T(1)) return stream << r.p;
return stream << reinterpret_cast<ostringstream&>(ostringstream() << r.p << '/' << r.q).str();
}
friend istream& operator>>(istream& stream, rational & r)
{
T p;
if (!(stream >> p)) return stream;
if (stream.peek() != '/')
{
r = rational(p);
return stream;
}
stream.get();
T q;
stream >> q;
r = rational(p, q);
return stream;
}
};
template<typename T>
rational<T> power_series(rational<T> const & r, int n)
{
rational<T> result = 0;
if (n < 0) return result;
// using non-inclusive < n (instead of <= n), results are correct even for for n == INT_MAX.
for (int k = 1; k < n; ++k) result += k * pow(r, k);
return result + (n * pow(r, n));
}
template<typename T>
rational<T> continued_fraction(vector<T> const & a, vector<T> const & b)
{
if (b.size() == 0u) throw invalid_argument("continued_fraction: vector of coefficients must not be empty");
if (a.size() != b.size() - 1u) throw invalid_argument("continued_fraction: vector sizes mismatch");
rational<T> result = b.back();
for
(
auto a_it = a.crbegin(), b_it = b.crbegin() + 1;
a_it != a.crend() && b_it != b.crend();
++a_it, ++b_it
) {
result = *b_it + *a_it / result;
}
return result;
}
template<typename T>
rational<T> continued_fraction(vector<T> const & b)
{
if (b.size() == 0u) throw invalid_argument("continued_fraction: vector of coefficients must not be empty");
return continued_fraction(vector<T>(b.size() - 1, 1), b);
}
// testing arithmetic
/*
int main()
{
try
{
rational<long> s, t;
READ(s);
// if (s == 0) return 0;
// READ(t);
// INSPECT(s);
// INSPECT(t);
// INSPECT(s + t);
// EXPECT(s + t - t, s);
// EXPECT(s + t - s, t);
// INSPECT(s - t);
// EXPECT(s - t + t, s);
// EXPECT(-(s - t) + s, t);
// INSPECT(s * t);
// if (t != 0)
// {
// EXPECT(s * t / t, s);
// }
// EXPECT(s * t / s, t);
// if (t != 0)
// {
// INSPECT(s / t);
// EXPECT(s / t * t, s);
// EXPECT(1/(s / t) * s, t);
// }
// INSPECT(pow(s, 5));
// EXPECT(pow(s, 5) / pow(s, 4), s);
using rat = rational<long>;
// rat s(3, 4), t(1, 2);
// INSPECT_RAT(s % t);
for (rat a : { rat(+10, 3), rat(-10, 3) })
for (rat b : { rat(+3, 2), rat(-3, 2) })
{
INSPECT(a);
INSPECT(b);
INSPECT(div(a, b));
INSPECT(a % b);
EXPECT(a - (div(a, b) * b + (a % b)), 0);
}
}
catch (overflow_error e)
{
cout << e.what() << endl;
}
return 0;
}
*/
// for debugging vectors
ostream& operator<<(ostream& stream, vector<int> const & v)
{
stream << '[';
if (!v.empty())
{
stream << v[0];
for (vector<int>::size_type i = 1; i < v.size(); ++i)
stream << ", " << v[i];
}
stream << ']';
return stream;
}
long double diff(long double a, long double b) { return a < b ? b - a : a - b; }
int main()
{
using T = int_least64_t;
using rat = rational<T>;
cout << "\n-- (a) and (b) ----------" << endl;
for (rat r : { rat(2, 3), rat(-10, 7) })
{
const int n = 8;
rat ps = power_series(r, n);
cout << "sum[k = 1 .. " << n << "] k*(" << r << ")^k = " << ps << " = " << static_cast<long double>(ps) << endl;
INSPECT_RAT((n * pow(r, n + 2) - (n + 1) * pow(r, n + 1) + r) / pow(r - 1, 2));
}
cout << "\n-- (c) ------------------" << endl;
using vec = vector<T>;
INSPECT_RAT(continued_fraction(vec{ 3, 7, 15, 1, 292 }));
cout << "\n-- (d) ------------------" << endl;
const long double pi = 3.141592653589793238462643383279L;
for (int n = 1; n < 25; ++n)
try
{
vector<T> a(n, 4);
vector<T> b(n + 1, 0);
for (int i = 1; i < n; ++i)
{
a[i] = i * i;
b[i] = 2 * i - 1;
}
b[n] = 2 * n - 1;
auto r = continued_fraction(a, b);
long double lr = static_cast<long double>(r);
cout << "for n = " << setw(2) << n << ": ";
cout << "continued_fraction(a, b) = " << setw(19) << fixed << setprecision(numeric_limits<long double>::digits10) << lr << " (diff = " << diff(lr, pi) << ")";
cout << " = " << r << endl;
}
catch (overflow_error)
{
cout << "for n = " << setw(2) << n << ": *** overflow ***" << endl;
}
return 0;
}