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Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2013-01-12 17:08:58 -08:00
parent a9fa232f11
commit 1e362e6fec
1 changed files with 129 additions and 56 deletions
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185
src/math/realclosure/realclosure.cpp
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@ -1280,6 +1280,12 @@ namespace realclosure {
inc_ref(m_e); inc_ref(m_e);
} }
} }
// ---------------------------------
//
// Root isolation
//
// ---------------------------------
/** /**
\brief r <- magnitude of the lower bound of |i|. \brief r <- magnitude of the lower bound of |i|.
@ -2305,6 +2311,12 @@ namespace realclosure {
} }
} }
// ---------------------------------
//
// Basic operations
//
// ---------------------------------
void reset(numeral & a) { void reset(numeral & a) {
del(a); del(a);
SASSERT(is_zero(a)); SASSERT(is_zero(a));
@ -2520,62 +2532,11 @@ namespace realclosure {
a.m_value = n.m_value; a.m_value = n.m_value;
} }
/** // ---------------------------------
\brief a <- b^{1/k} //
*/ // Polynomial arithmetic in RCF
void root(numeral const & a, unsigned k, numeral & b) { //
if (k == 0) // ---------------------------------
throw exception("0-th root is indeterminate");
if (k == 1 || is_zero(a)) {
set(b, a);
return;
}
if (sign(a) < 0 && k % 2 == 0)
throw exception("even root of negative number");
// create the polynomial p of the form x^k - a
value_ref_buffer p(*this);
value_ref neg_a(*this);
neg(a.m_value, neg_a);
p.push_back(neg_a);
for (unsigned i = 0; i < k - 1; i++)
p.push_back(0);
p.push_back(one());
numeral_vector roots;
nz_isolate_roots(p.size(), p.c_ptr(), roots);
SASSERT(roots.size() == 1 || roots.size() == 2);
if (roots.size() == 1 || sign(roots[0].m_value) > 0) {
set(b, roots[0]);
}
else {
SASSERT(roots.size() == 2);
SASSERT(sign(roots[1].m_value) > 0);
set(b, roots[1]);
}
del(roots);
}
/**
\brief a <- b^k
*/
void power(numeral const & a, unsigned k, numeral & b) {
unsigned mask = 1;
value_ref power(*this);
value_ref _b(*this);
power = a.m_value;
_b = one();
while (mask <= k) {
checkpoint();
if (mask & k)
mul(_b, power, _b);
mul(power, power, power);
mask = mask << 1;
}
set(b, _b);
}
/** /**
\brief Remove 0s \brief Remove 0s
@ -2947,6 +2908,12 @@ namespace realclosure {
if (d % 2 == 0 || (sz2 > 0 && sign(p2[sz2-1]) < 0)) if (d % 2 == 0 || (sz2 > 0 && sign(p2[sz2-1]) < 0))
neg(r); neg(r);
} }
// ---------------------------------
//
// GCD
//
// ---------------------------------
/** /**
\brief Force the leading coefficient of p to be 1. \brief Force the leading coefficient of p to be 1.
@ -3006,6 +2973,12 @@ namespace realclosure {
} }
} }
// ---------------------------------
//
// Derivatives and Sturm-Tarski Sequences
//
// ---------------------------------
/** /**
\brief r <- dp/dx \brief r <- dp/dx
*/ */
@ -3107,6 +3080,13 @@ namespace realclosure {
seq.push(p1_prime_p2.size(), p1_prime_p2.c_ptr()); seq.push(p1_prime_p2.size(), p1_prime_p2.c_ptr());
sturm_seq_core(seq); sturm_seq_core(seq);
} }
// ---------------------------------
//
// Sign evaluation for polynomials
// That is, sign of p(x) at b
//
// ---------------------------------
/** /**
\brief Return the sign of p(0) \brief Return the sign of p(0)
@ -3285,6 +3265,12 @@ namespace realclosure {
} }
} }
// ---------------------------------
//
// Sign variations in polynomial sequences.
//
// ---------------------------------
enum location { enum location {
ZERO, ZERO,
MINUS_INF, MINUS_INF,
@ -3380,6 +3366,12 @@ namespace realclosure {
return sign_variations_at(seq, interval.upper()); return sign_variations_at(seq, interval.upper());
} }
// ---------------------------------
//
// Tarski-Queries (see BPR book)
//
// ---------------------------------
/** /**
\brief Given a polynomial Sturm sequence seq for (P; P' * Q) and an interval (a, b], it returns \brief Given a polynomial Sturm sequence seq for (P; P' * Q) and an interval (a, b], it returns
TaQ(Q, P; a, b) = TaQ(Q, P; a, b) =
@ -3425,6 +3417,12 @@ namespace realclosure {
sturm_seq(p_sz, p, seq); sturm_seq(p_sz, p, seq);
return TaQ(seq, interval); return TaQ(seq, interval);
} }
// ---------------------------------
//
// Interval refinement
//
// ---------------------------------
void refine_rational_interval(rational_value * v, unsigned prec) { void refine_rational_interval(rational_value * v, unsigned prec) {
mpbqi & i = interval(v); mpbqi & i = interval(v);
@ -3687,6 +3685,12 @@ namespace realclosure {
} }
} }
// ---------------------------------
//
// Sign determination
//
// ---------------------------------
/** /**
\brief Return the position of the first non-zero coefficient of p. \brief Return the position of the first non-zero coefficient of p.
*/ */
@ -4156,6 +4160,12 @@ namespace realclosure {
return determine_sign(to_rational_function(r.get())); return determine_sign(to_rational_function(r.get()));
} }
// ---------------------------------
//
// Arithmetic operations
//
// ---------------------------------
/** /**
\brief Set new_p1 and new_p2 using the following normalization rules: \brief Set new_p1 and new_p2 using the following normalization rules:
- new_p1 <- p1/p2[0]; new_p2 <- one IF sz2 == 1 - new_p1 <- p1/p2[0]; new_p2 <- one IF sz2 == 1
@ -4735,6 +4745,69 @@ namespace realclosure {
set(c, r); set(c, r);
} }
/**
\brief a <- b^{1/k}
*/
void root(numeral const & a, unsigned k, numeral & b) {
if (k == 0)
throw exception("0-th root is indeterminate");
if (k == 1 || is_zero(a)) {
set(b, a);
return;
}
if (sign(a) < 0 && k % 2 == 0)
throw exception("even root of negative number");
// create the polynomial p of the form x^k - a
value_ref_buffer p(*this);
value_ref neg_a(*this);
neg(a.m_value, neg_a);
p.push_back(neg_a);
for (unsigned i = 0; i < k - 1; i++)
p.push_back(0);
p.push_back(one());
numeral_vector roots;
nz_isolate_roots(p.size(), p.c_ptr(), roots);
SASSERT(roots.size() == 1 || roots.size() == 2);
if (roots.size() == 1 || sign(roots[0].m_value) > 0) {
set(b, roots[0]);
}
else {
SASSERT(roots.size() == 2);
SASSERT(sign(roots[1].m_value) > 0);
set(b, roots[1]);
}
del(roots);
}
/**
\brief a <- b^k
*/
void power(numeral const & a, unsigned k, numeral & b) {
unsigned mask = 1;
value_ref power(*this);
value_ref _b(*this);
power = a.m_value;
_b = one();
while (mask <= k) {
checkpoint();
if (mask & k)
mul(_b, power, _b);
mul(power, power, power);
mask = mask << 1;
}
set(b, _b);
}
// ---------------------------------
//
// Comparison
//
// ---------------------------------
int compare(value * a, value * b) { int compare(value * a, value * b) {
if (a == 0) if (a == 0)
return -sign(b); return -sign(b);