#include "tutil.h"
#include "tmathutil.h"
#include "tconvert.h"
#include <iterator>
using TConsts::epsilon;
TMathException::TMathException(std::string msg) : m_msg(::to_wstring(msg)) {}
namespace {
//-------------------------------------------------------------------------
//! maximum order for a polynomial
const int MAX_ORDER = 12;
//! smallest relative error we want
const double RELERROR = 1.0e-14;
//! max power of 10 we wish to search to
const int MAXPOW = 32;
//! max number of iterations
const int MAXIT = 800;
//! a coefficient smaller than SMALL_ENOUGH is considered to be zero (0.0).
const double SMALL_ENOUGH = 1.0e-12;
//-------------------------------------------------------------------------
inline int getEl(int i, int j, int n) { return (j - 1) + (i - 1) * n; }
//-------------------------------------------------------------------------
//! structure type for representing a polynomial
typedef struct {
int ord;
double coef[MAX_ORDER + 1];
} poly;
//-------------------------------------------------------------------------
// compute number of root in (min,max)
double evalpoly(int ord, double *coef, double x);
int numchanges(int np, poly *sseq, double a);
int buildsturm(int ord, poly *sseq);
int modrf(int ord, double *coef, double a, double b, double *val);
//-------------------------------------------------------------------------
void convert(const std::vector<double> &v, poly &p) {
assert((int)v.size() <= MAX_ORDER);
if ((int)v.size() > MAX_ORDER) return;
p.ord = v.size() - 1;
std::copy(v.begin(), v.end(), p.coef);
}
//-------------------------------------------------------------------------
void convert(const poly &p, std::vector<double> &v) {
v.resize(p.ord);
std::copy(p.coef, p.coef + p.ord, v.begin());
}
//-------------------------------------------------------------------------
/*
* modp
*
* calculates the modulus of u(x) / v(x) leaving it in r, it
* returns 0 if r(x) is a constant.
* note: this function assumes the leading coefficient of v
* is 1 or -1
*/
int modp(poly *u, poly *v, poly *r) {
int k, j;
double *nr, *end, *uc;
nr = r->coef;
end = &u->coef[u->ord];
uc = u->coef;
while (uc <= end) *nr++ = *uc++;
if (v->coef[v->ord] < 0.0) {
for (k = u->ord - v->ord - 1; k >= 0; k -= 2) r->coef[k] = -r->coef[k];
for (k = u->ord - v->ord; k >= 0; k--)
for (j = v->ord + k - 1; j >= k; j--)
r->coef[j] = -r->coef[j] - r->coef[v->ord + k] * v->coef[j - k];
} else {
for (k = u->ord - v->ord; k >= 0; k--)
for (j = v->ord + k - 1; j >= k; j--)
r->coef[j] -= r->coef[v->ord + k] * v->coef[j - k];
}
k = v->ord - 1;
while (k >= 0 && fabs(r->coef[k]) < SMALL_ENOUGH) {
r->coef[k] = 0.0;
k--;
}
r->ord = (k < 0) ? 0 : k;
return (r->ord);
}
//-------------------------------------------------------------------------
/*
* buildsturm
*
* build up a sturm sequence for a polynomial in sseq, returning
* the number of polynomials in the sequence
*/
int buildsturm(int ord, poly *sseq) {
int i;
double f, *fp, *fc;
poly *sp;
sseq[0].ord = ord;
sseq[1].ord = ord - 1;
// calculate the derivative and normalise the leadingcoefficient.
f = fabs(sseq[0].coef[ord] * ord);
fp = sseq[1].coef;
fc = sseq[0].coef + 1;
for (i = 1; i <= ord; i++) *fp++ = *fc++ * i / f;
// construct the rest of the Sturm sequence
for (sp = sseq + 2; modp(sp - 2, sp - 1, sp); sp++) {
// reverse the sign and normalise
f = -fabs(sp->coef[sp->ord]);
for (fp = &sp->coef[sp->ord]; fp >= sp->coef; fp--) *fp /= f;
}
sp->coef[0] = -sp->coef[0]; // reverse the sign
return (sp - sseq);
}
//-------------------------------------------------------------------------
/*
return the number of distinct real roots of the polynomial
described in sseq
*/
int numroots(int np, poly *sseq, int &atneg, int &atpos) {
int atposinf = 0, atneginf = 0;
poly *s;
double f, lf;
// change at positive infinity
lf = sseq[0].coef[sseq[0].ord];
for (s = sseq + 1; s <= sseq + np; ++s) {
f = s->coef[s->ord];
if (0.0 == lf || lf * f < 0.0) ++atposinf;
lf = f;
}
// change at negative infinity
if (sseq[0].ord & 1)
lf = -sseq[0].coef[sseq[0].ord];
else
lf = sseq[0].coef[sseq[0].ord];
for (s = sseq + 1; s <= sseq + np; ++s) {
if (s->ord & 1)
f = -s->coef[s->ord];
else
f = s->coef[s->ord];
if (0.0 == lf || lf * f < 0.0) ++atneginf;
lf = f;
}
atneg = atneginf;
atpos = atposinf;
return atneginf - atposinf;
}
//-------------------------------------------------------------------------
/*
* numchanges
*
* return the number of sign changes in the Sturm sequence in
* sseq at the value a.
*/
int numchanges(int np, poly *sseq, double a) {
int changes;
double f, lf;
poly *s;
changes = 0;
lf = evalpoly(sseq[0].ord, sseq[0].coef, a);
for (s = sseq + 1; s <= sseq + np; s++) {
f = evalpoly(s->ord, s->coef, a);
if (lf == 0.0 || lf * f < 0) changes++;
lf = f;
}
return (changes);
}
//-------------------------------------------------------------------------
/*
* sbisect
*
* uses a bisection based on the sturm sequence for the polynomial
* described in sseq to isolate intervals in which roots occur,
* the roots are returned in the roots array in order of magnitude.
*/
void sbisect(int np, poly *sseq, double min, double max, int atmin, int atmax,
double *roots) {
double mid;
int n1 = 0, n2 = 0, its, atmid;
if ((atmin - atmax) == 1) {
// first try a less expensive technique.
if (modrf(sseq->ord, sseq->coef, min, max, &roots[0])) return;
/*
* if we get here we have to evaluate the root the hard
* way by using the Sturm sequence.
*/
for (its = 0; its < MAXIT; its++) {
mid = (min + max) / 2;
atmid = numchanges(np, sseq, mid);
if (fabs(mid) > RELERROR) {
if (fabs((max - min) / mid) < RELERROR) {
roots[0] = mid;
return;
}
} else if (fabs(max - min) < RELERROR) {
roots[0] = mid;
return;
}
if ((atmin - atmid) == 0)
min = mid;
else
max = mid;
}
if (its == MAXIT) {
/*
fprintf(stderr, "sbisect: overflow min %f max %f\
diff %e nroot %d n1 %d n2 %d\n",
min, max, max - min, nroot, n1, n2);
*/
roots[0] = mid;
}
return;
}
/*
* more than one root in the interval, we have to bisect...
*/
for (its = 0; its < MAXIT; its++) {
mid = (min + max) / 2;
atmid = numchanges(np, sseq, mid);
n1 = atmin - atmid;
n2 = atmid - atmax;
if (n1 != 0 && n2 != 0) {
sbisect(np, sseq, min, mid, atmin, atmid, roots);
sbisect(np, sseq, mid, max, atmid, atmax, &roots[n1]);
break;
}
if (n1 == 0)
min = mid;
else
max = mid;
}
if (its == MAXIT) {
/*
fprintf(stderr, "sbisect: roots too close together\n");
fprintf(stderr, "sbisect: overflow min %f max %f diff %e\
nroot %d n1 %d n2 %d\n",
min, max, max - min, nroot, n1, n2);
*/
for (n1 = atmax; n1 < atmin; n1++) roots[n1 - atmax] = mid;
}
}
//-------------------------------------------------------------------------
/*
* evalpoly
*
* evaluate polynomial defined in coef returning its value.
*/
double evalpoly(int ord, double *coef, double x) {
double *fp, f;
fp = &coef[ord];
f = *fp;
for (fp--; fp >= coef; fp--) f = x * f + *fp;
return (f);
}
//-------------------------------------------------------------------------
/*
* modrf
*
* uses the modified regula-falsi method to evaluate the root
* in interval [a,b] of the polynomial described in coef. The
* root is returned is returned in *val. The routine returns zero
* if it can't converge.
*/
int modrf(int ord, double *coef, double a, double b, double *val) {
int its;
double fa, fb, x, fx, lfx;
double *fp, *scoef, *ecoef;
scoef = coef;
ecoef = &coef[ord];
fb = fa = *ecoef;
for (fp = ecoef - 1; fp >= scoef; fp--) {
fa = a * fa + *fp;
fb = b * fb + *fp;
}
// if there is no sign difference the method won't work
if (fa * fb > 0.0) return (0);
if (fabs(fa) < RELERROR) {
*val = a;
return (1);
}
if (fabs(fb) < RELERROR) {
*val = b;
return (1);
}
lfx = fa;
for (its = 0; its < MAXIT; its++) {
x = (fb * a - fa * b) / (fb - fa);
fx = *ecoef;
for (fp = ecoef - 1; fp >= scoef; fp--) fx = x * fx + *fp;
if (fabs(x) > RELERROR) {
if (fabs(fx / x) < RELERROR) {
*val = x;
return (1);
}
} else if (fabs(fx) < RELERROR) {
*val = x;
return (1);
}
if ((fa * fx) < 0) {
b = x;
fb = fx;
if ((lfx * fx) > 0) fa /= 2;
} else {
a = x;
fa = fx;
if ((lfx * fx) > 0) fb /= 2;
}
lfx = fx;
}
// fprintf(stderr, "modrf overflow %f %f %f\n", a, b, fx);
return (0);
}
//-------------------------------------------------------------------------
/*!
a x^2 + b x + c
Remark:
poly[0] = c
poly[1] = b
poly[2] = a
*/
int rootForQuadraticEquation(const std::vector<double> &v,
std::vector<double> &sol) {
double q, delta;
/*
if( isAlmostZero(v[2]))
{
if ( isAlmostZero(v[1]) )
return -1;
sol.push_back(-v[0]/v[1]);
return 1;
}
*/
if (isAlmostZero(v[1])) {
q = -v[0] / v[2];
if (q < 0)
return 0;
else if (isAlmostZero(q)) {
sol.push_back(0);
return 1;
}
q = sqrt(q);
sol.push_back(-q);
sol.push_back(q);
return 2;
}
delta = sq(v[1]) - 4.0 * v[0] * v[2];
if (delta < 0.0) return 0;
assert(v[2] != 0);
if (isAlmostZero(delta)) {
sol.push_back(-v[1] / (v[2] * 2.0));
return 1;
}
q = -0.5 * (v[1] + tsign(v[1]) * sqrt(delta));
assert(q != 0);
sol.push_back(v[0] / q);
sol.push_back(q / v[2]);
return 2;
}
//-----------------------------------------------------------------------------
/*
a x^3+b x^2 + c x + d
Remark:
poly[0] = d
poly[1] = c
poly[2] = b
poly[3] = a
*/
int rootForCubicEquation(const std::vector<double> &p,
std::vector<double> &sol) {
/*
if( isAlmostZero(p[3]) )
return rootForQuadraticEquation(p,sol);
*/
if (isAlmostZero(p[0])) {
int numberOfSol;
std::vector<double> redPol(3);
redPol[0] = p[1];
redPol[1] = p[2];
redPol[2] = p[3];
numberOfSol = rootForQuadraticEquation(redPol, sol);
for (int i = 0; i < numberOfSol; ++i)
if (0.0 == sol[i]) return numberOfSol;
// altrimenti devo contare la soluzione nulla
++numberOfSol;
sol.push_back(0);
return numberOfSol;
}
double inv_v3 = 1.0 / p[3], a = p[2] * inv_v3, b = p[1] * inv_v3,
c = p[0] * inv_v3;
static const double inv_3 = 1.0 / 3.0;
static const double inv_9 = 1.0 / 9.0;
static const double inv_54 = 1.0 / 54.0;
double Q = (sq(a) - 3.0 * b) * inv_9,
R = (2.0 * sq(a) * a - 9.0 * a * b + 27.0 * c) * inv_54;
double R_2 = sq(R), Q_3 = sq(Q) * Q;
if (R_2 < Q_3) {
double Q_sqrt = sqrt(Q);
double theta = acos(R / (Q * Q_sqrt));
sol.push_back(-2 * Q_sqrt * cos(theta * inv_3) - a * inv_3);
sol.push_back(-2 * Q_sqrt * cos((theta - M_2PI) * inv_3) - a * inv_3);
sol.push_back(-2 * Q_sqrt * cos((theta + M_2PI) * inv_3) - a * inv_3);
std::sort(sol.begin(), sol.end());
return 3;
}
double A = -tsign(R) * pow((fabs(R) + sqrt(R_2 - Q_3)), inv_3);
double B = A != 0 ? Q / A : 0;
sol.push_back((A + B) - a * inv_3);
return 1;
}
//-----------------------------------------------------------------------------
int rootForGreaterThanThreeEquation(const std::vector<double> &p,
std::vector<double> &sol) {
poly sseq[MAX_ORDER];
convert(p, sseq[0]);
int np = buildsturm(p.size() - 1, sseq);
int atmin, atmax;
int nroot = numroots(np, sseq, atmin, atmax);
if (nroot == 0) return 0;
double minVal = -1.0;
UINT i = 0;
int nchanges = numchanges(np, sseq, minVal);
for (i = 0; nchanges != atmin && i != (UINT)MAXPOW; ++i) {
minVal *= 10.0;
nchanges = numchanges(np, sseq, minVal);
}
if (nchanges != atmin) {
atmin = nchanges;
}
double maxVal = 1.0;
nchanges = numchanges(np, sseq, maxVal);
for (i = 0; nchanges != atmax && i != (UINT)MAXPOW; ++i) {
maxVal *= 10.0;
nchanges = numchanges(np, sseq, maxVal);
}
if (nchanges != atmax) {
atmax = nchanges;
}
nroot = atmin - atmax;
assert(nroot > 0);
poly outPoly;
outPoly.ord = nroot;
sbisect(np, sseq, minVal, maxVal, atmin, atmax, outPoly.coef);
convert(outPoly, sol);
return nroot < 0 ? -1 : nroot;
}
//-----------------------------------------------------------------------------
} // end of unnamed namespace
//-----------------------------------------------------------------------------
void tLUDecomposition(double *a, int n, int *indx, double &d) {
int i, imax, j, k;
double big, dum, sum, temp;
std::vector<double> vv(n);
d = 1.0;
for (i = 1; i <= n; i++) {
big = 0.0;
for (j = 1; j <= n; j++)
if ((temp = fabs(a[getEl(i, j, n)])) > big) big = temp;
if (big == 0.0)
throw TMathException("Singular matrix in routine tLUDecomposition\n");
vv[i - 1] = 1.0 / big;
}
for (j = 1; j <= n; j++) {
for (i = 1; i < j; i++) {
sum = a[getEl(i, j, n)];
for (k = 1; k < i; k++) sum -= a[getEl(i, k, n)] * a[getEl(k, j, n)];
a[getEl(i, j, n)] = sum;
}
big = 0.0;
for (i = j; i <= n; i++) {
sum = a[getEl(i, j, n)];
for (k = 1; k < j; k++) sum -= a[getEl(i, k, n)] * a[getEl(k, j, n)];
a[getEl(i, j, n)] = sum;
if ((dum = vv[i - 1] * fabs(sum)) >= big) {
big = dum;
imax = i;
}
}
if (j != imax) {
for (k = 1; k <= n; k++) {
dum = a[getEl(imax, k, n)];
a[getEl(imax, k, n)] = a[getEl(j, k, n)];
a[getEl(j, k, n)] = dum;
}
d = -(d);
vv[imax - 1] = vv[j - 1];
}
indx[j - 1] = imax;
if (a[getEl(j, j, n)] == 0.0) a[getEl(j, j, n)] = TConsts::epsilon;
if (j != n) {
dum = 1.0 / (a[getEl(j, j, n)]);
for (i = j + 1; i <= n; i++) a[getEl(i, j, n)] *= dum;
}
}
}
//-----------------------------------------------------------------------------
void tbackSubstitution(double *a, int n, int *indx, double *b) {
int i, ii = 0, ip, j;
double sum;
for (i = 1; i <= n; i++) {
ip = indx[i - 1];
sum = b[ip - 1];
b[ip - 1] = b[i - 1];
if (ii)
for (j = ii; j <= i - 1; j++) sum -= a[getEl(i, j, n)] * b[j - 1];
else if (sum)
ii = i;
b[i - 1] = sum;
}
for (i = n; i >= 1; i--) {
sum = b[i - 1];
for (j = i + 1; j <= n; j++) sum -= a[getEl(i, j, n)] * b[j - 1];
b[i - 1] = sum / a[getEl(i, i, n)];
}
}
//-----------------------------------------------------------------------------
double tdet(double *LUa, int n, double d) {
for (int i = 1; i <= n; ++i) d *= LUa[getEl(i, i, n)];
return d;
}
//-----------------------------------------------------------------------------
double tdet(double *a, int n) {
double d;
std::vector<int> indx(n);
tLUDecomposition(a, n, &indx[0], d);
for (int i = 1; i <= n; ++i) d *= a[getEl(i, i, n)];
return d;
}
//-----------------------------------------------------------------------------
void tsolveSistem(double *a, int n, double *res) {
double d;
std::vector<int> indx(n);
tLUDecomposition(a, n, &indx[0], d);
assert(tdet(a, n, d) != 0);
/*
if( isAlmostZero(tdet(a, n, d)) )
throw TMathException("Singular matrix in routine tLUDecomposition\n");
*/
tbackSubstitution(a, n, &indx[0], res);
}
//-----------------------------------------------------------------------------
int rootFinding(const std::vector<double> &in_poly, std::vector<double> &sol) {
// per ora risolvo solo i polinomi di grado al piu' pari a 3
assert((int)in_poly.size() <= MAX_ORDER);
if (in_poly.empty() || (int)in_poly.size() > MAX_ORDER) return -1;
std::vector<double> p;
std::copy(in_poly.begin(), in_poly.end(), std::back_inserter(p));
// eat zero in poly
while (!p.empty() && isAlmostZero(p.back())) p.pop_back();
sol.clear();
while (!p.empty() && p.front() == 0) {
sol.push_back(0.0);
p.erase(p.begin()); // se i coefficienti bassi sono zero, ci sono soluzioni
// pari a 0.0. le metto,
} // e abbasso il grado del polinomio(piu' veloce)
switch (p.size()) {
case 0:
if (sol.empty()) return INFINITE_SOLUTIONS;
break;
case 1: // no solutions
break;
case 2:
sol.push_back(-p[0] / p[1]);
break;
case 3:
rootForQuadraticEquation(p, sol);
break;
case 4:
rootForCubicEquation(p, sol);
break;
default:
rootForGreaterThanThreeEquation(p, sol);
}
return sol.size();
}
//-----------------------------------------------------------------------------
/*
*/
int numberOfRootsInInterval(int order, const double *polyH, double min,
double max) {
poly sseq[MAX_ORDER];
int i, nchanges_0, nchanges_1, np;
if (order > MAX_ORDER) return -1;
while (polyH[order] == 0.0 && order > 0) --order;
// init a sturm's sequence with our polynomious
for (i = order; i >= 0; --i) sseq[0].coef[i] = polyH[i];
// build the Sturm sequence
np = buildsturm(order, sseq);
// compute number of variation on 0.0
nchanges_0 = numchanges(np, sseq, min);
nchanges_1 = numchanges(np, sseq, max);
return (nchanges_0 - nchanges_1);
}
//-----------------------------------------------------------------------------
double quadraticRoot(double a, double b, double c) {
double bb, q;
// caso lineare
if (fabs(a) < epsilon) {
if (fabs(b) >= epsilon) return -c / b;
return 1;
}
bb = b * b;
q = bb - 4.0 * a * c;
if (q < 0.0) return 1;
q = sqrt(q);
if (b < 0.0) q = -q;
q = -0.5 * (b + q);
double root1 = -1;
double root2 = -1;
if (fabs(q) >= epsilon) root1 = c / q;
if (fabs(a) >= epsilon) root2 = q / a;
if (0.0 - epsilon <= root1 && root1 <= 1.0 + epsilon) return root1;
if (0.0 - epsilon <= root2 && root2 <= 1.0 + epsilon) return root2;
return 1;
}
//-----------------------------------------------------------------------------
double cubicRoot(double a, double b, double c, double d) {
double A, Q, R, QQQ, RR;
double theta;
/* Test for a quadratic or linear degeneracy */
if (fabs(a) < epsilon) return quadraticRoot(b, c, d);
/* Normalize */
b /= a;
c /= a;
d /= a;
a = 1.0;
/* Compute discriminants */
Q = (b * b - 3.0 * c) / 9.0;
QQQ = Q * Q * Q;
R = (2.0 * b * b * b - 9.0 * b * c + 27.0 * d) / 54.0;
RR = R * R;
/* Three real roots */
if (RR < QQQ) {
theta = acos(R / sqrt(QQQ));
double root[3];
root[0] = root[1] = root[2] = -2.0 * sqrt(Q);
root[0] *= cos(theta / 3.0);
root[1] *= cos((theta + M_2PI) / 3.0);
root[2] *= cos((theta - M_2PI) / 3.0);
root[0] -= b / 3.0;
root[1] -= b / 3.0;
root[2] -= b / 3.0;
if (0.0 - epsilon < root[0] && root[0] < 1.0 + epsilon) return root[0];
if (0.0 - epsilon < root[1] && root[1] < 1.0 + epsilon) return root[1];
if (0.0 - epsilon < root[2] && root[2] < 1.0 + epsilon) return root[2];
return 1;
}
/* One real root */
else {
double root = 0;
A = -pow(fabs(R) + sqrt(RR - QQQ), 1.0 / 3.0);
if (A != 0.0) {
if (R < 0.0) A = -A;
root = A + Q / A;
}
root -= b / 3.0;
if (0.0 - epsilon < root && root < 1.0 + epsilon) return root;
return 1;
}
}
//-----------------------------------------------------------------------------
// End Of File
//-----------------------------------------------------------------------------