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cpoly.C
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cpoly.C
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/****************************************************************************
*
* cpoly.C René Hartung
* Laurent Bartholdi
*
* @(#)$id: fr_dll.c,v 1.18 2010/10/26 05:19:40 gap exp $
*
* Copyright (c) 2011, Laurent Bartholdi
*
****************************************************************************
*
* template for root-finding of complex polynomial
*
****************************************************************************/
// CAUCHY COMPUTES A LOWER BOUND ON THE MODULI OF THE ZEROS OF A
// POLYNOMIAL - PT IS THE MODULUS OF THE COEFFICIENTS.
//
static xcomplex cauchy(const int deg, xcomplex *P)
{
xreal x, xm, f, dx, df, tmp[deg+1];
for(int i = 0; i<=deg; i++){ tmp[i] = xabs(P[i]); };
// Compute upper estimate bound
x = xroot(tmp[deg],deg) / xroot(tmp[0],deg);
if(tmp[deg - 1] != 0.0) {
// Newton step at the origin is better, use it
xm = tmp[deg] / tmp[deg-1];
if (xm < x) x = xm;
}
tmp[deg] = -tmp[deg];
// Chop the interval (0,x) until f < 0
while(1) {
xm = x * 0.1;
// Evaluate the polynomial <tmp> at <xm>
f = tmp[0];
for(int i = 1; i <= deg; i++)
f = f * xm + tmp[i];
if(f <= 0.0) break;
x = xm;
}
dx = x;
// Do Newton iteration until x converges to two decimal places
while(fabsl(dx / x) > 0.005) {
f = tmp[0];
df = 0.0;
for(int i = 1; i <= deg; i++){
df = df * x + f;
f = f * x + tmp[i];
}
dx = f / df;
x -= dx; // Newton step
}
return (xcomplex)(x);
}
// RETURNS A SCALE FACTOR TO MULTIPLY THE COEFFICIENTS OF THE POLYNOMIAL.
// THE SCALING IS DONE TO AVOID OVERFLOW AND TO AVOID UNDETECTED UNDERFLOW
// INTERFERING WITH THE CONVERGENCE CRITERION. THE FACTOR IS A POWER OF THE
//int BASE.
// PT - MODULUS OF COEFFICIENTS OF P
// ETA, INFIN, SMALNO, BASE - CONSTANTS DESCRIBING THE FLOATING POINT ARITHMETIC.
//
static void scale(const int deg, xcomplex* P)
{
int hi, lo, max, min, x, sc;
// Find largest and smallest moduli of coefficients
hi = (int)(xdata.MAX_EXP / 2.0);
lo = (int)(xdata.MIN_EXP - xbits(P[0]));
max = xlogb(P[0]); // leading coefficient does not vanish!
min = xlogb(P[0]);
for(int i = 0; i <= deg; i++) {
if (P[i] != xdata.ZERO){
x = xlogb(P[i]);
if(x > max) max = x;
if(x < min) min = x;
}
}
// Scale only if there are very large or very small components
if(min >= lo && max <= hi) return;
x = lo - min;
if(x <= 0)
sc = -(max+min) / 2;
else {
sc = x;
if(xdata.MAX_EXP - sc > max) sc = 0;
}
// Scale the polynomial
for(int i = 0; i<= deg; i++){ xscalbln(&P[i],sc); }
}
// COMPUTES THE DERIVATIVE POLYNOMIAL AS THE INITIAL H
// POLYNOMIAL AND COMPUTES L1 NO-SHIFT H POLYNOMIALS.
//
static void noshft(const int l1, int deg, xcomplex *P, xcomplex *H)
{
int i, j, jj;
xcomplex t;
// compute the first H-polynomial as the (normed) derivative of P
for(i = 0; i < deg; i++)
H[i] = (P[i] * (deg-i)) / deg;
for(jj = 1; jj <= l1; jj++) {
if(xnorm(H[deg - 1]) > xeta(P[deg-1])*xeta(P[deg-1])* 10*10 * xnorm(P[deg - 1])) {
t = -P[deg] / H[deg-1];
for(i = 0; i < deg-1; i++){
j = deg - i - 1;
H[j] = t * H[j-1] + P[j];
}
H[0] = P[0];
} else {
// if the constant term is essentially zero, shift H coefficients
for(i = 0; i < deg-1; i++) {
j = deg - i - 1;
H[j] = H[j - 1];
}
H[0] = xdata.ZERO;
}
}
}
// EVALUATES A POLYNOMIAL P AT S BY THE HORNER RECURRENCE
// PLACING THE PARTIAL SUMS IN Q AND THE COMPUTED VALUE IN PV.
//
static xcomplex polyev(const int deg, const xcomplex s, const xcomplex *Q, xcomplex *q) {
q[0] = Q[0];
for(int i = 1; i <= deg; i++){ q[i] = q[i-1] * s + Q[i]; };
return q[deg];
}
// COMPUTES T = -P(S)/H(S).
// BOOL - LOGICAL, SET TRUE IF H(S) IS ESSENTIALLY ZERO.
//
static xcomplex calct(bool *bol, int deg, xcomplex Ps, xcomplex *H, xcomplex *h, xcomplex s){
xcomplex Hs;
Hs = polyev(deg-1, s, H, h);
*bol = xnorm(Hs) <= xeta(H[deg-1])*xeta(H[deg-1]) * 10*10 * xnorm(H[deg-1]);
if(!*bol)
return -Ps / Hs;
else
return xdata.ZERO;
}
// CALCULATES THE NEXT SHIFTED H POLYNOMIAL.
// BOOL - LOGICAL, IF .TRUE. H(S) IS ESSENTIALLY ZERO
//
static void nexth(const bool bol, int deg, xcomplex t, xcomplex *H, xcomplex *h, xcomplex *p){
if(!bol){
for(int j = 1; j < deg; j++)
H[j] = t * h[j-1] + p[j];
H[0] = p[0];
}
else {
// If h[s] is zero replace H with qh
for(int j = 1; j < deg; j++)
H[j] = h[j - 1];
h[0] = xdata.ZERO;
}
}
// BOUNDS THE ERROR IN EVALUATING THE POLYNOMIAL BY THE HORNER RECURRENCE.
// QR,QI - THE PARTIAL SUMS
// MS -MODULUS OF THE POINT
// MP -MODULUS OF POLYNOMIAL VALUE
// ARE, MRE -ERROR BOUNDS ON COMPLEX ADDITION AND MULTIPLICATION
//
static xreal errev(const int deg, const xcomplex *p, const xreal ms, const xreal mp){
xreal MRE = 2.0 * sqrt(2.0) * xeta(p[0]);
xreal e = xabs(p[0]) * MRE / (xeta(p[0]) + MRE);
for(int i = 0; i <= deg; i++)
e = e * ms + xabs(p[i]);
return e * (xeta(p[0]) + MRE) - MRE * mp;
}
// CARRIES OUT THE THIRD STAGE ITERATION.
// L3 - LIMIT OF STEPS IN STAGE 3.
// ZR,ZI - ON ENTRY CONTAINS THE INITIAL ITERATE, IF THE
// ITERATION CONVERGES IT CONTAINS THE FINAL ITERATE ON EXIT.
// CONV - .TRUE. IF ITERATION CONVERGES
//
static bool vrshft(const int l3, int deg, xcomplex *P, xcomplex *p, xcomplex *H, xcomplex *h, xcomplex *zero, xcomplex *s){
bool bol, conv, b;
int i, j;
xcomplex Ps, t;
xreal mp, ms, omp = 0.0, relstp = 0.0, tp;
conv = b = false;
*s = *zero;
// Main loop for stage three
for(i = 1; i <= l3; i++) {
// Evaluate P at S and test for convergence
Ps = polyev(deg, *s, P, p);
mp = xabs(Ps);
ms = xabs(*s);
if(mp <= 20 * errev(deg, p, ms, mp)) {
// Polynomial value is smaller in value than a bound on the error
// in evaluating P, terminate the iteration
conv = true;
*zero = *s;
return conv;
}
if(i != 1) {
if(!(b || mp < omp || relstp >= 0.05)){
// if(!(b || xlogb(mp) < omp || real(relstp) >= 0.05)){
// Iteration has stalled. Probably a cluster of zeros. Do 5 fixed
// shift steps into the cluster to force one zero to dominate
tp = relstp;
b = true;
if(relstp < xeta(P[0])) tp = xeta(P[0]);
*s *= 1.0 + (1.0+1.0i)*sqrt(tp);
Ps = polyev(deg, *s, P, p);
for(j = 1; j <= 5; j++){
t = calct(&bol, deg, Ps, H, h, *s);
nexth(bol, deg, t, H, h, p);
}
omp = xdata.INFIN;
goto _20;
}
// Exit if polynomial value increase significantly
if(mp * 0.1 > omp) return conv;
}
omp = mp;
// Calculate next iterate
_20: t = calct(&bol, deg, Ps, H, h, *s);
nexth(bol, deg, t, H, h, p);
t = calct(&bol, deg, Ps, H, h, *s);
if(!bol) {
relstp = xabs(t) / xabs(*s);
*s += t;
}
} // end for
return conv;
}
// COMPUTES L2 FIXED-SHIFT H POLYNOMIALS AND TESTS FOR CONVERGENCE.
// INITIATES A VARIABLE-SHIFT ITERATION AND RETURNS WITH THE
// APPROXIMATE ZERO IF SUCCESSFUL.
// L2 - LIMIT OF FIXED SHIFT STEPS
// ZR,ZI - APPROXIMATE ZERO IF CONV IS .TRUE.
// CONV - LOGICAL INDICATING CONVERGENCE OF STAGE 3 ITERATION
//
static bool fxshft(const int l2, int deg, xcomplex *P, xcomplex *p, xcomplex *H, xcomplex *h, xcomplex *zero, xcomplex *s){
bool bol, conv; // boolean for convergence of stage 2
bool test, pasd;
xcomplex old_T, old_S, Ps, t;
xcomplex Tmp[deg+1];
Ps = polyev(deg, *s, P, p);
test = true;
pasd = false;
// Calculate first T = -P(S)/H(S)
t = calct(&bol, deg, Ps, H, h, *s);
// Main loop for second stage
for(int j = 1; j <= l2; j++){
old_T = t;
// Compute the next H Polynomial and new t
nexth(bol, deg, t, H, h, p);
t = calct(&bol, deg, Ps, H, h, *s);
*zero = *s + t;
// Test for convergence unless stage 3 has failed once or this
// is the last H Polynomial
if(!(bol || !test || j == l2)){
if(xabs(t - old_T) < 0.5 * xabs(*zero)) {
if(pasd) {
// The weak convergence test has been passwed twice, start the third stage
// Iteration, after saving the current H polynomial and shift
for(int i = 0; i < deg; i++)
Tmp[i] = H[i];
old_S = *s;
conv = vrshft(10, deg, P, p, H, h, zero, s);
if(conv) return conv;
//The iteration failed to converge. Turn off testing and restore h,s,pv and T
test = false;
for(int i = 0; i < deg; i++)
H[i] = Tmp[i];
*s = old_S;
Ps = polyev(deg, *s, P, p);
t = calct(&bol, deg, Ps, H, h, *s);
continue;
}
pasd = true;
}
else
pasd = false;
}
}
// Attempt an iteration with final H polynomial from second stage
conv = vrshft(10, deg, P, p, H, h, zero, s);
return conv;
}
// Main function
//
int cpoly(int degree, const xcomplex poly[], xcomplex Roots[])
{
xcomplex PhiDiff = -0.069756473 + 0.99756405i;
xcomplex PhiRand = (1.0-1.0i) /sqrt(2.0);
xcomplex P[degree+1], H[degree+1], h[degree+1], p[degree+1], zero, s, bnd;
unsigned int conv = 0;
for (int i = 0; i <= degree; i++)
if (isnan(poly[i]))
return -1; // otherwise we may get stuck in infinite loops
while(poly[0] == xdata.ZERO) {
poly++;
degree--;
if (degree < 0)
return -1;
};
int deg = degree;
// Remove the zeros at the origin if any
while(poly[deg] == xdata.ZERO){
Roots[degree - deg] = xdata.ZERO;
deg--;
}
if (deg == 0) return degree;
// Make a copy of the coefficients
for(int i = 0; i <= deg; i++) { P[i] = poly[i]; }
scale(deg, P);
search:
if(deg <= 1){
Roots[degree-1] = - P[1] / P[0];
return degree;
}
// compute a bound of the moduli of the roots (Newton-Raphson)
bnd = cauchy(deg, P);
// Outer loop to control 2 Major passes with different sequences of shifts
for(int cnt1 = 1; cnt1 <= 2; cnt1++) {
// First stage calculation , no shift
noshft(5, deg, P, H);
// Inner loop to select a shift
for(int cnt2 = 1; cnt2 <= 9; cnt2++) {
// Shift is chosen with modulus bnd and amplitude rotated by 94 degree from the previous shif
PhiRand = PhiDiff * PhiRand;
s = bnd * PhiRand;
// Second stage calculation, fixed shift
conv = fxshft(10 * cnt2, deg, P, p, H, h, &zero, &s);
if(conv) {
// The second stage jumps directly to the third stage iteration
// If successful the zero is stored and the polynomial deflated
Roots[degree - deg] = zero;
// continue with the remaining polynomial
deg--;
for(int i = 0; i <= deg; i++){ P[i] = p[i]; };
goto search;
}
// if the iteration is unsuccessful another shift is chosen
}
// if 9 shifts fail, the outer loop is repeated with another sequence of shifts
}
// The zerofinder has failed on two major passes
// return empty handed with the number of roots found (less than the original degree)
return degree - deg;
}