/*
    polynom.cpp
    POLYNOM 0.11, complex polynomial factoring and expansion library
    implementation. See API specification in headerfile polynom.hpp.
    Latest version available at: https://ecomaan.nl/cpp/polynom

    Copyright (c) 2006, 2020 - Pieter Suurmond

    Permission is hereby granted, free of charge, to any person obtaining
    a copy of this software and associated documentation files
    (the "Software"), to deal in the Software without restriction,
    including without limitation the rights to use, copy, modify, merge,
    publish, distribute, sublicense, and/or sell copies of the Software,
    and to permit persons to whom the Software is furnished to do so,
    subject to the following conditions:

    The above copyright notice and this permission notice shall be
    included in all copies or substantial portions of the Software.

    Any person wishing to distribute modifications to the Software is
    requested to send the modifications to the original developer so that
    they can be incorporated into the canonical version.

    THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND,
    EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF
    MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.
    IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR
    ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF
    CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION
    WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE.  
*/

#define kPOLYNOM_DEBUG  (0)  // Make this constant 0 to surpress logging of
                             // function factor(). Make 1 for debugging.
#if (kPOLYNOM_DEBUG)
 #include <fstream>
#endif

#include <complex>
#include <vector>

using namespace std;

#include "polynom.hpp"      // Header needs std namespace prior to inclusion.

#define kEPSILON (1e-6)     // Remainders after synthetic division below this
                            // value are regarded zero (maximum error allowed).
/*
    Function laguerre() tries to find one single root. Took Fortran code from
    http://www.netlib.org/kincaid-cheney/laguerre.f as a starting point:   

      c     Numerical Analysis: The Mathematics of Scientific Computing
      c     D.R. Kincaid & E.W. Cheney. Brooks/Cole Publ., 1990.
      c     Section 3.5 Example of Laguerre's Method
      c
            implicit complex (a-h,o-z)
            parameter (n=4, M=7)
            dimension a(0:n)
            data a/(-72.,68.),(1140.,636.),(-216.,190.),(-10.,-26.),(1.,0.)/
            data z/(1.,1.)/
      c
            sn = real(n)
            do 3 k=1,m
               alpha = a(n)
               beta = (0.0,0.0)
               gamma = (0.0,0.0)
      c
               do 2 j=n-1,0,-1
                  gamma = z*gamma + beta
                  beta  = z*beta  + alpha
                  alpha = z*alpha + a(j)
       2       continue
      c                                             Pieter: you will notice:
               ca = -beta/alpha                     DIV BY ZERO POSSIBLE HERE
               cb =  ca*ca - 2.0*gamma/alpha        AND HERE (converged & done?)
               c1 = sqrt((sn-1.0)*(sn*cb - ca*ca))
               cc1 = (ca + c1)/sn
               cc2 = (ca - c1)/sn
               cc = cc2
               if (abs(cc1) .gt. abs(cc2)) then cc = cc1
	           c2 = 1.0/cc
	           z = z + c2
	           d = (ca - cc)/(sn - 1.0)
	           rho1 = sqrt(sn)*c2
	           rho2 = sqrt(sn/(cc*cc + (sn-1)*d*d))
	           print *,'values of z, c2, rho1, and rho2:'
	           print *,z,c2,rho1,rho2
       3    continue
      c
*/
static int laguerre(int                    degree, // Number of coefficients-1.
                    const complex<double>* a,      // Array of coefficients.
                    complex<double>*       zp)     // Starting and end point.
{
#if (kPOLYNOM_DEBUG)
    ofstream        t("POLYNOM.DEBUG"); // Some stream for debugging.
#endif                                  // cout, cerr, '/dev/null', etc.
    static const double f[8] = { 1.0, 0.5, 0.25, 0.75, 0.13, 0.38, 0.62, 0.88 };
    complex<double>     z = *zp;    // Copy for faster access.
    double              sn(degree); // degree = sn = number of coefficients - 1.
    double              snm(degree-1);
    long                k;

    for (k = 0; k < 10000; k++)  // Not more than ten thousand iterations.
        {
        complex<double> ca, caca, cb, c1, dz, cc, cp, cm, zz;
        complex<double> alpha(a[degree]);
        complex<double> beta (0.0, 0.0);
        complex<double> gamma(0.0, 0.0);
        for (int j = degree - 1; j >= 0; j--)   // Horner's scheme.
            {
            gamma = (z * gamma) + beta;
            beta  = (z * beta)  + alpha;
            alpha = (z * alpha) + a[j];
            }
        if (alpha == 0.0)
            {
            *zp = z;
#if (kPOLYNOM_DEBUG)
            t << "Laguerre: (alpha=0) converged after " << k << " iterations.\n";
#endif
            return 0;
            }
        ca   = -beta / alpha;
        caca = ca * ca;
        cb   = caca - (2.0 * gamma / alpha);
        c1   = sqrt(snm * ((sn * cb) - caca));
        cp   = ca + c1;
        cm   = ca - c1;
        if (norm(cm) > norm(cp))      // norm() is faster than abs().
            cp = cm;
        if (cp == 0.0)                // Re-init trick from 'Numerical Recipes'.
            dz = (1.0 + abs(z)) * polar(1.0, static_cast<double>(k));
        else
            dz = sn / cp;
        short kk = k % 9;               // Once every 9 iterations.
        if (!kk)
            {
            dz *= f[(k / 9) & 7];       // Cycle breaking trick (but f[0]=1.0).
            }
	    z += dz;
        if (kk && (norm(dz) < 1e-31))   // OK, converged: almost no change.
            {                           // THIS 1e-31 IS RELATED TO kEPSILON !
            *zp = z;
#if (kPOLYNOM_DEBUG)
            t << "Laguerre: (dz=0) converged after " << k << " iterations.\n";
#endif
            return 0;                        
            }
        if      (k == 2000) { z *= -1.0;    }           // Kick z around when it
        else if (k == 4000) { z = conj(z);  }           // is taking too long.
        else if (k == 6000) { z = -conj(z); }
        else if (k == 8000) { z *= complex<double>(0.0, 1.0); }
        }
#if (kPOLYNOM_DEBUG)
    t << "Laguerre: failed to converge within " << k << " iterations.\n";
#endif
    return 1;
}

/*
    In most cases, the number of roots will equal the number of input coeffi-
    cients minus one. Only if the last (or latter) coefficient(s) is (are) zero,
    the number of resulting roots may be less than expected (coeffs.size() - 1).
    Coefficients expected in ascending order of degree.
*/
int factor(const vector<complex<double> >&  coeffs,
           vector<complex<double> >&        roots)
{
    int             attempt = 0;
    const int       n_attempts = 4;
    static const 
    complex<double> guess[n_attempts] = { complex<double>( 8.0),
                                          complex<double>(-6.0),
                                          complex<double>( 0.0,  6.0),
                                          complex<double>( 0.0, -8.0) };
    complex<double> *cc = NULL, // Tmp copy of coefficients.
                    *c  = NULL; // Itarator within it.
    vector<complex<double> >::const_iterator  coeff  = coeffs.begin();
    int                                       degree = coeffs.size() - 1;
    int                                       num_roots = 0;
    int                                       e = 0;

    // Find the ACTUAL degree of the polynomial. This guarentees last coeff is non-zero.
    while ((degree > 0) && (coeff[degree] == 0.0)) // Use kEPSILON instead of 0.0?
        degree--;

    if (degree < 1)                 // There must at least be 2 coefficients, of which
        return 1;                   // only the first may be zero.

    if (roots.size() != static_cast<unsigned int>(degree)) // Prepare output vector.
        roots.resize(degree);

    // Factor-out roots at the origin.
    while ((degree > 0) && (coeff[0] == 0.0)) // Use kEPSILON instead of 0.0?
        {
        roots[num_roots++] = 0.0;       // Insert root at 0.
        coeff++;                        // Never reuse this coeff (INC ITERATOR).
        degree--;
        }
    int degreeH = degree; // Remember for retries.
    // From here on, first and last coeff are non-zero.
    if (degree > 0)                   // Copy input array so we can write in it.
        {                             // (Only necessary for Laguerre, actually).
        cc = c = new complex<double>[degree + 1]; // MAY THROW AN EXCEPTION.
        for (int n = 0; n <= degree; n++)
            c[n] = coeff[n];
        }
    while (degree > 2)
        {        
        // It is perhaps a bit stupid to investigate this EACH time: the chance that
        // such a simple form appears in between Laguerre iterations is almost nil(?).
        int n;                          // See whether all coeffs in between are zero.
        for (n = 1; (n < degree) && (c[n] == 0.0); n++) // Use kEPSILON instead of 0.0?
            ;                           // Then we have a complex    88
        if (n == degree)                // binomial in this form:   x  = -a0 / a88 
            {                           // Now we have 88 solutions all at once:
            complex<double> p = -c[0] / c[degree];
            double          magnitude = pow(norm(p), 0.5 / static_cast<double>(degree));
            double          angle     = arg(p) / static_cast<double>(degree);
            double          angle_inc = 8.0 * atan(1.0) / static_cast<double>(degree);
            do  {
                roots[num_roots++] = polar(magnitude, angle);
                angle += angle_inc;
                } while (--degree);     // degree = 0 returns immediately.
            }
        else
            {                         // Solve numerically.
            complex<double>  z = guess[attempt]; // Select a starting point.
            if (laguerre(degree,      // Number of coefficients - 1.
                         c,           // Ptr to array of complex coefficients.
                         &z))         // Starting point and found root.
                e = 2;                // 2 means Laguerre did not converge.
            else
                { // Synthetic division (and check if z is really a root).
	            for (n = degree - 1; n >= 0; n--) // c[degree] stays the same.
                    c[n] += c[n+1] * z;
                if (abs(c[0]) < kEPSILON)               // Must be (almost) zero.
                    {
                    degree--;
                    c++;                                // Remove first coefficient.
                    roots[num_roots++] = z;             // Store the divisor.
                    e = 0;
                    }
                else
                    e = 3; // 3 means synthetic division left non-zero remainder.
                }
            if (e)
                {
                if (++attempt < n_attempts) // Try all over again.
                    {
                    num_roots -= (degreeH - degree);    // Remove only the non-
                    degree = degreeH;                   // zero roots found.
                    c = cc;                             // Restore coeffs.
                    for (n = 0; n <= degree; n++)
                        c[n] = coeff[n];
                    }
                else
                    degree = 0; /* Exit the loop, e=2 or e=3. */
                }
            }
        }
    if (degree == 2)                // Solve quadratic equation analytically:
        {
        complex<double> a2 = c[2] * 2.0;                //        --------
        complex<double> D  = sqrt(c[1]*c[1]             //       / 2
                           -4.0 * c[2]*c[0]);           // D = \/ b - 4ac
        roots[num_roots++] = (-c[1] + D) / a2;          // x1 = (-b+D)/2a
        roots[num_roots++] = (-c[1] - D) / a2;          // x2 = (-b-D)/2a
        }
    else if (degree == 1)                   // Solve linear equation analytically:
        {                                   // a0 + a1 x = 0  <=>  x = -a0 / a1.
        roots[num_roots++] = -c[0]/c[1];    // Division by zero can never occur.
        }
    // else (if (degree == 0)) we're done.
    delete[] cc;                            // Release temporary workspace.
    return e;                               // 0 = Ok.
}

/*
    Same as factor() except that coefficients are expected in descending order
    of degree (and this function works slightly slower :-).
*/
int factor_d(const vector<complex<double> >&    coeffs,
             vector<complex<double> >&          roots)
{
    int                         n;
    int                         num_coeffs = coeffs.size(); // MAY THROW.
    vector<complex<double> >    c(num_coeffs);
    
    // Work with temporary copy of the given coefficients in reverse order.
    for (n = 0; n < num_coeffs; n++)
        c[n] = coeffs[num_coeffs - n - 1];
    n = factor(c, roots);
    return n;
}

/*
    Expand the polynome. Recursive multiplication of its' roots.
*/
int expand(const vector<complex<double> >&  roots,  // Input array.
           vector<complex<double> >&        coeffs) // Output, ascending degree.
{
    int i;
    int num_roots = roots.size();

    if (num_roots < 1)                  // At least one root should be there.
        return 1;

    if (coeffs.size() != static_cast<unsigned int>(num_roots + 1))
        coeffs.resize(num_roots + 1);   // Prepare output vector.

    for (i = 0; i < num_roots; i++)     // Initialise coefficient vector.
        coeffs[i] = 0.0;
    coeffs[i] = 1.0;                    // Last coeff = 1.0.
    
    for (i = num_roots; i > 0; i--)     // Recursive multiplication.
        {
        coeffs[i - 1] = -roots[i - 1] * coeffs[i];
        for (int j = i; j < num_roots; j++)
            coeffs[j] -= roots[i - 1] * coeffs[j + 1];
        }
    return 0;                           // 0 = Ok.
}

/*
    Same as expand() except that coefficients are delivered in descending order
    of degree.
*/
int expand_d(const vector<complex<double> >&  roots,  // Input array.
             vector<complex<double> >&        coeffs) // Output, descending degree.
{
    int i;
    int num_roots = roots.size();
    
    if (num_roots < 1)                  // At least one root should be there.
        return 1;

    if (coeffs.size() != static_cast<unsigned int>(num_roots + 1))
        coeffs.resize(num_roots + 1);   // Prepare output vector.

    coeffs[0] = 1.0;                    // Init coefficient vector.
    for (i = 1; i <= num_roots; i++)
        coeffs[i] = 0.0;

    for (i = 0; i < num_roots; i++)     // Recursive multiplication.
        {
        coeffs[i + 1] = -roots[i] * coeffs[i];
        for (int j = i; j > 0; j--)
            coeffs[j] -= roots[i] * coeffs[j - 1];
        }
    return 0;                           // 0 = Ok.
}