391 lines
No EOL
13 KiB
C++
Executable file
391 lines
No EOL
13 KiB
C++
Executable file
// méthodes issues de quartic.c
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// on a laissé les informations des programmes originaux
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// l'ensemble est organisée sous forme d'une classe pour encapsuler
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//********************************************************************************
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// l'écriture initiale a été modifiée -> C++
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//
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// calcul :
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// - racine d'un polynome du second degré
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// - racine d'un polynome du 3ieme degré
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// - racine d'un polynome du 4ième degré avec différentes méthodes
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// . Descartes' algorithm
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// . Ferrari's algorithm
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// . Neumark's algorithm
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// . Yacoub & Fraidenraich's algorithm
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// . Christianson's algorithm
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//********************************************************************************
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#ifndef QUARTIC_H
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#define QUARTIC_H
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#include <stdio.h>
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class Quartic
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{ public:
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/* quartic.c
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version 54
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a test of relative accuracy of various quartic routines.
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use:
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quartic [-a] [-c n] [-q n] [-d n]
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(no parameters) : a loop through 10,000 prechosen quartic coefficients
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-a : improve cubic roots (default: no iteration)
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-c n : keep reading n cubic roots from standard input
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(if n=0 read coefficients), terminate with 'q'.
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-d n : set debug value to 'n' (default: 5)
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-q n : keep reading n quartic roots from standard input
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(if n=0 read coefficients), terminate with 'q'.
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debug <= 1 : print cases used
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method-
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http://linus.socs.uts.edu.au/~don/pubs/solving.html
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4 Apr 2004 bringing yacfraid notation into line with solving.html
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4 Apr 2004 fixed bug in final root calculation in yacfraid
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8 Mar 2004 corrected modified yacfraid algorithm
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8 Mar 2004 modified yacfraid algorithm
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31 Jan 2004 printing results when number of roots vary
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30 Jan 2004 printing error of chosen cubic if debug<1
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27 Jan 2004 seeking worst coefficients for each algorithm
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27 Jan 2004 choosing best route for k in chris
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26 Jan 2004 conforming variables to solving.html
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26 Jan 2004 fixed bug in yacfraid for multiplicity 3
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25 Jan 2004 speeding up chris
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24 Jan 2004 fixed chris error (A <-> B)
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23 Jan 2004 use debug<1 for diagnostics
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21 Jan 2004 solve cubic in neumark, yacfraid and chris if d=0
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20 Jan 2004 3 roots to cubic in chris if e1=0
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19 Jan 2004 debugging Christianson routine
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14 Jan 2004 adding Christianson, cut determinant in quadratic call
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5 Jan 2004 improving cubic roots by Newton-Raphson iteration
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23 Dec 2003 seeking snag in yacfraid
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21 Dec 2003 putting diagnostic printout in yacfraid
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18 Dec 2003 allowing input of equation coefficients
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18 Dec 2003 recording cubic root giving least worst error
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17 Dec 2003 recording cubic root giving the most quartic roots
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16 Dec 2003 using the cubic root giving the most quartic roots
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16 Dec 2003 trying consistency of 3 cubic roots where available
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15 Dec 2003 trying all 3 cubic roots where available
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13 Dec 2003 trying to fix Neumark
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13 Dec 2003 cleaning up diagnostic format
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12 Dec 2003 initialising n,m,po3 in cubic
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12 Dec 2003 allow cubic to return 3 zero roots if p=q=r=0
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10 Dec 2003 added setargs
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2 Dec 2003 finding worst cases
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2 Dec 2003 negating j if p>0 in cubic
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1 Dec 2003 changing v in cubic from (sinsqk > doub0) to (sinsqk >= doub0)
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1 Dec 2003 test changing v in cubic from po3sq+po3sq to doub2*po3sq
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30 Nov 2003 counting cases in all solving routines
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29 Nov 2003 testing wsq >= doub0
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29 Nov 2003 mult by doub2 for v in cubic
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29 Nov 2003 cut po3cu from cubic
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29 Nov 2003 better quadratic
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29 Nov 2003 count agreements
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17 Nov 2003 count special cases
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16 Nov 2003 option of loop or read coefficients from input
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15 Nov 2003 fixed cubic() bug
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11 Nov 2003 added Brown and Yacoub & Fraidenraich's algorithms
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21 Jan 1989 quartic selecting Descartes, Ferrari, or Neumark algorithms
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16 Jul 1981 Don Herbison-Evans
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"Solving Quartics and Cubics for Graphics", D. Herbison-Evans,
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Graphics Gems V (ed.: A. Paeth) Academic Press (Chesnut Hill), pp. 3-15 (1995).
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"Solving Quartics and Cubics for Graphics", D. Herbison-Evans,
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Research Report CS-86-56, Department of Computer Science, University of Waterloo (1986)
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"Caterpillars and the Inaccurate Solution of Cubic and Quartic Equations",
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D. Herbison-Evans, Australian Computer Science Communications,
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Vol. 5, No. 1, pp. 80-90 (1983)
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subroutines:
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errors - find errors in a set of roots
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acos3 - find arcos(x/3)
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curoot - find cube root
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quadratic - solve a quadratic
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cubic - solve a cubic
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cubnewton - improve cubic roots by iteration
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quartic - solve a quartic
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descartes - use Descartes' algorithm to solve a quartic
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ferrari - use Ferrari's algorithm to solve a quartic
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neumark - use Neumark's algorithm to solve a quartic
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yacfraid - use Yacoub & Fraidenraich's algorithm to solve a quartic
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chris - use Christianson's algorithm to solve a quartic
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*/
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/***********************************/
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// constructeur permettant d'initialiser les variables
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Quartic();
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// modification du paramètre de débug et d'itération pour cubic
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void Change(int debuge, bool iteratee) {debug=debuge;iterate=iteratee;};
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//****************************************************
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//
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// find the errors
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//
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// called by quartictest, docoeff, compare,
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// chris, descartes, ferrari, neumark, yacfraid.
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//****************************************************
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double errors(double a,double b,double c,double d,double rts[4],double rterr[4],int nrts);
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//double a,b,c,d,rts[4],rterr[4];
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//int nrts;
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/**********************************************/
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// find cos(acos(x)/3)
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//
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// 16 Jul 1981 Don Herbison-Evans
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//
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// called by cubic .
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/**********************************************/
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double acos3(double x);
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// double x ;
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/***************************************/
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// find cube root of x.
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//
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// 30 Jan 1989 Don Herbison-Evans
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//
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// called by cubic .
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/***************************************/
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double curoot(double x);
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// double x ;
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/****************************************************/
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// solve the quadratic equation -
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//
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// x**2 + b*x + c = 0
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//
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// 14 Jan 2004 cut determinant in quadratic call
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// 29 Nov 2003 improved
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// 16 Jul 1981 Don Herbison-Evans
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//
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// called by cubic,quartic,chris,descartes,ferrari,neumark.
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/****************************************************/
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int quadratic(double b,double c,double rts[4]);
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// double b,c,rts[4];
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/**************************************************/
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// find the real roots of the cubic -
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// x**3 + p*x**2 + q*x + r = 0
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//
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// 12 Dec 2003 initialising n,m,po3
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// 12 Dec 2003 allow return of 3 zero roots if p=q=r=0
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// 2 Dec 2003 negating j if p>0
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// 1 Dec 2003 changing v from (sinsqk > doub0) to (sinsqk >= doub0)
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// 1 Dec 2003 test changing v from po3sq+po3sq to doub2*po3sq
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// 16 Jul 1981 Don Herbison-Evans
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//
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// input parameters -
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// p,q,r - coeffs of cubic equation.
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// iterate : booléen indiquant si l'on veut améliorer les racines via du newton (4 fois)
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// debug : si <1 impression d'un paquet de paramètre ??
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//
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// output-
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// the number of real roots
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// v3 - the roots.
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//
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// global constants -
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// rt3 - sqrt(3)
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// inv3 - 1/3
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// doubmax - square root of largest number held by machine
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//
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// method -
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// see D.E. Littlewood, "A University Algebra" pp.173 - 6
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//
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// 15 Nov 2003 output 3 real roots: Don Herbison-Evans
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// Apr 1981 initial version: Charles Prineas
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//
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// called by cubictest,quartic,chris,yacfraid,neumark,descartes,ferrari.
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// calls quadratic,acos3,curoot,cubnewton.
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/**************************************************/
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int cubic(double p,double q,double r, double v3[4]) ;
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/****************************************************/
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// improve roots of cubic by Newton-Raphson iteration
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//
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// 5 Jan 2004 Don Herbison-Evans
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//
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// called by cubic.
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/****************************************************/
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void cubnewton(double p,double q,double r,int n3,double v3[4]);
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//int n3;
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//double p,q,r,v3[4];
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//
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//****************************************************
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// Solve quartic equation using either
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// quadratic, Ferrari's or Neumark's algorithm.
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//
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// called by
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// calls descartes, ferrari, neumark, yacfraid.
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//
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// 15 Dec 2003 added yacfraid
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// 10 Dec 2003 added descartes with neg coeffs
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// 21 Jan 1989 Don Herbison-Evans
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//****************************************************
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int quartic(double a,double b,double c,double d,double rts[4]);
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//double a,b,c,d,rts[4];
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//*****************************************/
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// Solve quartic equation using
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// Descartes-Euler-Cardano algorithm
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//
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// called by quartic, compare, quartictest.
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//
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// Strong, T. "Elemementary and Higher Algebra"
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// Pratt and Oakley, p. 469 (1859)
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//
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// 16 Jul 1981 Don Herbison-Evans
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//*****************************************/
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int descartes(double a,double b,double c,double d,double rts[4]);
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//double a,b,c,d,rts[4];
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//****************************************************/
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// solve the quartic equation -
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//
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// x**4 + a*x**3 + b*x**2 + c*x + d = 0
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//
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// calls cubic, quadratic.
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//
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// input -
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// a,b,c,e - coeffs of equation.
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//
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// output -
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// n4 - number of real roots.
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// rts - array of root values.
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//
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// method : Ferrari - Lagrange
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// Theory of Equations, H.W. Turnbull p. 140 (1947)
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//
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// 16 Jul 1981 Don Herbison-Evans
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//
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// calls cubic, quadratic
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//****************************************************/
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int ferrari(double a,double b,double c,double d,double rts[4]);
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// double a,b,c,d,rts[4];
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//*****************************************/
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// solve the quartic equation -
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//
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// x**4 + a*x**3 + b*x**2 + c*x + d = 0
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//
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// calls cubic, quadratic.
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//
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// input parameters -
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// a,b,c,e - coeffs of equation.
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//
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// output parameters -
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// n4 - number of real roots.
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// rts - array of root values.
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//
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// method - S. Neumark
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// "Solution of Cubic and Quartic Equations" - Pergamon 1965
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//
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// 1 Dec 1985 translated to C with help of Shawn Neely
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// 16 Jul 1981 Don Herbison-Evans
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//*****************************************/
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int neumark(double a,double b,double c,double d,double rts[4]);
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// double a,b,c,d,rts[4];
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//****************************************************/
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// solve the quartic equation -
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// x**4 + a*x**3 + b*x**2 + c*x + d = 0
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//
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// calls cubic, quadratic.
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//
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// input parameters -
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// a,b,c,e - coeffs of equation.
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//
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// output parameters -
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// n4 - number of real roots.
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// rts - array of root values.
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//
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// method -
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// K.S. Brown
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// Reducing Quartics to Cubics,
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// http://www.seanet.com/~ksbrown/kmath296.htm (1967)
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//
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// Michael Daoud Yacoub & Gustavo Fraidenraich
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// "A new simple solution of the general quartic equation"
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// Revised 16 Feb 2004
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//
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// 14 Nov 2003 Don Herbison-Evans
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//*****************************************/
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int yacfraid(double a,double b,double c,double d,double rts[4]);
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// double a,b,c,d,rts[4];
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//*****************************************/
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// Solve quartic equation using
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// Christianson's algorithm
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// calls errors, quadratic, cubic.
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//
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// Bruce Christianson, Solving Quartics Using Palindromes,
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// Mathematical Gazette, Vol. 75, pp. 327-328 (1991)
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//
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// 14 Jan 2004 Don Herbison-Evans
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int chris(double a,double b,double c,double d,double rts[4]);
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//double a,b,c,d,rts[4];
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// données propres
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protected:
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#define NCASES 40
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double d3o8,d3o256; /* double precision constants */
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double doub0;
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double doub1,doub2;
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double doub3,doub4;
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double doub5,doub6;
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double doub8,doub9,doub12;
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double doub16,doub24;
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double doub27,doub64;
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double doubmax; /* approx square root of max double number */
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double doubmin; /* smallest double number */
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double doubtol; /* tolerance of double numbers */
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double inv2,inv3,inv4;
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double inv8,inv16,inv32,inv64,inv128;
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double qrts[4][3]; /* quartic roots for each cubic root */
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double rt3; /* square root of 3 */
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double rterc[4],rterd[4],rterf[4],rtern[4],rterq[4],rtery[4]; /* errors of roots */
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double worst3[3]; /* worst error for a given cubic root */
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int debug; /* <1 for lots of diagnostics, >5 for none */
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bool iterate; // indique si l'on veut itérer on pas
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int j3;
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int n1,n2,n3,n4[3];
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int nqud[NCASES];
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int ncub[NCASES];
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int nchr[NCASES];
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int ndes[NCASES];
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int nfer[NCASES];
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int nneu[NCASES];
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int nyac[NCASES];
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};
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#endif |