1/*
2 * Copyright (C) 2011 The Android Open Source Project
3 *
4 * Licensed under the Apache License, Version 2.0 (the "License");
5 * you may not use this file except in compliance with the License.
6 * You may obtain a copy of the License at
7 *
8 *      http://www.apache.org/licenses/LICENSE-2.0
9 *
10 * Unless required by applicable law or agreed to in writing, software
11 * distributed under the License is distributed on an "AS IS" BASIS,
12 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
13 * See the License for the specific language governing permissions and
14 * limitations under the License.
15 */
16
17/* $Id: db_utilities_poly.cpp,v 1.2 2010/09/03 12:00:10 bsouthall Exp $ */
18
19#include "db_utilities_poly.h"
20#include "db_utilities.h"
21
22
23
24/*****************************************************************
25*    Lean and mean begins here                                   *
26*****************************************************************/
27
28void db_SolveCubic(double *roots,int *nr_roots,double a,double b,double c,double d)
29{
30    double bp,bp2,cp,dp,q,r,srq;
31    double r2_min_q3,theta,bp_through3,theta_through3;
32    double cos_theta_through3,sin_theta_through3,min2_cos_theta_plu,min2_cos_theta_min;
33    double si_r_srq,A;
34
35    /*For nondegenerate cubics with three roots
36    [24 mult 9 add 2sqrt 1acos 1cos=33flops 4func]
37    For nondegenerate cubics with one root
38    [16 mult 6 add 1sqrt 1qbrt=24flops 3func]*/
39
40    if(a==0.0) db_SolveQuadratic(roots,nr_roots,b,c,d);
41    else
42    {
43        bp=b/a;
44        bp2=bp*bp;
45        cp=c/a;
46        dp=d/a;
47
48        q=(bp2-3.0*cp)/9.0;
49        r=(2.0*bp2*bp-9.0*bp*cp+27.0*dp)/54.0;
50        r2_min_q3=r*r-q*q*q;
51        if(r2_min_q3<0.0)
52        {
53            *nr_roots=3;
54            /*q has to be > 0*/
55            srq=sqrt(q);
56            theta=acos(db_maxd(-1.0,db_mind(1.0,r/(q*srq))));
57            bp_through3=bp/3.0;
58            theta_through3=theta/3.0;
59            cos_theta_through3=cos(theta_through3);
60            sin_theta_through3=sqrt(db_maxd(0.0,1.0-cos_theta_through3*cos_theta_through3));
61
62            /*cos(theta_through3+2*pi/3)=cos_theta_through3*cos(2*pi/3)-sin_theta_through3*sin(2*pi/3)
63            = -0.5*cos_theta_through3-sqrt(3)/2.0*sin_theta_through3
64            = -0.5*(cos_theta_through3+sqrt(3)*sin_theta_through3)*/
65            min2_cos_theta_plu=cos_theta_through3+DB_SQRT3*sin_theta_through3;
66            min2_cos_theta_min=cos_theta_through3-DB_SQRT3*sin_theta_through3;
67
68            roots[0]= -2.0*srq*cos_theta_through3-bp_through3;
69            roots[1]=srq*min2_cos_theta_plu-bp_through3;
70            roots[2]=srq*min2_cos_theta_min-bp_through3;
71        }
72        else if(r2_min_q3>0.0)
73        {
74            *nr_roots=1;
75            A= -db_sign(r)*db_CubRoot(db_absd(r)+sqrt(r2_min_q3));
76            bp_through3=bp/3.0;
77            if(A!=0.0) roots[0]=A+q/A-bp_through3;
78            else roots[0]= -bp_through3;
79        }
80        else
81        {
82            *nr_roots=2;
83            bp_through3=bp/3.0;
84            /*q has to be >= 0*/
85            si_r_srq=db_sign(r)*sqrt(q);
86            /*Single root*/
87            roots[0]= -2.0*si_r_srq-bp_through3;
88            /*Double root*/
89            roots[1]=si_r_srq-bp_through3;
90        }
91    }
92}
93
94void db_SolveQuartic(double *roots,int *nr_roots,double a,double b,double c,double d,double e)
95{
96    /*Normalized coefficients*/
97    double c0,c1,c2,c3;
98    /*Temporary coefficients*/
99    double c3through2,c3through4,c3c3through4_min_c2,min4_c0;
100    double lz,ms,ns,mn,m,n,lz_through2;
101    /*Cubic polynomial roots, nr of roots and coefficients*/
102    double c_roots[3];
103    int nr_c_roots;
104    double k0,k1;
105    /*nr additional roots from second quadratic*/
106    int addroots;
107
108    /*For nondegenerate quartics
109    [16mult 11add 2sqrt 1cubic 2quadratic=74flops 8funcs]*/
110
111    if(a==0.0) db_SolveCubic(roots,nr_roots,b,c,d,e);
112    else if(e==0.0)
113    {
114        db_SolveCubic(roots,nr_roots,a,b,c,d);
115        roots[*nr_roots]=0.0;
116        *nr_roots+=1;
117    }
118    else
119    {
120        /*Compute normalized coefficients*/
121        c3=b/a;
122        c2=c/a;
123        c1=d/a;
124        c0=e/a;
125        /*Compute temporary coefficients*/
126        c3through2=c3/2.0;
127        c3through4=c3/4.0;
128        c3c3through4_min_c2=c3*c3through4-c2;
129        min4_c0= -4.0*c0;
130        /*Compute coefficients of cubic*/
131        k0=min4_c0*c3c3through4_min_c2-c1*c1;
132        k1=c1*c3+min4_c0;
133        /*k2= -c2*/
134        /*k3=1.0*/
135
136        /*Solve it for roots*/
137        db_SolveCubic(c_roots,&nr_c_roots,1.0,-c2,k1,k0);
138
139        if(nr_c_roots>0)
140        {
141            lz=c_roots[0];
142            lz_through2=lz/2.0;
143            ms=lz+c3c3through4_min_c2;
144            ns=lz_through2*lz_through2-c0;
145            mn=lz*c3through4-c1/2.0;
146
147            if((ms>=0.0)&&(ns>=0.0))
148            {
149                m=sqrt(ms);
150                n=sqrt(ns)*db_sign(mn);
151
152                db_SolveQuadratic(roots,nr_roots,
153                    1.0,c3through2+m,lz_through2+n);
154
155                db_SolveQuadratic(&roots[*nr_roots],&addroots,
156                    1.0,c3through2-m,lz_through2-n);
157
158                *nr_roots+=addroots;
159            }
160            else *nr_roots=0;
161        }
162        else *nr_roots=0;
163    }
164}
165
166void db_SolveQuarticForced(double *roots,int *nr_roots,double a,double b,double c,double d,double e)
167{
168    /*Normalized coefficients*/
169    double c0,c1,c2,c3;
170    /*Temporary coefficients*/
171    double c3through2,c3through4,c3c3through4_min_c2,min4_c0;
172    double lz,ms,ns,mn,m,n,lz_through2;
173    /*Cubic polynomial roots, nr of roots and coefficients*/
174    double c_roots[3];
175    int nr_c_roots;
176    double k0,k1;
177    /*nr additional roots from second quadratic*/
178    int addroots;
179
180    /*For nondegenerate quartics
181    [16mult 11add 2sqrt 1cubic 2quadratic=74flops 8funcs]*/
182
183    if(a==0.0) db_SolveCubic(roots,nr_roots,b,c,d,e);
184    else if(e==0.0)
185    {
186        db_SolveCubic(roots,nr_roots,a,b,c,d);
187        roots[*nr_roots]=0.0;
188        *nr_roots+=1;
189    }
190    else
191    {
192        /*Compute normalized coefficients*/
193        c3=b/a;
194        c2=c/a;
195        c1=d/a;
196        c0=e/a;
197        /*Compute temporary coefficients*/
198        c3through2=c3/2.0;
199        c3through4=c3/4.0;
200        c3c3through4_min_c2=c3*c3through4-c2;
201        min4_c0= -4.0*c0;
202        /*Compute coefficients of cubic*/
203        k0=min4_c0*c3c3through4_min_c2-c1*c1;
204        k1=c1*c3+min4_c0;
205        /*k2= -c2*/
206        /*k3=1.0*/
207
208        /*Solve it for roots*/
209        db_SolveCubic(c_roots,&nr_c_roots,1.0,-c2,k1,k0);
210
211        if(nr_c_roots>0)
212        {
213            lz=c_roots[0];
214            lz_through2=lz/2.0;
215            ms=lz+c3c3through4_min_c2;
216            ns=lz_through2*lz_through2-c0;
217            mn=lz*c3through4-c1/2.0;
218
219            if(ms<0.0) ms=0.0;
220            if(ns<0.0) ns=0.0;
221
222            m=sqrt(ms);
223            n=sqrt(ns)*db_sign(mn);
224
225            db_SolveQuadratic(roots,nr_roots,
226                1.0,c3through2+m,lz_through2+n);
227
228            db_SolveQuadratic(&roots[*nr_roots],&addroots,
229                1.0,c3through2-m,lz_through2-n);
230
231            *nr_roots+=addroots;
232        }
233        else *nr_roots=0;
234    }
235}
236