/*
* (C) Vladislav Malyshkin 2010
* This file is under GPL version 3.
*
*/
/** Polynomial root.
* @version $Id: PolynomialRoot.java,v 1.105 2012/08/18 00:00:05 mal Exp $
* @author Vladislav Malyshkin mal@gromco.com
*/
/**
* @test
* @bug 8005956
* @summary C2: assert(!def_outside->member(r)) failed: Use of external LRG overlaps the same LRG defined in this block
* @library /test/lib
* @modules java.base/jdk.internal.misc
* java.management
*
* @run main/timeout=300 compiler.c2.PolynomialRoot
*/
package compiler.c2;
import jdk.test.lib.Utils;
import java.util.Arrays;
import java.util.Random;
public class PolynomialRoot {
public static int findPolynomialRoots(final int n,
final double [] p,
final double [] re_root,
final double [] im_root)
{
if(n==4)
{
return root4(p,re_root,im_root);
}
else if(n==3)
{
return root3(p,re_root,im_root);
}
else if(n==2)
{
return root2(p,re_root,im_root);
}
else if(n==1)
{
return root1(p,re_root,im_root);
}
else
{
throw new RuntimeException("n="+n+" is not supported yet");
}
}
static final double SQRT3=Math.sqrt(3.0),SQRT2=Math.sqrt(2.0);
private static final boolean PRINT_DEBUG=false;
public static int root4(final double [] p,final double [] re_root,final double [] im_root)
{
if (PRINT_DEBUG) { System.err.println("=====================root4:p=" + Arrays.toString(p)); }
final double vs=p[4];
if(PRINT_DEBUG) System.err.println("p[4]="+p[4]);
if(!(Math.abs(vs)>EPS))
{
re_root[0]=re_root[1]=re_root[2]=re_root[3]=
im_root[0]=im_root[1]=im_root[2]=im_root[3]=Double.NaN;
return -1;
}
/* zsolve_quartic.c - finds the complex roots of
* x^4 + a x^3 + b x^2 + c x + d = 0
*/
final double a=p[3]/vs,b=p[2]/vs,c=p[1]/vs,d=p[0]/vs;
if(PRINT_DEBUG) System.err.println("input a="+a+" b="+b+" c="+c+" d="+d);
final double r4 = 1.0 / 4.0;
final double q2 = 1.0 / 2.0, q4 = 1.0 / 4.0, q8 = 1.0 / 8.0;
final double q1 = 3.0 / 8.0, q3 = 3.0 / 16.0;
final int mt;
/* Deal easily with the cases where the quartic is degenerate. The
* ordering of solutions is done explicitly. */
if (0 == b && 0 == c)
{
if (0 == d)
{
re_root[0]=-a;
im_root[0]=im_root[1]=im_root[2]=im_root[3]=0;
re_root[1]=re_root[2]=re_root[3]=0;
return 4;
}
else if (0 == a)
{
if (d > 0)
{
final double sq4 = Math.sqrt(Math.sqrt(d));
re_root[0]=sq4*SQRT2/2;
im_root[0]=re_root[0];
re_root[1]=-re_root[0];
im_root[1]=re_root[0];
re_root[2]=-re_root[0];
im_root[2]=-re_root[0];
re_root[3]=re_root[0];
im_root[3]=-re_root[0];
if(PRINT_DEBUG) System.err.println("Path a=0 d>0");
}
else
{
final double sq4 = Math.sqrt(Math.sqrt(-d));
re_root[0]=sq4;
im_root[0]=0;
re_root[1]=0;
im_root[1]=sq4;
re_root[2]=0;
im_root[2]=-sq4;
re_root[3]=-sq4;
im_root[3]=0;
if(PRINT_DEBUG) System.err.println("Path a=0 d<0");
}
return 4;
}
}
if (0.0 == c && 0.0 == d)
{
root2(new double []{p[2],p[3],p[4]},re_root,im_root);
re_root[2]=im_root[2]=re_root[3]=im_root[3]=0;
return 4;
}
if(PRINT_DEBUG) System.err.println("G Path c="+c+" d="+d);
final double [] u=new double[3];
if(PRINT_DEBUG) System.err.println("Generic Path");
/* For non-degenerate solutions, proceed by constructing and
* solving the resolvent cubic */
final double aa = a * a;
final double pp = b - q1 * aa;
final double qq = c - q2 * a * (b - q4 * aa);
final double rr = d - q4 * a * (c - q4 * a * (b - q3 * aa));
final double rc = q2 * pp , rc3 = rc / 3;
final double sc = q4 * (q4 * pp * pp - rr);
final double tc = -(q8 * qq * q8 * qq);
if(PRINT_DEBUG) System.err.println("aa="+aa+" pp="+pp+" qq="+qq+" rr="+rr+" rc="+rc+" sc="+sc+" tc="+tc);
final boolean flag_realroots;
/* This code solves the resolvent cubic in a convenient fashion
* for this implementation of the quartic. If there are three real
* roots, then they are placed directly into u[]. If two are
* complex, then the real root is put into u[0] and the real
* and imaginary part of the complex roots are placed into
* u[1] and u[2], respectively. */
{
final double qcub = (rc * rc - 3 * sc);
final double rcub = (rc*(2 * rc * rc - 9 * sc) + 27 * tc);
final double Q = qcub / 9;
final double R = rcub / 54;
final double Q3 = Q * Q * Q;
final double R2 = R * R;
final double CR2 = 729 * rcub * rcub;
final double CQ3 = 2916 * qcub * qcub * qcub;
if(PRINT_DEBUG) System.err.println("CR2="+CR2+" CQ3="+CQ3+" R="+R+" Q="+Q);
if (0 == R && 0 == Q)
{
flag_realroots=true;
u[0] = -rc3;
u[1] = -rc3;
u[2] = -rc3;
}
else if (CR2 == CQ3)
{
flag_realroots=true;
final double sqrtQ = Math.sqrt (Q);
if (R > 0)
{
u[0] = -2 * sqrtQ - rc3;
u[1] = sqrtQ - rc3;
u[2] = sqrtQ - rc3;
}
else
{
u[0] = -sqrtQ - rc3;
u[1] = -sqrtQ - rc3;
u[2] = 2 * sqrtQ - rc3;
}
}
else if (R2 < Q3)
{
flag_realroots=true;
final double ratio = (R >= 0?1:-1) * Math.sqrt (R2 / Q3);
final double theta = Math.acos (ratio);
final double norm = -2 * Math.sqrt (Q);
u[0] = norm * Math.cos (theta / 3) - rc3;
u[1] = norm * Math.cos ((theta + 2.0 * Math.PI) / 3) - rc3;
u[2] = norm * Math.cos ((theta - 2.0 * Math.PI) / 3) - rc3;
}
else
{
flag_realroots=false;
final double A = -(R >= 0?1:-1)*Math.pow(Math.abs(R)+Math.sqrt(R2-Q3),1.0/3.0);
final double B = Q / A;
u[0] = A + B - rc3;
u[1] = -0.5 * (A + B) - rc3;
u[2] = -(SQRT3*0.5) * Math.abs (A - B);
}
if(PRINT_DEBUG) System.err.println("u[0]="+u[0]+" u[1]="+u[1]+" u[2]="+u[2]+" qq="+qq+" disc="+((CR2 - CQ3) / 2125764.0));
}
/* End of solution to resolvent cubic */
/* Combine the square roots of the roots of the cubic
* resolvent appropriately. Also, calculate 'mt' which
* designates the nature of the roots:
* mt=1 : 4 real roots
* mt=2 : 0 real roots
* mt=3 : 2 real roots
*/
final double w1_re,w1_im,w2_re,w2_im,w3_re,w3_im,mod_w1w2,mod_w1w2_squared;
if (flag_realroots)
{
mod_w1w2=-1;
mt = 2;
int jmin=0;
double vmin=Math.abs(u[jmin]);
for(int j=1;j<3;j++)
{
final double vx=Math.abs(u[j]);
if(vx<vmin)
{
vmin=vx;
jmin=j;
}
}
final double u1=u[(jmin+1)%3],u2=u[(jmin+2)%3];
mod_w1w2_squared=Math.abs(u1*u2);
if(u1>=0)
{
w1_re=Math.sqrt(u1);
w1_im=0;
}
else
{
w1_re=0;
w1_im=Math.sqrt(-u1);
}
if(u2>=0)
{
w2_re=Math.sqrt(u2);
w2_im=0;
}
else
{
w2_re=0;
w2_im=Math.sqrt(-u2);
}
if(PRINT_DEBUG) System.err.println("u1="+u1+" u2="+u2+" jmin="+jmin);
}
else
{
mt = 3;
final double w_mod2_sq=u[1]*u[1]+u[2]*u[2],w_mod2=Math.sqrt(w_mod2_sq),w_mod=Math.sqrt(w_mod2);
if(w_mod2_sq<=0)
{
w1_re=w1_im=0;
}
else
{
// calculate square root of a complex number (u[1],u[2])
// the result is in the (w1_re,w1_im)
final double absu1=Math.abs(u[1]),absu2=Math.abs(u[2]),w;
if(absu1>=absu2)
{
final double t=absu2/absu1;
w=Math.sqrt(absu1*0.5 * (1.0 + Math.sqrt(1.0 + t * t)));
if(PRINT_DEBUG) System.err.println(" Path1 ");
}
else
{
final double t=absu1/absu2;
w=Math.sqrt(absu2*0.5 * (t + Math.sqrt(1.0 + t * t)));
if(PRINT_DEBUG) System.err.println(" Path1a ");
}
if(u[1]>=0)
{
w1_re=w;
w1_im=u[2]/(2*w);
if(PRINT_DEBUG) System.err.println(" Path2 ");
}
else
{
final double vi = (u[2] >= 0) ? w : -w;
w1_re=u[2]/(2*vi);
w1_im=vi;
if(PRINT_DEBUG) System.err.println(" Path2a ");
}
}
final double absu0=Math.abs(u[0]);
if(w_mod2>=absu0)
{
mod_w1w2=w_mod2;
mod_w1w2_squared=w_mod2_sq;
w2_re=w1_re;
w2_im=-w1_im;
}
else
{
mod_w1w2=-1;
mod_w1w2_squared=w_mod2*absu0;
if(u[0]>=0)
{
w2_re=Math.sqrt(absu0);
w2_im=0;
}
else
{
w2_re=0;
w2_im=Math.sqrt(absu0);
}
}
if(PRINT_DEBUG) System.err.println("u[0]="+u[0]+"u[1]="+u[1]+" u[2]="+u[2]+" absu0="+absu0+" w_mod="+w_mod+" w_mod2="+w_mod2);
}
/* Solve the quadratic in order to obtain the roots
* to the quartic */
if(mod_w1w2>0)
{
// a shorcut to reduce rounding error
w3_re=qq/(-8)/mod_w1w2;
w3_im=0;
}
else if(mod_w1w2_squared>0)
{
// regular path
final double mqq8n=qq/(-8)/mod_w1w2_squared;
w3_re=mqq8n*(w1_re*w2_re-w1_im*w2_im);
w3_im=-mqq8n*(w1_re*w2_im+w2_re*w1_im);
}
else
{
// typically occur when qq==0
w3_re=w3_im=0;
}
final double h = r4 * a;
if(PRINT_DEBUG) System.err.println("w1_re="+w1_re+" w1_im="+w1_im+" w2_re="+w2_re+" w2_im="+w2_im+" w3_re="+w3_re+" w3_im="+w3_im+" h="+h);
re_root[0]=w1_re+w2_re+w3_re-h;
im_root[0]=w1_im+w2_im+w3_im;
re_root[1]=-(w1_re+w2_re)+w3_re-h;
im_root[1]=-(w1_im+w2_im)+w3_im;
re_root[2]=w2_re-w1_re-w3_re-h;
im_root[2]=w2_im-w1_im-w3_im;
re_root[3]=w1_re-w2_re-w3_re-h;
im_root[3]=w1_im-w2_im-w3_im;
return 4;
}
static void setRandomP(final double [] p, final int n, Random r)
{
if(r.nextDouble()<0.1)
{
// integer coefficiens
for(int j=0;j<p.length;j++)
{
if(j<=n)
{
p[j]=(r.nextInt(2)<=0?-1:1)*r.nextInt(10);
}
else
{
p[j]=0;
}
}
}
else
{
// real coefficiens
for(int j=0;j<p.length;j++)
{
if(j<=n)
{
p[j]=-1+2*r.nextDouble();
}
else
{
p[j]=0;
}
}
}
if(Math.abs(p[n])<1e-2)
{
p[n]=(r.nextInt(2)<=0?-1:1)*(0.1+r.nextDouble());
}
}
static void checkValues(final double [] p,
final int n,
final double rex,
final double imx,
final double eps,
final String txt)
{
double res=0,ims=0,sabs=0;
final double xabs=Math.abs(rex)+Math.abs(imx);
for(int k=n;k>=0;k--)
{
final double res1=(res*rex-ims*imx)+p[k];
final double ims1=(ims*rex+res*imx);
res=res1;
ims=ims1;
sabs+=xabs*sabs+p[k];
}
sabs=Math.abs(sabs);
if(false && sabs>1/eps?
(!(Math.abs(res/sabs)<=eps)||!(Math.abs(ims/sabs)<=eps))
:
(!(Math.abs(res)<=eps)||!(Math.abs(ims)<=eps)))
{
throw new RuntimeException(
getPolinomTXT(p)+"\n"+
"\t x.r="+rex+" x.i="+imx+"\n"+
"res/sabs="+(res/sabs)+" ims/sabs="+(ims/sabs)+
" sabs="+sabs+
"\nres="+res+" ims="+ims+" n="+n+" eps="+eps+" "+
" sabs>1/eps="+(sabs>1/eps)+
" f1="+(!(Math.abs(res/sabs)<=eps)||!(Math.abs(ims/sabs)<=eps))+
" f2="+(!(Math.abs(res)<=eps)||!(Math.abs(ims)<=eps))+
" "+txt);
}
}
static String getPolinomTXT(final double [] p)
{
final StringBuilder buf=new StringBuilder();
buf.append("order="+(p.length-1)+"\t");
for(int k=0;k<p.length;k++)
{
buf.append("p["+k+"]="+p[k]+";");
}
return buf.toString();
}
static String getRootsTXT(int nr,final double [] re,final double [] im)
{
final StringBuilder buf=new StringBuilder();
for(int k=0;k<nr;k++)
{
buf.append("x."+k+"("+re[k]+","+im[k]+")\n");
}
return buf.toString();
}
static void testRoots(final int n,
final int n_tests,
final Random rn,
final double eps)
{
final double [] p=new double [n+1];
final double [] rex=new double [n],imx=new double [n];
for(int i=0;i<n_tests;i++)
{
for(int dg=n;dg-->-1;)
{
for(int dr=3;dr-->0;)
{
setRandomP(p,n,rn);
for(int j=0;j<=dg;j++)
{
p[j]=0;
}
if(dr==0)
{
p[0]=-1+2.0*rn.nextDouble();
}
else if(dr==1)
{
p[0]=p[1]=0;
}
findPolynomialRoots(n,p,rex,imx);
for(int j=0;j<n;j++)
{
//System.err.println("j="+j);
checkValues(p,n,rex[j],imx[j],eps," t="+i);
}
}
}
}
System.err.println("testRoots(): n_tests="+n_tests+" OK, dim="+n);
}
static final double EPS=0;
public static int root1(final double [] p,final double [] re_root,final double [] im_root)
{
if(!(Math.abs(p[1])>EPS))
{
re_root[0]=im_root[0]=Double.NaN;
return -1;
}
re_root[0]=-p[0]/p[1];
im_root[0]=0;
return 1;
}
public static int root2(final double [] p,final double [] re_root,final double [] im_root)
{
if(!(Math.abs(p[2])>EPS))
{
re_root[0]=re_root[1]=im_root[0]=im_root[1]=Double.NaN;
return -1;
}
final double b2=0.5*(p[1]/p[2]),c=p[0]/p[2],d=b2*b2-c;
if(d>=0)
{
final double sq=Math.sqrt(d);
if(b2<0)
{
re_root[1]=-b2+sq;
re_root[0]=c/re_root[1];
}
else if(b2>0)
{
re_root[0]=-b2-sq;
re_root[1]=c/re_root[0];
}
else
{
re_root[0]=-b2-sq;
re_root[1]=-b2+sq;
}
im_root[0]=im_root[1]=0;
}
else
{
final double sq=Math.sqrt(-d);
re_root[0]=re_root[1]=-b2;
im_root[0]=sq;
im_root[1]=-sq;
}
return 2;
}
public static int root3(final double [] p,final double [] re_root,final double [] im_root)
{
final double vs=p[3];
if(!(Math.abs(vs)>EPS))
{
re_root[0]=re_root[1]=re_root[2]=
im_root[0]=im_root[1]=im_root[2]=Double.NaN;
return -1;
}
final double a=p[2]/vs,b=p[1]/vs,c=p[0]/vs;
/* zsolve_cubic.c - finds the complex roots of x^3 + a x^2 + b x + c = 0
*/
final double q = (a * a - 3 * b);
final double r = (a*(2 * a * a - 9 * b) + 27 * c);
final double Q = q / 9;
final double R = r / 54;
final double Q3 = Q * Q * Q;
final double R2 = R * R;
final double CR2 = 729 * r * r;
final double CQ3 = 2916 * q * q * q;
final double a3=a/3;
if (R == 0 && Q == 0)
{
re_root[0]=re_root[1]=re_root[2]=-a3;
im_root[0]=im_root[1]=im_root[2]=0;
return 3;
}
else if (CR2 == CQ3)
{
/* this test is actually R2 == Q3, written in a form suitable
for exact computation with integers */
/* Due to finite precision some double roots may be missed, and
will be considered to be a pair of complex roots z = x +/-
epsilon i close to the real axis. */
final double sqrtQ = Math.sqrt (Q);
if (R > 0)
{
re_root[0] = -2 * sqrtQ - a3;
re_root[1]=re_root[2]=sqrtQ - a3;
im_root[0]=im_root[1]=im_root[2]=0;
}
else
{
re_root[0]=re_root[1] = -sqrtQ - a3;
re_root[2]=2 * sqrtQ - a3;
im_root[0]=im_root[1]=im_root[2]=0;
}
return 3;
}
else if (R2 < Q3)
{
final double sgnR = (R >= 0 ? 1 : -1);
final double ratio = sgnR * Math.sqrt (R2 / Q3);
final double theta = Math.acos (ratio);
final double norm = -2 * Math.sqrt (Q);
final double r0 = norm * Math.cos (theta/3) - a3;
final double r1 = norm * Math.cos ((theta + 2.0 * Math.PI) / 3) - a3;
final double r2 = norm * Math.cos ((theta - 2.0 * Math.PI) / 3) - a3;
re_root[0]=r0;
re_root[1]=r1;
re_root[2]=r2;
im_root[0]=im_root[1]=im_root[2]=0;
return 3;
}
else
{
final double sgnR = (R >= 0 ? 1 : -1);
final double A = -sgnR * Math.pow (Math.abs (R) + Math.sqrt (R2 - Q3), 1.0 / 3.0);
final double B = Q / A;
re_root[0]=A + B - a3;
im_root[0]=0;
re_root[1]=-0.5 * (A + B) - a3;
im_root[1]=-(SQRT3*0.5) * Math.abs(A - B);
re_root[2]=re_root[1];
im_root[2]=-im_root[1];
return 3;
}
}
static void root3a(final double [] p,final double [] re_root,final double [] im_root)
{
if(Math.abs(p[3])>EPS)
{
final double v=p[3],
a=p[2]/v,b=p[1]/v,c=p[0]/v,
a3=a/3,a3a=a3*a,
pd3=(b-a3a)/3,
qd2=a3*(a3a/3-0.5*b)+0.5*c,
Q=pd3*pd3*pd3+qd2*qd2;
if(Q<0)
{
// three real roots
final double SQ=Math.sqrt(-Q);
final double th=Math.atan2(SQ,-qd2);
im_root[0]=im_root[1]=im_root[2]=0;
final double f=2*Math.sqrt(-pd3);
re_root[0]=f*Math.cos(th/3)-a3;
re_root[1]=f*Math.cos((th+2*Math.PI)/3)-a3;
re_root[2]=f*Math.cos((th+4*Math.PI)/3)-a3;
//System.err.println("3r");
}
else
{
// one real & two complex roots
final double SQ=Math.sqrt(Q);
final double r1=-qd2+SQ,r2=-qd2-SQ;
final double v1=Math.signum(r1)*Math.pow(Math.abs(r1),1.0/3),
v2=Math.signum(r2)*Math.pow(Math.abs(r2),1.0/3),
sv=v1+v2;
// real root
re_root[0]=sv-a3;
im_root[0]=0;
// complex roots
re_root[1]=re_root[2]=-0.5*sv-a3;
im_root[1]=(v1-v2)*(SQRT3*0.5);
im_root[2]=-im_root[1];
//System.err.println("1r2c");
}
}
else
{
re_root[0]=re_root[1]=re_root[2]=im_root[0]=im_root[1]=im_root[2]=Double.NaN;
}
}
static void printSpecialValues()
{
for(int st=0;st<6;st++)
{
//final double [] p=new double []{8,1,3,3.6,1};
final double [] re_root=new double [4],im_root=new double [4];
final double [] p;
final int n;
if(st<=3)
{
if(st<=0)
{
p=new double []{2,-4,6,-4,1};
//p=new double []{-6,6,-6,8,-2};
}
else if(st==1)
{
p=new double []{0,-4,8,3,-9};
}
else if(st==2)
{
p=new double []{-1,0,2,0,-1};
}
else
{
p=new double []{-5,2,8,-2,-3};
}
root4(p,re_root,im_root);
n=4;
}
else
{
p=new double []{0,2,0,1};
if(st==4)
{
p[1]=-p[1];
}
root3(p,re_root,im_root);
n=3;
}
System.err.println("======== n="+n);
for(int i=0;i<=n;i++)
{
if(i<n)
{
System.err.println(String.valueOf(i)+"\t"+
p[i]+"\t"+
re_root[i]+"\t"+
im_root[i]);
}
else
{
System.err.println(String.valueOf(i)+"\t"+p[i]+"\t");
}
}
}
}
public static void main(final String [] args)
{
if (System.getProperty("os.arch").equals("x86") ||
System.getProperty("os.arch").equals("amd64") ||
System.getProperty("os.arch").equals("x86_64")){
final long t0=System.currentTimeMillis();
final double eps=1e-6;
//checkRoots();
final Random r = Utils.getRandomInstance();
printSpecialValues();
final int n_tests=100000;
//testRoots(2,n_tests,r,eps);
//testRoots(3,n_tests,r,eps);
testRoots(4,n_tests,r,eps);
final long t1=System.currentTimeMillis();
System.err.println("PolynomialRoot.main: "+n_tests+" tests OK done in "+(t1-t0)+" milliseconds. ver=$Id: PolynomialRoot.java,v 1.105 2012/08/18 00:00:05 mal Exp $");
System.out.println("PASSED");
} else {
System.out.println("PASS test for non-x86");
}
}
}