Can someone please tell me that what is the root of polynomials and how to calculate them???
and whats this API?
3  3979  
The root of a polynomial of the form:
P(x) = ax**n + bx**(n-1) + cx**(n-2) + ... + yx + z (in which a, b, c, ...y, and z are all real numbers, n is an integer, and x is the variable of interest) (x**n means x to the power of n)
is a value of x such that P(x) = 0.
For simple polynomials, there are easy roots. For instance, a polynomial of degree 0 (i.e. P(x) = a, where a is a real number) has infinite solutions if a = 0, and no solutions if a != 0.
A polynomial of degree 1 (i.e. P(x) = ax + b) has a root x = -(b/a) if and only if a != 0 (if a = 0, P(x) has no root).
A polynomial of degree 2 has roots according to the quadratic formula (Google it if you're interested).
I believe algorithms for computing roots of 3rd degree and 4th degree polynomials have been found, but they are non-trivial.
No algorithm exists to find roots of 5th or higher degree polynomials.
You may be able to use Newton's Method to approximate roots of any polynomial - however, Newton's method requires a 'good' initial guess.
I'm not sure what API has to do with polynomials.
The root of a polynomial of the form:
P(x) = ax**n + bx**(n-1) + cx**(n-2) + ... + yx + z (in which a, b, c, ...y, and z are all real numbers, n is an integer, and x is the variable of interest) (x**n means x to the power of n)
is a value of x such that P(x) = 0.
For simple polynomials, there are easy roots. For instance, a polynomial of degree 0 (i.e. P(x) = a, where a is a real number) has infinite solutions if a = 0, and no solutions if a != 0.
A polynomial of degree 1 (i.e. P(x) = ax + b) has a root x = -(b/a) if and only if a != 0 (if a = 0, P(x) has no root).
A polynomial of degree 2 has roots according to the quadratic formula (Google it if you're interested).
I believe algorithms for computing roots of 3rd degree and 4th degree polynomials have been found, but they are non-trivial.
No algorithm exists to find roots of 5th or higher degree polynomials.
You may be able to use Newton's Method to approximate roots of any polynomial - however, Newton's method requires a 'good' initial guess.
I'm not sure what API has to do with polynomials.
ok......thanks....
thanks a lot.....
I have the following C code for the solution of a Polynomial of degree DIM, for DIM up to degree 4. The Mathematician Galois (1700's) showed there can be no direct computational method of solution of polynomials above degree 4. API probably means 'Advanced Programmer Interface'. An interface between the programmer and the computer, to enable the programmer to easily develop programs. Code follows:- - if(DIM==0){
-
if(coeffs[4]==coeffs[5])MessageBox(hwnd,"Equation is a Tautology!","Message from Polynomial",MB_OK);
-
else MessageBox(hwnd,"Equation is a Impossible!","Error from Polynomial",MB_OK);
-
EigR[0]=0.0;
-
EigR[1]=0.0;
-
EigR[2]=0.0;
-
EigR[3]=0.0;
-
EigI[0]=0.0;
-
EigI[1]=0.0;
-
EigI[2]=0.0;
-
EigI[3]=0.0;
-
} else if(DIM==1){EigR[0]=(coeffs[5]-coeffs[4])/coeffs[3];
-
EigR[1]=0.0;
-
EigR[2]=0.0;
-
EigR[3]=0.0;
-
EigI[0]=0.0;
-
EigI[1]=0.0;
-
EigI[2]=0.0;
-
EigI[3]=0.0;
-
} else if(DIM==2){a=1.0;b=coeffs[3]/coeffs[2];c=(coeffs[4]-coeffs[5])/coeffs[2];discrim=b*b-4*a*c;
-
if(discrim>0.0)
-
{discrim=sqrt(discrim)/2;
-
EigR[0]=-b/2+discrim;
-
EigR[1]=-b/2-discrim;
-
EigR[2]=0.0;
-
EigR[3]=0.0;
-
EigI[0]=0.0;
-
EigI[1]=0.0;
-
EigI[2]=0.0;
-
EigI[3]=0.0;
-
} else if(discrim==0.0)
-
{EigR[0]=-b/2;
-
EigR[1]=-b/2;
-
EigR[2]=0.0;
-
EigR[3]=0.0;
-
EigI[0]=0.0;
-
EigI[1]=0.0;
-
EigI[2]=0.0;
-
EigI[3]=0.0;
-
} else {discrim=sqrt(-discrim)/2;
-
EigR[0]=-b/2;
-
EigR[1]=-b/2;
-
EigR[2]=0.0;
-
EigR[3]=0.0;
-
EigI[0]=discrim;
-
EigI[1]=-discrim;
-
EigI[2]=0.0;
-
EigI[3]=0.0;
-
}
-
} else if(DIM==3)
-
{a=coeffs[2]/coeffs[1];
-
b=coeffs[3]/coeffs[1];
-
c=(coeffs[4]-coeffs[5])/coeffs[1];
-
q=(b/3.0)-((a*a)/9.0);r=(((a*b)-(3.0*c))/6.0)-((a*a*a)/27.0);
-
discrim=q*q*q+r*r;
-
third=1.0/3.0;
-
if(fabs(discrim)==0.0)discrim=0.0;
-
if(discrim>=0)
-
{temp1=r+sqrt(discrim);
-
if(temp1>=0)s1=pow(temp1,third);
-
else s1=-pow(-temp1,third);
-
temp2=r-sqrt(discrim);
-
if(temp2>=0)s2=pow(temp2,third);
-
else s2=-pow(-temp2,third);
-
EigR[0]=(s1+s2)-a/3.0;EigI[0]=0.0;
-
EigR[1]=-(s1+s2)/2.0-a/3.0;if(s1!=s2)EigI[1]=sqrt(3.0)*(s1-s2)/2.0; else EigI[1]=0.0;
-
EigR[2]=-(s1+s2)/2.0-a/3.0;if(s1!=s2)EigI[2]=-sqrt(3.0)*(s1-s2)/2.0; else EigI[2]=0.0;
-
EigR[3]=0.0;EigI[3]=0.0;
-
if(fabs(EigR[0])<1e-6)EigR[0]=0.0;
-
if(fabs(EigR[1])<1e-6)EigR[1]=0.0;
-
if(fabs(EigR[2])<1e-6)EigR[2]=0.0;
-
if(fabs(EigR[3])<1e-6)EigR[3]=0.0;
-
}
-
else {SQRT=sqrt(-discrim);
-
if(fabs(r)==0.0)theta=PIBY2; else if(r>0.0)theta=atan(SQRT/r); else theta=2.0*PIBY2+atan(SQRT/r);
-
theta/=3.0;R=pow(sqrt(r*r+SQRT*SQRT),third);
-
COS=cos(theta)*R;SIN=sin(theta)*R;
-
EigR[0]=2.0*COS-a/3.0;EigI[0]=0.0;
-
EigR[1]=-COS-a/3.0+sqrt(3.0)*SIN;EigI[1]=0.0;
-
EigR[2]=-COS-a/3.0-sqrt(3.0)*SIN;EigI[2]=0.0;
-
EigR[3]=0.0;EigI[3]=0.0;
-
if(fabs(EigR[0])<1e-6)EigR[0]=0.0;
-
if(fabs(EigR[1])<1e-6)EigR[1]=0.0;
-
if(fabs(EigR[2])<1e-6)EigR[2]=0.0;
-
if(fabs(EigR[3])<1e-6)EigR[3]=0.0;
-
}
-
} else if(DIM==4){a3=coeffs[1]/coeffs[0];
-
a2=coeffs[2]/coeffs[0];
-
a1=coeffs[3]/coeffs[0];
-
a0=(coeffs[4]-coeffs[5])/coeffs[0];
-
a=-a2;b=(a1*a3-4.0*a0);c=-(a1*a1+a0*a3*a3-4.0*a0*a2);
-
q=(b/3.0)-((a*a)/9.0);r=(((a*b)-(3.0*c))/6.0)-((a*a*a)/27.0);
-
discrim=q*q*q+r*r;
-
third=1.0/3.0;
-
if(fabs(discrim)==0.0)discrim=0.0;
-
if(discrim>=0.0)
-
{temp1=r+sqrt(discrim);
-
if(temp1>=0)s1=pow(temp1,third);
-
else s1=-pow(-temp1,third);
-
temp2=r-sqrt(discrim);
-
if(temp2>=0)s2=pow(temp2,third);
-
else s2=-pow(-temp2,third);
-
u1=(s1+s2)-a/3.0;
-
SQRT1=a3*a3/4.0+u1-a2;
-
if(fabs(SQRT1)<1e-6)SQRT1=0.0;
-
if(SQRT1<0.0 && discrim==0.0){u1=-(s1+s2)/2.0-a/3.0;
-
SQRT1=a3*a3/4.0+u1-a2;};
-
if(SQRT1<0.0){MessageBox(hwnd,"Sorry! Something has gone Wrong with the Calculation of Quartic Roots!","Error from Polynomial",MB_OK);
-
for(j=0;j<4;j++){EigR[j]=0.0;EigI[j]=0.0;};}
-
else {SQRT1=sqrt(SQRT1);
-
SQRT2=u1*u1/4.0-a0;I=-1;
-
if(fabs(SQRT2)<1e-6)SQRT2=0.0;
-
if(SQRT2<0.0){MessageBox(hwnd,"Sorry! Something has gone Wrong with the Calculation of Quartic Roots!","Error from Polynomial",MB_OK);
-
for(j=0;j<4;j++){EigR[j]=0.0;EigI[j]=0.0;};}
-
else {SQRT2=sqrt(SQRT2);
-
AA[0]=1.0;BB[0]=a3/2.0-SQRT1;CC[0]=u1/2.0-SQRT2;
-
AA[1]=1.0;BB[1]=a3/2.0+SQRT1;CC[1]=u1/2.0+SQRT2;
-
AA[2]=1.0;BB[2]=a3/2.0-SQRT1;CC[2]=u1/2.0+SQRT2;
-
AA[3]=1.0;BB[3]=a3/2.0+SQRT1;CC[3]=u1/2.0-SQRT2;
-
for(m=0;m<4;m++){
-
for(n=m;n<4;n++)
-
{A1=AA[m];B1=BB[m];C1=CC[m];
-
A2=AA[n];B2=BB[n];C2=CC[n];
-
if(fabs(C1*C2-a0)<0.01 && fabs(B1*C2+B2*C1-a1)<0.01 && fabs(A1*C2+A2*C1+B1*B2-a2)<0.01 && fabs(A1*B2+A2*B1-a3)<0.01){I=0;break;};
-
}
-
if(I!=-1)break;
-
}
-
if(I==-1){MessageBox(hwnd,"Sorry! Something has gone Wrong with the Calculation of Quartic Roots!","Error from Polynomial",MB_OK);for(j=0;j<4;j++){EigR[j]=0.0;EigI[j]=0.0;};
-
} else {
-
discrim1=B1*B1-4.0*A1*C1;
-
discrim2=B2*B2-4.0*A2*C2;
-
if(fabs(discrim1)<1e-6)discrim1=0.0;
-
if(fabs(discrim2)<1e-6)discrim2=0.0;
-
if(discrim1>=0.0)
-
{SQRT=sqrt(discrim1);
-
EigR[0]=(-B1+SQRT)/2.0;EigI[0]=0.0;
-
EigR[1]=(-B1-SQRT)/2.0;EigI[1]=0.0;
-
if(fabs(EigR[0])<1e-6)EigR[0]=0.0;
-
if(fabs(EigR[1])<1e-6)EigR[1]=0.0;
-
} else {
-
SQRT=sqrt(-discrim1);
-
EigR[0]=-B1/2.0;EigI[0]=SQRT/2.0;
-
EigR[1]=-B1/2.0;EigI[1]=-SQRT/2.0;
-
if(fabs(EigR[0])<1e-6)EigR[0]=0.0;
-
if(fabs(EigR[1])<1e-6)EigR[1]=0.0;
-
}
-
if(discrim2>=0.0)
-
{SQRT=sqrt(discrim2);
-
EigR[2]=(-B2+SQRT)/2.0;EigI[2]=0.0;
-
EigR[3]=(-B2-SQRT)/2.0;EigI[3]=0.0;
-
if(fabs(EigR[2])<1e-6)EigR[2]=0.0;
-
if(fabs(EigR[3])<1e-6)EigR[3]=0.0;
-
} else {
-
SQRT=sqrt(-discrim2);
-
EigR[2]=-B2/2.0;EigI[2]=SQRT/2.0;
-
EigR[3]=-B2/2.0;EigI[3]=-SQRT/2.0;
-
if(fabs(EigR[2])<1e-6)EigR[2]=0.0;
-
if(fabs(EigR[3])<1e-6)EigR[3]=0.0;
-
}
-
}
-
}
-
}
-
} else {SQRT=sqrt(-discrim);
-
if(fabs(r)==0.0)theta=PIBY2; else if(r>0.0)theta=atan(SQRT/r); else theta=2.0*PIBY2+atan(SQRT/r);
-
theta/=3.0;R=pow(sqrt(r*r+SQRT*SQRT),third);
-
COS=cos(theta)*R;SIN=sin(theta)*R;
-
u[0]=2.0*COS-a/3.0;
-
u[1]=-COS-a/3.0+sqrt(3.0)*SIN;
-
u[2]=-COS-a/3.0-sqrt(3.0)*SIN;
-
I=-1;error=0;
-
for(i=0;i<3 && !error;i++)
-
{
-
SQRT1=a3*a3/4.0+u[i]-a2;
-
if(fabs(SQRT1)<1e-6)SQRT1=0.0;
-
if(SQRT1<0.0);
-
else {SQRT1=sqrt(SQRT1);
-
SQRT2=u[i]*u[i]/4.0-a0;
-
if(fabs(SQRT2)<1e-6)SQRT2=0.0;
-
if(SQRT2<0.0);
-
else {SQRT2=sqrt(SQRT2);
-
AA[0]=1;BB[0]=a3/2.0-SQRT1;CC[0]=u[i]/2.0-SQRT2;
-
AA[1]=1;BB[1]=a3/2.0+SQRT1;CC[1]=u[i]/2.0+SQRT2;
-
AA[2]=1;BB[2]=a3/2.0-SQRT1;CC[2]=u[i]/2.0+SQRT2;
-
AA[3]=1;BB[3]=a3/2.0+SQRT1;CC[3]=u[i]/2.0-SQRT2;
-
for(m=0;m<4;m++){
-
for(n=m;n<4;n++)
-
{A1=AA[m];B1=BB[m];C1=CC[m];
-
A2=AA[n];B2=BB[n];C2=CC[n];
-
if(fabs(C1*C2-a0)<0.01 && fabs(B1*C2+B2*C1-a1)<0.01 && fabs(A1*C2+A2*C1+B1*B2-a2)<0.01 && fabs(A1*B2+A2*B1-a3)<0.01){I=i;break;};
-
}
-
if(I!=-1)break;
-
}
-
}
-
}
-
if(I!=-1)break;
-
}
-
if(I==-1){MessageBox(hwnd,"Sorry! Something has gone Wrong with the Calculation of Quartic Roots!","Error from Polynomial",MB_OK);
-
for(j=0;j<4;j++){EigR[j]=0.0;EigI[j]=0.0;}
-
} else {
-
discrim1=B1*B1-4.0*A1*C1;
-
discrim2=B2*B2-4.0*A2*C2;
-
if(fabs(discrim1)<1e-6)discrim1=0.0;
-
if(fabs(discrim2)<1e-6)discrim2=0.0;
-
if(discrim1>=0.0)
-
{SQRT=sqrt(discrim1);
-
EigR[0]=(-B1+SQRT)/2.0;EigI[0]=0.0;
-
EigR[1]=(-B1-SQRT)/2.0;EigI[1]=0.0;
-
if(fabs(EigR[0])<1e-6)EigR[0]=0.0;
-
if(fabs(EigR[1])<1e-6)EigR[1]=0.0;
-
} else {
-
SQRT=sqrt(-discrim1);
-
EigR[0]=-B1/2.0;EigI[0]=SQRT/2.0;
-
EigR[1]=-B1/2.0;EigI[1]=-SQRT/2.0;
-
if(fabs(EigR[0])<1e-6)EigR[0]=0.0;
-
if(fabs(EigR[1])<1e-6)EigR[1]=0.0;
-
}
-
if(discrim2>=0.0)
-
{SQRT=sqrt(discrim2);
-
EigR[2]=(-B2+SQRT)/2.0;EigI[2]=0.0;
-
EigR[3]=(-B2-SQRT)/2.0;EigI[3]=0.0;
-
if(fabs(EigR[2])<1e-6)EigR[2]=0.0;
-
if(fabs(EigR[3])<1e-6)EigR[3]=0.0;
-
} else {
-
SQRT=sqrt(-discrim2);
-
EigR[2]=-B2/2.0;EigI[2]=SQRT/2.0;
-
EigR[3]=-B2/2.0;EigI[3]=-SQRT/2.0;
-
if(fabs(EigR[2])<1e-6)EigR[2]=0.0;
-
if(fabs(EigR[3])<1e-6)EigR[3]=0.0;
-
}
-
}
-
}
-
}
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