eigen_sym.h

经典numerical receip 配套代码

void eigsrt(VecDoub_IO &d, MatDoub_IO *v=NULL)
{
	Int k;
	Int n=d.size();
	for (Int i=0;i<n-1;i++) {
		Doub p=d[k=i];
		for (Int j=i;j<n;j++)
			if (d[j] >= p) p=d[k=j];
		if (k != i) {
			d[k]=d[i];
			d[i]=p;
			if (v != NULL)
				for (Int j=0;j<n;j++) {
					p=(*v)[j][i];
					(*v)[j][i]=(*v)[j][k];
					(*v)[j][k]=p;
				}
		}
	}
}
struct Jacobi {
	const Int n;
	MatDoub a,v;
	VecDoub d;
	Int nrot;
	const Doub EPS;

	Jacobi(MatDoub_I &aa) : n(aa.nrows()), a(aa), v(n,n), d(n), nrot(0),
		EPS(numeric_limits<Doub>::epsilon())
	{
		Int i,j,ip,iq;
		Doub tresh,theta,tau,t,sm,s,h,g,c;
		VecDoub b(n),z(n);
		for (ip=0;ip<n;ip++) {
			for (iq=0;iq<n;iq++) v[ip][iq]=0.0;
			v[ip][ip]=1.0;
		}
		for (ip=0;ip<n;ip++) {
			b[ip]=d[ip]=a[ip][ip];
			z[ip]=0.0;
		}
		for (i=1;i<=50;i++) {
			sm=0.0;
			for (ip=0;ip<n-1;ip++) {
				for (iq=ip+1;iq<n;iq++)
					sm += abs(a[ip][iq]);
			}
			if (sm == 0.0) {
				eigsrt(d,&v);
				return;
			}
			if (i < 4)
				tresh=0.2*sm/(n*n);
			else
				tresh=0.0;
			for (ip=0;ip<n-1;ip++) {
				for (iq=ip+1;iq<n;iq++) {
					g=100.0*abs(a[ip][iq]);
					if (i > 4 && g <= EPS*abs(d[ip]) && g <= EPS*abs(d[iq]))
							a[ip][iq]=0.0;
					else if (abs(a[ip][iq]) > tresh) {
						h=d[iq]-d[ip];
						if (g <= EPS*abs(h))
							t=(a[ip][iq])/h;
						else {
							theta=0.5*h/(a[ip][iq]);
							t=1.0/(abs(theta)+sqrt(1.0+theta*theta));
							if (theta < 0.0) t = -t;
						}
						c=1.0/sqrt(1+t*t);
						s=t*c;
						tau=s/(1.0+c);
						h=t*a[ip][iq];
						z[ip] -= h;
						z[iq] += h;
						d[ip] -= h;
						d[iq] += h;
						a[ip][iq]=0.0;
						for (j=0;j<ip;j++)
							rot(a,s,tau,j,ip,j,iq);
						for (j=ip+1;j<iq;j++)
							rot(a,s,tau,ip,j,j,iq);
						for (j=iq+1;j<n;j++)
							rot(a,s,tau,ip,j,iq,j);
						for (j=0;j<n;j++)
							rot(v,s,tau,j,ip,j,iq);
						++nrot;
					}
				}
			}
			for (ip=0;ip<n;ip++) {
				b[ip] += z[ip];
				d[ip]=b[ip];
				z[ip]=0.0;
			}
		}
		throw("Too many iterations in routine jacobi");
	}
	inline void rot(MatDoub_IO &a, const Doub s, const Doub tau, const Int i,
		const Int j, const Int k, const Int l)
	{
		Doub g=a[i][j];
		Doub h=a[k][l];
		a[i][j]=g-s*(h+g*tau);
		a[k][l]=h+s*(g-h*tau);
	}
};
struct Symmeig {
	Int n;
	MatDoub z;
	VecDoub d,e;
	Bool yesvecs;

	Symmeig(MatDoub_I &a, Bool yesvec=true) : n(a.nrows()), z(a), d(n),
		e(n), yesvecs(yesvec)
	{
		tred2();
		tqli();
		sort();
	}
	Symmeig(VecDoub_I &dd, VecDoub_I &ee, Bool yesvec=true) :
		n(dd.size()), d(dd), e(ee), z(n,n,0.0), yesvecs(yesvec)
	{
		for (Int i=0;i<n;i++) z[i][i]=1.0;
		tqli();
		sort();
	}
	void sort() {
		if (yesvecs)
			eigsrt(d,&z);
		else
			eigsrt(d);
	}
	void tred2();
	void tqli();
	Doub pythag(const Doub a, const Doub b);
};
void Symmeig::tred2()
{
	Int l,k,j,i;
	Doub scale,hh,h,g,f;
	for (i=n-1;i>0;i--) {
		l=i-1;
		h=scale=0.0;
		if (l > 0) {
			for (k=0;k<i;k++)
				scale += abs(z[i][k]);
			if (scale == 0.0)
				e[i]=z[i][l];
			else {
				for (k=0;k<i;k++) {
					z[i][k] /= scale;
					h += z[i][k]*z[i][k];
				}
				f=z[i][l];
				g=(f >= 0.0 ? -sqrt(h) : sqrt(h));
				e[i]=scale*g;
				h -= f*g;
				z[i][l]=f-g;
				f=0.0;
				for (j=0;j<i;j++) {
					if (yesvecs)
						z[j][i]=z[i][j]/h;
					g=0.0;
					for (k=0;k<j+1;k++)
						g += z[j][k]*z[i][k];
					for (k=j+1;k<i;k++)
						g += z[k][j]*z[i][k];
					e[j]=g/h;
					f += e[j]*z[i][j];
				}
				hh=f/(h+h);
				for (j=0;j<i;j++) {
					f=z[i][j];
					e[j]=g=e[j]-hh*f;
					for (k=0;k<j+1;k++)
						z[j][k] -= (f*e[k]+g*z[i][k]);
				}
			}
		} else
			e[i]=z[i][l];
		d[i]=h;
	}
	if (yesvecs) d[0]=0.0;
	e[0]=0.0;
	for (i=0;i<n;i++) {
		if (yesvecs) {
			if (d[i] != 0.0) {
				for (j=0;j<i;j++) {
					g=0.0;
					for (k=0;k<i;k++)
						g += z[i][k]*z[k][j];
					for (k=0;k<i;k++)
						z[k][j] -= g*z[k][i];
				}
			}
			d[i]=z[i][i];
			z[i][i]=1.0;
			for (j=0;j<i;j++) z[j][i]=z[i][j]=0.0;
		} else {
			d[i]=z[i][i];
		}
	}
}
void Symmeig::tqli()
{
	Int m,l,iter,i,k;
	Doub s,r,p,g,f,dd,c,b;
	const Doub EPS=numeric_limits<Doub>::epsilon();
	for (i=1;i<n;i++) e[i-1]=e[i];
	e[n-1]=0.0;
	for (l=0;l<n;l++) {
		iter=0;
		do {
			for (m=l;m<n-1;m++) {
				dd=abs(d[m])+abs(d[m+1]);
				if (abs(e[m]) <= EPS*dd) break;
			}
			if (m != l) {
				if (iter++ == 30) throw("Too many iterations in tqli");
				g=(d[l+1]-d[l])/(2.0*e[l]);
				r=pythag(g,1.0);
				g=d[m]-d[l]+e[l]/(g+SIGN(r,g));
				s=c=1.0;
				p=0.0;
				for (i=m-1;i>=l;i--) {
					f=s*e[i];
					b=c*e[i];
					e[i+1]=(r=pythag(f,g));
					if (r == 0.0) {
						d[i+1] -= p;
						e[m]=0.0;
						break;
					}
					s=f/r;
					c=g/r;
					g=d[i+1]-p;
					r=(d[i]-g)*s+2.0*c*b;
					d[i+1]=g+(p=s*r);
					g=c*r-b;
					if (yesvecs) {
						for (k=0;k<n;k++) {
							f=z[k][i+1];
							z[k][i+1]=s*z[k][i]+c*f;
							z[k][i]=c*z[k][i]-s*f;
						}
					}
				}
				if (r == 0.0 && i >= l) continue;
				d[l] -= p;
				e[l]=g;
				e[m]=0.0;
			}
		} while (m != l);
	}
}

Doub Symmeig::pythag(const Doub a, const Doub b) {
	Doub absa=abs(a), absb=abs(b);
	return (absa > absb ? absa*sqrt(1.0+SQR(absb/absa)) :
		(absb == 0.0 ? 0.0 : absb*sqrt(1.0+SQR(absa/absb))));
}