svdcmp.cpp

这是C++数值算法(第二版)的源代码,其中包含了目前一些比较常用的数值计算的算法.

#include <cmath>
#include "nr.h"
using namespace std;

void NR::svdcmp(Mat_IO_DP &a, Vec_O_DP &w, Mat_O_DP &v)
{
	bool flag;
	int i,its,j,jj,k,l,nm;
	DP anorm,c,f,g,h,s,scale,x,y,z;

	int m=a.nrows();
	int n=a.ncols();
	Vec_DP rv1(n);
	g=scale=anorm=0.0;
	for (i=0;i<n;i++) {
		l=i+2;
		rv1[i]=scale*g;
		g=s=scale=0.0;
		if (i < m) {
			for (k=i;k<m;k++) scale += fabs(a[k][i]);
			if (scale != 0.0) {
				for (k=i;k<m;k++) {
					a[k][i] /= scale;
					s += a[k][i]*a[k][i];
				}
				f=a[i][i];
				g = -SIGN(sqrt(s),f);
				h=f*g-s;
				a[i][i]=f-g;
				for (j=l-1;j<n;j++) {
					for (s=0.0,k=i;k<m;k++) s += a[k][i]*a[k][j];
					f=s/h;
					for (k=i;k<m;k++) a[k][j] += f*a[k][i];
				}
				for (k=i;k<m;k++) a[k][i] *= scale;
			}
		}
		w[i]=scale *g;
		g=s=scale=0.0;
		if (i+1 <= m && i != n) {
			for (k=l-1;k<n;k++) scale += fabs(a[i][k]);
			if (scale != 0.0) {
				for (k=l-1;k<n;k++) {
					a[i][k] /= scale;
					s += a[i][k]*a[i][k];
				}
				f=a[i][l-1];
				g = -SIGN(sqrt(s),f);
				h=f*g-s;
				a[i][l-1]=f-g;
				for (k=l-1;k<n;k++) rv1[k]=a[i][k]/h;
				for (j=l-1;j<m;j++) {
					for (s=0.0,k=l-1;k<n;k++) s += a[j][k]*a[i][k];
					for (k=l-1;k<n;k++) a[j][k] += s*rv1[k];
				}
				for (k=l-1;k<n;k++) a[i][k] *= scale;
			}
		}
		anorm=MAX(anorm,(fabs(w[i])+fabs(rv1[i])));
	}
	for (i=n-1;i>=0;i--) {
		if (i < n-1) {
			if (g != 0.0) {
				for (j=l;j<n;j++)
					v[j][i]=(a[i][j]/a[i][l])/g;
				for (j=l;j<n;j++) {
					for (s=0.0,k=l;k<n;k++) s += a[i][k]*v[k][j];
					for (k=l;k<n;k++) v[k][j] += s*v[k][i];
				}
			}
			for (j=l;j<n;j++) v[i][j]=v[j][i]=0.0;
		}
		v[i][i]=1.0;
		g=rv1[i];
		l=i;
	}
	for (i=MIN(m,n)-1;i>=0;i--) {
		l=i+1;
		g=w[i];
		for (j=l;j<n;j++) a[i][j]=0.0;
		if (g != 0.0) {
			g=1.0/g;
			for (j=l;j<n;j++) {
				for (s=0.0,k=l;k<m;k++) s += a[k][i]*a[k][j];
				f=(s/a[i][i])*g;
				for (k=i;k<m;k++) a[k][j] += f*a[k][i];
			}
			for (j=i;j<m;j++) a[j][i] *= g;
		} else for (j=i;j<m;j++) a[j][i]=0.0;
		++a[i][i];
	}
	for (k=n-1;k>=0;k--) {
		for (its=0;its<30;its++) {
			flag=true;
			for (l=k;l>=0;l--) {
				nm=l-1;
				if (fabs(rv1[l])+anorm == anorm) {
					flag=false;
					break;
				}
				if (fabs(w[nm])+anorm == anorm) break;
			}
			if (flag) {
				c=0.0;
				s=1.0;
				for (i=l-1;i<k+1;i++) {
					f=s*rv1[i];
					rv1[i]=c*rv1[i];
					if (fabs(f)+anorm == anorm) break;
					g=w[i];
					h=pythag(f,g);
					w[i]=h;
					h=1.0/h;
					c=g*h;
					s = -f*h;
					for (j=0;j<m;j++) {
						y=a[j][nm];
						z=a[j][i];
						a[j][nm]=y*c+z*s;
						a[j][i]=z*c-y*s;
					}
				}
			}
			z=w[k];
			if (l == k) {
				if (z < 0.0) {
					w[k] = -z;
					for (j=0;j<n;j++) v[j][k] = -v[j][k];
				}
				break;
			}
			if (its == 29) nrerror("no convergence in 30 svdcmp iterations");
			x=w[l];
			nm=k-1;
			y=w[nm];
			g=rv1[nm];
			h=rv1[k];
			f=((y-z)*(y+z)+(g-h)*(g+h))/(2.0*h*y);
			g=pythag(f,1.0);
			f=((x-z)*(x+z)+h*((y/(f+SIGN(g,f)))-h))/x;
			c=s=1.0;
			for (j=l;j<=nm;j++) {
				i=j+1;
				g=rv1[i];
				y=w[i];
				h=s*g;
				g=c*g;
				z=pythag(f,h);
				rv1[j]=z;
				c=f/z;
				s=h/z;
				f=x*c+g*s;
				g=g*c-x*s;
				h=y*s;
				y *= c;
				for (jj=0;jj<n;jj++) {
					x=v[jj][j];
					z=v[jj][i];
					v[jj][j]=x*c+z*s;
					v[jj][i]=z*c-x*s;
				}
				z=pythag(f,h);
				w[j]=z;
				if (z) {
					z=1.0/z;
					c=f*z;
					s=h*z;
				}
				f=c*g+s*y;
				x=c*y-s*g;
				for (jj=0;jj<m;jj++) {
					y=a[jj][j];
					z=a[jj][i];
					a[jj][j]=y*c+z*s;
					a[jj][i]=z*c-y*s;
				}
			}
			rv1[l]=0.0;
			rv1[k]=f;
			w[k]=x;
		}
	}
}