void SvdCmp(double **a, int m, int n, double w[], double **v)
{
// SVD Composition
int flag,i,its,j,jj,k,l,nm;
double anorm,c,f,g,h,s,scale,x,y,z,*rv1;
rv1=dVector(1,n);
g=scale=anorm=0.0;
for (i=1;i<=n;i++) {
l=i+1;
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) {
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;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 <= m && i != n) {
for (k=l;k<=n;k++) scale += fabs(a[i][k]);
if (scale) {
for (k=l;k<=n;k++) {
a[i][k] /= scale;
s += a[i][k]*a[i][k];
}
f=a[i][l];
g = -SIGN(sqrt(s),f);
h=f*g-s;
a[i][l]=f-g;
for (k=l;k<=n;k++) rv1[k]=a[i][k]/h;
for (j=l;j<=m;j++) {
for (s=0.0,k=l;k<=n;k++) s += a[j][k]*a[i][k];
for (k=l;k<=n;k++) a[j][k] += s*rv1[k];
}
for (k=l;k<=n;k++) a[i][k] *= scale;
}
}
anorm=__max(anorm,(fabs(w[i])+fabs(rv1[i])));
}
for (i=n;i>=1;i--) {
if (i < n) {
if (g) {
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);i>=1;i--) {
l=i+1;
g=w[i];
for (j=l;j<=n;j++) a[i][j]=0.0;
if (g) {
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;k>=1;k--) {
for (its=1;its<=100;its++) {
flag=1;
for (l=k;l>=1;l--) {
nm=l-1;
if ((double)(fabs(rv1[l])+anorm) == anorm) {
flag=0;
break;
}
if ((double)(fabs(w[nm])+anorm) == anorm) break;
}
if (flag) {
c=0.0;
s=1.0;
for (i=l;i<=k;i++) {
f=s*rv1[i];
rv1[i]=c*rv1[i];
if ((double)(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=1;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=1;j<=n;j++) v[j][k] = -v[j][k];
}
break;
}
if (its == 100) 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=1;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=1;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;
}
}
Free_dVector(rv1,1,n);
}
//==========================================================================
// Class: Matrix
// Function: DiagonalizeBidiagonalForm
//
// Description: Part of SVD algorithm. "Diagonalizes the bidiagonal form."
//
// Input Arguments:
// U = Matrix&
// V = Matrix&
// W = Matrix&
// rv1 = double*
// anorm = const double&
//
// Output Arguments:
// None
//
// Return Value:
// bool, false if iteration limit was reached, true otherwise
//
//==========================================================================
bool Matrix::DiagonalizeBidiagonalForm(Matrix &U, Matrix &V, Matrix &W, double *rv1, const double &anorm) const
{
int i, j, its, jj, k, l(0), nm(0);
double c, f, g, h, s, x, y, z;
bool finished;
double eps = 1.0e-6;
int its_limit = 30;
for (k = V.rows - 1; k >= 0; k--)
{
for (its = 0; its < its_limit; its++)
{
finished = false;
for (l = k; l >= 0; l--)
{
nm = l - 1;
if (l == 0 || fabs(rv1[l]) <= eps * anorm)
{
finished = true;
break;
}
if (fabs(W.elements[nm][nm]) <= eps * anorm)
break;
}
if (!finished)
{
c = 0.0;
s = 1.0;
for (i = l; i <= k; i++)
{
f = s * rv1[i];
rv1[i] = c * rv1[i];
if (fabs(f) <= eps * anorm)
break;
g = W.elements[i][i];
h = Pythag(f, g);
W.elements[i][i] = h;
h = 1.0 / h;
c = g * h;
s = -f * h;
for (j = 0; j < (int)U.rows; j++)
{
y = U.elements[j][nm];
z = U.elements[j][i];
U.elements[j][nm] = y * c + z * s;
U.elements[j][i] = z * c - y * s;
}
}
}
z = W.elements[k][k];
if (l == k)
{
if (z < 0.0)
{
W.elements[k][k] = -z;
for (j = 0; j < (int)V.rows; j++)
V.elements[j][k] = -V.elements[j][k];
}
break;
}
if (its == its_limit - 1)// Reached iteration limit
return false;
x = W.elements[l][l];
nm = k - 1;
y = W.elements[nm][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);
if (f >= 0.0)
f = ((x-z) * (x+z) + h * ((y / (f + fabs(g))) - h)) / x;
else
f = ((x-z) * (x+z) + h * ((y / (f - fabs(g))) - h)) / x;
c = 1.0;
s = 1.0;
for (j = l; j <= nm; j++)
{
i = j + 1;
g = rv1[i];
y = W.elements[i][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 < (int)V.rows; jj++)
{
x = V.elements[jj][j];
z = V.elements[jj][i];
V.elements[jj][j] = x * c + z * s;
V.elements[jj][i] = z * c - x * s;
}
z = Pythag(f, h);
W.elements[j][j] = z;
if (z != 0.0)
{
z = 1.0 / z;
c = f * z;
s = h * z;
}
f = c * g + s * y;
x = c * y - s * g;
for (jj = 0; jj < (int)U.rows; jj++)
{
y = U.elements[jj][j];
z = U.elements[jj][i];
U.elements[jj][j] = y * c + z * s;
U.elements[jj][i] = z * c - y * s;
}
}
rv1[l] = 0.0;
rv1[k] = f;
W.elements[k][k] = x;
}
}
return true;
}