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SVD.CXX
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1992-11-30
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5KB
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161 lines
//$$svd.cxx singular value decomposition
// Copyright (C) 1991,2: R B Davies
#define WANT_MATH
#include "include.h"
#include "newmat.h"
#include "newmatrm.h"
#include "precisio.h"
void SVD(const Matrix& A, DiagonalMatrix& Q, Matrix& U, Matrix& V,
Boolean withU, Boolean withV)
// from Wilkinson and Reinsch: "Handbook of Automatic Computation"
{
Tracer trace("SVD");
Real eps = FloatingPointPrecision::Epsilon();
Real tol = FloatingPointPrecision::Minimum()/eps;
int m = A.Nrows(); int n = A.Ncols();
if (m<n)
Throw(ProgramException("Want no. Rows >= no. Cols", A));
U = A; Real g = 0.0; Real f,h; Real x = 0.0;
RowVector E(n); RectMatrixRow EI(E,0); Q.ReDimension(n);
RectMatrixCol UCI(U,0); RectMatrixRow URI(U,0,1,n-1);
for (int i=0; i<n; i++)
{
EI.First() = g; Real ei = g; EI.Right(); Real s = UCI.SumSquare();
if (s<tol) Q.element(i) = 0.0;
else
{
f = UCI.First(); g = -sign(sqrt(s), f); h = f*g-s; UCI.First() = f-g;
Q.element(i) = g; RectMatrixCol UCJ = UCI; int j=n-i;
while (--j) { UCJ.Right(); UCJ.AddScaled(UCI, (UCI*UCJ)/h); }
}
s = URI.SumSquare();
if (s<tol) g = 0.0;
else
{
f = URI.First(); g = -sign(sqrt(s), f); URI.First() = f-g;
EI.Divide(URI,f*g-s); RectMatrixRow URJ = URI; int j=m-i;
while (--j) { URJ.Down(); URJ.AddScaled(EI, URI*URJ); }
}
Real y = fabs(Q.element(i)) + fabs(ei); if (x<y) x = y;
UCI.DownDiag(); URI.DownDiag();
}
if (withV)
{
V.ReDimension(n,n); V = 0.0; RectMatrixCol VCI(V,n,n,0);
for (i=n-1; i>=0; i--)
{
URI.UpDiag(); VCI.Left();
if (g!=0.0)
{
VCI.Divide(URI, URI.First()*g); int j = n-i;
RectMatrixCol VCJ = VCI;
while (--j) { VCJ.Right(); VCJ.AddScaled( VCI, (URI*VCJ) ); }
}
VCI.Zero(); VCI.Up(); VCI.First() = 1.0; g=E.element(i);
}
}
if (withU)
{
for (i=n-1; i>=0; i--)
{
UCI.UpDiag(); g = Q.element(i); URI.Reset(U,i,i+1,n-i-1); URI.Zero();
if (g!=0.0)
{
h=UCI.First()*g; int j=n-i; RectMatrixCol UCJ = UCI;
while (--j)
{
UCJ.Right(); UCI.Down(); UCJ.Down(); Real s = UCI*UCJ;
UCI.Up(); UCJ.Up(); UCJ.AddScaled(UCI,s/h);
}
UCI.Divide(g);
}
else UCI.Zero();
UCI.First() += 1.0;
}
}
eps *= x;
for (int k=n-1; k>=0; k--)
{
Real z; Real y; int limit = 50; int l;
while (limit--)
{
Real c=0.0; Real s=1.0; int i; int l1=k; Boolean tfc=FALSE;
for (l=k; l>=0; l--)
{
// if (fabs(E.element(l))<=eps) goto test_f_convergence;
if (fabs(E.element(l))<=eps) { tfc=TRUE; break; }
if (fabs(Q.element(l-1))<=eps) { l1=l; break; }
}
if (!tfc)
{
l=l1; l1=l-1;
for (i=l; i<=k; i++)
{
f = s * E.element(i); E.element(i) *= c;
// if (fabs(f)<=eps) goto test_f_convergence;
if (fabs(f)<=eps) break;
g = Q.element(i); h = sqrt(f*f + g*g); Q.element(i) = h;
if (withU)
{
RectMatrixCol UCI(U,i); RectMatrixCol UCJ(U,l1);
ComplexScale(UCI, UCJ, g/h, -f/h);
}
}
}
// test_f_convergence: z = Q.element(k); if (l==k) goto convergence;
z = Q.element(k); if (l==k) break;
x = Q.element(l); y = Q.element(k-1);
g = E.element(k-1); h = E.element(k);
f = ((y-z)*(y+z) + (g-h)*(g+h)) / (2*h*y); g = sqrt(f*f + 1);
f = ((x-z)*(x+z) + h*(y / ((f<0.0) ? f-g : f+g)-h)) / x;
c = 1.0; s = 1.0;
for (i=l+1; i<=k; i++)
{
g = E.element(i); y = Q.element(i); h = s*g; g *= c;
z = sqrt(f*f + h*h); E.element(i-1) = z; c = f/z; s = h/z;
f = x*c + g*s; g = -x*s + g*c; h = y*s; y *= c;
if (withV)
{
RectMatrixCol VCI(V,i); RectMatrixCol VCJ(V,i-1);
ComplexScale(VCI, VCJ, c, s);
}
z = sqrt(f*f + h*h); Q.element(i-1) = z; c = f/z; s = h/z;
f = c*g + s*y; x = -s*g + c*y;
if (withU)
{
RectMatrixCol UCI(U,i); RectMatrixCol UCJ(U,i-1);
ComplexScale(UCI, UCJ, c, s);
}
}
E.element(l) = 0.0; E.element(k) = f; Q.element(k) = x;
}
if (l!=k) { Throw(ConvergenceException(A)); }
// convergence:
if (z < 0.0)
{
Q.element(k) = -z;
if (withV) { RectMatrixCol VCI(V,k); VCI.Negate(); }
}
}
}
void SVD(const Matrix& A, DiagonalMatrix& D)
{ Matrix U; SVD(A, D, U, U, FALSE, FALSE); }