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//$$svd.cpp singular value decomposition
// Copyright (C) 1991,2,3,4,5: R B Davies
// Updated 17 July, 1995
#define WANT_MATH
#include "include.h"
#include "config.h"
#include "newmatap.h"
#include "newmatrm.h"
#include "precisio.h"
#ifdef use_namespace
namespace NEWMAT {
#endif
#ifdef DO_REPORT
#define REPORT { static ExeCounter ExeCount(__LINE__,15); ++ExeCount; }
#else
#define REPORT {}
#endif
static Real pythag(Real f, Real g, Real& c, Real& s)
// return z=sqrt(f*f+g*g), c=f/z, s=g/z
// set c=1,s=0 if z==0
// avoid floating point overflow or divide by zero
{
if (f==0 && g==0) { c=1.0; s=0.0; return 0.0; }
Real af = f>=0 ? f : -f;
Real ag = g>=0 ? g : -g;
if (ag<af)
{
REPORT
Real h = g/f; Real sq = sqrt(1.0+h*h);
if (f<0) sq = -sq; // make return value non-negative
c = 1.0/sq; s = h/sq; return sq*f;
}
else
{
REPORT
Real h = f/g; Real sq = sqrt(1.0+h*h);
if (g<0) sq = -sq;
s = 1.0/sq; c = h/sq; return sq*g;
}
}
void SVD(const Matrix& A, DiagonalMatrix& Q, Matrix& U, Matrix& V,
bool withU, bool withV)
// from Wilkinson and Reinsch: "Handbook of Automatic Computation"
{
REPORT
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));
if (withV && &U == &V)
Throw(ProgramException("Need different matrices for U and V", U, V));
U = A; Real g = 0.0; Real f,h; Real x = 0.0; int i;
RowVector E(n); RectMatrixRow EI(E,0); Q.ReSize(n);
RectMatrixCol UCI(U,0); RectMatrixRow URI(U,0,1,n-1);
if (n) for (i=0;;)
{
EI.First() = g; Real ei = g; EI.Right(); Real s = UCI.SumSquare();
if (s<tol) { REPORT Q.element(i) = 0.0; }
else
{
REPORT
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) { REPORT g = 0.0; }
else
{
REPORT
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) { REPORT x = y; }
if (++i == n) { REPORT break; }
UCI.DownDiag(); URI.DownDiag();
}
if (withV)
{
REPORT
V.ReSize(n,n); V = 0.0; RectMatrixCol VCI(V,n-1,n-1,1);
if (n) { VCI.First() = 1.0; g=E.element(n-1); if (n!=1) URI.UpDiag(); }
for (i=n-2; i>=0; i--)
{
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 (i==0) break;
URI.UpDiag();
}
}
if (withU)
{
REPORT
for (i=n-1; i>=0; i--)
{
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;
if (i==0) break;
UCI.UpDiag();
}
}
eps *= x;
for (int k=n-1; k>=0; k--)
{
Real z = -FloatingPointPrecision::Maximum(); // to keep Gnu happy
Real y; int limit = 50; int l = 0;
while (limit--)
{
Real c, s; int i; int l1=k; bool tfc=false;
for (l=k; l>=0; l--)
{
// if (fabs(E.element(l))<=eps) goto test_f_convergence;
if (fabs(E.element(l))<=eps) { REPORT tfc=true; break; }
if (fabs(Q.element(l-1))<=eps) { REPORT l1=l; break; }
REPORT
}
if (!tfc)
{
REPORT
l=l1; l1=l-1; s = -1.0; c = 0.0;
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) { REPORT break; }
g = Q.element(i); h = pythag(g,f,c,s); Q.element(i) = h;
if (withU)
{
REPORT
RectMatrixCol UCI(U,i); RectMatrixCol UCJ(U,l1);
ComplexScale(UCJ, UCI, c, s);
}
}
}
// test_f_convergence: z = Q.element(k); if (l==k) goto convergence;
z = Q.element(k); if (l==k) { REPORT 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);
if (f>1) { REPORT g = f * sqrt(1 + square(1/f)); }
else if (f<-1) { REPORT g = -f * sqrt(1 + square(1/f)); }
else { REPORT g = sqrt(f*f + 1); }
{ REPORT 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 = pythag(f,h,c,s); E.element(i-1) = z;
f = x*c + g*s; g = -x*s + g*c; h = y*s; y *= c;
if (withV)
{
REPORT
RectMatrixCol VCI(V,i); RectMatrixCol VCJ(V,i-1);
ComplexScale(VCI, VCJ, c, s);
}
z = pythag(f,h,c,s); Q.element(i-1) = z;
f = c*g + s*y; x = -s*g + c*y;
if (withU)
{
REPORT
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)
{
REPORT
Q.element(k) = -z;
if (withV) { RectMatrixCol VCI(V,k); VCI.Negate(); }
}
}
if (withU & withV) SortSV(Q, U, V);
else if (withU) SortSV(Q, U);
else if (withV) SortSV(Q, V);
else SortDescending(Q);
}
void SVD(const Matrix& A, DiagonalMatrix& D)
{ REPORT Matrix U; SVD(A, D, U, U, false, false); }
#ifdef use_namespace
}
#endif
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