//$$evalue.cpp                           eigen-value decomposition// Copyright (C) 1991,2,3,4: R B Davies#define WANT_MATH#include "include.h"#include "newmat.h"#include "newmatrm.h"#include "precisio.h"#ifdef use_namespacenamespace NEWMAT {#endifstatic void tred2(const SymmetricMatrix& A, DiagonalMatrix& D,   DiagonalMatrix& E, Matrix& Z){   Tracer et("Evalue(tred2)");   Real tol =      FloatingPointPrecision::Minimum()/FloatingPointPrecision::Epsilon();   int n = A.Nrows(); Z.ReSize(n,n); Z.Inject(A);   D.ReSize(n); E.ReSize(n);   Real* z = Z.Store(); int i;   for (i=n-1; i > 0; i--)                   // i=0 is excluded   {      Real f = Z.element(i,i-1); Real g = 0.0;      int k = i-1; Real* zik = z + i*n;      while (k--) g += square(*zik++);      Real h = g + square(f);      if (g <= tol) { E.element(i) = f; h = 0.0; }      else      {         g = sign(-sqrt(h), f); E.element(i) = g; h -= f*g;         Z.element(i,i-1) = f-g; f = 0.0;         Real* zji = z + i; Real* zij = z + i*n; Real* ej = E.Store();         int j;         for (j=0; j<i; j++)         {            *zji = (*zij++)/h; g = 0.0;            Real* zjk = z + j*n; zik = z + i*n;            k = j; while (k--) g += *zjk++ * (*zik++);            k = i-j;            if (k) for(;;)               { g += *zjk * (*zik++); if (!(--k)) break; zjk += n; }            *ej++ = g/h; f += g * (*zji); zji += n;         }         Real hh = f / (h + h); zij = z + i*n; ej = E.Store();         for (j=0; j<i; j++)         {            f = *zij++; g = *ej - hh * f; *ej++ = g;            Real* zjk = z + j*n; Real* zik = z + i*n;            Real* ek = E.Store(); k = j+1;            while (k--)  *zjk++ -= ( f*(*ek++) + g*(*zik++) );          }      }      D.element(i) = h;   }   D.element(0) = 0.0; E.element(0) = 0.0;   for (i=0; i<n; i++)   {      if (D.element(i) != 0.0)      {         for (int j=0; j<i; j++)         {            Real g = 0.0;            Real* zik = z + i*n; Real* zkj = z + j;            int k = i;            if (k) for (;;)               { g += *zik++ * (*zkj); if (!(--k)) break; zkj += n; }            Real* zki = z + i; zkj = z + j;            k = i;            if (k) for (;;)               { *zkj -= g * (*zki); if (!(--k)) break; zkj += n; zki += n; }         }      }      Real* zij = z + i*n; Real* zji = z + i;      int j = i;      if (j) for (;;)         { *zij++ = 0.0; *zji = 0.0; if (!(--j)) break; zji += n; }      D.element(i) = *zij; *zij = 1.0;   }}static void tql2(DiagonalMatrix& D, DiagonalMatrix& E, Matrix& Z){   Tracer et("Evalue(tql2)");   Real eps = FloatingPointPrecision::Epsilon();   int n = D.Nrows(); Real* z = Z.Store(); int l;   for (l=1; l<n; l++) E.element(l-1) = E.element(l);   Real b = 0.0; Real f = 0.0; E.element(n-1) = 0.0;   for (l=0; l<n; l++)   {      int i,j;      Real& dl = D.element(l); Real& el = E.element(l);      Real h = eps * ( fabs(dl) + fabs(el) );      if (b < h) b = h;      int m;      for (m=l; m<n; m++) if (fabs(E.element(m)) <= b) break;      bool test = false;      for (j=0; j<30; j++)      {         if (m==l) { test = true; break; }         Real& dl1 = D.element(l+1);         Real g = dl; Real p = (dl1-g) / (2.0*el); Real r = sqrt(p*p + 1.0);         dl = el / (p < 0.0 ? p-r : p+r); Real h = g - dl; f += h;         Real* dlx = &dl1; i = n-l-1; while (i--) *dlx++ -= h;         p = D.element(m); Real c = 1.0; Real s = 0.0;         for (i=m-1; i>=l; i--)         {            Real ei = E.element(i); Real di = D.element(i);            Real& ei1 = E.element(i+1);            g = c * ei; h = c * p;            if ( fabs(p) >= fabs(ei))            {               c = ei / p; r = sqrt(c*c + 1.0);               ei1 = s*p*r; s = c/r; c = 1.0/r;            }            else            {               c = p / ei; r = sqrt(c*c + 1.0);               ei1 = s * ei * r; s = 1.0/r; c /= r;            }            p = c * di - s*g; D.element(i+1) = h + s * (c*g + s*di);            Real* zki = z + i; Real* zki1 = zki + 1; int k = n;            if (k) for (;;)            {               h = *zki1; *zki1 = s*(*zki) + c*h; *zki = c*(*zki) - s*h;               if (!(--k)) break;               zki += n; zki1 += n;            }         }         el = s*p; dl = c*p;         if (fabs(el) <= b) { test = true; break; }      }      if (!test) Throw ( ConvergenceException(D) );      dl += f;   }   for (int i=0; i<n; i++)   {      int k = i; Real p = D.element(i);      for (int j=i+1; j<n; j++)         { if (D.element(j) < p) { k = j; p = D.element(j); } }      if (k != i)      {         D.element(k) = D.element(i); D.element(i) = p; int j = n;         Real* zji = z + i; Real* zjk = z + k;         if (j) for(;;)         {            p = *zji; *zji = *zjk; *zjk = p;            if (!(--j)) break;            zji += n; zjk += n;         }      }   }}static void tred3(const SymmetricMatrix& X, DiagonalMatrix& D,   DiagonalMatrix& E, SymmetricMatrix& A){   Tracer et("Evalue(tred3)");   Real tol =      FloatingPointPrecision::Minimum()/FloatingPointPrecision::Epsilon();   int n = X.Nrows(); A = X; D.ReSize(n); E.ReSize(n);   Real* ei = E.Store() + n;   for (int i = n-1; i >= 0; i--)   {      Real h = 0.0; Real f = - FloatingPointPrecision::Maximum();      Real* d = D.Store(); Real* a = A.Store() + (i*(i+1))/2; int k = i;      while (k--) { f = *a++; *d++ = f; h += square(f); }      if (h <= tol) { *(--ei) = 0.0; h = 0.0; }      else      {         Real g = sign(-sqrt(h), f); *(--ei) = g; h -= f*g;         f -= g; *(d-1) = f; *(a-1) = f; f = 0.0;         Real* dj = D.Store(); Real* ej = E.Store(); int j;         for (j = 0; j < i; j++)         {            Real* dk = D.Store(); Real* ak = A.Store()+(j*(j+1))/2;            Real g = 0.0; k = j;            while (k--)  g += *ak++ * *dk++;            k = i-j; int l = j;             if (k) for (;;) { g += *ak * *dk++; if (!(--k)) break; ak += ++l; }            g /= h; *ej++ = g; f += g * *dj++;         }           Real hh = f / (2 * h); Real* ak = A.Store();         dj = D.Store(); ej = E.Store();         for (j = 0; j < i; j++)         {            f = *dj++; g = *ej - hh * f; *ej++ = g;            Real* dk = D.Store(); Real* ek = E.Store(); k = j+1;            while (k--) { *ak++ -= (f * *ek++ + g * *dk++); }         }      }      *d = *a; *a = h;   }}static void tql1(DiagonalMatrix& D, DiagonalMatrix& E){   Tracer et("Evalue(tql1)");   Real eps = FloatingPointPrecision::Epsilon();   int n = D.Nrows(); int l;   for (l=1; l<n; l++) E.element(l-1) = E.element(l);   Real b = 0.0; Real f = 0.0; E.element(n-1) = 0.0;   for (l=0; l<n; l++)   {      int i,j;      Real& dl = D.element(l); Real& el = E.element(l);      Real h = eps * ( fabs(dl) + fabs(el) );      if (b < h) b = h;      int m;      for (m=l; m<n; m++) if (fabs(E.element(m)) <= b) break;      bool test = false;      for (j=0; j<30; j++)      {         if (m==l) { test = true; break; }         Real& dl1 = D.element(l+1);         Real g = dl; Real p = (dl1-g) / (2.0*el); Real r = sqrt(p*p + 1.0);         dl = el / (p < 0.0 ? p-r : p+r); Real h = g - dl; f += h;         Real* dlx = &dl1; i = n-l-1; while (i--) *dlx++ -= h;         p = D.element(m); Real c = 1.0; Real s = 0.0;         for (i=m-1; i>=l; i--)         {            Real ei = E.element(i); Real di = D.element(i);            Real& ei1 = E.element(i+1);            g = c * ei; h = c * p;            if ( fabs(p) >= fabs(ei))            {               c = ei / p; r = sqrt(c*c + 1.0);               ei1 = s*p*r; s = c/r; c = 1.0/r;            }            else            {               c = p / ei; r = sqrt(c*c + 1.0);               ei1 = s * ei * r; s = 1.0/r; c /= r;            }            p = c * di - s*g; D.element(i+1) = h + s * (c*g + s*di);         }         el = s*p; dl = c*p;         if (fabs(el) <= b) { test = true; break; }      }      if (!test) Throw ( ConvergenceException(D) );      Real p = dl + f;      test = false;      for (i=l; i>0; i--)      {         if (p < D.element(i-1)) D.element(i) = D.element(i-1);         else { test = true; break; }      }      if (!test) i=0;      D.element(i) = p;   }}void EigenValues(const SymmetricMatrix& A, DiagonalMatrix& D, Matrix& Z){ DiagonalMatrix E; tred2(A, D, E, Z); tql2(D, E, Z); }void EigenValues(const SymmetricMatrix& X, DiagonalMatrix& D){ DiagonalMatrix E; SymmetricMatrix A; tred3(X,D,E,A); tql1(D,E); }void EigenValues(const SymmetricMatrix& X, DiagonalMatrix& D,   SymmetricMatrix& A){ DiagonalMatrix E; tred3(X,D,E,A); tql1(D,E); }#ifdef use_namespace}#endif