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00006 #define WANT_MATH
00007
00008 #include "include.h"
00009 #include "newmatap.h"
00010 #include "newmatrm.h"
00011 #include "precisio.h"
00012
00013 #ifdef use_namespace
00014 namespace NEWMAT {
00015 #endif
00016
00017 #ifdef DO_REPORT
00018 #define REPORT { static ExeCounter ExeCount(__LINE__,15); ++ExeCount; }
00019 #else
00020 #define REPORT {}
00021 #endif
00022
00023
00024
00025
00026 void SVD(const Matrix& A, DiagonalMatrix& Q, Matrix& U, Matrix& V,
00027 bool withU, bool withV)
00028
00029 {
00030 REPORT
00031 Tracer trace("SVD");
00032 Real eps = FloatingPointPrecision::Epsilon();
00033 Real tol = FloatingPointPrecision::Minimum()/eps;
00034
00035 int m = A.Nrows(); int n = A.Ncols();
00036 if (m<n)
00037 Throw(ProgramException("Want no. Rows >= no. Cols", A));
00038 if (withV && &U == &V)
00039 Throw(ProgramException("Need different matrices for U and V", U, V));
00040 U = A; Real g = 0.0; Real f,h; Real x = 0.0; int i;
00041 RowVector E(n); RectMatrixRow EI(E,0); Q.ReSize(n);
00042 RectMatrixCol UCI(U,0); RectMatrixRow URI(U,0,1,n-1);
00043
00044 if (n) for (i=0;;)
00045 {
00046 EI.First() = g; Real ei = g; EI.Right(); Real s = UCI.SumSquare();
00047 if (s<tol) { REPORT Q.element(i) = 0.0; }
00048 else
00049 {
00050 REPORT
00051 f = UCI.First(); g = -sign(sqrt(s), f); h = f*g-s; UCI.First() = f-g;
00052 Q.element(i) = g; RectMatrixCol UCJ = UCI; int j=n-i;
00053 while (--j) { UCJ.Right(); UCJ.AddScaled(UCI, (UCI*UCJ)/h); }
00054 }
00055
00056 s = URI.SumSquare();
00057 if (s<tol) { REPORT g = 0.0; }
00058 else
00059 {
00060 REPORT
00061 f = URI.First(); g = -sign(sqrt(s), f); URI.First() = f-g;
00062 EI.Divide(URI,f*g-s); RectMatrixRow URJ = URI; int j=m-i;
00063 while (--j) { URJ.Down(); URJ.AddScaled(EI, URI*URJ); }
00064 }
00065
00066 Real y = fabs(Q.element(i)) + fabs(ei); if (x<y) { REPORT x = y; }
00067 if (++i == n) { REPORT break; }
00068 UCI.DownDiag(); URI.DownDiag();
00069 }
00070
00071 if (withV)
00072 {
00073 REPORT
00074 V.ReSize(n,n); V = 0.0; RectMatrixCol VCI(V,n-1,n-1,1);
00075 if (n) { VCI.First() = 1.0; g=E.element(n-1); if (n!=1) URI.UpDiag(); }
00076 for (i=n-2; i>=0; i--)
00077 {
00078 VCI.Left();
00079 if (g!=0.0)
00080 {
00081 VCI.Divide(URI, URI.First()*g); int j = n-i;
00082 RectMatrixCol VCJ = VCI;
00083 while (--j) { VCJ.Right(); VCJ.AddScaled( VCI, (URI*VCJ) ); }
00084 }
00085 VCI.Zero(); VCI.Up(); VCI.First() = 1.0; g=E.element(i);
00086 if (i==0) break;
00087 URI.UpDiag();
00088 }
00089 }
00090
00091 if (withU)
00092 {
00093 REPORT
00094 for (i=n-1; i>=0; i--)
00095 {
00096 g = Q.element(i); URI.Reset(U,i,i+1,n-i-1); URI.Zero();
00097 if (g!=0.0)
00098 {
00099 h=UCI.First()*g; int j=n-i; RectMatrixCol UCJ = UCI;
00100 while (--j)
00101 {
00102 UCJ.Right(); UCI.Down(); UCJ.Down(); Real s = UCI*UCJ;
00103 UCI.Up(); UCJ.Up(); UCJ.AddScaled(UCI,s/h);
00104 }
00105 UCI.Divide(g);
00106 }
00107 else UCI.Zero();
00108 UCI.First() += 1.0;
00109 if (i==0) break;
00110 UCI.UpDiag();
00111 }
00112 }
00113
00114 eps *= x;
00115 for (int k=n-1; k>=0; k--)
00116 {
00117 Real z = -FloatingPointPrecision::Maximum();
00118 Real y; int limit = 50; int l = 0;
00119 while (limit--)
00120 {
00121 Real c, s; int i; int l1=k; bool tfc=false;
00122 for (l=k; l>=0; l--)
00123 {
00124
00125 if (fabs(E.element(l))<=eps) { REPORT tfc=true; break; }
00126 if (fabs(Q.element(l-1))<=eps) { REPORT l1=l; break; }
00127 REPORT
00128 }
00129 if (!tfc)
00130 {
00131 REPORT
00132 l=l1; l1=l-1; s = -1.0; c = 0.0;
00133 for (i=l; i<=k; i++)
00134 {
00135 f = - s * E.element(i); E.element(i) *= c;
00136
00137 if (fabs(f)<=eps) { REPORT break; }
00138 g = Q.element(i); h = pythag(g,f,c,s); Q.element(i) = h;
00139 if (withU)
00140 {
00141 REPORT
00142 RectMatrixCol UCI(U,i); RectMatrixCol UCJ(U,l1);
00143 ComplexScale(UCJ, UCI, c, s);
00144 }
00145 }
00146 }
00147
00148 z = Q.element(k); if (l==k) { REPORT break; }
00149
00150 x = Q.element(l); y = Q.element(k-1);
00151 g = E.element(k-1); h = E.element(k);
00152 f = ((y-z)*(y+z) + (g-h)*(g+h)) / (2*h*y);
00153 if (f>1) { REPORT g = f * sqrt(1 + square(1/f)); }
00154 else if (f<-1) { REPORT g = -f * sqrt(1 + square(1/f)); }
00155 else { REPORT g = sqrt(f*f + 1); }
00156 { REPORT f = ((x-z)*(x+z) + h*(y / ((f<0.0) ? f-g : f+g)-h)) / x; }
00157
00158 c = 1.0; s = 1.0;
00159 for (i=l+1; i<=k; i++)
00160 {
00161 g = E.element(i); y = Q.element(i); h = s*g; g *= c;
00162 z = pythag(f,h,c,s); E.element(i-1) = z;
00163 f = x*c + g*s; g = -x*s + g*c; h = y*s; y *= c;
00164 if (withV)
00165 {
00166 REPORT
00167 RectMatrixCol VCI(V,i); RectMatrixCol VCJ(V,i-1);
00168 ComplexScale(VCI, VCJ, c, s);
00169 }
00170 z = pythag(f,h,c,s); Q.element(i-1) = z;
00171 f = c*g + s*y; x = -s*g + c*y;
00172 if (withU)
00173 {
00174 REPORT
00175 RectMatrixCol UCI(U,i); RectMatrixCol UCJ(U,i-1);
00176 ComplexScale(UCI, UCJ, c, s);
00177 }
00178 }
00179 E.element(l) = 0.0; E.element(k) = f; Q.element(k) = x;
00180 }
00181 if (l!=k) { Throw(ConvergenceException(A)); }
00182
00183 if (z < 0.0)
00184 {
00185 REPORT
00186 Q.element(k) = -z;
00187 if (withV) { RectMatrixCol VCI(V,k); VCI.Negate(); }
00188 }
00189 }
00190 if (withU & withV) SortSV(Q, U, V);
00191 else if (withU) SortSV(Q, U);
00192 else if (withV) SortSV(Q, V);
00193 else SortDescending(Q);
00194 }
00195
00196 void SVD(const Matrix& A, DiagonalMatrix& D)
00197 { REPORT Matrix U; SVD(A, D, U, U, false, false); }
00198
00199
00200
00201 #ifdef use_namespace
00202 }
00203 #endif
00204