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## root / tmp / org.txm.analec.rcp / src / JamaPlus / SingularValueDecomposition.java0 @ 1726

 1 package JamaPlus;  import JamaPlus.util.*;   /** Singular Value Decomposition.  
  For an m-by-n matrix A with m >= n, the singular value decomposition is   an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and   an n-by-n orthogonal matrix V so that A = U*S*V'.  
  The singular values, sigma[k] = S[k][k], are ordered so that   sigma[0] >= sigma[1] >= ... >= sigma[n-1].  
  The singular value decompostion always exists, so the constructor will   never fail. The matrix condition number and the effective numerical   rank can be computed from this decomposition.   */  public class SingularValueDecomposition implements java.io.Serializable {  /* ------------------------   Class variables   * ------------------------ */   /** Arrays for internal storage of U and V.   @serial internal storage of U.   @serial internal storage of V.   */   private double[][] U, V;   /** Array for internal storage of singular values.   @serial internal storage of singular values.   */   private double[] s;   /** Row and column dimensions.   @serial row dimension.   @serial column dimension.   */   private int m, n;  /* ------------------------   Constructor   * ------------------------ */   /** Construct the singular value decomposition   @param A Rectangular matrix   @return Structure to access U, S and V.   */   public SingularValueDecomposition (Matrix Arg) {   // Derived from LINPACK code.   // Initialize.   double[][] A = Arg.getArrayCopy();   m = Arg.getRowDimension();   n = Arg.getColumnDimension();   /* Apparently the failing cases are only a proper subset of (m= n"); }   */   int nu = Math.min(m,n);   s = new double [Math.min(m+1,n)];   U = new double [m][nu];   V = new double [n][n];   double[] e = new double [n];   double[] work = new double [m];   boolean wantu = true;   boolean wantv = true;   // Reduce A to bidiagonal form, storing the diagonal elements   // in s and the super-diagonal elements in e.   int nct = Math.min(m-1,n);   int nrt = Math.max(0,Math.min(n-2,m));   for (int k = 0; k < Math.max(nct,nrt); k++) {   if (k < nct) {   // Compute the transformation for the k-th column and   // place the k-th diagonal in s[k].   // Compute 2-norm of k-th column without under/overflow.   s[k] = 0;   for (int i = k; i < m; i++) {   s[k] = Math.hypot(s[k],A[i][k]);   }   if (s[k] != 0.0) {   if (A[k][k] < 0.0) {   s[k] = -s[k];   }   for (int i = k; i < m; i++) {   A[i][k] /= s[k];   }   A[k][k] += 1.0;   }   s[k] = -s[k];   }   for (int j = k+1; j < n; j++) {   if ((k < nct) & (s[k] != 0.0)) {   // Apply the transformation.   double t = 0;   for (int i = k; i < m; i++) {   t += A[i][k]*A[i][j];   }   t = -t/A[k][k];   for (int i = k; i < m; i++) {   A[i][j] += t*A[i][k];   }   }   // Place the k-th row of A into e for the   // subsequent calculation of the row transformation.   e[j] = A[k][j];   }   if (wantu & (k < nct)) {   // Place the transformation in U for subsequent back   // multiplication.   for (int i = k; i < m; i++) {   U[i][k] = A[i][k];   }   }   if (k < nrt) {   // Compute the k-th row transformation and place the   // k-th super-diagonal in e[k].   // Compute 2-norm without under/overflow.   e[k] = 0;   for (int i = k+1; i < n; i++) {   e[k] = Math.hypot(e[k],e[i]);   }   if (e[k] != 0.0) {   if (e[k+1] < 0.0) {   e[k] = -e[k];   }   for (int i = k+1; i < n; i++) {   e[i] /= e[k];   }   e[k+1] += 1.0;   }   e[k] = -e[k];   if ((k+1 < m) & (e[k] != 0.0)) {   // Apply the transformation.   for (int i = k+1; i < m; i++) {   work[i] = 0.0;   }   for (int j = k+1; j < n; j++) {   for (int i = k+1; i < m; i++) {   work[i] += e[j]*A[i][j];   }   }   for (int j = k+1; j < n; j++) {   double t = -e[j]/e[k+1];   for (int i = k+1; i < m; i++) {   A[i][j] += t*work[i];   }   }   }   if (wantv) {   // Place the transformation in V for subsequent   // back multiplication.   for (int i = k+1; i < n; i++) {   V[i][k] = e[i];   }   }   }   }   // Set up the final bidiagonal matrix or order p.   int p = Math.min(n,m+1);   if (nct < n) {   s[nct] = A[nct][nct];   }   if (m < p) {   s[p-1] = 0.0;   }   if (nrt+1 < p) {   e[nrt] = A[nrt][p-1];   }   e[p-1] = 0.0;   // If required, generate U.   if (wantu) {   for (int j = nct; j < nu; j++) {   for (int i = 0; i < m; i++) {   U[i][j] = 0.0;   }   U[j][j] = 1.0;   }   for (int k = nct-1; k >= 0; k--) {   if (s[k] != 0.0) {   for (int j = k+1; j < nu; j++) {   double t = 0;   for (int i = k; i < m; i++) {   t += U[i][k]*U[i][j];   }   t = -t/U[k][k];   for (int i = k; i < m; i++) {   U[i][j] += t*U[i][k];   }   }   for (int i = k; i < m; i++ ) {   U[i][k] = -U[i][k];   }   U[k][k] = 1.0 + U[k][k];   for (int i = 0; i < k-1; i++) {   U[i][k] = 0.0;   }   } else {   for (int i = 0; i < m; i++) {   U[i][k] = 0.0;   }   U[k][k] = 1.0;   }   }   }   // If required, generate V.   if (wantv) {   for (int k = n-1; k >= 0; k--) {   if ((k < nrt) & (e[k] != 0.0)) {   for (int j = k+1; j < nu; j++) {   double t = 0;   for (int i = k+1; i < n; i++) {   t += V[i][k]*V[i][j];   }   t = -t/V[k+1][k];   for (int i = k+1; i < n; i++) {   V[i][j] += t*V[i][k];   }   }   }   for (int i = 0; i < n; i++) {   V[i][k] = 0.0;   }   V[k][k] = 1.0;   }   }   // Main iteration loop for the singular values.   int pp = p-1;   int iter = 0;   double eps = Math.pow(2.0,-52.0);   double tiny = Math.pow(2.0,-966.0);   while (p > 0) {   int k,kase;   // Here is where a test for too many iterations would go.   // This section of the program inspects for   // negligible elements in the s and e arrays. On   // completion the variables kase and k are set as follows.   // kase = 1 if s(p) and e[k-1] are negligible and k
= -1; k--) {   if (k == -1) {   break;   }   if (Math.abs(e[k]) <=   tiny + eps*(Math.abs(s[k]) + Math.abs(s[k+1]))) {   e[k] = 0.0;   break;   }   }   if (k == p-2) {   kase = 4;   } else {   int ks;   for (ks = p-1; ks >= k; ks--) {   if (ks == k) {   break;   }   double t = (ks != p ? Math.abs(e[ks]) : 0.) +   (ks != k+1 ? Math.abs(e[ks-1]) : 0.);   if (Math.abs(s[ks]) <= tiny + eps*t) {   s[ks] = 0.0;   break;   }   }   if (ks == k) {   kase = 3;   } else if (ks == p-1) {   kase = 1;   } else {   kase = 2;   k = ks;   }   }   k++;   // Perform the task indicated by kase.   switch (kase) {   // Deflate negligible s(p).   case 1: {   double f = e[p-2];   e[p-2] = 0.0;   for (int j = p-2; j >= k; j--) {   double t = Math.hypot(s[j],f);   double cs = s[j]/t;   double sn = f/t;   s[j] = t;   if (j != k) {   f = -sn*e[j-1];   e[j-1] = cs*e[j-1];   }   if (wantv) {   for (int i = 0; i < n; i++) {   t = cs*V[i][j] + sn*V[i][p-1];   V[i][p-1] = -sn*V[i][j] + cs*V[i][p-1];   V[i][j] = t;   }   }   }   }   break;   // Split at negligible s(k).   case 2: {   double f = e[k-1];   e[k-1] = 0.0;   for (int j = k; j < p; j++) {   double t = Math.hypot(s[j],f);   double cs = s[j]/t;   double sn = f/t;   s[j] = t;   f = -sn*e[j];   e[j] = cs*e[j];   if (wantu) {   for (int i = 0; i < m; i++) {   t = cs*U[i][j] + sn*U[i][k-1];   U[i][k-1] = -sn*U[i][j] + cs*U[i][k-1];   U[i][j] = t;   }   }   }   }   break;   // Perform one qr step.   case 3: {   // Calculate the shift.     double scale = Math.max(Math.max(Math.max(Math.max(   Math.abs(s[p-1]),Math.abs(s[p-2])),Math.abs(e[p-2])),   Math.abs(s[k])),Math.abs(e[k]));   double sp = s[p-1]/scale;   double spm1 = s[p-2]/scale;   double epm1 = e[p-2]/scale;   double sk = s[k]/scale;   double ek = e[k]/scale;   double b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;   double c = (sp*epm1)*(sp*epm1);   double shift = 0.0;   if ((b != 0.0) | (c != 0.0)) {   shift = Math.sqrt(b*b + c);   if (b < 0.0) {   shift = -shift;   }   shift = c/(b + shift);   }   double f = (sk + sp)*(sk - sp) + shift;   double g = sk*ek;     // Chase zeros.     for (int j = k; j < p-1; j++) {   double t = Math.hypot(f,g);   double cs = f/t;   double sn = g/t;   if (j != k) {   e[j-1] = t;   }   f = cs*s[j] + sn*e[j];   e[j] = cs*e[j] - sn*s[j];   g = sn*s[j+1];   s[j+1] = cs*s[j+1];   if (wantv) {   for (int i = 0; i < n; i++) {   t = cs*V[i][j] + sn*V[i][j+1];   V[i][j+1] = -sn*V[i][j] + cs*V[i][j+1];   V[i][j] = t;   }   }   t = Math.hypot(f,g);   cs = f/t;   sn = g/t;   s[j] = t;   f = cs*e[j] + sn*s[j+1];   s[j+1] = -sn*e[j] + cs*s[j+1];   g = sn*e[j+1];   e[j+1] = cs*e[j+1];   if (wantu && (j < m-1)) {   for (int i = 0; i < m; i++) {   t = cs*U[i][j] + sn*U[i][j+1];   U[i][j+1] = -sn*U[i][j] + cs*U[i][j+1];   U[i][j] = t;   }   }   }   e[p-2] = f;   iter = iter + 1;   }   break;   // Convergence.   case 4: {   // Make the singular values positive.     if (s[k] <= 0.0) {   s[k] = (s[k] < 0.0 ? -s[k] : 0.0);   if (wantv) {   for (int i = 0; i <= pp; i++) {   V[i][k] = -V[i][k];   }   }   }     // Order the singular values.     while (k < pp) {   if (s[k] >= s[k+1]) {   break;   }   double t = s[k];   s[k] = s[k+1];   s[k+1] = t;   if (wantv && (k < n-1)) {   for (int i = 0; i < n; i++) {   t = V[i][k+1]; V[i][k+1] = V[i][k]; V[i][k] = t;   }   }   if (wantu && (k < m-1)) {   for (int i = 0; i < m; i++) {   t = U[i][k+1]; U[i][k+1] = U[i][k]; U[i][k] = t;   }   }   k++;   }   iter = 0;   p--;   }   break;   }   }   }  /* ------------------------   Public Methods   * ------------------------ */   /** Return the left singular vectors   @return U   */   public Matrix getU () {   return new Matrix(U,m,Math.min(m+1,n));   }   /** Return the right singular vectors   @return V   */   public Matrix getV () {   return new Matrix(V,n,n);   }   /** Return the one-dimensional array of singular values   @return diagonal of S.   */   public double[] getSingularValues () {   return s;   }   /** Return the diagonal matrix of singular values   @return S   */   public Matrix getS () {   Matrix X = new Matrix(n,n);   double[][] S = X.getArray();   for (int i = 0; i < n; i++) {   for (int j = 0; j < n; j++) {   S[i][j] = 0.0;   }   S[i][i] = this.s[i];   }   return X;   }   /** Two norm   @return max(S)   */   public double norm2 () {   return s[0];   }   /** Two norm condition number   @return max(S)/min(S)   */   public double cond () {   return s[0]/s[Math.min(m,n)-1];   }   /** Effective numerical matrix rank   @return Number of nonnegligible singular values.   */   public int rank () {   double eps = Math.pow(2.0,-52.0);   double tol = Math.max(m,n)*s[0]*eps;   int r = 0;   for (int i = 0; i < s.length; i++) {   if (s[i] > tol) {   r++;   }   }   return r;   }  }