468 lines
14 KiB
Java
468 lines
14 KiB
Java
package math.matrix;
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import java.io.Serializable;
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import math.Maths;
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/** Singular Value Decomposition.
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* <p>
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* For an m-by-n matrix A with m ≥ n, the singular value decomposition is
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* an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
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* an n-by-n orthogonal matrix V so that A = U*S*V'.
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* <p>
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* The singular values, σ<sub>k</sub> = S<sub>kk</sub>, are ordered so that
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* σ<sub>1</sub> ≥ σ<sub>2</sub> ≥ ... ≥ σ<sub>n</sub>.
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* <p>
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* The singular value decompostion always exists, so the constructor will
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* never fail. The matrix condition number and the effective numerical
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* rank can be computed from this decomposition.
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*/
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public class SingularValueDecomposition implements Serializable {
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/**
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* UID
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*/
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private static final long serialVersionUID = -4816488865370991490L;
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/**
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* Array for internal storage of U.
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* @serial internal storage of U.
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*/
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private double[][] U;
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/**
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* Array for internal storage of V.
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* @serial internal storage of V.
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*/
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private double[][] V;
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/**
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* Array for internal storage of singular values.
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* @serial internal storage of singular values.
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*/
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private double[] s;
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/**
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* Row dimensions.
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* @serial row dimension.
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*/
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private int m;
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/**
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* Column dimensions.
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* @serial column dimension.
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*/
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private int n;
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/* ------------------------
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Constructor
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* ------------------------ */
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/**
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* Construct the singular value decomposition Structure to access U, S and V.
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* @param Arg rectangular matrix
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*/
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public SingularValueDecomposition(Matrix Arg){
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int i, j, k;
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// Derived from LINPACK code.
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// Initialize.
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double[][] A = Arg.getCopy();
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m = Arg.getM();
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n = Arg.getN();
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/* TODO: Apparently the failing cases are only a proper subset of (m<n),
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* so let's not throw error. Correct fix to come later?
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* if (m<n) {
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* throw new IllegalArgumentException("SVD only works for m >= n"); }
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*/
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int nu = Math.min(m,n);
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s = new double [Math.min(m+1,n)];
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U = new double [m][nu];
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V = new double [n][n];
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double[] e = new double [n];
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double[] work = new double [m];
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boolean wantu = true;
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boolean wantv = true;
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// Reduce A to bidiagonal form, storing the diagonal elements
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// in s and the super-diagonal elements in e.
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int nct = Math.min(m-1,n);
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int nrt = Math.max(0,Math.min(n-2,m));
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for(k=0; k<Math.max(nct,nrt); k++){
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if(k<nct){
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// Compute the transformation for the k-th column and
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// place the k-th diagonal in s[k].
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// Compute 2-norm of k-th column without under/overflow.
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s[k] = 0;
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for(i=k; i<m; i++)
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s[k] = Maths.hypot(s[k],A[i][k]);
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if(s[k] != 0.0){
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if(A[k][k] < 0.0)
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s[k] = -s[k];
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for(i=k; i<m; i++)
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A[i][k] /= s[k];
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A[k][k] += 1.0;
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}
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s[k] = -s[k];
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}
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for(j=k+1; j<n; j++){
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if((k < nct) & (s[k] != 0.0)){
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// Apply the transformation.
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double t = 0;
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for(i=k; i<m; i++)
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t += A[i][k]*A[i][j];
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t = -t/A[k][k];
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for(i=k; i<m; i++)
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A[i][j] += t*A[i][k];
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}
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// Place the k-th row of A into e for the
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// subsequent calculation of the row transformation.
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e[j] = A[k][j];
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}
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if(wantu & (k < nct)){
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// Place the transformation in U for subsequent back
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// multiplication.
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for(i=k; i<m; i++)
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U[i][k] = A[i][k];
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}
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if(k < nrt){
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// Compute the k-th row transformation and place the
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// k-th super-diagonal in e[k].
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// Compute 2-norm without under/overflow.
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e[k] = 0;
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for(i=k+1; i<n; i++)
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e[k] = Maths.hypot(e[k],e[i]);
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if(e[k] != 0.0){
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if(e[k+1] < 0.0)
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e[k] = -e[k];
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for(i=k+1; i<n; i++)
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e[i] /= e[k];
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e[k+1] += 1.0;
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}
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e[k] = -e[k];
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if((k+1 < m) & (e[k] != 0.0)){
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// Apply the transformation.
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for(i=k+1; i<m; i++)
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work[i] = 0.0;
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for(j=k+1; j<n; j++)
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for(i=k+1; i<m; i++)
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work[i] += e[j]*A[i][j];
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for(j=k+1; j<n; j++){
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double t = -e[j]/e[k+1];
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for(i=k+1; i<m; i++)
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A[i][j] += t*work[i];
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}
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}
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if(wantv){
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// Place the transformation in V for subsequent
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// back multiplication.
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for(i=k+1; i<n; i++)
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V[i][k] = e[i];
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}
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}
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}
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// Set up the final bidiagonal matrix or order p.
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int p = Math.min(n,m+1);
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if(nct < n) s[nct] = A[nct][nct];
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if(m < p) s[p-1] = 0.0;
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if(nrt+1 < p) e[nrt] = A[nrt][p-1];
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e[p-1] = 0.0;
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// If required, generate U.
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if(wantu){
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for(j=nct; j<nu; j++){
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for(i=0; i<m; i++)
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U[i][j] = 0.0;
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U[j][j] = 1.0;
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}
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for(k=nct-1; k>=0; k--){
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if(s[k] != 0.0) {
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for(j=k+1; j<nu; j++){
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double t = 0;
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for(i=k; i < m; i++)
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t += U[i][k]*U[i][j];
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t = -t/U[k][k];
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for(i=k; i<m; i++)
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U[i][j] += t*U[i][k];
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}
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for(i=k; i < m; i++)
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U[i][k] = -U[i][k];
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U[k][k] = 1.0 + U[k][k];
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for(i=0; i<k-1; i++)
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U[i][k] = 0.0;
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} else {
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for(i=0; i < m; i++)
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U[i][k] = 0.0;
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U[k][k] = 1.0;
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}
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}
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}
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// If required, generate V.
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if(wantv)
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for(k=n-1; k>=0; k--){
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if((k < nrt) & (e[k] != 0.0))
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for(j=k+1; j<nu; j++){
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double t = 0;
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for(i=k+1; i < n; i++)
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t += V[i][k]*V[i][j];
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t = -t/V[k+1][k];
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for(i=k+1; i<n; i++)
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V[i][j] += t*V[i][k];
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}
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for(i=0; i<n; i++)
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V[i][k] = 0.0;
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V[k][k] = 1.0;
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}
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// Main iteration loop for the singular values.
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int pp = p-1;
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int iter = 0;
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double eps = Math.pow(2.0,-52.0);
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double tiny = Math.pow(2.0,-966.0);
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while(p > 0){
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int kase;
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// Here is where a test for too many iterations would go.
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//
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// This section of the program inspects for
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// negligible elements in the s and e arrays. On
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// completion the variables kase and k are set as follows.
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// kase = 1 if s(p) and e[k-1] are negligible and k<p
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// kase = 2 if s(k) is negligible and k<p
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// kase = 3 if e[k-1] is negligible, k<p, and
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// s(k), ..., s(p) are not negligible (qr step).
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// kase = 4 if e(p-1) is negligible (convergence).
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for(k=p-2; k>=-1; k--){
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if(k == -1) break;
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if(Math.abs(e[k]) <=
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tiny + eps*(Math.abs(s[k]) + Math.abs(s[k+1]))){
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e[k] = 0.0;
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break;
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}
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}
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if(k == p-2)
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kase = 4;
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else {
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int ks;
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for(ks=p-1; ks>=k; ks--){
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if (ks == k) break;
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double t = (ks != p ? Math.abs(e[ks]) : 0.) +
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(ks != k+1 ? Math.abs(e[ks-1]) : 0.);
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if(Math.abs(s[ks]) <= tiny + eps*t){
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s[ks] = 0.0;
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break;
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}
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}
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if(ks == k)
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kase = 3;
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else if(ks == p-1)
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kase = 1;
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else {
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kase = 2;
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k = ks;
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}
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}
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k++;
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// Perform the task indicated by kase.
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switch (kase) {
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// Deflate negligible s(p).
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case 1: {
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double f = e[p-2];
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e[p-2] = 0.0;
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for(j=p-2; j>=k; j--){
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double t = Maths.hypot(s[j],f);
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double cs = s[j]/t;
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double sn = f/t;
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s[j] = t;
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if(j != k){
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f = -sn*e[j-1];
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e[j-1] = cs*e[j-1];
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}
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if(wantv)
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for(i=0; i<n; i++){
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t = cs*V[i][j] + sn*V[i][p-1];
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V[i][p-1] = -sn*V[i][j] + cs*V[i][p-1];
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V[i][j] = t;
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}
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}
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}
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break;
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// Split at negligible s(k).
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case 2: {
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double f = e[k-1];
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e[k-1] = 0.0;
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for(j=k; j<p; j++){
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double t = Maths.hypot(s[j],f);
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double cs = s[j]/t;
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double sn = f/t;
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s[j] = t;
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f = -sn*e[j];
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e[j] = cs*e[j];
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if(wantu)
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for(i=0; i<m; i++){
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t = cs*U[i][j] + sn*U[i][k-1];
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U[i][k-1] = -sn*U[i][j] + cs*U[i][k-1];
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U[i][j] = t;
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}
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}
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}
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break;
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// Perform one qr step.
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case 3: {
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// Calculate the shift.
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double scale = Math.max(Math.max(Math.max(Math.max(
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Math.abs(s[p-1]),Math.abs(s[p-2])),Math.abs(e[p-2])),
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Math.abs(s[k])),Math.abs(e[k]));
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double sp = s[p-1]/scale;
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double spm1 = s[p-2]/scale;
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double epm1 = e[p-2]/scale;
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double sk = s[k]/scale;
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double ek = e[k]/scale;
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double b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0;
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double c = (sp*epm1)*(sp*epm1);
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double shift = 0.0;
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if((b != 0.0) | (c != 0.0)){
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shift = Math.sqrt(b*b + c);
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if(b < 0.0)
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shift = -shift;
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shift = c/(b + shift);
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}
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double f = (sk + sp)*(sk - sp) + shift;
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double g = sk*ek;
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// Chase zeros.
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for(j=k; j<p-1; j++){
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double t = Maths.hypot(f,g);
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double cs = f/t;
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double sn = g/t;
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if(j != k)
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e[j-1] = t;
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f = cs*s[j] + sn*e[j];
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e[j] = cs*e[j] - sn*s[j];
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g = sn*s[j+1];
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s[j+1] = cs*s[j+1];
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if(wantv)
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for(i=0; i<n; i++){
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t = cs*V[i][j] + sn*V[i][j+1];
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V[i][j+1] = -sn*V[i][j] + cs*V[i][j+1];
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V[i][j] = t;
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}
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t = Maths.hypot(f,g);
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cs = f/t;
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sn = g/t;
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s[j] = t;
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f = cs*e[j] + sn*s[j+1];
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s[j+1] = -sn*e[j] + cs*s[j+1];
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g = sn*e[j+1];
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e[j+1] = cs*e[j+1];
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if(wantu && (j < m-1))
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for(i=0; i<m; i++){
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t = cs*U[i][j] + sn*U[i][j+1];
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U[i][j+1] = -sn*U[i][j] + cs*U[i][j+1];
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U[i][j] = t;
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}
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}
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e[p-2] = f;
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iter = iter + 1;
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}
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break;
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// Convergence.
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case 4: {
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// Make the singular values positive.
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if(s[k] <= 0.0){
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s[k] = (s[k] < 0.0 ? -s[k] : 0.0);
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if(wantv)
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for(i=0; i<=pp; i++)
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V[i][k] = -V[i][k];
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}
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// Order the singular values.
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while(k < pp){
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if (s[k] >= s[k+1])
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break;
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double t = s[k];
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s[k] = s[k+1];
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s[k+1] = t;
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if(wantv && (k < n-1)){
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for(i=0; i<n; i++){
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t = V[i][k+1];
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V[i][k+1] = V[i][k];
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V[i][k] = t;
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}
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}
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if(wantu && (k < m-1)){
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for(i=0; i<m; i++){
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t = U[i][k+1];
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U[i][k+1] = U[i][k];
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U[i][k] = t;
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}
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}
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k++;
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}
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iter = 0;
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p--;
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}
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break;
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}
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}
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}
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/**
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* @return the left singular vectors U
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*/
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public Matrix getU(){
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return new Matrix(U);
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}
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/**
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* @return the right singular vectors V
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*/
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public Matrix getV(){
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return new Matrix(V);
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}
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/**
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* @return the one-dimensional array of singular values, diagonal of S.
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*/
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public double[] getSingularValues(){
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return s;
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}
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/**
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* @return the diagonal matrix of singular values S
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*/
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public Matrix getS(){
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int i, j;
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Matrix S = new Matrix(n,n);
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for(i=0; i<n; i++){
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for(j=0; j<n; j++)
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S.set(i,j, 0.0);
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S.set(i,i, this.s[i]);
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}
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return S;
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}
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/**
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* Two norm
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* @return max(S)
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*/
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public double norm2(){
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return s[0];
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}
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/**
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* Two norm condition number
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* @return max(S)/min(S)
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*/
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public double cond(){
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return s[0]/s[Math.min(m,n)-1];
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}
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/**
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* Effective numerical matrix rank
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* @return number of nonnegligible singular values.
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*/
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public int rank(){
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double eps = Math.pow(2.0,-52.0);
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double tol = Math.max(m,n)*s[0]*eps;
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int i, r = 0;
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for(i=0; i<s.length; i++)
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if(s[i] > tol)
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r++;
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return r;
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}
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}
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