package agents.Jama; import agents.Jama.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 Structure to access U, S and V. @param Arg Rectangular matrix */ 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] = Maths.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] = Maths.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 = Maths.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 = Maths.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 = Maths.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 = Maths.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; } private static final long serialVersionUID = 1; }