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#ifndef JAMA_SVD_H |
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#define JAMA_SVD_H |
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|
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#include "math/DynamicRectMatrix.hpp" |
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|
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#include <algorithm> |
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// for min(), max() below |
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#include <cmath> |
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// for abs() below |
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|
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using namespace OpenMD; |
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using namespace std; |
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|
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namespace JAMA |
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{ |
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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, sigma(k) = S(k,k), are ordered so that |
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sigma(0) >= sigma(1) >= ... >= sigma(n-1). |
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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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<p> |
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(Adapted from JAMA, a Java Matrix Library, developed by jointly |
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by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama). |
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*/ |
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template <class Real> |
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class SVD |
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{ |
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|
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DynamicRectMatrix<Real> U, V; |
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DynamicVector<Real> s; |
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int m, n; |
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|
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public: |
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|
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|
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SVD (const DynamicRectMatrix<Real> &Arg) { |
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m = Arg.getNRow(); |
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n = Arg.getNCol(); |
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int nu = min(m,n); |
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s = DynamicVector<Real>(min(m+1,n)); |
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U = DynamicRectMatrix<Real>(m, nu, Real(0)); |
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V = DynamicRectMatrix<Real>(n,n); |
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DynamicVector<Real> e(n); |
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DynamicVector<Real> work(m); |
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DynamicRectMatrix<Real> A(Arg); |
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|
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int wantu = 1; /* boolean */ |
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int wantv = 1; /* boolean */ |
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int i=0, j=0, k=0; |
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|
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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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|
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int nct = min(m-1,n); |
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int nrt = max(0,min(n-2,m)); |
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|
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for (k = 0; k < max(nct,nrt); k++) { |
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if (k < nct) { |
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|
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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) = hypot(s(k),A(i,k)); |
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} |
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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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} |
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for (i = k; i < m; i++) { |
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A(i,k) /= s(k); |
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} |
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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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|
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// Apply the transformation. |
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|
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Real t(0.0); |
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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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} |
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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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} |
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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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|
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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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|
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// Place the transformation in U for subsequent back |
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// multiplication. |
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|
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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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} |
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if (k < nrt) { |
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|
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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) = hypot(e(k),e(i)); |
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} |
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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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} |
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for (i = k+1; i < n; i++) { |
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e(i) /= e(k); |
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} |
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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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|
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// Apply the transformation. |
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|
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for (i = k+1; i < m; i++) { |
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work(i) = 0.0; |
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} |
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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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} |
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} |
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for (j = k+1; j < n; j++) { |
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Real 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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} |
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if (wantv) { |
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|
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// Place the transformation in V for subsequent |
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// back multiplication. |
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|
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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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} |
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|
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// Set up the final bidiagonal matrix or order p. |
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|
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int p = min(n,m+1); |
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if (nct < n) { |
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s(nct) = A(nct,nct); |
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} |
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if (m < p) { |
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s(p-1) = 0.0; |
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} |
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if (nrt+1 < p) { |
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e(nrt) = A(nrt,p-1); |
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} |
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e(p-1) = 0.0; |
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|
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// If required, generate U. |
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|
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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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} |
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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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Real t(0.0); |
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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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} |
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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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} |
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for (i = k; i < m; i++ ) { |
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U(i,k) = -U(i,k); |
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} |
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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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} |
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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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} |
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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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|
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// If required, generate V. |
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|
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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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Real t(0.0); |
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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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} |
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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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} |
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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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} |
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V(k,k) = 1.0; |
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} |
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} |
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|
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// Main iteration loop for the singular values. |
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|
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int pp = p-1; |
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int iter = 0; |
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Real eps(pow(2.0,-52.0)); |
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while (p > 0) { |
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int k=0; |
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int kase=0; |
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|
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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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|
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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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|
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for (k = p-2; k >= -1; k--) { |
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if (k == -1) { |
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break; |
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} |
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if (abs(e(k)) <= eps*(abs(s(k)) + 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) { |
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break; |
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} |
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Real t( (ks != p ? abs(e(ks)) : 0.) + |
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(ks != k+1 ? abs(e(ks-1)) : 0.)); |
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if (abs(s(ks)) <= 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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|
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// Perform the task indicated by kase. |
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|
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switch (kase) { |
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|
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// Deflate negligible s(p). |
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|
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case 1: { |
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Real 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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Real t( hypot(s(j),f)); |
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Real cs(s(j)/t); |
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Real 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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} |
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break; |
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|
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// Split at negligible s(k). |
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|
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case 2: { |
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Real 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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Real t(hypot(s(j),f)); |
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Real cs( s(j)/t); |
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Real 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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} |
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break; |
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|
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// Perform one qr step. |
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|
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case 3: { |
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|
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// Calculate the shift. |
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|
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Real scale = max(max(max(max( |
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abs(s(p-1)),abs(s(p-2))),abs(e(p-2))), |
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abs(s(k))),abs(e(k))); |
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Real sp = s(p-1)/scale; |
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Real spm1 = s(p-2)/scale; |
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Real epm1 = e(p-2)/scale; |
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Real sk = s(k)/scale; |
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Real ek = e(k)/scale; |
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Real b = ((spm1 + sp)*(spm1 - sp) + epm1*epm1)/2.0; |
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Real c = (sp*epm1)*(sp*epm1); |
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Real shift = 0.0; |
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if ((b != 0.0) || (c != 0.0)) { |
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shift = sqrt(b*b + c); |
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if (b < 0.0) { |
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shift = -shift; |
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} |
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shift = c/(b + shift); |
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} |
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Real f = (sk + sp)*(sk - sp) + shift; |
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Real g = sk*ek; |
| 373 |
|
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// Chase zeros. |
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|
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for (j = k; j < p-1; j++) { |
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Real t = hypot(f,g); |
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Real cs = f/t; |
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Real sn = g/t; |
| 380 |
if (j != k) { |
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e(j-1) = t; |
| 382 |
} |
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f = cs*s(j) + sn*e(j); |
| 384 |
e(j) = cs*e(j) - sn*s(j); |
| 385 |
g = sn*s(j+1); |
| 386 |
s(j+1) = cs*s(j+1); |
| 387 |
if (wantv) { |
| 388 |
for (i = 0; i < n; i++) { |
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t = cs*V(i,j) + sn*V(i,j+1); |
| 390 |
V(i,j+1) = -sn*V(i,j) + cs*V(i,j+1); |
| 391 |
V(i,j) = t; |
| 392 |
} |
| 393 |
} |
| 394 |
t = hypot(f,g); |
| 395 |
cs = f/t; |
| 396 |
sn = g/t; |
| 397 |
s(j) = t; |
| 398 |
f = cs*e(j) + sn*s(j+1); |
| 399 |
s(j+1) = -sn*e(j) + cs*s(j+1); |
| 400 |
g = sn*e(j+1); |
| 401 |
e(j+1) = cs*e(j+1); |
| 402 |
if (wantu && (j < m-1)) { |
| 403 |
for (i = 0; i < m; i++) { |
| 404 |
t = cs*U(i,j) + sn*U(i,j+1); |
| 405 |
U(i,j+1) = -sn*U(i,j) + cs*U(i,j+1); |
| 406 |
U(i,j) = t; |
| 407 |
} |
| 408 |
} |
| 409 |
} |
| 410 |
e(p-2) = f; |
| 411 |
iter = iter + 1; |
| 412 |
} |
| 413 |
break; |
| 414 |
|
| 415 |
// Convergence. |
| 416 |
|
| 417 |
case 4: { |
| 418 |
|
| 419 |
// Make the singular values positive. |
| 420 |
|
| 421 |
if (s(k) <= 0.0) { |
| 422 |
s(k) = (s(k) < 0.0 ? -s(k) : 0.0); |
| 423 |
if (wantv) { |
| 424 |
for (i = 0; i <= pp; i++) { |
| 425 |
V(i,k) = -V(i,k); |
| 426 |
} |
| 427 |
} |
| 428 |
} |
| 429 |
|
| 430 |
// Order the singular values. |
| 431 |
|
| 432 |
while (k < pp) { |
| 433 |
if (s(k) >= s(k+1)) { |
| 434 |
break; |
| 435 |
} |
| 436 |
Real t = s(k); |
| 437 |
s(k) = s(k+1); |
| 438 |
s(k+1) = t; |
| 439 |
if (wantv && (k < n-1)) { |
| 440 |
for (i = 0; i < n; i++) { |
| 441 |
t = V(i,k+1); V(i,k+1) = V(i,k); V(i,k) = t; |
| 442 |
} |
| 443 |
} |
| 444 |
if (wantu && (k < m-1)) { |
| 445 |
for (i = 0; i < m; i++) { |
| 446 |
t = U(i,k+1); U(i,k+1) = U(i,k); U(i,k) = t; |
| 447 |
} |
| 448 |
} |
| 449 |
k++; |
| 450 |
} |
| 451 |
iter = 0; |
| 452 |
p--; |
| 453 |
} |
| 454 |
break; |
| 455 |
} |
| 456 |
} |
| 457 |
} |
| 458 |
|
| 459 |
|
| 460 |
void getU (DynamicRectMatrix<Real> &A) { |
| 461 |
|
| 462 |
int minm = min(m+1,n); |
| 463 |
|
| 464 |
A = DynamicRectMatrix<Real>(m, minm); |
| 465 |
|
| 466 |
for (int i=0; i<m; i++) |
| 467 |
for (int j=0; j<minm; j++) |
| 468 |
A(i,j) = U(i,j); |
| 469 |
} |
| 470 |
|
| 471 |
/* Return the right singular vectors */ |
| 472 |
void getV (DynamicRectMatrix<Real> &A) { |
| 473 |
A = V; |
| 474 |
} |
| 475 |
|
| 476 |
/** Return the one-dimensional array of singular values */ |
| 477 |
void getSingularValues (DynamicVector<Real> &x) { |
| 478 |
x = s; |
| 479 |
} |
| 480 |
|
| 481 |
/** Return the diagonal matrix of singular values |
| 482 |
@return S |
| 483 |
*/ |
| 484 |
void getS (DynamicRectMatrix<Real> &A) { |
| 485 |
A = DynamicRectMatrix<Real>(n,n); |
| 486 |
for (int i = 0; i < n; i++) { |
| 487 |
for (int j = 0; j < n; j++) { |
| 488 |
A(i,j) = 0.0; |
| 489 |
} |
| 490 |
A(i,i) = s(i); |
| 491 |
} |
| 492 |
} |
| 493 |
|
| 494 |
/** Two norm (max(S)) */ |
| 495 |
Real norm2 () { |
| 496 |
return s(0); |
| 497 |
} |
| 498 |
|
| 499 |
/** Two norm of condition number (max(S)/min(S)) */ |
| 500 |
Real cond () { |
| 501 |
return s(0)/s(min(m,n)-1); |
| 502 |
} |
| 503 |
|
| 504 |
/** Effective numerical matrix rank |
| 505 |
@return Number of nonnegligible singular values. |
| 506 |
*/ |
| 507 |
int rank () { |
| 508 |
Real eps = pow(2.0,-52.0); |
| 509 |
Real tol = max(m,n)*s(0)*eps; |
| 510 |
int r = 0; |
| 511 |
for (int i = 0; i < s.dim(); i++) { |
| 512 |
if (s(i) > tol) { |
| 513 |
r++; |
| 514 |
} |
| 515 |
} |
| 516 |
return r; |
| 517 |
} |
| 518 |
}; |
| 519 |
} |
| 520 |
#endif |
| 521 |
// JAMA_SVD_H |
