488 lines
15 KiB
C++
488 lines
15 KiB
C++
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// Authors: Unknown. Please, if you are the author of this file, or if you
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// know who are the authors of this file, let us know, so we can give the
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// adequate credits and/or get the adequate authorizations.
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#include "numerics.h"
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#include <cmath>
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#include <vector>
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#include <limits>
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#include <algorithm>
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namespace libNumerics {
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const flnum MinLM::DEFAULT_RELATIVE_TOL = 1E-3;
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const flnum MinLM::DEFAULT_LAMBDA_INIT = 1E-3;
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const flnum MinLM::DEFAULT_LAMBDA_FACT = 10.0;
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const flnum MinLM::EPSILON_KERNEL = 1E-9;
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inline flnum ABS(flnum x)
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{ return (x >= 0)? x: -x; }
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/// Resolution by LU decomposition with pivot.
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bool solveLU(const matrix<flnum>& A, const vector<flnum>& B, vector<flnum>& X)
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{
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X = B;
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return solveLU(A, X);
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}
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/// Replace X by A^{-1}X, by LU solver.
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bool solveLU(matrix<flnum> A, vector<flnum>& X)
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{
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assert(A.nrow() == A.ncol());
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int n = A.nrow();
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vector<flnum> rowscale(n); // Implicit scaling of each row
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std::vector<int> permut(n,0); // Permutation of rows
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// Get the implicit scaling information of each row
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for(int i=0; i< n; i++) {
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flnum max = 0.0;
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for(int j=0; j< n; j++) {
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flnum tmp = ABS(A(i,j));
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if (tmp> max)
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max = tmp;
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}
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if(max == 0.0)
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return false;
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rowscale(i) = 1.0/max;
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}
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// Perform the decomposition
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for(int k=0; k < n; k++) {
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// Search for largest pivot element
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flnum max = rowscale(k)*ABS(A(k,k));
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int imax = k;
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for(int i=k+1; i < n; i++) {
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flnum tmp = rowscale(i)*ABS(A(i,k));
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if(tmp > max) {
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max = tmp;
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imax = i;
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}
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}
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if(max == 0.0)
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return false;
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// Interchange rows if needed
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if(k != imax) {
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A.swapRows(k, imax);
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rowscale(imax) = rowscale(k); // Scale of row k no longer needed
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}
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permut[k] = imax; // permut(k) was not initialized before
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flnum Akk = 1/A(k,k);
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for(int i=k+1; i < n; i++) {
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flnum tmp = A(i,k) *= Akk; // Divide by pivot
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for (int j=k+1;j < n; j++) // Reduce the row
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A(i,j) -= tmp*A(k,j);
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}
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}
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// Forward substitution
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for (int k=0; k < n; k++) {
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flnum sum = X(permut[k]);
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X(permut[k]) = X(k);
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for(int j = 0; j < k; j++)
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sum -= A(k,j)*X(j);
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X(k) = sum;
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}
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// Backward substitution
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for(int k=n-1; k >= 0; k--) {
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flnum sum = X(k);
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for(int j=k+1; j < n; j++)
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sum -= A(k,j)*X(j);
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X(k) = sum/A(k,k);
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}
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return true;
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}
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/// Decompose A into U diag(W) V^T with U(m,n) and V(n,n) having orthonormal
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/// vectors.
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SVD::SVD(const matrix<flnum>& A)
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: m_U(A), m_V(A.ncol(),A.ncol()), m_W(A.ncol())
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{
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compute();
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sort();
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}
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/// SVD computation. Initial matrix stored in m_U as input.
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void SVD::compute()
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{
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const flnum EPSILON = std::numeric_limits<flnum>::epsilon();
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const int SVD_MAX_ITS = 30;
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int rows = m_U.nrow();
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int cols = m_U.ncol();
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flnum g, scale, anorm;
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vector<flnum> RV1(cols);
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// Householder reduction to bidiagonal form:
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anorm = g = scale = 0.0;
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for (int i=0; i< cols; i++) {
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int l = i + 1;
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RV1(i) = scale*g;
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g = scale = 0.0;
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if(i< rows) {
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for (int k=i; k< rows; k++)
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scale += ABS(m_U(k,i));
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if (scale != 0.0) {
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flnum invScale=1.0/scale, s=0.0;
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for (int k=i; k< rows; k++) {
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m_U(k,i) *= invScale;
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s += m_U(k,i) * m_U(k,i);
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}
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flnum f = m_U(i,i);
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g = - withSignOf(std::sqrt(s),f);
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flnum h = 1.0 / (f*g - s);
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m_U(i,i) = f - g;
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for (int j=l; j< cols; j++) {
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s = 0.0;
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for (int k=i; k< rows; k++)
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s += m_U(k,i) * m_U(k,j);
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f = s * h;
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for (int k=i; k< rows; k++)
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m_U(k,j) += f * m_U(k,i);
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}
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for (int k=i; k< rows; k++)
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m_U(k,i) *= scale;
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}
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}
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m_W(i) = scale * g;
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g = scale = 0.0;
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if ( i< rows && i< cols-1 ) {
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for (int k=l; k< cols; k++)
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scale += ABS(m_U(i,k));
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if (scale != 0.0) {
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flnum invScale=1.0/scale, s=0.0;
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for (int k=l; k< cols; k++) {
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m_U(i,k) *= invScale;
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s += m_U(i,k) * m_U(i,k);
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}
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flnum f = m_U(i,l);
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g = - withSignOf(std::sqrt(s),f);
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flnum h = 1.0 / (f*g - s);
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m_U(i,l) = f - g;
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for (int k=l; k< cols; k++)
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RV1(k) = m_U(i,k) * h;
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for (int j=l; j< rows; j++) {
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s = 0.0;
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for (int k=l; k< cols; k++)
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s += m_U(j,k) * m_U(i,k);
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for (int k=l; k< cols; k++)
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m_U(j,k) += s * RV1(k);
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}
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for (int k=l; k< cols; k++)
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m_U(i,k) *= scale;
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}
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}
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anorm = std::max(anorm, ABS(m_W(i)) + ABS(RV1(i)) );
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}
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// Accumulation of right-hand transformations:
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m_V(cols-1,cols-1) = 1.0;
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for (int i= cols-2; i>=0; i--) {
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m_V(i,i) = 1.0;
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int l = i+1;
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g = RV1(l);
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if (g != 0.0) {
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flnum invgUil = 1.0 / (m_U(i,l)*g);
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for (int j=l; j< cols; j++)
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m_V(j,i) = m_U(i,j) * invgUil;
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for (int j=l; j< cols; j++){
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flnum s = 0.0;
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for (int k=l; k< cols; k++)
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s += m_U(i,k) * m_V(k,j);
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for (int k=l; k< cols; k++)
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m_V(k,j) += s * m_V(k,i);
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}
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}
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for (int j=l; j< cols; j++)
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m_V(i,j) = m_V(j,i) = 0.0;
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}
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// Accumulation of left-hand transformations:
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for (int i=std::min(rows,cols)-1; i>=0; i--) {
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int l = i+1;
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g = m_W(i);
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for (int j=l; j< cols; j++)
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m_U(i,j) = 0.0;
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if (g != 0.0) {
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g = 1.0 / g;
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flnum invUii = 1.0 / m_U(i,i);
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for (int j=l; j< cols; j++) {
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flnum s = 0.0;
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for (int k=l; k< rows; k++)
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s += m_U(k,i) * m_U(k,j);
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flnum f = (s * invUii) * g;
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for (int k=i; k< rows; k++)
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m_U(k,j) += f * m_U(k,i);
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}
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for (int j=i; j< rows; j++)
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m_U(j,i) *= g;
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} else
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for (int j=i; j< rows; j++)
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m_U(j,i) = 0.0;
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m_U(i,i) = m_U(i,i) + 1.0;
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}
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// Diagonalization of the bidiagonal form:
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for (int k=cols-1; k>=0; k--) { // Loop over singular values
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for (int its=1; its<=SVD_MAX_ITS; its++) {
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bool flag = false;
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int l = k;
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int nm = k-1;
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while(l>0 && ABS(RV1(l)) > EPSILON*anorm) { // Test for splitting
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if(ABS(m_W(nm)) <= EPSILON*anorm) {
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flag = true;
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break;
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}
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l--;
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nm--;
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}
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if (flag) { // Cancellation of RV1(l), if l > 0
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flnum c=0.0, s=1.0;
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for (int i=l; i< k+1; i++) {
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flnum f = s * RV1(i);
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RV1(i) = c * RV1(i);
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if (ABS(f)<=EPSILON*anorm)
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break;
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g = m_W(i);
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flnum h = SVD::hypot(f,g);
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m_W(i) = h;
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h = 1.0 / h;
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c = g * h;
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s = - f * h;
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for (int j=0; j< rows; j++)
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rotate(m_U(j,nm),m_U(j,i), c,s);
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}
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}
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flnum z = m_W(k);
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if (l==k) { // Convergence of the singular value
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if (z< 0.0) { // Singular value is made nonnegative
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m_W(k) = -z;
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for (int j=0; j< cols; j++)
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m_V(j,k) = - m_V(j,k);
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}
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break;
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}
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// Exception if convergence to the singular value not reached:
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if(its==SVD_MAX_ITS) throw SvdConvergenceError();
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flnum x = m_W(l); // Get QR shift value from bottom 2x2 minor
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nm = k-1;
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flnum y = m_W(nm);
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g = RV1(nm);
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flnum h = RV1(k);
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flnum f = ( (y-z)*(y+z) + (g-h)*(g+h) ) / ( 2.0*h*y );
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g = SVD::hypot(f,1.0);
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f = ( (x-z)*(x+z) + h*(y/(f+withSignOf(g,f)) - h) ) / x;
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// Next QR transformation (through Givens reflections)
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flnum c=1.0, s=1.0;
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for (int j=l; j<=nm; j++) {
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int i = j+1;
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g = RV1(i);
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y = m_W(i);
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h = s * g;
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g = c * g;
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z = SVD::hypot(f,h);
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RV1(j) = z;
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z = 1.0 / z;
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c = f * z;
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s = h * z;
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f = x*c + g*s;
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g = g*c - x*s;
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h = y * s;
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y *= c;
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for(int jj=0; jj < cols; jj++)
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rotate(m_V(jj,j),m_V(jj,i), c,s);
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z = SVD::hypot(f,h);
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m_W(j) = z;
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if (z!=0.0) { // Rotation can be arbitrary if z = 0.0
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z = 1.0 / z;
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c = f * z;
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s = h * z;
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}
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f = c*g + s*y;
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x = c*y - s*g;
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for(int jj=0; jj < rows; jj++)
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rotate(m_U(jj,j),m_U(jj,i), c,s);
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}
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RV1(l) = 0.0;
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RV1(k) = f;
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m_W(k) = x;
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}
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}
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}
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/// Recompose from SVD. This should be the initial matrix.
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matrix<flnum> SVD::compose() const
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{
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return m_U * m_W.diag() * m_V.t();
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}
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flnum SVD::withSignOf(flnum a, flnum b)
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{ return b >= 0 ? (a >= 0 ? a : -a) : (a >= 0 ? -a : a); }
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/// Replace hypot of math.h by robust numeric implementation.
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flnum SVD::hypot(flnum a, flnum b)
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{
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a = ABS(a);
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b = ABS(b);
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if(a > b) {
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b /= a;
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return a*std::sqrt(1.0 + b*b);
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} else if(b) {
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a /= b;
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return b*std::sqrt(1.0 + a*a);
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}
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return 0.0;
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}
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/// Utility function used while computing SVD.
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void SVD::rotate(flnum& a, flnum& b, flnum c, flnum s)
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{
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flnum d = a;
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a = +d*c +b*s;
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b = -d*s +b*c;
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}
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class SVDElement {
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|
public:
|
||
|
|
SVDElement(const vector<flnum>& W, int i)
|
||
|
|
: m_val(W(i)), m_i(i) {}
|
||
|
|
bool operator<(const SVDElement& e) const
|
||
|
|
{ return (m_val>e.m_val); }
|
||
|
|
|
||
|
|
flnum m_val;
|
||
|
|
int m_i;
|
||
|
|
};
|
||
|
|
|
||
|
|
/// Sort SVD by decreasing order of singular value.
|
||
|
|
void SVD::sort()
|
||
|
|
{
|
||
|
|
std::vector<SVDElement> vec;
|
||
|
|
for(int i=0; i < m_U.ncol(); i++)
|
||
|
|
vec.push_back( SVDElement(m_W, i) );
|
||
|
|
std::sort(vec.begin(), vec.end());
|
||
|
|
// Apply permutation
|
||
|
|
for(int i=m_U.ncol()-1; i >=0; i--)
|
||
|
|
if(vec[i].m_i != i) { // Find cycle of i
|
||
|
|
const vector<flnum> colU = m_U.col(i);
|
||
|
|
const vector<flnum> colV = m_V.col(i);
|
||
|
|
const flnum w = m_W(i);
|
||
|
|
int j = i;
|
||
|
|
while(vec[j].m_i != i) {
|
||
|
|
m_U.paste(0,j, m_U.col(vec[j].m_i));
|
||
|
|
m_V.paste(0,j, m_V.col(vec[j].m_i));
|
||
|
|
m_W(j) = m_W(vec[j].m_i);
|
||
|
|
std::swap(j,vec[j].m_i);
|
||
|
|
}
|
||
|
|
vec[j].m_i = j;
|
||
|
|
m_U.paste(0,j, colU);
|
||
|
|
m_V.paste(0,j, colV);
|
||
|
|
m_W(j) = w;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
/// Constructor.
|
||
|
|
MinLM::MinLM()
|
||
|
|
: iterations(0), relativeTol(DEFAULT_RELATIVE_TOL),
|
||
|
|
lambdaInit(DEFAULT_LAMBDA_INIT), lambdaFact(DEFAULT_LAMBDA_FACT)
|
||
|
|
{}
|
||
|
|
|
||
|
|
/// In equation JtJ X = B, remove columns of J close to 0, so that JtJ can be
|
||
|
|
/// invertible
|
||
|
|
void MinLM::compress(matrix<flnum>& JtJ, vector<flnum>& B)
|
||
|
|
{
|
||
|
|
flnum max=0;
|
||
|
|
for(int i=0; i < JtJ.nrow(); i++)
|
||
|
|
if(JtJ(i,i) > max)
|
||
|
|
max = JtJ(i,i);
|
||
|
|
max *= EPSILON_KERNEL;
|
||
|
|
m_nullCols.clear();
|
||
|
|
for(int i=0; i < JtJ.nrow(); i++)
|
||
|
|
if(JtJ(i,i) <= max)
|
||
|
|
m_nullCols.push_back(i);
|
||
|
|
if( m_nullCols.empty() )
|
||
|
|
return;
|
||
|
|
int n=(int)m_nullCols.size();
|
||
|
|
matrix<flnum> JtJ2(JtJ.nrow()-m_nullCols.size(),
|
||
|
|
JtJ.ncol()-m_nullCols.size());
|
||
|
|
vector<flnum> B2(B.nrow()-(int)m_nullCols.size());
|
||
|
|
for(int i=0,i2=0; i < JtJ.nrow(); i++) {
|
||
|
|
if(i-i2 < n && m_nullCols[i-i2]==i)
|
||
|
|
continue;
|
||
|
|
for(int j=0,j2=0; j < JtJ.ncol(); j++) {
|
||
|
|
if(j-j2 < n && m_nullCols[j-j2]==j)
|
||
|
|
continue;
|
||
|
|
JtJ2(i2,j2) = JtJ(i,j);
|
||
|
|
j2++;
|
||
|
|
}
|
||
|
|
B2(i2) = B(i);
|
||
|
|
i2++;
|
||
|
|
}
|
||
|
|
swap(JtJ,JtJ2);
|
||
|
|
swap(B,B2);
|
||
|
|
}
|
||
|
|
|
||
|
|
/// Insert 0 in rows of B that were removed by \c compress()
|
||
|
|
void MinLM::uncompress(vector<flnum>& B)
|
||
|
|
{
|
||
|
|
if(m_nullCols.empty())
|
||
|
|
return;
|
||
|
|
int n=(int)m_nullCols.size();
|
||
|
|
vector<flnum> B2(B.nrow()+(int)m_nullCols.size());
|
||
|
|
for(int i=0,i2=0; i2 < B2.nrow(); i2++)
|
||
|
|
if(i2-i < n && m_nullCols[i2-i]==i2)
|
||
|
|
B2(i2)=0;
|
||
|
|
else
|
||
|
|
B2(i2) = B(i++);
|
||
|
|
swap(B,B2);
|
||
|
|
}
|
||
|
|
|
||
|
|
/// Perform minimization.
|
||
|
|
/// \a targetRMSE is the root mean square error aimed at.
|
||
|
|
/// Return the reached RMSE. Since the class does not know the dimension, the
|
||
|
|
/// real RMSE should be this value multiplied by sqrt(dim). For example, for 2-D
|
||
|
|
/// points this would be sqrt(2) times the returned value.
|
||
|
|
flnum MinLM::minimize(vector<flnum>& P, const vector<flnum>& yData,
|
||
|
|
flnum targetRMSE, int maxIters)
|
||
|
|
{
|
||
|
|
flnum errorMax = targetRMSE*targetRMSE*yData.nrow();
|
||
|
|
vector<flnum> yModel( yData.nrow() );
|
||
|
|
modelData(P, yModel);
|
||
|
|
vector<flnum> E( yData-yModel );
|
||
|
|
flnum error = E.qnorm();
|
||
|
|
matrix<flnum> J( yData.nrow(), P.nrow() );
|
||
|
|
modelJacobian(P, J);
|
||
|
|
matrix<flnum> Jt = J.t();
|
||
|
|
matrix<flnum> JtJ = Jt*J;
|
||
|
|
vector<flnum> B = Jt*E;
|
||
|
|
compress(JtJ, B);
|
||
|
|
|
||
|
|
flnum lambda = lambdaInit;
|
||
|
|
for(iterations=0; iterations < maxIters && error > errorMax; iterations++) {
|
||
|
|
matrix<flnum> H(JtJ);
|
||
|
|
for(int i = 0; i < H.nrow(); i++)
|
||
|
|
H(i,i) *= 1+lambda;
|
||
|
|
vector<flnum> dP( P.nrow() );
|
||
|
|
solveLU(H, B, dP);
|
||
|
|
uncompress(dP);
|
||
|
|
vector<flnum> tryP = P + dP;
|
||
|
|
modelData(tryP, yModel);
|
||
|
|
E = yData - yModel;
|
||
|
|
flnum tryError = E.qnorm();
|
||
|
|
if(ABS(tryError-error) <= relativeTol*error)
|
||
|
|
break;
|
||
|
|
if(tryError > error)
|
||
|
|
lambda *= lambdaFact;
|
||
|
|
else {
|
||
|
|
lambda /= lambdaFact;
|
||
|
|
error = tryError;
|
||
|
|
P = tryP;
|
||
|
|
modelJacobian(P, J);
|
||
|
|
Jt = J.t();
|
||
|
|
JtJ = Jt*J;
|
||
|
|
B = Jt*E;
|
||
|
|
compress(JtJ, B);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
return sqrt(error/yData.nrow());
|
||
|
|
}
|
||
|
|
|
||
|
|
} // namespace libNumerics
|