/// <summary> /// QR Decomposition, computed by Householder reflections. /// </summary> /// <param name="A">Structure to access R and the Householder vectors and compute Q.</param> public QRDecomposition(Matrix A) { // Initialize. QR = A.GetArrayCopy(); m = A.Rows; n = A.Cols; Rdiag = new double[n]; // Main loop. for (int k = 0; k < n; k++) { // Compute 2-norm of k-th column without under/overflow. double nrm = 0; for (int i = k; i < m; i++) { nrm = EncogMath.Hypot(nrm, QR[i][k]); } if (nrm != 0.0) { // Form k-th Householder vector. if (QR[k][k] < 0) { nrm = -nrm; } for (int i = k; i < m; i++) { QR[i][k] /= nrm; } QR[k][k] += 1.0; // Apply transformation to remaining columns. for (int j = k + 1; j < n; j++) { double s = 0.0; for (int i = k; i < m; i++) { s += QR[i][k] * QR[i][j]; } s = -s / QR[k][k]; for (int i = k; i < m; i++) { QR[i][j] += s * QR[i][k]; } } } Rdiag[k] = -nrm; } }
/// <summary> /// Least squares solution of A*X = B /// </summary> /// <param name="B">A Matrix with as many rows as A and any number of columns.</param> /// <returns>that minimizes the two norm of Q*R*X-B.</returns> public Matrix Solve(Matrix B) { if (B.Rows != m) { throw new MatrixError( "Matrix row dimensions must agree."); } if (!this.IsFullRank() ) { throw new MatrixError("Matrix is rank deficient."); } // Copy right hand side int nx = B.Cols; double[][] X = B.GetArrayCopy(); // Compute Y = transpose(Q)*B for (int k = 0; k < n; k++) { for (int j = 0; j < nx; j++) { double s = 0.0; for (int i = k; i < m; i++) { s += QR[i][k] * X[i][j]; } s = -s / QR[k][k]; for (int i = k; i < m; i++) { X[i][j] += s * QR[i][k]; } } } // Solve R*X = Y; for (int k = n - 1; k >= 0; k--) { for (int j = 0; j < nx; j++) { X[k][j] /= Rdiag[k]; } for (int i = 0; i < k; i++) { for (int j = 0; j < nx; j++) { X[i][j] -= X[k][j] * QR[i][k]; } } } return (new Matrix(X).GetMatrix(0, n - 1, 0, nx - 1)); }
/// <summary> /// Construct the singular value decomposition /// </summary> /// <param name="Arg">Rectangular matrix</param> public SingularValueDecomposition(Matrix Arg) { // Derived from LINPACK code. // Initialize. double[][] A = Arg.GetArrayCopy(); m = Arg.Rows; n = Arg.Cols; /* * Apparently the failing cases are only a proper subset of (m<n), so * let's not throw error. Correct fix to come later? if (m<n) { throw * new IllegalArgumentException("Jama SVD only works for m >= n"); } */ int nu = Math.Min(m, n); s = new double[Math.Min(m + 1, n)]; umatrix = EngineArray.AllocateDouble2D(m, nu); vmatrix = EngineArray.AllocateDouble2D(n, n); var e = new double[n]; var work = new double[m]; bool wantu = true; bool 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] = EncogMath.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++) { umatrix[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] = EncogMath.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++) { vmatrix[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++) { umatrix[i][j] = 0.0; } umatrix[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 += umatrix[i][k]*umatrix[i][j]; } t = -t/umatrix[k][k]; for (int i = k; i < m; i++) { umatrix[i][j] += t*umatrix[i][k]; } } for (int i = k; i < m; i++) { umatrix[i][k] = -umatrix[i][k]; } umatrix[k][k] = 1.0 + umatrix[k][k]; for (int i = 0; i < k - 1; i++) { umatrix[i][k] = 0.0; } } else { for (int i = 0; i < m; i++) { umatrix[i][k] = 0.0; } umatrix[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 += vmatrix[i][k]*vmatrix[i][j]; } t = -t/vmatrix[k + 1][k]; for (int i = k + 1; i < n; i++) { vmatrix[i][j] += t*vmatrix[i][k]; } } } for (int i = 0; i < n; i++) { vmatrix[i][k] = 0.0; } vmatrix[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<p // kase = 2 if s(k) is negligible and k<p // kase = 3 if e[k-1] is negligible, k<p, and // s(k), ..., s(p) are not negligible (qr step). // kase = 4 if e(p-1) is negligible (convergence). for (k = p - 2; 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]) : 0.0) + (ks != k + 1 ? Math.Abs(e[ks - 1]) : 0.0); 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 = EncogMath.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*vmatrix[i][j] + sn*vmatrix[i][p - 1]; vmatrix[i][p - 1] = -sn*vmatrix[i][j] + cs*vmatrix[i][p - 1]; vmatrix[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 = EncogMath.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*umatrix[i][j] + sn*umatrix[i][k - 1]; umatrix[i][k - 1] = -sn*umatrix[i][j] + cs*umatrix[i][k - 1]; umatrix[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 = EncogMath.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*vmatrix[i][j] + sn*vmatrix[i][j + 1]; vmatrix[i][j + 1] = -sn*vmatrix[i][j] + cs*vmatrix[i][j + 1]; vmatrix[i][j] = t; } } t = EncogMath.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*umatrix[i][j] + sn*umatrix[i][j + 1]; umatrix[i][j + 1] = -sn*umatrix[i][j] + cs*umatrix[i][j + 1]; umatrix[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++) { vmatrix[i][k] = -vmatrix[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 = vmatrix[i][k + 1]; vmatrix[i][k + 1] = vmatrix[i][k]; vmatrix[i][k] = t; } } if (wantu && (k < m - 1)) { for (int i = 0; i < m; i++) { t = umatrix[i][k + 1]; umatrix[i][k + 1] = umatrix[i][k]; umatrix[i][k] = t; } } k++; } iter = 0; p--; } break; } } }
/// <summary> /// LU Decomposition /// </summary> /// <param name="A">Rectangular matrix</param> public LUDecomposition(Matrix A) { // Use a "left-looking", dot-product, Crout/Doolittle algorithm. LU = A.GetArrayCopy(); m = A.Rows; n = A.Cols; piv = new int[m]; for (int i = 0; i < m; i++) { piv[i] = i; } pivsign = 1; double[] LUrowi; double[] LUcolj = new double[m]; // Outer loop. for (int j = 0; j < n; j++) { // Make a copy of the j-th column to localize references. for (int i = 0; i < m; i++) { LUcolj[i] = LU[i][j]; } // Apply previous transformations. for (int i = 0; i < m; i++) { LUrowi = LU[i]; // Most of the time is spent in the following dot product. int kmax = Math.Min(i, j); double s = 0.0; for (int k = 0; k < kmax; k++) { s += LUrowi[k] * LUcolj[k]; } LUrowi[j] = LUcolj[i] -= s; } // Find pivot and exchange if necessary. int p = j; for (int i = j + 1; i < m; i++) { if (Math.Abs(LUcolj[i]) > Math.Abs(LUcolj[p])) { p = i; } } if (p != j) { for (int k = 0; k < n; k++) { double t = LU[p][k]; LU[p][k] = LU[j][k]; LU[j][k] = t; } int temp = piv[p]; piv[p] = piv[j]; piv[j] = temp; pivsign = -pivsign; } // Compute multipliers. if (j < m & LU[j][j] != 0.0) { for (int i = j + 1; i < m; i++) { LU[i][j] /= LU[j][j]; } } } }
/// <summary> /// Solve A*X = B. /// </summary> /// <param name="b">A Matrix with as many rows as A and any number of columns.</param> /// <returns>X so that L*L'*X = b.</returns> public Matrix Solve(Matrix b) { if (b.Rows != n) { throw new MatrixError( "Matrix row dimensions must agree."); } if (!isspd) { throw new MatrixError( "Matrix is not symmetric positive definite."); } // Copy right hand side. double[][] x = b.GetArrayCopy(); int nx = b.Cols; // Solve L*Y = B; for (int k = 0; k < n; k++) { for (int j = 0; j < nx; j++) { for (int i = 0; i < k; i++) { x[k][j] -= x[i][j] * l[k][i]; } x[k][j] /= l[k][k]; } } // Solve L'*X = Y; for (int k = n - 1; k >= 0; k--) { for (int j = 0; j < nx; j++) { for (int i = k + 1; i < n; i++) { x[k][j] -= x[i][j] * l[i][k]; } x[k][j] /= l[k][k]; } } return new Matrix(x); }