// ------------------------
// Constructor
// ------------------------
// QR Decomposition, computed by Householder reflections.
// @param A Rectangular matrix
public QRDecomposition (Matrix A)
{
// Initialize.
QR = A.GetArrayCopy();
m = A.GetRowDimension();
n = A.GetColumnDimension();
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 = MathUtils.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;
}
}
// ------------------------
// Constructor
// ------------------------
// LU Decomposition
// @param A Rectangular matrix
public LUDecomposition (Matrix A)
{
// Use a "left-looking", dot-product, Crout/Doolittle algorithm.
LU = A.GetArrayCopy();
m = A.GetRowDimension();
n = A.GetColumnDimension();
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 k1 = piv[p];
piv[p] = piv[j];
piv[j] = k1;
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.GetRowDimension() != n)
{
throw new ArgumentException("Matrix row dimensions must agree.");
}
if (!isspd)
{
throw new Exception("Matrix is not symmetric positive definite.");
}
// Copy right hand side.
double[][] X = B.GetArrayCopy();
int nx = B.GetColumnDimension();
// 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,n,nx);
}
// Least squares solution of A*X = B
// @param B A Matrix with as many rows as A and any number of columns.
// @return X that minimizes the two norm of Q*R*X-B.
// @exception ArgumentException Matrix row dimensions must agree.
// @exception Exception Matrix is rank deficient.
public Matrix Solve (Matrix B)
{
if (B.GetRowDimension() != m)
{
throw new ArgumentException("Matrix row dimensions must agree.");
}
if (!this.IsFullRank())
{
throw new Exception("Matrix is rank deficient.");
}
// Copy right hand side
int nx = B.GetColumnDimension();
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,n,nx).GetMatrix(0,n-1,0,nx-1));
}
// ------------------------
// Constructor
// ------------------------
// Construct the singular value decomposition
// @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),
// so let's not throw error. Correct fix to come later?
// if (m<n) {
// throw new ArgumentException("Jama SVD only works for m >= n"); }
//
int nu = Math.Min(m,n);
s = new double [Math.Min(m+1,n)];
U = new double[m][];
for (int i=0;i<m;i++)
U[i] = new double[nu];
V = new double [n][];
for (int i=0;i<n;i++)
V[i] = new double[n];
double[] e = new double [n];
double[] 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] = MathUtils.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] = MathUtils.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<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 = MathUtils.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 = MathUtils.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 = MathUtils.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 = MathUtils.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;
}
}
}
