/* ------------------------
Constructor
* ------------------------ */
/** QR Decomposition, computed by Householder reflections.
@param A Rectangular matrix
@return Structure to access R and the Householder vectors and compute Q.
*/
public QRDecomposition(CoreMatrix 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 = Maths.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;
}
}
/** Get a subCoreMatrix.
@param r Array of row indices.
@param c Array of column indices.
@return A(r(:),c(:))
@exception Exception SubCoreMatrix indices
*/
public CoreMatrix getMatrix(int[] r, int[] c)
{
CoreMatrix X = new CoreMatrix(r.Length, c.Length);
double[,] B = X.getArray();
try
{
for (int i = 0; i < r.Length; i++)
{
for (int j = 0; j < c.Length; j++)
{
B[i, j] = A[r[i], c[j]];
}
}
}
catch (Exception)
{
throw new Exception("SubCoreMatrix indices");
}
return X;
}
/** 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 IllegalArgumentException Matrix row dimensions must agree.
@exception RuntimeException Matrix is rank deficient.
*/
public CoreMatrix solve(CoreMatrix B)
{
if (B.getRowDimension() != m)
{
throw new Exception("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 CoreMatrix(X, n, nx).getMatrix(0, n - 1, 0, nx - 1));
}
/** Return the upper triangular factor
@return R
*/
public CoreMatrix getR()
{
CoreMatrix X = new CoreMatrix(n, n);
double[,] R = X.getArray();
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
if (i < j)
{
R[i, j] = QR[i, j];
}
else if (i == j)
{
R[i, j] = Rdiag[i];
}
else
{
R[i, j] = 0.0;
}
}
}
return X;
}
/** Return the block diagonal eigenvalue matrix
@return D
*/
public CoreMatrix getD()
{
CoreMatrix X = new CoreMatrix(n, n);
double[,] D = X.getArray();
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
D[i, j] = 0.0;
}
D[i, i] = d[i];
if (e[i] > 0)
{
D[i, i + 1] = e[i];
}
else if (e[i] < 0)
{
D[i, i - 1] = e[i];
}
}
return X;
}
/* ------------------------
Constructor
* ------------------------ */
/** LU Decomposition
@param A Rectangular matrix
@return Structure to access L, U and piv.
*/
public LUDecomposition(CoreMatrix 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 = new double[m];
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++)
{
for (int h = 0; i < m; i++)
LUrowi[h] = LU[i, h];
// 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 ki = 0; ki < n; ki++)
{
double t = LU[p, ki]; LU[p, ki] = LU[j, ki]; LU[j, ki] = t;
}
int k = piv[p]; piv[p] = piv[j]; piv[j] = k;
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];
}
}
}
}
/** Return upper triangular factor
@return U
*/
public CoreMatrix getU()
{
CoreMatrix X = new CoreMatrix(n, n);
double[,] U = X.getArray();
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
if (i <= j)
{
U[i, j] = LU[i, j];
}
else
{
U[i, j] = 0.0;
}
}
}
return X;
}
/* ------------------------
Private Methods
* ------------------------ */
/** Check if size(A) == size(B) **/
private void checkCoreMatrixDimensions(CoreMatrix B)
{
if (B.m != m || B.n != n)
{
throw new Exception("CoreMatrix dimensions must agree.");
}
}
/** CoreMatrix transpose.
@return A'
*/
public CoreMatrix transpose()
{
CoreMatrix X = new CoreMatrix(n, m);
double[,] C = X.getArray();
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
C[j, i] = A[i, j];
}
}
return X;
}
/** Set a subCoreMatrix.
@param i0 Initial row index
@param i1 Final row index
@param c Array of column indices.
@param X A(i0:i1,c(:))
@exception Exception SubCoreMatrix indices
*/
public void setMatrix(int i0, int i1, int[] c, CoreMatrix X)
{
try
{
for (int i = i0; i <= i1; i++)
{
for (int j = 0; j < c.Length; j++)
{
A[i, c[j]] = X.get(i - i0, j);
}
}
}
catch (Exception)
{
throw new Exception("SubCoreMatrix indices");
}
}
/** Set a subCoreMatrix.
@param r Array of row indices.
@param j0 Initial column index
@param j1 Final column index
@param X A(r(:),j0:j1)
@exception Exception SubCoreMatrix indices
*/
public void setMatrix(int[] r, int j0, int j1, CoreMatrix X)
{
try
{
for (int i = 0; i < r.Length; i++)
{
for (int j = j0; j <= j1; j++)
{
A[r[i], j] = X.get(i, j - j0);
}
}
}
catch (Exception)
{
throw new Exception("SubCoreMatrix indices");
}
}
/** Set a subCoreMatrix.
@param r Array of row indices.
@param c Array of column indices.
@param X A(r(:),c(:))
@exception Exception SubCoreMatrix indices
*/
public void setMatrix(int[] r, int[] c, CoreMatrix X)
{
try
{
for (int i = 0; i < r.Length; i++)
{
for (int j = 0; j < c.Length; j++)
{
A[r[i], c[j]] = X.get(i, j);
}
}
}
catch (Exception)
{
throw new Exception("SubCoreMatrix indices");
}
}
/** Set a subCoreMatrix.
@param i0 Initial row index
@param i1 Final row index
@param j0 Initial column index
@param j1 Final column index
@param X A(i0:i1,j0:j1)
@exception Exception SubCoreMatrix indices
*/
public void setMatrix(int i0, int i1, int j0, int j1, CoreMatrix X)
{
try
{
for (int i = i0; i <= i1; i++)
{
for (int j = j0; j <= j1; j++)
{
A[i, j] = X.get(i - i0, j - j0);
}
}
}
catch (Exception)
{
throw new Exception("SubCoreMatrix indices");
}
}
/** Get a subCoreMatrix.
@param r Array of row indices.
@param i0 Initial column index
@param i1 Final column index
@return A(r(:),j0:j1)
@exception Exception SubCoreMatrix indices
*/
public CoreMatrix getMatrix(int[] r, int j0, int j1)
{
CoreMatrix X = new CoreMatrix(r.Length, j1 - j0 + 1);
double[,] B = X.getArray();
try
{
for (int i = 0; i < r.Length; i++)
{
for (int j = j0; j <= j1; j++)
{
B[i, j - j0] = A[r[i], j];
}
}
}
catch (Exception)
{
throw new Exception("SubCoreMatrix indices");
}
return X;
}
/** Get a subCoreMatrix.
@param i0 Initial row index
@param i1 Final row index
@param c Array of column indices.
@return A(i0:i1,c(:))
@exception Exception SubCoreMatrix indices
*/
public CoreMatrix getMatrix(int i0, int i1, int[] c)
{
CoreMatrix X = new CoreMatrix(i1 - i0 + 1, c.Length);
double[,] B = X.getArray();
try
{
for (int i = i0; i <= i1; i++)
{
for (int j = 0; j < c.Length; j++)
{
B[i - i0, j] = A[i, c[j]];
}
}
}
catch (Exception)
{
throw new Exception("SubCoreMatrix indices");
}
return X;
}
/** Solve X*A = B, which is also A'*X' = B'
@param B right hand side
@return solution if A is square, least squares solution otherwise.
*/
public CoreMatrix solveTranspose(CoreMatrix B)
{
return transpose().solve(B.transpose());
}
/** Generate identity CoreMatrix
@param m Number of rows.
@param n Number of colums.
@return An m-by-n CoreMatrix with ones on the diagonal and zeros elsewhere.
*/
public static CoreMatrix identity(int m, int n)
{
CoreMatrix A = new CoreMatrix(m, n);
double[,] X = A.getArray();
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
X[i, j] = (i == j ? 1.0 : 0.0);
}
}
return A;
}
/** Unary minus
@return -A
*/
public CoreMatrix uminus()
{
CoreMatrix X = new CoreMatrix(m, n);
double[,] C = X.getArray();
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
C[i, j] = -A[i, j];
}
}
return X;
}
/** Return lower triangular factor
@return L
*/
public CoreMatrix getL()
{
CoreMatrix X = new CoreMatrix(m, n);
double[,] L = X.getArray();
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
if (i > j)
{
L[i, j] = LU[i, j];
}
else if (i == j)
{
L[i, j] = 1.0;
}
else
{
L[i, j] = 0.0;
}
}
}
return X;
}
/** Element-by-element multiplication in place, A = A.*B
@param B another CoreMatrix
@return A.*B
*/
public CoreMatrix arrayTimesEquals(CoreMatrix B)
{
checkCoreMatrixDimensions(B);
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
A[i, j] = A[i, j] * B.A[i, j];
}
}
return this;
}
/** Solve A*X = B
@param B A Matrix with as many rows as A and any number of columns.
@return X so that L*U*X = B(piv,:)
@exception IllegalArgumentException Matrix row dimensions must agree.
@exception RuntimeException Matrix is singular.
*/
public CoreMatrix solve(CoreMatrix B)
{
if (B.getRowDimension() != m)
{
throw new Exception("Matrix row dimensions must agree.");
}
if (!this.isNonsingular())
{
throw new Exception("Matrix is singular.");
}
// Copy right hand side with pivoting
int nx = B.getColumnDimension();
CoreMatrix Xmat = B.getMatrix(piv, 0, nx - 1);
double[,] X = Xmat.getArray();
// Solve L*Y = B(piv,:)
for (int k = 0; k < n; k++)
{
for (int i = k + 1; i < n; i++)
{
for (int j = 0; j < nx; j++)
{
X[i, j] -= X[k, j] * LU[i, k];
}
}
}
// Solve U*X = Y;
for (int k = n - 1; k >= 0; k--)
{
for (int j = 0; j < nx; j++)
{
X[k, j] /= LU[k, k];
}
for (int i = 0; i < k; i++)
{
for (int j = 0; j < nx; j++)
{
X[i, j] -= X[k, j] * LU[i, k];
}
}
}
return Xmat;
}
/** Element-by-element left division, C = A.\B
@param B another CoreMatrix
@return A.\B
*/
public CoreMatrix arrayLeftDivide(CoreMatrix B)
{
checkCoreMatrixDimensions(B);
CoreMatrix X = new CoreMatrix(m, n);
double[,] C = X.getArray();
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
C[i, j] = B.A[i, j] / A[i, j];
}
}
return X;
}
/* ------------------------
Constructor
* ------------------------ */
/** Check for symmetry, then construct the eigenvalue decomposition
@param A Square matrix
@return Structure to access D and V.
*/
public EigenvalueDecomposition(CoreMatrix Arg)
{
double[,] A = Arg.getArray();
n = Arg.getColumnDimension();
V = new double[n, n];
d = new double[n];
e = new double[n];
issymmetric = true;
for (int j = 0; (j < n) & issymmetric; j++)
{
for (int i = 0; (i < n) & issymmetric; i++)
{
issymmetric = (A[i, j] == A[j, i]);
}
}
if (issymmetric)
{
for (int i = 0; i < n; i++)
{
for (int j = 0; j < n; j++)
{
V[i, j] = A[i, j];
}
}
// Tridiagonalize.
tred2();
// Diagonalize.
tql2();
}
else
{
H = new double[n, n];
ort = new double[n];
for (int j = 0; j < n; j++)
{
for (int i = 0; i < n; i++)
{
H[i, j] = A[i, j];
}
}
// Reduce to Hessenberg form.
orthes();
// Reduce Hessenberg to real Schur form.
hqr2();
}
}
/** Element-by-element left division in place, A = A.\B
@param B another CoreMatrix
@return A.\B
*/
public CoreMatrix arrayLeftDivideEquals(CoreMatrix B)
{
checkCoreMatrixDimensions(B);
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
A[i, j] = B.A[i, j] / A[i, j];
}
}
return this;
}
/** Return the Householder vectors
@return Lower trapezoidal matrix whose columns define the reflections
*/
public CoreMatrix getH()
{
CoreMatrix X = new CoreMatrix(m, n);
double[,] H = X.getArray();
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
if (i >= j)
{
H[i, j] = QR[i, j];
}
else
{
H[i, j] = 0.0;
}
}
}
return X;
}
/** Multiply a CoreMatrix by a scalar, C = s*A
@param s scalar
@return s*A
*/
public CoreMatrix times(double s)
{
CoreMatrix X = new CoreMatrix(m, n);
double[,] C = X.getArray();
for (int i = 0; i < m; i++)
{
for (int j = 0; j < n; j++)
{
C[i, j] = s * A[i, j];
}
}
return X;
}
/** Generate and return the (economy-sized) orthogonal factor
@return Q
*/
public CoreMatrix getQ()
{
CoreMatrix X = new CoreMatrix(m, n);
double[,] Q = X.getArray();
for (int k = n - 1; k >= 0; k--)
{
for (int i = 0; i < m; i++)
{
Q[i, k] = 0.0;
}
Q[k, k] = 1.0;
for (int j = k; j < n; j++)
{
if (QR[k, k] != 0)
{
double s = 0.0;
for (int i = k; i < m; i++)
{
s += QR[i, k] * Q[i, j];
}
s = -s / QR[k, k];
for (int i = k; i < m; i++)
{
Q[i, j] += s * QR[i, k];
}
}
}
}
return X;
}
/** Linear algebraic CoreMatrix multiplication, A * B
@param B another CoreMatrix
@return CoreMatrix product, A * B
@exception IllegalArgumentException CoreMatrix inner dimensions must agree.
*/
public CoreMatrix times(CoreMatrix B)
{
if (B.m != n)
{
throw new Exception("CoreMatrix inner dimensions must agree.");
}
CoreMatrix X = new CoreMatrix(m, B.n);
double[,] C = X.getArray();
double[] Bcolj = new double[n];
for (int j = 0; j < B.n; j++)
{
for (int k = 0; k < n; k++)
{
Bcolj[k] = B.A[k, j];
}
for (int i = 0; i < m; i++)
{
double[] Arowi = new double[n];
for (int z = 0; z < n; z++)
Arowi[z] = A[i, z];
double s = 0;
for (int k = 0; k < n; k++)
{
s += Arowi[k] * Bcolj[k];
}
C[i, j] = s;
}
}
return X;
}
/** Solve A*X = B
@param B right hand side
@return solution if A is square, least squares solution otherwise
*/
public CoreMatrix solve(CoreMatrix B)
{
return (m == n ? (new LUDecomposition(this)).solve(B) :
(new QRDecomposition(this)).solve(B));
}
/** Get a subCoreMatrix.
@param i0 Initial row index
@param i1 Final row index
@param j0 Initial column index
@param j1 Final column index
@return A(i0:i1,j0:j1)
@exception Exception SubCoreMatrix indices
*/
public CoreMatrix getMatrix(int i0, int i1, int j0, int j1)
{
CoreMatrix X = new CoreMatrix(i1 - i0 + 1, j1 - j0 + 1);
double[,] B = X.getArray();
try
{
for (int i = i0; i <= i1; i++)
{
for (int j = j0; j <= j1; j++)
{
B[i - i0, j - j0] = A[i, j];
}
}
}
catch (Exception)
{
throw new Exception("SubCoreMatrix indices");
}
return X;
}
