package jhplot.math;
/**
* QR Decomposition.
*
* For an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n
* orthogonal matrix Q and an n-by-n upper triangular matrix R so that A = Q*R.
*
* The QR decompostion always exists, even if the matrix does not have full
* rank, so the constructor will never fail. The primary use of the QR
* decomposition is in the least squares solution of nonsquare systems of
* simultaneous linear equations. This will fail if isFullRank() returns false.
*/
public class QRDecomposition {
/*
* ------------------------ Class variables ------------------------
*/
/**
* Array for internal storage of decomposition.
*
* @serial internal array storage.
*/
private double[][] QR;
/**
* Row and column dimensions.
*
* @serial column dimension.
* @serial row dimension.
*/
private int m, n;
/**
* Array for internal storage of diagonal of R.
*
* @serial diagonal of R.
*/
private double[] Rdiag;
/*
* ------------------------ 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(double[][] A) {
// Initialize.
QR = A;
m = A.length;
n = A[0].length;
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 = Mathfun.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;
}
}
/*
* ------------------------ Public Methods ------------------------
*/
/**
* Is the matrix full rank?
*
* @return true if R, and hence A, has full rank.
*/
public boolean isFullRank() {
for (int j = 0; j < n; j++) {
if (Rdiag[j] == 0)
return false;
}
return true;
}
/**
* Return the Householder vectors
*
* @return Lower trapezoidal matrix whose columns define the reflections
*/
public double[][] getH() {
double[][] X = new double[m][n];
for (int i = 0; i < m; i++) {
for (int j = 0; j < n; j++) {
if (i >= j) {
X[i][j] = QR[i][j];
} else {
X[i][j] = 0.0;
}
}
}
return X;
}
/**
* Return the upper triangular factor
*
* @return R
*/
public double[][] getR() {
double[][] X = new double[n][n];
for (int i = 0; i < n; i++) {
for (int j = 0; j < n; j++) {
if (i < j) {
X[i][j] = QR[i][j];
} else if (i == j) {
X[i][j] = Rdiag[i];
} else {
X[i][j] = 0.0;
}
}
}
return X;
}
/**
* Generate and return the (economy-sized) orthogonal factor
*
* @return Q
*/
public double[][] getQ() {
double[][] X = new double[m][n];
for (int k = n - 1; k >= 0; k--) {
for (int i = 0; i < m; i++) {
X[i][k] = 0.0;
}
X[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] * X[i][j];
}
s = -s / QR[k][k];
for (int i = k; i < m; i++) {
X[i][j] = X[i][j] + s * QR[i][k];
}
}
}
}
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 double[][] solve(double[][] B) {
if (B.length != m) {
throw new IllegalArgumentException("Matrix row dimensions must agree.");
}
if (!this.isFullRank()) {
throw new RuntimeException("Matrix is rank deficient.");
}
// Copy right hand side
int nx = B[0].length;
double[][] X = B;
// 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 DoubleArray.getSubMatrixRangeCopy(X, 0, n - 1, 0, nx - 1);
}
}