#ifndef JAMA_QR_H
#define JAMA_QR_H
#include "tnt_array1d.h"
#include "tnt_array2d.h"
#include "tnt_math_utils.h"
namespace JAMA
{
/**
Classical QR Decompisition:
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 0 (false).
The Q and R factors can be retrived via the getQ() and getR()
methods. Furthermore, a solve() method is provided to find the
least squares solution of Ax=b using the QR factors.
(Adapted from JAMA, a Java Matrix Library, developed by jointly
by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).
*/
template
class QR {
/** Array for internal storage of decomposition.
@serial internal array storage.
*/
TNT::Array2D QR_;
/** Row and column dimensions.
@serial column dimension.
@serial row dimension.
*/
int m, n;
/** Array for internal storage of diagonal of R.
@serial diagonal of R.
*/
TNT::Array1D Rdiag;
public:
/**
Create a QR factorization object for A.
@param A rectangular (m>=n) matrix.
*/
QR(const TNT::Array2D &A) /* constructor */
{
QR_ = A.copy();
m = A.dim1();
n = A.dim2();
Rdiag = TNT::Array1D(n);
int i=0, j=0, k=0;
// Main loop.
for (k = 0; k < n; k++) {
// Compute 2-norm of k-th column without under/overflow.
Real nrm = 0;
for (i = k; i < m; i++) {
nrm = TNT::hypot(nrm,QR_[i][k]);
}
if (nrm != 0.0) {
// Form k-th Householder vector.
if (QR_[k][k] < 0) {
nrm = -nrm;
}
for (i = k; i < m; i++) {
QR_[i][k] /= nrm;
}
QR_[k][k] += 1.0;
// Apply transformation to remaining columns.
for (j = k+1; j < n; j++) {
Real s = 0.0;
for (i = k; i < m; i++) {
s += QR_[i][k]*QR_[i][j];
}
s = -s/QR_[k][k];
for (i = k; i < m; i++) {
QR_[i][j] += s*QR_[i][k];
}
}
}
Rdiag[k] = -nrm;
}
}
/**
Flag to denote the matrix is of full rank.
@return 1 if matrix is full rank, 0 otherwise.
*/
int isFullRank() const
{
for (int j = 0; j < n; j++)
{
if (Rdiag[j] == 0)
return 0;
}
return 1;
}
/**
Retreive the Householder vectors from QR factorization
@returns lower trapezoidal matrix whose columns define the reflections
*/
TNT::Array2D getHouseholder (void) const
{
TNT::Array2D H(m,n);
/* note: H is completely filled in by algorithm, so
initializaiton of H is not necessary.
*/
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 H;
}
/** Return the upper triangular factor, R, of the QR factorization
@return R
*/
TNT::Array2D getR() const
{
TNT::Array2D R(n,n);
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 R;
}
/**
Generate and return the (economy-sized) orthogonal factor
@param Q the (ecnomy-sized) orthogonal factor (Q*R=A).
*/
TNT::Array2D getQ() const
{
int i=0, j=0, k=0;
TNT::Array2D Q(m,n);
for (k = n-1; k >= 0; k--) {
for (i = 0; i < m; i++) {
Q[i][k] = 0.0;
}
Q[k][k] = 1.0;
for (j = k; j < n; j++) {
if (QR_[k][k] != 0) {
Real s = 0.0;
for (i = k; i < m; i++) {
s += QR_[i][k]*Q[i][j];
}
s = -s/QR_[k][k];
for (i = k; i < m; i++) {
Q[i][j] += s*QR_[i][k];
}
}
}
}
return Q;
}
/** Least squares solution of A*x = b
@param B m-length array (vector).
@return x n-length array (vector) that minimizes the two norm of Q*R*X-B.
If B is non-conformant, or if QR.isFullRank() is false,
the routine returns a null (0-length) vector.
*/
TNT::Array1D solve(const TNT::Array1D &b) const
{
if (b.dim1() != m) /* arrays must be conformant */
return TNT::Array1D();
if ( !isFullRank() ) /* matrix is rank deficient */
{
return TNT::Array1D();
}
TNT::Array1D x = b.copy();
// Compute Y = transpose(Q)*b
for (int k = 0; k < n; k++)
{
Real s = 0.0;
for (int i = k; i < m; i++)
{
s += QR_[i][k]*x[i];
}
s = -s/QR_[k][k];
for (int i = k; i < m; i++)
{
x[i] += s*QR_[i][k];
}
}
// Solve R*X = Y;
for (int k = n-1; k >= 0; k--)
{
x[k] /= Rdiag[k];
for (int i = 0; i < k; i++) {
x[i] -= x[k]*QR_[i][k];
}
}
/* return n x nx portion of X */
TNT::Array1D x_(n);
for (int i=0; i solve(const TNT::Array2D &B) const
{
if (B.dim1() != m) /* arrays must be conformant */
return TNT::Array2D(0,0);
if ( !isFullRank() ) /* matrix is rank deficient */
{
return TNT::Array2D(0,0);
}
int nx = B.dim2();
TNT::Array2D X = B.copy();
int i=0, j=0, k=0;
// Compute Y = transpose(Q)*B
for (k = 0; k < n; k++) {
for (j = 0; j < nx; j++) {
Real s = 0.0;
for (i = k; i < m; i++) {
s += QR_[i][k]*X[i][j];
}
s = -s/QR_[k][k];
for (i = k; i < m; i++) {
X[i][j] += s*QR_[i][k];
}
}
}
// Solve R*X = Y;
for (k = n-1; k >= 0; k--) {
for (j = 0; j < nx; j++) {
X[k][j] /= Rdiag[k];
}
for (i = 0; i < k; i++) {
for (j = 0; j < nx; j++) {
X[i][j] -= X[k][j]*QR_[i][k];
}
}
}
/* return n x nx portion of X */
TNT::Array2D X_(n,nx);
for (i=0; i