org.ejml.alg.dense.decomposition.qr.QRDecompositionHouseholder Maven / Gradle / Ivy
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/*
* Copyright (c) 2009-2011, Peter Abeles. All Rights Reserved.
*
* This file is part of Efficient Java Matrix Library (EJML).
*
* EJML is free software: you can redistribute it and/or modify
* it under the terms of the GNU Lesser General Public License as
* published by the Free Software Foundation, either version 3
* of the License, or (at your option) any later version.
*
* EJML is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU Lesser General Public License for more details.
*
* You should have received a copy of the GNU Lesser General Public
* License along with EJML. If not, see
* This variation of QR decomposition uses reflections to compute the Q matrix.
* Each reflection uses a householder operations, hence its name. To provide a meaningful solution
* the original matrix must have full rank. This is intended for processing of small to medium matrices.
*
*
* Both Q and R are stored in the same m by n matrix. Q is not stored directly, instead the u from
* Qk=(I-γ*u*uT) is stored. Decomposition requires about 2n*m2-2m2/3 flops.
*
*
*
* See the QR reflections algorithm described in:
* David S. Watkins, "Fundamentals of Matrix Computations" 2nd Edition, 2002
*
*
*
* For the most part this is a straight forward implementation. To improve performance on large matrices a column is writen to an array and the order
* of some of the loops has been changed. This will degrade performance noticeably on small matrices. Since
* it is unlikely that the QR decomposition would be a bottle neck when small matrices are involved only
* one implementation is provided.
*
*
* @author Peter Abeles
*/
public class QRDecompositionHouseholder implements QRDecomposition {
/**
* Where the Q and R matrices are stored. R is stored in the
* upper triangular portion and Q on the lower bit. Lower columns
* are where u is stored. Q_k = (I - gamma_k*u_k*u_k^T).
*/
protected DenseMatrix64F QR;
// used internally to store temporary data
protected double u[],v[];
// dimension of the decomposed matrices
protected int numCols; // this is 'n'
protected int numRows; // this is 'm'
protected int minLength;
protected double dataQR[];
// the computed gamma for Q_k matrix
protected double gammas[];
// local variables
protected double gamma;
protected double tau;
// did it encounter an error?
protected boolean error;
public void setExpectedMaxSize( int numRows , int numCols ) {
error = false;
this.numCols = numCols;
this.numRows = numRows;
minLength = Math.min(numRows,numCols);
int maxLength = Math.max(numRows,numCols);
if( QR == null ) {
QR = new DenseMatrix64F(numRows,numCols);
u = new double[ maxLength ];
v = new double[ maxLength ];
gammas = new double[ minLength ];
} else {
QR.reshape(numRows,numCols,false);
}
dataQR = QR.data;
if( u.length < maxLength ) {
u = new double[ maxLength ];
v = new double[ maxLength ];
}
if( gammas.length < minLength ) {
gammas = new double[ minLength ];
}
}
/**
* Returns a single matrix which contains the combined values of Q and R. This
* is possible since Q is symmetric and R is upper triangular.
*
* @return The combined Q R matrix.
*/
public DenseMatrix64F getQR() {
return QR;
}
/**
* Computes the Q matrix from the imformation stored in the QR matrix. This
* operation requires about 4(m2n-mn2+n3/3) flops.
*
* @param Q The orthogonal Q matrix.
*/
@Override
public DenseMatrix64F getQ( DenseMatrix64F Q , boolean compact ) {
if( compact ) {
if( Q == null ) {
Q = CommonOps.identity(numRows,minLength);
} else {
if( Q.numRows != numRows || Q.numCols != minLength ) {
throw new IllegalArgumentException("Unexpected matrix dimension.");
} else {
CommonOps.setIdentity(Q);
}
}
} else {
if( Q == null ) {
Q = CommonOps.identity(numRows);
} else {
if( Q.numRows != numRows || Q.numCols != numRows ) {
throw new IllegalArgumentException("Unexpected matrix dimension.");
} else {
CommonOps.setIdentity(Q);
}
}
}
for( int j = minLength-1; j >= 0; j-- ) {
u[j] = 1;
for( int i = j+1; i < numRows; i++ ) {
u[i] = QR.get(i,j);
}
QrHelperFunctions.rank1UpdateMultR(Q,u,gammas[j],j,j,numRows,v);
}
return Q;
}
/**
* Returns an upper triangular matrix which is the R in the QR decomposition.
*
* @param R An upper triangular matrix.
* @param compact
*/
@Override
public DenseMatrix64F getR(DenseMatrix64F R, boolean compact) {
if( R == null ) {
if( compact ) {
R = new DenseMatrix64F(minLength,numCols);
} else
R = new DenseMatrix64F(numRows,numCols);
} else {
if( compact ) {
if( R.numCols != numCols || R.numRows != minLength )
throw new IllegalArgumentException("Unexpected dimensions");
} else {
if( R.numCols != numCols || R.numRows != numRows )
throw new IllegalArgumentException("Unexpected dimensions");
}
for( int i = 0; i < R.numRows; i++ ) {
int min = Math.min(i,R.numCols);
for( int j = 0; j < min; j++ ) {
R.set(i,j,0);
}
}
}
for( int i = 0; i < minLength; i++ ) {
for( int j = i; j < numCols; j++ ) {
double val = QR.get(i,j);
R.set(i,j,val);
}
}
return R;
}
/**
*
* In order to decompose the matrix 'A' it must have full rank. 'A' is a 'm' by 'n' matrix.
* It requires about 2n*m2-2m2/3 flops.
*
*
*
* The matrix provided here can be of different
* dimension than the one specified in the constructor. It just has to be smaller than or equal
* to it.
*
*/
@Override
public boolean decompose( DenseMatrix64F A ) {
commonSetup(A);
for( int j = 0; j < minLength; j++ ) {
householder(j);
updateA(j);
}
return !error;
}
@Override
public boolean inputModified() {
return false;
}
/**
*
* Computes the householder vector "u" for the first column of submatrix j. Note this is
* a specialized householder for this problem. There is some protection against
* overflow and underflow.
*
*
* Q = I - γuuT
*
*
* This function finds the values of 'u' and 'γ'.
*
*
* @param j Which submatrix to work off of.
*/
protected void householder( int j )
{
// find the element with the largest absolute value in the column and make a copy
int index = j+j*numCols;
double max = 0;
for( int i = j; i < numRows; i++ ) {
double d = u[i] = dataQR[index];
// absolute value of d
if( d < 0 ) d = -d;
if( max < d ) {
max = d;
}
index += numCols;
}
if( max == 0.0 ) {
gamma = 0;
error = true;
} else {
// compute the norm2 of the matrix, with each element
// normalized by the max value to avoid overflow problems
tau = 0;
for( int i = j; i < numRows; i++ ) {
u[i] /= max;
double d = u[i];
tau += d*d;
}
tau = Math.sqrt(tau);
if( u[j] < 0 )
tau = -tau;
double u_0 = u[j] + tau;
gamma = u_0/tau;
for( int i = j+1; i < numRows; i++ ) {
u[i] /= u_0;
}
u[j] = 1;
tau *= max;
}
gammas[j] = gamma;
}
/**
*
* Takes the results from the householder computation and updates the 'A' matrix.
*
* A = (I - γ*u*uT)A
*
*
* @param w The submatrix.
*/
protected void updateA( int w )
{
// much of the code below is equivalent to the rank1Update function
// however, since τ has already been computed there is no need to
// recompute it, saving a few multiplication operations
// for( int i = w+1; i < numCols; i++ ) {
// double val = 0;
//
// for( int k = w; k < numRows; k++ ) {
// val += u[k]*dataQR[k*numCols +i];
// }
// v[i] = gamma*val;
// }
// This is functionally the same as the above code but the order has been changed
// to avoid jumping the cpu cache
for( int i = w+1; i < numCols; i++ ) {
v[i] = u[w]*dataQR[w*numCols +i];
}
for( int k = w+1; k < numRows; k++ ) {
int indexQR = k*numCols+w+1;
for( int i = w+1; i < numCols; i++ ) {
// v[i] += u[k]*dataQR[k*numCols +i];
v[i] += u[k]*dataQR[indexQR++];
}
}
for( int i = w+1; i < numCols; i++ ) {
v[i] *= gamma;
}
// end of reordered code
for( int i = w; i < numRows; i++ ) {
double valU = u[i];
int indexQR = i*numCols+w+1;
for( int j = w+1; j < numCols; j++ ) {
// dataQR[i*numCols+j] -= valU*v[j];
dataQR[indexQR++] -= valU*v[j];
}
}
if( w < numCols ) {
dataQR[w+w*numCols] = -tau;
}
// save the Q matrix in the lower portion of QR
for( int i = w+1; i < numRows; i++ ) {
dataQR[w+i*numCols] = u[i];
}
}
/**
* This function performs sanity check on the input for decompose and sets up the QR matrix.
*
* @param A
*/
protected void commonSetup(DenseMatrix64F A) {
setExpectedMaxSize(A.numRows,A.numCols);
QR.set(A);
}
public double[] getGammas() {
return gammas;
}
}