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A fast and easy to use dense and sparse matrix linear algebra library written in Java.
/*
* Copyright (c) 2009-2018, Peter Abeles. All Rights Reserved.
*
* This file is part of Efficient Java Matrix Library (EJML).
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.ejml.dense.row;
import org.ejml.UtilEjml;
import org.ejml.data.FMatrixRMaj;
import org.ejml.dense.row.linsol.qr.SolveNullSpaceQRP_FDRM;
import org.ejml.dense.row.linsol.qr.SolveNullSpaceQR_FDRM;
import org.ejml.dense.row.linsol.svd.SolveNullSpaceSvd_FDRM;
import org.ejml.interfaces.SolveNullSpace;
import org.ejml.interfaces.decomposition.SingularValueDecomposition_F32;
/**
* Operations related to singular value decomposition.
*
* @author Peter Abeles
*/
public class SingularOps_FDRM {
/**
*
* Adjusts the matrices so that the singular values are in descending order.
*
*
*
* In most implementations of SVD the singular values are automatically arranged in in descending
* order. In EJML this is not the case since it is often not needed and some computations can
* be saved by not doing that.
*
*
* @param U Matrix. Modified.
* @param tranU is U transposed or not.
* @param W Diagonal matrix with singular values. Modified.
* @param V Matrix. Modified.
* @param tranV is V transposed or not.
*/
// TODO the number of copies can probably be reduced here
public static void descendingOrder(FMatrixRMaj U , boolean tranU ,
FMatrixRMaj W ,
FMatrixRMaj V , boolean tranV )
{
int numSingular = Math.min(W.numRows,W.numCols);
checkSvdMatrixSize(U, tranU, W, V, tranV);
for( int i = 0; i < numSingular; i++ ) {
float bigValue=-1;
int bigIndex=-1;
// find the smallest singular value in the submatrix
for( int j = i; j < numSingular; j++ ) {
float v = W.get(j,j);
if( v > bigValue ) {
bigValue = v;
bigIndex = j;
}
}
// only swap if the current index is not the smallest
if( bigIndex == i)
continue;
if( bigIndex == -1 ) {
// there is at least one uncountable singular value. just stop here
break;
}
float tmp = W.get(i,i);
W.set(i,i,bigValue);
W.set(bigIndex,bigIndex,tmp);
if( V != null ) {
swapRowOrCol(V, tranV, i, bigIndex);
}
if( U != null ) {
swapRowOrCol(U, tranU, i, bigIndex);
}
}
}
/**
*
* Similar to {@link #descendingOrder(FMatrixRMaj, boolean, FMatrixRMaj, FMatrixRMaj, boolean)}
* but takes in an array of singular values instead.
*
*
* @param U Matrix. Modified.
* @param tranU is U transposed or not.
* @param singularValues Array of singular values. Modified.
* @param singularLength Number of elements in singularValues array
* @param V Matrix. Modified.
* @param tranV is V transposed or not.
*/
public static void descendingOrder(FMatrixRMaj U , boolean tranU ,
float singularValues[] ,
int singularLength ,
FMatrixRMaj V , boolean tranV )
{
// checkSvdMatrixSize(U, tranU, W, V, tranV);
for( int i = 0; i < singularLength; i++ ) {
float bigValue=-1;
int bigIndex=-1;
// find the smallest singular value in the submatrix
for( int j = i; j < singularLength; j++ ) {
float v = singularValues[j];
if( v > bigValue ) {
bigValue = v;
bigIndex = j;
}
}
// only swap if the current index is not the smallest
if( bigIndex == i)
continue;
if( bigIndex == -1 ) {
// there is at least one uncountable singular value. just stop here
break;
}
float tmp = singularValues[i];
singularValues[i] = bigValue;
singularValues[bigIndex] = tmp;
if( V != null ) {
swapRowOrCol(V, tranV, i, bigIndex);
}
if( U != null ) {
swapRowOrCol(U, tranU, i, bigIndex);
}
}
}
/**
* Checks to see if all the provided matrices are the expected size for an SVD. If an error is encountered
* then an exception is thrown. This automatically handles compact and non-compact formats
*/
public static void checkSvdMatrixSize(FMatrixRMaj U, boolean tranU, FMatrixRMaj W, FMatrixRMaj V, boolean tranV ) {
int numSingular = Math.min(W.numRows,W.numCols);
boolean compact = W.numRows == W.numCols;
if( compact ) {
if( U != null ) {
if( tranU && U.numRows != numSingular )
throw new IllegalArgumentException("Unexpected size of matrix U");
else if( !tranU && U.numCols != numSingular )
throw new IllegalArgumentException("Unexpected size of matrix U");
}
if( V != null ) {
if( tranV && V.numRows != numSingular )
throw new IllegalArgumentException("Unexpected size of matrix V");
else if( !tranV && V.numCols != numSingular )
throw new IllegalArgumentException("Unexpected size of matrix V");
}
} else {
if( U != null && U.numRows != U.numCols )
throw new IllegalArgumentException("Unexpected size of matrix U");
if( V != null && V.numRows != V.numCols )
throw new IllegalArgumentException("Unexpected size of matrix V");
if( U != null && U.numRows != W.numRows )
throw new IllegalArgumentException("Unexpected size of W");
if( V != null && V.numRows != W.numCols )
throw new IllegalArgumentException("Unexpected size of W");
}
}
private static void swapRowOrCol(FMatrixRMaj M, boolean tran, int i, int bigIndex) {
float tmp;
if( tran ) {
// swap the rows
for( int col = 0; col < M.numCols; col++ ) {
tmp = M.get(i,col);
M.set(i,col,M.get(bigIndex,col));
M.set(bigIndex,col,tmp);
}
} else {
// swap the columns
for( int row = 0; row < M.numRows; row++ ) {
tmp = M.get(row,i);
M.set(row,i,M.get(row,bigIndex));
M.set(row,bigIndex,tmp);
}
}
}
/**
*
* Returns the null-space from the singular value decomposition. The null space is a set of non-zero vectors that
* when multiplied by the original matrix return zero.
*
*
*
* The null space is found by extracting the columns in V that are associated singular values less than
* or equal to the threshold. In some situations a non-compact SVD is required.
*
*
* @param svd A precomputed decomposition. Not modified.
* @param nullSpace Storage for null space. Will be reshaped as needed. Modified.
* @param tol Threshold for selecting singular values. Try UtilEjml.F_EPS.
* @return The null space.
*/
public static FMatrixRMaj nullSpace(SingularValueDecomposition_F32 svd ,
FMatrixRMaj nullSpace , float tol )
{
int N = svd.numberOfSingularValues();
float s[] = svd.getSingularValues();
FMatrixRMaj V = svd.getV(null,true);
if( V.numRows != svd.numCols() ) {
throw new IllegalArgumentException("Can't compute the null space using a compact SVD for a matrix of this size.");
}
// first determine the size of the null space
int numVectors = svd.numCols()-N;
for( int i = 0; i < N; i++ ) {
if( s[i] <= tol ) {
numVectors++;
}
}
// declare output data
if( nullSpace == null ) {
nullSpace = new FMatrixRMaj(numVectors,svd.numCols());
} else {
nullSpace.reshape(numVectors,svd.numCols());
}
// now extract the vectors
int count = 0;
for( int i = 0; i < N; i++ ) {
if( s[i] <= tol ) {
CommonOps_FDRM.extract(V, i,i+1,0, V.numCols,nullSpace,count++,0);
}
}
for( int i = N; i < svd.numCols(); i++ ) {
CommonOps_FDRM.extract(V, i,i+1,0, V.numCols,nullSpace,count++,0);
}
CommonOps_FDRM.transpose(nullSpace);
return nullSpace;
}
/**
* Computes the null space using QR decomposition. This is much faster than using SVD
* @param A (Input) Matrix
* @param totalSingular Number of singular values
* @return Null space
*/
public static FMatrixRMaj nullspaceQR( FMatrixRMaj A , int totalSingular ) {
SolveNullSpaceQR_FDRM solver = new SolveNullSpaceQR_FDRM();
FMatrixRMaj nullspace = new FMatrixRMaj(1,1);
if( !solver.process(A,totalSingular,nullspace))
throw new RuntimeException("Solver failed. try SVD based method instead?");
return nullspace;
}
/**
* Computes the null space using QRP decomposition. This is faster than using SVD but slower than QR.
* Much more stable than QR though.
* @param A (Input) Matrix
* @param totalSingular Number of singular values
* @return Null space
*/
public static FMatrixRMaj nullspaceQRP( FMatrixRMaj A , int totalSingular ) {
SolveNullSpaceQRP_FDRM solver = new SolveNullSpaceQRP_FDRM();
FMatrixRMaj nullspace = new FMatrixRMaj(1,1);
if( !solver.process(A,totalSingular,nullspace))
throw new RuntimeException("Solver failed. try SVD based method instead?");
return nullspace;
}
/**
* Computes the null space using SVD. Slowest bust most stable way to find the solution
*
* @param A (Input) Matrix
* @param totalSingular Number of singular values
* @return Null space
*/
public static FMatrixRMaj nullspaceSVD( FMatrixRMaj A , int totalSingular ) {
SolveNullSpace solver = new SolveNullSpaceSvd_FDRM();
FMatrixRMaj nullspace = new FMatrixRMaj(1,1);
if( !solver.process(A,totalSingular,nullspace))
throw new RuntimeException("Solver failed. try SVD based method instead?");
return nullspace;
}
/**
*
* The vector associated will the smallest singular value is returned as the null space
* of the decomposed system. A right null space is returned if 'isRight' is set to true,
* and a left null space if false.
*
*
* @param svd A precomputed decomposition. Not modified.
* @param isRight true for right null space and false for left null space. Right is more commonly used.
* @param nullVector Optional storage for a vector for the null space. Modified.
* @return Vector in V associated with smallest singular value..
*/
public static FMatrixRMaj nullVector(SingularValueDecomposition_F32 svd ,
boolean isRight ,
FMatrixRMaj nullVector )
{
int N = svd.numberOfSingularValues();
float s[] = svd.getSingularValues();
FMatrixRMaj A = isRight ? svd.getV(null,true) : svd.getU(null,false);
if( isRight ) {
if( A.numRows != svd.numCols() ) {
throw new IllegalArgumentException("Can't compute the null space using a compact SVD for a matrix of this size.");
}
if( nullVector == null ) {
nullVector = new FMatrixRMaj(svd.numCols(),1);
}
} else {
if( A.numCols != svd.numRows() ) {
throw new IllegalArgumentException("Can't compute the null space using a compact SVD for a matrix of this size.");
}
if( nullVector == null ) {
nullVector = new FMatrixRMaj(svd.numRows(),1);
}
}
int smallestIndex = -1;
if( isRight && svd.numCols() > svd.numRows() )
smallestIndex = svd.numCols()-1;
else if( !isRight && svd.numCols() < svd.numRows() )
smallestIndex = svd.numRows()-1;
else {
// find the smallest singular value
float smallestValue = Float.MAX_VALUE;
for( int i = 0; i < N; i++ ) {
if( s[i] < smallestValue ) {
smallestValue = s[i];
smallestIndex = i;
}
}
}
// extract the null space
if( isRight )
SpecializedOps_FDRM.subvector(A,smallestIndex,0,A.numRows,true,0,nullVector);
else
SpecializedOps_FDRM.subvector(A,0,smallestIndex,A.numRows,false,0,nullVector);
return nullVector;
}
/**
* Returns a reasonable threshold for singular values.
*
* tol = max (size (A)) * largest sigma * eps;
*
* @param svd A precomputed decomposition. Not modified.
* @return threshold for singular values
*/
public static float singularThreshold( SingularValueDecomposition_F32 svd ) {
float largest = 0;
float w[]= svd.getSingularValues();
int N = svd.numberOfSingularValues();
for( int j = 0; j < N; j++ ) {
if( w[j] > largest)
largest = w[j];
}
int M = Math.max(svd.numCols(),svd.numRows());
return M*largest* UtilEjml.F_EPS;
}
/**
* Extracts the rank of a matrix using a preexisting decomposition and default threshold.
*
* @see #singularThreshold(SingularValueDecomposition_F32)
*
* @param svd A precomputed decomposition. Not modified.
* @return The rank of the decomposed matrix.
*/
public static int rank( SingularValueDecomposition_F32 svd ) {
float threshold = singularThreshold(svd);
return rank(svd,threshold);
}
/**
* Extracts the rank of a matrix using a preexisting decomposition.
*
* @see #singularThreshold(SingularValueDecomposition_F32)
*
* @param svd A precomputed decomposition. Not modified.
* @param threshold Tolerance used to determine of a singular value is singular.
* @return The rank of the decomposed matrix.
*/
public static int rank(SingularValueDecomposition_F32 svd , float threshold ) {
int numRank=0;
float w[]= svd.getSingularValues();
int N = svd.numberOfSingularValues();
for( int j = 0; j < N; j++ ) {
if( w[j] > threshold)
numRank++;
}
return numRank;
}
/**
* Extracts the nullity of a matrix using a preexisting decomposition and default threshold.
*
* @see #singularThreshold(SingularValueDecomposition_F32)
*
* @param svd A precomputed decomposition. Not modified.
* @return The nullity of the decomposed matrix.
*/
public static int nullity( SingularValueDecomposition_F32 svd ) {
float threshold = singularThreshold(svd);
return nullity(svd, threshold);
}
/**
* Extracts the nullity of a matrix using a preexisting decomposition.
*
* @see #singularThreshold(SingularValueDecomposition_F32)
*
* @param svd A precomputed decomposition. Not modified.
* @param threshold Tolerance used to determine of a singular value is singular.
* @return The nullity of the decomposed matrix.
*/
public static int nullity(SingularValueDecomposition_F32 svd , float threshold ) {
int ret = 0;
float w[]= svd.getSingularValues();
int N = svd.numberOfSingularValues();
int numCol = svd.numCols();
for( int j = 0; j < N; j++ ) {
if( w[j] <= threshold) ret++;
}
return ret + numCol-N;
}
}