/**
* <p>
* 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.
* </p>
*
* @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 DMatrixRMaj nullVector(SingularValueDecomposition_F64 <DMatrixRMaj> svd,
bool isRight,
DMatrixRMaj nullVector)
{
int N = svd.numberOfSingularValues();
double[] s = svd.getSingularValues();
DMatrixRMaj A = isRight ? svd.getV(null, true) : svd.getU(null, false);
if (isRight)
{
if (A.numRows != svd.numCols())
{
throw new ArgumentException(
"Can't compute the null space using a compact SVD for a matrix of this size.");
}
if (nullVector == null)
{
nullVector = new DMatrixRMaj(svd.numCols(), 1);
}
}
else
{
if (A.numCols != svd.numRows())
{
throw new ArgumentException(
"Can't compute the null space using a compact SVD for a matrix of this size.");
}
if (nullVector == null)
{
nullVector = new DMatrixRMaj(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
double smallestValue = double.MaxValue;
for (int i = 0; i < N; i++)
{
if (s[i] < smallestValue)
{
smallestValue = s[i];
smallestIndex = i;
}
}
}
// extract the null space
if (isRight)
{
SpecializedOps_DDRM.subvector(A, smallestIndex, 0, A.numRows, true, 0, nullVector);
}
else
{
SpecializedOps_DDRM.subvector(A, 0, smallestIndex, A.numRows, false, 0, nullVector);
}
return(nullVector);
}
/**
* <p>
* Given an eigenvalue it computes an eigenvector using inverse iteration:
* <br>
* for i=1:MAX {<br>
* (A - μI)z<sup>(i)</sup> = q<sup>(i-1)</sup><br>
* q<sup>(i)</sup> = z<sup>(i)</sup> / ||z<sup>(i)</sup>||<br>
* λ<sup>(i)</sup> = q<sup>(i)</sup><sup>T</sup> A q<sup>(i)</sup><br>
* }<br>
* </p>
* <p>
* NOTE: If there is another eigenvalue that is very similar to the provided one then there
* is a chance of it converging towards that one instead. The larger a matrix is the more
* likely this is to happen.
* </p>
* @param A Matrix whose eigenvector is being computed. Not modified.
* @param eigenvalue The eigenvalue in the eigen pair.
* @return The eigenvector or null if none could be found.
*/
public static DEigenpair computeEigenVector(DMatrixRMaj A, double eigenvalue)
{
if (A.numRows != A.numCols)
{
throw new ArgumentException("Must be a square matrix.");
}
DMatrixRMaj M = new DMatrixRMaj(A.numRows, A.numCols);
DMatrixRMaj x = new DMatrixRMaj(A.numRows, 1);
DMatrixRMaj b = new DMatrixRMaj(A.numRows, 1);
CommonOps_DDRM.fill(b, 1);
// perturb the eigenvalue slightly so that its not an exact solution the first time
// eigenvalue -= eigenvalue*UtilEjml.EPS*10;
double origEigenvalue = eigenvalue;
SpecializedOps_DDRM.addIdentity(A, M, -eigenvalue);
double threshold = NormOps_DDRM.normPInf(A) * UtilEjml.EPS;
double prevError = double.MaxValue;
bool hasWorked = false;
LinearSolverDense <DMatrixRMaj> solver = LinearSolverFactory_DDRM.linear(M.numRows);
double perp = 0.0001;
for (int i = 0; i < 200; i++)
{
bool failed = false;
// if the matrix is singular then the eigenvalue is within machine precision
// of the true value, meaning that x must also be.
if (!solver.setA(M))
{
failed = true;
}
else
{
solver.solve(b, x);
}
// see if solve silently failed
if (MatrixFeatures_DDRM.hasUncountable(x))
{
failed = true;
}
if (failed)
{
if (!hasWorked)
{
// if it failed on the first trial try perturbing it some more
double val = i % 2 == 0 ? 1.0 - perp : 1.0 + perp;
// maybe this should be turn into a parameter allowing the user
// to configure the wise of each step
eigenvalue = origEigenvalue * Math.Pow(val, i / 2 + 1);
SpecializedOps_DDRM.addIdentity(A, M, -eigenvalue);
}
else
{
// otherwise assume that it was so accurate that the matrix was singular
// and return that result
return(new DEigenpair(eigenvalue, b));
}
}
else
{
hasWorked = true;
b.set(x);
NormOps_DDRM.normalizeF(b);
// compute the residual
CommonOps_DDRM.mult(M, b, x);
double error = NormOps_DDRM.normPInf(x);
if (error - prevError > UtilEjml.EPS * 10)
{
// if the error increased it is probably converging towards a different
// eigenvalue
// CommonOps.set(b,1);
prevError = double.MaxValue;
hasWorked = false;
double val = i % 2 == 0 ? 1.0 - perp : 1.0 + perp;
eigenvalue = origEigenvalue * Math.Pow(val, 1);
}
else
{
// see if it has converged
if (error <= threshold || Math.Abs(prevError - error) <= UtilEjml.EPS)
{
return(new DEigenpair(eigenvalue, b));
}
// update everything
prevError = error;
eigenvalue = VectorVectorMult_DDRM.innerProdA(b, A, b);
}
SpecializedOps_DDRM.addIdentity(A, M, -eigenvalue);
}
}
return(null);
}