/**
* Computes the p=∞ norm. If A is a matrix then the induced norm is computed.
*
* @param A Matrix or vector.
* @return The norm.
*/
public static float normPInf(FMatrixRMaj A)
{
if (MatrixFeatures_FDRM.isVector(A))
{
return(CommonOps_FDRM.elementMaxAbs(A));
}
else
{
return(inducedPInf(A));
}
}
/**
* Computes the p=2 norm. If A is a matrix then the induced norm is computed. This
* implementation is faster, but more prone to buffer overflow or underflow problems.
*
* @param A Matrix or vector.
* @return The norm.
*/
public static float fastNormP2(FMatrixRMaj A)
{
if (MatrixFeatures_FDRM.isVector(A))
{
return(fastNormF(A));
}
else
{
return(inducedP2(A));
}
}
/**
* <p>
* Creates a reflector from the provided vector and gamma.<br>
* <br>
* Q = I - γ u u<sup>T</sup><br>
* </p>
*
* <p>
* In practice {@link VectorVectorMult_FDRM#householder(float, FMatrixD1, FMatrixD1, FMatrixD1)} multHouseholder}
* should be used for performance reasons since there is no need to calculate Q explicitly.
* </p>
*
* @param u A vector. Not modified.
* @param gamma To produce a reflector gamma needs to be equal to 2/||u||.
* @return An orthogonal reflector.
*/
public static FMatrixRMaj createReflector(FMatrixRMaj u, float gamma)
{
if (!MatrixFeatures_FDRM.isVector(u))
{
throw new ArgumentException("u must be a vector");
}
FMatrixRMaj Q = CommonOps_FDRM.identity(u.getNumElements());
CommonOps_FDRM.multAddTransB(-gamma, u, u, Q);
return(Q);
}
/**
* <p>
* Creates a reflector from the provided vector.<br>
* <br>
* Q = I - γ u u<sup>T</sup><br>
* γ = 2/||u||<sup>2</sup>
* </p>
*
* <p>
* In practice {@link VectorVectorMult_FDRM#householder(float, FMatrixD1, FMatrixD1, FMatrixD1)} multHouseholder}
* should be used for performance reasons since there is no need to calculate Q explicitly.
* </p>
*
* @param u A vector. Not modified.
* @return An orthogonal reflector.
*/
public static FMatrixRMaj createReflector(FMatrix1Row u)
{
if (!MatrixFeatures_FDRM.isVector(u))
{
throw new ArgumentException("u must be a vector");
}
float norm = NormOps_FDRM.fastNormF(u);
float gamma = -2.0f / (norm * norm);
FMatrixRMaj Q = CommonOps_FDRM.identity(u.getNumElements());
CommonOps_FDRM.multAddTransB(gamma, u, u, Q);
return(Q);
}
/**
* Performs a variety of tests to see if the provided matrix is a valid
* covariance matrix.
*
* @return 0 = is valid 1 = failed positive diagonal, 2 = failed on symmetry, 2 = failed on positive definite
*/
public static int isValid(FMatrixRMaj cov)
{
if (!MatrixFeatures_FDRM.isDiagonalPositive(cov))
{
return(1);
}
if (!MatrixFeatures_FDRM.isSymmetric(cov, TOL))
{
return(2);
}
if (!MatrixFeatures_FDRM.isPositiveSemidefinite(cov))
{
return(3);
}
return(0);
}
/**
* An unsafe but faster version of {@link #normP} that calls routines which are faster
* but more prone to overflow/underflow problems.
*
* @param A Vector or matrix whose norm is to be computed.
* @param p The p value of the p-norm.
* @return The computed norm.
*/
public static float fastNormP(FMatrixRMaj A, float p)
{
if (p == 1)
{
return(normP1(A));
}
else if (p == 2)
{
return(fastNormP2(A));
}
else if (float.IsInfinity(p))
{
return(normPInf(A));
}
if (MatrixFeatures_FDRM.isVector(A))
{
return(fastElementP(A, p));
}
else
{
throw new ArgumentException("Doesn't support induced norms yet.");
}
}
/**
* <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 FEigenpair computeEigenVector(FMatrixRMaj A, float eigenvalue)
{
if (A.numRows != A.numCols)
{
throw new ArgumentException("Must be a square matrix.");
}
FMatrixRMaj M = new FMatrixRMaj(A.numRows, A.numCols);
FMatrixRMaj x = new FMatrixRMaj(A.numRows, 1);
FMatrixRMaj b = new FMatrixRMaj(A.numRows, 1);
CommonOps_FDRM.fill(b, 1);
// perturb the eigenvalue slightly so that its not an exact solution the first time
// eigenvalue -= eigenvalue*UtilEjml.F_EPS*10;
float origEigenvalue = eigenvalue;
SpecializedOps_FDRM.addIdentity(A, M, -eigenvalue);
float threshold = NormOps_FDRM.normPInf(A) * UtilEjml.F_EPS;
float prevError = float.MaxValue;
bool hasWorked = false;
LinearSolverDense <FMatrixRMaj> solver = LinearSolverFactory_FDRM.linear(M.numRows);
float perp = 0.0001f;
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_FDRM.hasUncountable(x))
{
failed = true;
}
if (failed)
{
if (!hasWorked)
{
// if it failed on the first trial try perturbing it some more
float val = i % 2 == 0 ? 1.0f - perp : 1.0f + perp;
// maybe this should be turn into a parameter allowing the user
// to configure the wise of each step
eigenvalue = origEigenvalue * (float)Math.Pow(val, i / 2 + 1);
SpecializedOps_FDRM.addIdentity(A, M, -eigenvalue);
}
else
{
// otherwise assume that it was so accurate that the matrix was singular
// and return that result
return(new FEigenpair(eigenvalue, b));
}
}
else
{
hasWorked = true;
b.set(x);
NormOps_FDRM.normalizeF(b);
// compute the residual
CommonOps_FDRM.mult(M, b, x);
float error = NormOps_FDRM.normPInf(x);
if (error - prevError > UtilEjml.F_EPS * 10)
{
// if the error increased it is probably converging towards a different
// eigenvalue
// CommonOps.set(b,1);
prevError = float.MaxValue;
hasWorked = false;
float val = i % 2 == 0 ? 1.0f - perp : 1.0f + perp;
eigenvalue = origEigenvalue * (float)Math.Pow(val, 1);
}
else
{
// see if it has converged
if (error <= threshold || Math.Abs(prevError - error) <= UtilEjml.F_EPS)
{
return(new FEigenpair(eigenvalue, b));
}
// update everything
prevError = error;
eigenvalue = VectorVectorMult_FDRM.innerProdA(b, A, b);
}
SpecializedOps_FDRM.addIdentity(A, M, -eigenvalue);
}
}
return(null);
}
/**
* This is a fairly light weight check to see of a covariance matrix is valid.
* It checks to see if the diagonal elements are all positive, which they should be
* if it is valid. Not all invalid covariance matrices will be caught by this method.
*
* @return true if valid and false if invalid
*/
public static bool isValidFast(FMatrixRMaj cov)
{
return(MatrixFeatures_FDRM.isDiagonalPositive(cov));
}