/**
* <p>
* Creates a randomly generated set of orthonormal vectors. At most it can generate the same
* number of vectors as the dimension of the vectors.
* </p>
*
* <p>
* This is done by creating random vectors then ensuring that they are orthogonal
* to all the ones previously created with reflectors.
* </p>
*
* <p>
* NOTE: This employs a brute force O(N<sup>3</sup>) algorithm.
* </p>
*
* @param dimen dimension of the space which the vectors will span.
* @param numVectors How many vectors it should generate.
* @param rand Used to create random vectors.
* @return Array of N random orthogonal vectors of unit length.
*/
// is there a faster algorithm out there? This one is a bit sluggish
public static DMatrixRMaj[] span(int dimen, int numVectors, IMersenneTwister rand)
{
if (dimen < numVectors)
{
throw new ArgumentException("The number of vectors must be less than or equal to the dimension");
}
DMatrixRMaj[] u = new DMatrixRMaj[numVectors];
u[0] = RandomMatrices_DDRM.rectangle(dimen, 1, -1, 1, rand);
NormOps_DDRM.normalizeF(u[0]);
for (int i = 1; i < numVectors; i++)
{
// Console.WriteLine(" i = "+i);
DMatrixRMaj a = new DMatrixRMaj(dimen, 1);
DMatrixRMaj r = null;
for (int j = 0; j < i; j++)
{
// Console.WriteLine("j = "+j);
if (j == 0)
{
r = RandomMatrices_DDRM.rectangle(dimen, 1, -1, 1, rand);
}
// find a vector that is normal to vector j
// u[i] = (1/2)*(r + Q[j]*r)
a.set(r);
VectorVectorMult_DDRM.householder(-2.0, u[j], r, a);
CommonOps_DDRM.add(r, a, a);
CommonOps_DDRM.scale(0.5, a);
// UtilEjml.print(a);
DMatrixRMaj t = a;
a = r;
r = t;
// normalize it so it doesn't get too small
double val = NormOps_DDRM.normF(r);
if (val == 0 || double.IsNaN(val) || double.IsInfinity(val))
{
throw new InvalidOperationException("Failed sanity check");
}
CommonOps_DDRM.divide(r, val);
}
u[i] = r;
}
return(u);
}
/**
* <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_DDRM#householder(double, DMatrixD1, DMatrixD1, DMatrixD1)} 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 DMatrixRMaj createReflector(DMatrix1Row u)
{
if (!MatrixFeatures_DDRM.isVector(u))
{
throw new ArgumentException("u must be a vector");
}
double norm = NormOps_DDRM.fastNormF(u);
double gamma = -2.0 / (norm * norm);
DMatrixRMaj Q = CommonOps_DDRM.identity(u.getNumElements());
CommonOps_DDRM.multAddTransB(gamma, u, u, Q);
return(Q);
}
/**
* <p>
* Computes the F norm of the difference between the two Matrices:<br>
* <br>
* Sqrt{∑<sub>i=1:m</sub> ∑<sub>j=1:n</sub> ( a<sub>ij</sub> - b<sub>ij</sub>)<sup>2</sup>}
* </p>
* <p>
* This is often used as a cost function.
* </p>
*
* @see NormOps_DDRM#fastNormF
*
* @param a m by n matrix. Not modified.
* @param b m by n matrix. Not modified.
*
* @return The F normal of the difference matrix.
*/
public static double diffNormF(DMatrixD1 a, DMatrixD1 b)
{
if (a.numRows != b.numRows || a.numCols != b.numCols)
{
throw new ArgumentException("Both matrices must have the same shape.");
}
int size = a.getNumElements();
DMatrixRMaj diff = new DMatrixRMaj(size, 1);
for (int i = 0; i < size; i++)
{
diff.set(i, b.get(i) - a.get(i));
}
return(NormOps_DDRM.normF(diff));
}
/**
* <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);
}
