/// <summary>Creates a 'n x r' matrix B such that B * B' is a correlation matrix and 'near' to the specified symmetric, normalized matrix of dimension n. A rank reduction will apply if r is strict less than n.
/// </summary>
/// <param name="rawCorrelationMatrix">The symmetric, normalized matrix where to find the 'nearest' correlation matrix.</param>
/// <param name="state">The state of the operation in its <see cref="PseudoSqrtMatrixDecomposer.State"/> representation (output).</param>
/// <param name="triangularMatrixType">A value indicating which part of <paramref name="rawCorrelationMatrix"/> to take into account.</param>
/// <param name="outputEntries">This argument will be used to store the matrix entries of the resulting matrix B, i.e. the return value array points to this array if != <c>null</c>; otherwise a memory allocation will be done.</param>
/// <param name="worksspaceContainer">A specific <see cref="PseudoSqrtMatrixDecomposer.WorkspaceContainer"/> object to reduce memory allocation; ignored if <c>null</c>.</param>
/// <returns>A <see cref="DenseMatrix"/> object that represents a matrix B such that B * B' is the 'nearest' correlation matrix with respect to <paramref name="rawCorrelationMatrix"/>.</returns>
/// <remarks>In general the return object does <b>not</b> represents the pseudo-root of <paramref name="rawCorrelationMatrix"/>, i.e. output of the Cholesky decomposition.
/// <para>The parameters <paramref name="outputEntries"/>, <paramref name="worksspaceContainer"/> allows to avoid memory allocation and to re-use arrays if the calculation of correlation matrices will be done often.</para></remarks>
public override DenseMatrix Create(DenseMatrix rawCorrelationMatrix, out State state, double[] outputEntries = null, PseudoSqrtMatrixDecomposer.WorkspaceContainer worksspaceContainer = null, BLAS.TriangularMatrixType triangularMatrixType = BLAS.TriangularMatrixType.LowerTriangularMatrix)
{
if (rawCorrelationMatrix.IsQuadratic == false)
{
throw new ArgumentException("rawCorrelationMatrix");
}
int n = rawCorrelationMatrix.RowCount;
if ((outputEntries == null) || (outputEntries.Length < n * n))
{
outputEntries = new double[n * n];
}
var ws = worksspaceContainer as Workspace;
if ((ws == null) || (ws.Dimension < n))
{
ws = new Workspace(n);
}
int m;
BLAS.Level1.dcopy(n * n, rawCorrelationMatrix.Data, ws.data);
LAPACK.EigenValues.Symmetric.driver_dsyevr(LapackEigenvalues.SymmetricGeneralJob.All, n, ws.data, out m, ws.eigenValues, outputEntries, ws.isuppz, ws.work, ws.iwork);
var originalEigenValueDataTable = InfoOutputDetailLevel.IsAtLeastAsComprehensiveAs(InfoOutputDetailLevel.High) ? CreateDataTableWithEigenvalues("Eigenvalues.Original", m, ws.eigenValues) : null;
int rank = n;
int minNumberOfEigenvaluesToSetZero = n - Math.Min(MaximalRank ?? n, n);
int i = 0;
while ((i < minNumberOfEigenvaluesToSetZero) || (ws.eigenValues[i] < 0.0))
{
ws.eigenValues[i] = 0.0;
i++;
rank--;
}
var adjustedEigenValueDataTable = InfoOutputDetailLevel.IsAtLeastAsComprehensiveAs(InfoOutputDetailLevel.High) ? CreateDataTableWithEigenvalues("Eigenvalues.Adjusted", m, ws.eigenValues) : null;
VectorUnit.Basics.Sqrt(n, ws.eigenValues); // calculate sqrt of eigenvalues only once, i.e. the array 'eigenValues' contains the sqrt of the eigenvalues!
for (i = 0; i < n; i++)
{
var t_i = 0.0;
for (int j = n - 1; j >= n - rank; j--)
{
t_i += outputEntries[i + j * n] * outputEntries[i + j * n] * ws.eigenValues[j] * ws.eigenValues[j];
outputEntries[i + j * n] *= ws.eigenValues[j];
}
BLAS.Level1.dscal(rank, 1.0 / Math.Sqrt(t_i), outputEntries, -n, i + (n - 1) * n); // [i+j*n] *= 1/Math.Sqrt(tempValue) for j = n-1, ..., n-rank
}
/* The eigenvalues are in ascending order. Thefore the first (and not last) 'rank' columns of the eigenvectors are not required. Therefore we swap the relevant part */
BLAS.Level1.dscal(n * (n - rank), 0.0, outputEntries);
BLAS.Level1.dswap(n * rank, outputEntries, 1, outputEntries, 1, n * (n - rank), 0);
state = State.Create(rank, detailProperties: new[] { InfoOutputProperty.Create("Eigenvalues set to 0.0", n - rank) }, detailDataTables: new[] { originalEigenValueDataTable, adjustedEigenValueDataTable }, iterationsNeeded: 1, infoOutputDetailLevel: InfoOutputDetailLevel);
return(new DenseMatrix(n, rank, outputEntries));
}
/// <summary>Finds the argmin of <see cref="IOneDimOptimizerAlgorithm.Function"/>.
/// </summary>
/// <param name="x">An initial guess of the algorithm (if applicable); on exit this argument contains the argmin.</param>
/// <param name="argMin">The estimated argmin of the objective function (output).</param>
/// <param name="minimum">The minimum, i.e. the function value with respect to <paramref name="argMin"/> which represents the argmin (output).</param>
/// <returns>The state of the algorithm, i.e. an indicating whether <paramref name="argMin"/> and <paramref name="minimum"/> contains valid data.</returns>
public OneDimOptimizer.State FindMinimum(double x, out double argMin, out double minimum)
{
if (m_Constraint.IntervalRepresentation.GetPointPosition(x) != PointRegionRelation.InsideOrBoundaryPoint)
{
throw new ArgumentException("Initial point is not a feasible point.");
}
var bracketingResultState = m_Optimizer.BracketingApproach.TryGetBracketingTriple(m_ObjectiveFunction.GetValue, m_Constraint.IntervalRepresentation.Infimum, m_Constraint.IntervalRepresentation.Supremum, x,
out BracketingTriple triple, out double fa, out double fb, out double fc, out int evaluationsNeeded);
switch (bracketingResultState)
{
case MinimumBracketingResultState.ProperResult:
return(FindMinimum(triple.A, triple.B, triple.C, evaluationsNeeded, out argMin, out minimum));
case MinimumBracketingResultState.FlatBracketingTriple:
argMin = triple.A;
minimum = fa;
return(State.Create(OptimizerErrorClassification.ProperResult, argMin, minimum, evaluationsNeeded: evaluationsNeeded, iterationsNeeded: 1, details: InfoOutputProperty.Create("Bracketing Result State", bracketingResultState)));
default: // return the smallest function values but indicating that no Bracketing triple has been found
if (fa < fb)
{
if (fa < fc) // fa=f(a) is smallest value
{
argMin = triple.A;
minimum = fa;
}
else // fc =f(c) is smallest value
{
argMin = triple.C;
minimum = fc;
}
}
else // fb <= fa
{
if (fc < fb)
{
argMin = triple.C;
minimum = fc;
}
else
{
argMin = triple.B;
minimum = fb;
}
}
double estimatedAbsoluteError = Math.Max(Math.Abs(triple.B - triple.A), Math.Max(Math.Abs(triple.C - triple.A), Math.Abs(triple.B - triple.C)));
double estimatedRelativeError = estimatedAbsoluteError / (MachineConsts.Epsilon + Math.Abs(triple.A) + Math.Abs(triple.B) + Math.Abs(triple.C));
return(State.Create(OptimizerErrorClassification.Unknown, argMin, minimum,
evaluationsNeeded, iterationsNeeded: 1,
details: new[] { InfoOutputProperty.Create("Bracketing Result State", bracketingResultState),
InfoOutputProperty.Create("estimatedAbsoluteError", estimatedAbsoluteError),
InfoOutputProperty.Create("estimatedRelativeError", estimatedRelativeError) }));
}
}
/// <summary>Creates a new <see cref="State"/> object.
/// </summary>
/// <param name="classification">The classification of the result.</param>
/// <param name="minimum">The estimated minimum of the specific algorithm.</param>
/// <param name="evaluationsNeeded">The number of function evaluations needed by the algorithm to reach the desired accuracy.</param>
/// <param name="iterationsNeeded">The number of iterations needed by the algorithm to reach the desired accuracy.</param>
/// <param name="details">Additional details in its <see cref="InfoOutputProperty"/> representation.</param>
/// <returns>A <see cref="State"/> object that represents the state of a specific calculation.</returns>
public static State Create(OptimizerErrorClassification classification, double minimum, int evaluationsNeeded = Int32.MaxValue, int iterationsNeeded = Int32.MaxValue, params InfoOutputProperty[] details)
{
var properties = new List <InfoOutputProperty>()
{
InfoOutputProperty.Create("Minimum", minimum)
};
if (details != null)
{
properties.AddRange(details);
}
return(new State(classification, properties, evaluationsNeeded, iterationsNeeded));
}
/// <summary>Creates a new <see cref="State"/> object.
/// </summary>
/// <param name="classification">The classification of the result.</param>
/// <param name="root">The estimated root of the specific algorithm.</param>
/// <param name="functionValue">The function value at <paramref name="root"/>; should be almost <c>0.0</c>.</param>
/// <param name="evaluationsNeeded">The number of function evaluations needed by the algorithm to reach the desired accuracy.</param>
/// <param name="iterationsNeeded">The number of iterations needed by the algorithm to reach the desired accuracy.</param>
/// <param name="details">Additional details in its <see cref="InfoOutputProperty"/> representation.</param>
/// <returns>A <see cref="State"/> object that represents the state of a specific calculation.</returns>
public static State Create(EquationSolverErrorClassification classification, double root, double functionValue, int evaluationsNeeded = Int32.MaxValue, int iterationsNeeded = Int32.MaxValue, params InfoOutputProperty[] details)
{
var properties = new List <InfoOutputProperty>()
{
InfoOutputProperty.Create("Root", root),
InfoOutputProperty.Create("FunctionValue", functionValue)
};
if (details != null)
{
properties.AddRange(details);
}
return(new State(classification, properties, evaluationsNeeded, iterationsNeeded));
}
/// <summary>Gets informations of the current object as a specific <see cref="InfoOutput" /> instance.
/// </summary>
/// <param name="infoOutput">The <see cref="InfoOutput" /> object which is to be filled with informations concering the current instance.</param>
/// <param name="categoryName">The name of the category, i.e. all informations will be added to these category.</param>
public void FillInfoOutput(InfoOutput infoOutput, string categoryName = InfoOutput.GeneralCategoryName)
{
var infoPackage = infoOutput.AcquirePackage(categoryName);
if (m_PropertyCollectionParentNode.HasChildNodes == true)
{
for (int j = 0; j < m_PropertyCollectionParentNode.ChildNodes.Count; j++)
{
var node = m_PropertyCollectionParentNode.ChildNodes[j];
var InfoProperty = new InfoOutputProperty(node.Name, node.InnerText);
infoPackage.Add(InfoProperty);
}
}
}
/// <summary>Finds the minimum of the objective function. This implementation does not take into account a probably given derivative.
/// </summary>
/// <param name="a">The first point of bracket triple.</param>
/// <param name="b">The second point of bracket triple.</param>
/// <param name="c">The third point of bracket triple.</param>
/// <param name="evaluationsNeeded">The number of function evaluations which are already needed.</param>
/// <param name="argMin">The estimated argmin of the objective function (output).</param>
/// <param name="minimum">The minimum, i.e. the function value with respect to <paramref name="argMin"/> which represents the argmin (output).</param>
/// <returns>The state of the algorithm.</returns>
/// <remarks>The implementation is based on Press, et al. (1992) "Numerical recipes in C", 2nd ed., p.401f.</remarks>
private State FindMinimum(double a, double b, double c, int evaluationsNeeded, out double argMin, out double minimum)
{
double x1, x2;
double x0 = a;
double x3 = c;
if (Math.Abs(c - b) > Math.Abs(b - a))
{
x1 = b;
x2 = b + MathConsts.TwoMinusGoldenRatio * (c - b);
}
else
{
x2 = b;
x1 = b - MathConsts.TwoMinusGoldenRatio * (b - a);
}
double f1 = m_ObjectiveFunction.GetValue(x1);
double f2 = m_ObjectiveFunction.GetValue(x2);
evaluationsNeeded += 2;
var resultStatus = OptimizerErrorClassification.NoResult;
int k = 1;
while (k <= AbortCondition.MaxIterations)
{
if (Math.Abs(x3 - x0) <= AbortCondition.Tolerance * (Math.Abs(x1) + Math.Abs(x2)))
{
resultStatus = OptimizerErrorClassification.ProperResult;
}
if (evaluationsNeeded >= m_Optimizer.AbortCondition.MaxEvaluations)
{
resultStatus = OptimizerErrorClassification.EvaluationLimitExceeded;
}
if (resultStatus != OptimizerErrorClassification.NoResult)
{
break; // exit loop
}
/* do some iteration step */
if (f2 < f1)
{
x0 = x1;
x1 = x2;
x2 = MathConsts.GoldenRatioMinusOne * x1 + MathConsts.TwoMinusGoldenRatio * x3;
f1 = f2;
f2 = m_ObjectiveFunction.GetValue(x2);
}
else
{
x3 = x2;
x2 = x1;
x1 = MathConsts.GoldenRatioMinusOne * x2 + MathConsts.TwoMinusGoldenRatio * x0;
f2 = f1;
f1 = m_ObjectiveFunction.GetValue(x1);
}
evaluationsNeeded++;
k++;
}
/* prepare output and state: */
if (f1 < f2)
{
argMin = x1;
minimum = f1;
}
else
{
argMin = x2;
minimum = f2;
}
if (k == AbortCondition.MaxIterations)
{
resultStatus = OptimizerErrorClassification.IterationLimitExceeded;
}
return(State.Create(resultStatus, argMin, minimum, evaluationsNeeded, k, InfoOutputProperty.Create("x0", x0), InfoOutputProperty.Create("x1", x1), InfoOutputProperty.Create("x2", x2), InfoOutputProperty.Create("f1", f1), InfoOutputProperty.Create("f2", f2)));
}
/// <summary>Finds the minimum of the objective function. This implementation does take into account the specified derivative.
/// </summary>
/// <param name="a">The first point of bracket triple.</param>
/// <param name="b">The second point of bracket triple.</param>
/// <param name="c">The third point of bracket triple.</param>
/// <param name="evaluationsNeeded">The number of function evaluations which are already needed.</param>
/// <param name="argMin">The estimated argmin of the objective function (output).</param>
/// <param name="minimum">The minimum, i.e. the function value with respect to <paramref name="argMin"/> which represents the argmin (output).</param>
/// <returns>The state of the algorithm.</returns>
/// <remarks>The implementation is based on Press, et al. (1992) "Numerical recipes in C", 2nd ed., p.402ff.</remarks>
private State OptimizeWithDerivative(double a, double b, double c, int evaluationsNeeded, out double argMin, out double minimum)
{
var objectiveFunction = (OneDimOptimizerDifferentiableFunction)m_ObjectiveFunction;
double tolerance, twoTimesTolerance;
double d = 0.0, e = 0.0;
double xm = 0.0, fb, fu, fv, fw, fx, u, v, w, x;
double dw, dv;
double aTemp = a < c ? a : c;
double bTemp = a > c ? a : c;
x = w = v = b;
fw = fv = fx = fb = objectiveFunction.GetValue(b, out double dx);
dw = dv = dx;
var resultStatus = OptimizerErrorClassification.NoResult;
for (int k = 1; k <= AbortCondition.MaxIterations; k++)
{
xm = 0.5 * (aTemp + bTemp);
tolerance = AbortCondition.Tolerance * Math.Abs(x) + MachineConsts.ExtremeTinyEpsilon;
twoTimesTolerance = 2.0 * tolerance;
if (Math.Abs(x - xm) <= (twoTimesTolerance - 0.5 * (bTemp - aTemp)))
{
resultStatus = OptimizerErrorClassification.ProperResult;
}
if (evaluationsNeeded >= AbortCondition.MaxEvaluations)
{
resultStatus = OptimizerErrorClassification.EvaluationLimitExceeded;
}
if (resultStatus != OptimizerErrorClassification.NoResult)
{
argMin = x;
minimum = fx;
return(State.Create(resultStatus, argMin, minimum, evaluationsNeeded, k,
InfoOutputProperty.Create("x", x), InfoOutputProperty.Create("fx", fx),
InfoOutputProperty.Create("w", w), InfoOutputProperty.Create("fw", fw),
InfoOutputProperty.Create("v", v), InfoOutputProperty.Create("fv", fv),
InfoOutputProperty.Create("b", b), InfoOutputProperty.Create("fb", fb)));
}
/* do some iteration step */
if (Math.Abs(e) > tolerance)
{
/* maybe there is no derivative given because of the given constraints: */
if (!Double.IsNaN(dw) && !Double.IsNaN(dv) && !Double.IsNaN(dx))
{
double d1 = 2.0 * (bTemp - aTemp);
double d2 = d1;
if (dw != dx)
{
d1 = (w - x) * dx / (dx - dw);
}
if (dv != dx)
{
d2 = (v - x) * dx / (dx - dv);
}
double u1 = x + d1;
double u2 = x + d2;
bool ok1 = (aTemp - u1) * (u1 - bTemp) > 0.0 && dx * d1 <= 0.0;
bool ok2 = (aTemp - u2) * (u2 - bTemp) > 0.0 && dx * d2 <= 0.0;
double olde = e;
e = d;
if (ok1 || ok2)
{
if (ok1 && ok2)
{
d = (Math.Abs(d1) < Math.Abs(d2) ? d1 : d2);
}
else if (ok1)
{
d = d1;
}
else
{
d = d2;
}
if (Math.Abs(d) <= Math.Abs(0.5 * olde))
{
u = x + d;
if (u - aTemp < twoTimesTolerance || bTemp - u < twoTimesTolerance)
{
d = (xm - x >= 0.0) ? tolerance : -tolerance;
}
}
else
{
e = dx >= 0.0 ? aTemp - x : bTemp - x;
d = 0.5 * e;
}
}
else
{
e = dx >= 0.0 ? aTemp - x : bTemp - x;
d = 0.5 * e;
}
}
else
{
double r = (x - w) * (fx - fv);
double q = (x - v) * (fx - fw);
double p = (x - v) * q - (x - w) * r;
q = 2.0 * (q - r);
if (q > 0.0)
{
p = -p;
}
q = Math.Abs(q);
double etemp = e;
e = d;
if (Math.Abs(p) >= Math.Abs(0.5 * q * etemp) || p <= q * (aTemp - x) || p >= q * (bTemp - x))
{
e = x >= xm ? aTemp - x : bTemp - x;
d = MathConsts.TwoMinusGoldenRatio * e;
}
else
{
d = p / q;
u = x + d;
if (u - aTemp < twoTimesTolerance || bTemp - u < twoTimesTolerance)
{
d = (xm - x >= 0) ? tolerance : -tolerance;
}
}
}
}
else
{
if (Double.IsNaN(dx))
{
e = x >= xm ? aTemp - x : bTemp - x;
d = MathConsts.TwoMinusGoldenRatio * e;
}
else
{
e = dx >= 0.0 ? aTemp - x : bTemp - x;
d = 0.5 * e;
}
}
double du;
if (Math.Abs(d) >= tolerance)
{
u = x + d;
fu = objectiveFunction.GetValue(u, out du);
evaluationsNeeded++;
}
else
{
u = x + (d >= 0.0 ? tolerance : -tolerance);
fu = objectiveFunction.GetValue(u, out du);
evaluationsNeeded++;
if (fu > fx)
{
argMin = x;
minimum = fx;
return(State.Create(OptimizerErrorClassification.ProperResult, argMin, minimum, evaluationsNeeded, k,
InfoOutputProperty.Create("x", x), InfoOutputProperty.Create("fx", fx),
InfoOutputProperty.Create("w", w), InfoOutputProperty.Create("fw", fw),
InfoOutputProperty.Create("v", v), InfoOutputProperty.Create("fv", fv),
InfoOutputProperty.Create("b", b), InfoOutputProperty.Create("fb", fb)));
}
}
if (fu <= fx)
{
if (u >= x)
{
aTemp = x;
}
else
{
bTemp = x;
}
Copy(ref v, ref fv, ref dv, w, fw, dw);
Copy(ref w, ref fw, ref dw, x, fx, dx);
Copy(ref x, ref fx, ref dx, u, fu, du);
}
else
{
if (u < x)
{
aTemp = u;
}
else
{
bTemp = u;
}
if (fu <= fw || w == x)
{
Copy(ref v, ref fv, ref dv, w, fw, dw);
Copy(ref w, ref fw, ref dw, u, fu, du);
}
else if (fu < fv || v == x || v == w)
{
Copy(ref v, ref fv, ref dv, u, fu, du);
}
}
}
argMin = x;
minimum = fx;
return(State.Create(OptimizerErrorClassification.IterationLimitExceeded, argMin, minimum, evaluationsNeeded, AbortCondition.MaxIterations,
InfoOutputProperty.Create("x", x), InfoOutputProperty.Create("fx", fx),
InfoOutputProperty.Create("w", w), InfoOutputProperty.Create("fw", fw),
InfoOutputProperty.Create("v", v), InfoOutputProperty.Create("fv", fv),
InfoOutputProperty.Create("b", b), InfoOutputProperty.Create("fb", fb)));
}
/// <summary>Finds the minimum of the objective function. This implementation does not take into account a probably given derivative.
/// </summary>
/// <param name="a">The first point of bracket triple.</param>
/// <param name="b">The second point of bracket triple.</param>
/// <param name="c">The third point of bracket triple.</param>
/// <param name="evaluationsNeeded">The number of function evaluations which are already needed.</param>
/// <param name="argMin">The estimated argmin of the objective function (output).</param>
/// <param name="minimum">The minimum, i.e. the function value with respect to <paramref name="argMin"/> which represents the argmin (output).</param>
/// <returns>The state of the algorithm.</returns>
/// <remarks>The implementation is based on Press, et al. (1992) "Numerical recipes in C", 2nd ed., p.402ff.</remarks>
private State OptimizeWithoutDerivative(double a, double b, double c, int evaluationsNeeded, out double argMin, out double minimum)
{
double tolerance, twoTimesTolerance;
double d = 0.0, e = 0.0;
double xm = 0.0, fb, fu, fv, fw, fx, u, v, w, x;
double aTemp = a < c ? a : c;
double bTemp = a > c ? a : c;
x = w = v = b;
fw = fv = fx = fb = m_ObjectiveFunction.GetValue(b);
var resultStatus = OptimizerErrorClassification.NoResult;
for (int k = 1; k <= AbortCondition.MaxIterations; k++)
{
xm = 0.5 * (aTemp + bTemp);
tolerance = AbortCondition.Tolerance * Math.Abs(x) + MachineConsts.ExtremeTinyEpsilon;
twoTimesTolerance = 2.0 * tolerance;
if (Math.Abs(x - xm) <= (twoTimesTolerance - 0.5 * (bTemp - aTemp)))
{
resultStatus = OptimizerErrorClassification.ProperResult;
}
if (evaluationsNeeded >= AbortCondition.MaxEvaluations)
{
resultStatus = OptimizerErrorClassification.EvaluationLimitExceeded;
}
if (resultStatus != OptimizerErrorClassification.NoResult)
{
argMin = x;
minimum = fx;
return(State.Create(resultStatus, argMin, minimum, evaluationsNeeded, k,
InfoOutputProperty.Create("x", x), InfoOutputProperty.Create("fx", fx),
InfoOutputProperty.Create("w", w), InfoOutputProperty.Create("fw", fw),
InfoOutputProperty.Create("v", v), InfoOutputProperty.Create("fv", fv),
InfoOutputProperty.Create("b", b), InfoOutputProperty.Create("fb", fb)));
}
/* do some iteration step */
if (Math.Abs(e) > tolerance)
{
double r = (x - w) * (fx - fv);
double q = (x - v) * (fx - fw);
double p = (x - v) * q - (x - w) * r;
q = 2.0 * (q - r);
if (q > 0.0)
{
p = -p;
}
q = Math.Abs(q);
double etemp = e;
e = d;
if (Math.Abs(p) >= Math.Abs(0.5 * q * etemp) || p <= q * (aTemp - x) || p >= q * (bTemp - x))
{
e = x >= xm ? aTemp - x : bTemp - x;
d = MathConsts.TwoMinusGoldenRatio * e;
}
else
{
d = p / q;
u = x + d;
if (u - aTemp < twoTimesTolerance || bTemp - u < twoTimesTolerance)
{
d = (xm - x >= 0) ? tolerance : -tolerance;
}
}
}
else
{
e = x >= xm ? aTemp - x : bTemp - x;
d = MathConsts.TwoMinusGoldenRatio * e;
}
u = Math.Abs(d) >= tolerance ? x + d : x + ((d >= 0) ? tolerance : -tolerance);
fu = m_ObjectiveFunction.GetValue(u);
evaluationsNeeded++;
if (fu <= fx)
{
if (u >= x)
{
aTemp = x;
}
else
{
bTemp = x;
}
v = w;
w = x;
x = u;
fv = fw;
fw = fx;
fx = fu;
}
else
{
if (u < x)
{
aTemp = u;
}
else
{
bTemp = u;
}
if (fu <= fw || w == x)
{
v = w;
w = u;
fv = fw;
fw = fu;
}
else if (fu <= fv || v == x || v == w)
{
v = u;
fv = fu;
}
}
}
argMin = x;
minimum = fx;
return(State.Create(OptimizerErrorClassification.IterationLimitExceeded, argMin, minimum, evaluationsNeeded, AbortCondition.MaxIterations,
InfoOutputProperty.Create("x", x), InfoOutputProperty.Create("fx", fx),
InfoOutputProperty.Create("w", w), InfoOutputProperty.Create("fw", fw),
InfoOutputProperty.Create("v", v), InfoOutputProperty.Create("fv", fv),
InfoOutputProperty.Create("b", b), InfoOutputProperty.Create("fb", fb)));
}
/// <summary>Creates a 'n x r' matrix B such that B * B' is a correlation matrix and 'near' to the specified symmetric, normalized matrix of dimension n. A rank reduction will apply if r is strict less than n.
/// </summary>
/// <param name="rawCorrelationMatrix">The symmetric, normalized matrix where to find the 'nearest' correlation matrix.</param>
/// <param name="state">The state of the operation in its <see cref="PseudoSqrtMatrixDecomposer.State"/> representation (output).</param>
/// <param name="triangularMatrixType">A value indicating which part of <paramref name="rawCorrelationMatrix"/> to take into account.</param>
/// <param name="outputEntries">This argument will be used to store the matrix entries of the resulting matrix B, i.e. the return value array points to this array if != <c>null</c>; otherwise a memory allocation will be done.</param>
/// <param name="worksspaceContainer">A specific <see cref="PseudoSqrtMatrixDecomposer.WorkspaceContainer"/> object to reduce memory allocation; ignored if <c>null</c>.</param>
/// <returns>A <see cref="DenseMatrix"/> object that represents a matrix B such that B * B' is the 'nearest' correlation matrix with respect to <paramref name="rawCorrelationMatrix"/>.</returns>
/// <remarks>In general the return object does <b>not</b> represents the pseudo-root of <paramref name="rawCorrelationMatrix"/>, i.e. output of the Cholesky decomposition.
/// <para>The parameters <paramref name="outputEntries"/>, <paramref name="worksspaceContainer"/> allows to avoid memory allocation and to re-use arrays if the calculation of correlation matrices will be done often.</para></remarks>
public override DenseMatrix Create(DenseMatrix rawCorrelationMatrix, out State state, double[] outputEntries = null, WorkspaceContainer worksspaceContainer = null, BLAS.TriangularMatrixType triangularMatrixType = BLAS.TriangularMatrixType.LowerTriangularMatrix)
{
if (rawCorrelationMatrix.IsQuadratic == false)
{
throw new ArgumentException("rawCorrelationMatrix");
}
int n = rawCorrelationMatrix.RowCount;
if ((outputEntries == null) || (outputEntries.Length < n * n))
{
outputEntries = new double[n * n];
}
var ws = worksspaceContainer as Workspace;
if ((ws == null) || (ws.Dimension < n))
{
ws = new Workspace(n);
}
/* calculate an initial value for the EZI approach, i.e. apply the Eigenvalue zeroing (without normalization): */
int initialRank;
BLAS.Level1.dcopy(n * n, rawCorrelationMatrix.Data, ws.p);
f1(n, ws.p, 0, ws, outputEntries, out initialRank);
var detailDataTables = new List <DataTable>();
if (InfoOutputDetailLevel.IsAtLeastAsComprehensiveAs(InfoOutputDetailLevel.High))
{
detailDataTables.Add(CreateDataTableWithEigenvalues("Eigenvalues.Adjusted[0]", initialRank, ws.eigenValues));
}
BLAS.Level3.dgemm(n, n, n, 1.0, outputEntries, outputEntries, 0.0, ws.p, transposeB: BLAS.MatrixTransposeState.Transpose);
BLAS.Level1.dcopy(n * n, ws.p, ws.q);
int minNumberOfEigenvaluesToSetZero = n - Math.Min(MaximalRank ?? n, initialRank);
var a = 1.0;
int rank = initialRank;
for (int k = 1; k < AbortCondition.MaxIterations; k++)
{
f1(n, ws.q, minNumberOfEigenvaluesToSetZero, ws, outputEntries, out rank);
if (InfoOutputDetailLevel.IsAtLeastAsComprehensiveAs(InfoOutputDetailLevel.Full))
{
detailDataTables.Add(CreateDataTableWithEigenvalues(String.Format("Eigenvalues.Adjusted[{0}", k), rank, ws.eigenValues));
}
BLAS.Level3.dgemm(n, n, n, 1.0, outputEntries, outputEntries, 0.0, ws.q, transposeB: BLAS.MatrixTransposeState.Transpose);
/* normalize the correlation matrix, i.e. calculate <q> */
for (int i = 0; i < n; i++)
{
var t_i = 0.0;
for (int j = n - 1; j >= n - rank; j--)
{
t_i += outputEntries[i + j * n] * outputEntries[i + j * n];
}
BLAS.Level1.dscal(rank, 1.0 / Math.Sqrt(t_i), outputEntries, -n, i + (n - 1) * n); // [i+j*n] *= 1/Math.Sqrt(t_i) for j = n-1, ..., n-rank
}
BLAS.Level3.dgemm(n, n, n, 1.0, outputEntries, outputEntries, 0.0, ws.qNormalized, transposeB: BLAS.MatrixTransposeState.Transpose);
if (AbortCondition.IsSatisfied(n, ws.p, ws.q, ws.qNormalized, ref a) == true)
{
/* The eigenvalues are in ascending order. Thefore the first (and not last) 'rank' columns of the eigenvectors are not required. Therefore we swap the relevant part: */
BLAS.Level1.dswap(n * rank, outputEntries, 1, outputEntries, 1, n * (n - rank), 0);
state = State.Create(rank, new InfoOutputProperty[] { InfoOutputProperty.Create("Eigenvalues set to 0.0", n - rank) }, detailDataTables, iterationsNeeded: k, infoOutputDetailLevel: InfoOutputDetailLevel);
return(new DenseMatrix(n, rank, outputEntries));
}
BLAS.Level1.dcopy(n * n, ws.q, ws.p); // set p := q
for (int j = 0; j < n; j++)
{
ws.q[j * n + j] = 1.0;
}
}
/* The eigenvalues are in ascending order. Thefore the first (and not last) 'rank' columns of the eigenvectors are not required. Therefore we swap the relevant part: */
BLAS.Level1.dswap(n * rank, outputEntries, 1, outputEntries, 1, n * (n - rank), 0);
state = State.Create(rank, new InfoOutputProperty[] { InfoOutputProperty.Create("Number of EigenValues set to 0.0", n - rank) }, detailDataTables, AbortCondition.MaxIterations, infoOutputDetailLevel: InfoOutputDetailLevel);
return(new DenseMatrix(n, rank, outputEntries));
}
/// <summary>Finds the minimum and argmin of <see cref="IMultiDimOptimizerAlgorithm.Function" />.
/// </summary>
/// <param name="x">An array with at least <see cref="IMultiDimOptimizerAlgorithm.Dimension" /> elements which is perhaps taken into account as an initial guess of the algorithm; on exit this argument contains the argmin.</param>
/// <param name="minimum">The minimum, i.e. the function value with respect to <paramref name="x" /> which represents the argmin (output).</param>
/// <returns>The state of the algorithm, i.e. an indicating whether <paramref name="x" /> and <paramref name="minimum" /> contain valid data.</returns>
/// <exception cref="NotOperableException">Thrown, if no objective function is specified.</exception>
public State FindMinimum(double[] x, out double minimum)
{
if (m_NLoptFunction == null)
{
throw new NotOperableException("No objective function specified.");
}
var nloptResultCode = m_NLoptPtr.FindMinimum(x, out minimum);
switch (nloptResultCode)
{
case NLoptResultCode.Success:
case NLoptResultCode.FToleranceReached:
case NLoptResultCode.XToleranceReached:
return(MultiDimOptimizer.State.Create(OptimizerErrorClassification.ProperResult, minimum, details: InfoOutputProperty.Create("NLopt result code", nloptResultCode)));
case NLoptResultCode.MaximalNumberOfFunctionEvaluationsReached:
return(MultiDimOptimizer.State.Create(OptimizerErrorClassification.EvaluationLimitExceeded, minimum));
case NLoptResultCode.MaximalTimeReached:
return(MultiDimOptimizer.State.Create(OptimizerErrorClassification.EvaluationTimeExceeded, minimum));
default:
return(MultiDimOptimizer.State.Create(OptimizerErrorClassification.Unknown, minimum, details: InfoOutputProperty.Create("NLopt result code", nloptResultCode)));
}
}