/// <summary>
/// Check the result.
/// </summary>
/// <param name="preconditioner">Specific preconditioner.</param>
/// <param name="matrix">Source matrix.</param>
/// <param name="vector">Initial vector.</param>
/// <param name="result">Result vector.</param>
protected override void CheckResult(IPreconditioner<float> preconditioner, SparseMatrix matrix, Vector<float> vector, Vector<float> result)
{
Assert.AreEqual(typeof (UnitPreconditioner<float>), preconditioner.GetType(), "#01");
// Unit preconditioner is doing nothing. Vector and result should be equal
for (var i = 0; i < vector.Count; i++)
{
Assert.IsTrue(vector[i] == result[i], "#02-" + i);
}
}
/// <summary>
/// Check the result.
/// </summary>
/// <param name="preconditioner">Specific preconditioner.</param>
/// <param name="matrix">Source matrix.</param>
/// <param name="vector">Initial vector.</param>
/// <param name="result">Result vector.</param>
protected override void CheckResult(IPreconditioner<double> preconditioner, SparseMatrix matrix, Vector<double> vector, Vector<double> result)
{
Assert.AreEqual(typeof (DiagonalPreconditioner), preconditioner.GetType(), "#01");
// Compute M * result = product
// compare vector and product. Should be equal
var product = new DenseVector(result.Count);
matrix.Multiply(result, product);
for (var i = 0; i < product.Count; i++)
{
Assert.IsTrue(vector[i].AlmostEqualNumbersBetween(product[i], -Epsilon.Magnitude()), "#02-" + i);
}
}
/// <summary>
/// Check the result.
/// </summary>
/// <param name="preconditioner">Specific preconditioner.</param>
/// <param name="matrix">Source matrix.</param>
/// <param name="vector">Initial vector.</param>
/// <param name="result">Result vector.</param>
protected override void CheckResult(IPreconditioner<Complex32> preconditioner, SparseMatrix matrix, Vector<Complex32> vector, Vector<Complex32> result)
{
Assert.AreEqual(typeof(Diagonal), preconditioner.GetType(), "#01");
// Compute M * result = product
// compare vector and product. Should be equal
var product = new DenseVector(result.Count);
matrix.Multiply(result, product);
for (var i = 0; i < product.Count; i++)
{
Assert.IsTrue(vector[i].Real.AlmostEqual(product[i].Real, -Epsilon.Magnitude()), "#02-" + i);
Assert.IsTrue(vector[i].Imaginary.AlmostEqual(product[i].Imaginary, -Epsilon.Magnitude()), "#03-" + i);
}
}
/// <summary>
/// Solves the matrix equation AX = B, where A is the coefficient matrix (this matrix), B is the solution matrix and X is the unknown matrix.
/// </summary>
/// <param name="input">The solution matrix <c>B</c>.</param>
/// <param name="solver">The iterative solver to use.</param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
/// <returns>The result matrix <c>X</c>.</returns>
public Matrix <T> SolveIterative(Matrix <T> input, IIterativeSolver <T> solver, Iterator <T> iterator = null, IPreconditioner <T> preconditioner = null)
{
var result = Build.Dense(input.RowCount, input.ColumnCount);
TrySolveIterative(input, result, solver, iterator, preconditioner);
return(result);
}
/// <summary>
/// Solves the matrix equation AX = B, where A is the coefficient matrix (this matrix), B is the solution matrix and X is the unknown matrix.
/// </summary>
/// <param name="input">The solution matrix <c>B</c>.</param>
/// <param name="result">The result matrix <c>X</c></param>
/// <param name="solver">The iterative solver to use.</param>
/// <param name="stopCriteria">Criteria to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
public IterationStatus TrySolveIterative(Matrix <T> input, Matrix <T> result, IIterativeSolver <T> solver, IPreconditioner <T> preconditioner, params IIterationStopCriterion <T>[] stopCriteria)
{
var iterator = new Iterator <T>(stopCriteria.Length == 0 ? Build.IterativeSolverStopCriteria() : stopCriteria);
return(TrySolveIterative(input, result, solver, iterator, preconditioner));
}
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution, ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
BCGStabParametrs ConGradParametrs = solverParametrs as BCGStabParametrs;
if (ConGradParametrs == null)
{
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in BSGStab");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in BSGSTab");
}
else
{
int oIter = 0;
double nPi, oPi, alpha, w, oNev, bNev, betta;
oPi = alpha = w = 1;
int size = rightPart.Size;
Vector x, r, rTab, s, t, z, sqt, y;
Vector v = new Vector(size);
Vector p = new Vector(size);
v.Nullify();
p.Nullify();
x = initialSolution;
r = rightPart - matrix.SourceMatrix.Multiply(x);
rTab = r;
bNev = rightPart.Norm();
oNev = r.Norm() / bNev;
solverLogger.AddIterationInfo(oIter, oNev);//logger
if (System.Double.IsInfinity(oNev))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by BSG Stab.");
return x;
}
if (System.Double.IsNaN(oNev))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by BSG Stab.");
return x;
}
while (oIter < ConGradParametrs.MaxIterations && oNev > ConGradParametrs.Epsilon)
{
nPi = rTab * r;
betta = (nPi / oPi) * (alpha / w);
p = r + (p - v * w) * betta;
y= matrix.QSolve(matrix.SSolve(p));
v = matrix.SourceMatrix.Multiply(y);
alpha = nPi / (rTab * v);
x = x + y * alpha;//УТОЧНИТЬ!!!!!!!!!!!!!!!
s = r - v * alpha;
if (s.Norm() / bNev < ConGradParametrs.Epsilon) return x;
z = matrix.QSolve(matrix.SSolve(s));
t = matrix.SourceMatrix.Multiply(z);
sqt = matrix.QSolve(matrix.SSolve(t));
w = (z * sqt) / (sqt * sqt);
x = x + z * w;//УТОЧНИТЬ!!!!!!!!!!!!!!!!!!!
r = s - t * w;
oPi = nPi;
oIter++;
oNev = r.Norm() / bNev;
if (System.Double.IsInfinity(oNev))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by BSG Stab.");
return x;
}
if (System.Double.IsNaN(oNev))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by BSG Stab.");
return x;
}
solverLogger.AddIterationInfo(oIter, oNev);//logger
}
return x;
}
}
/// <summary> /// Check the result. /// </summary> /// <param name="preconditioner">Specific preconditioner.</param> /// <param name="matrix">Source matrix.</param> /// <param name="vector">Initial vector.</param> /// <param name="result">Result vector.</param> protected abstract void CheckResult(IPreconditioner <double> preconditioner, SparseMatrix matrix, Vector <double> vector, Vector <double> result);
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution,
ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
if (solverParametrs is GaussSeidelParametrs)
{
GaussSeidelParametrs GZParametrs = solverParametrs as GaussSeidelParametrs;
Vector x, xnext, r, di;
int size, k;
double Residual, w;
Vector DEx, DEb, Dx, Fx, Ex;
size = initialSolution.Size;
x = new Vector(size);
xnext = new Vector(size);
r = new Vector(size);
di = new Vector(size);
x = initialSolution;
w = GZParametrs.Relaxation;
di = matrix.SourceMatrix.Diagonal;
Dx = Vector.Mult(di, x);
Fx = matrix.SourceMatrix.LMult(x, false);
Ex = matrix.SourceMatrix.UMult(x, false);
r = Dx + Fx + Ex - rightPart;
var rpnorm = rightPart.Norm();
Residual = r.Norm() / rpnorm;
DEb = Vector.Division(rightPart, di);
solverLogger.AddIterationInfo(0, Residual);
for (k = 1; k <= GZParametrs.MaxIterations && Residual > GZParametrs.Epsilon; k++)
{
DEx = Vector.Division(Ex + Fx, di);
xnext = (DEb - DEx) * w + x * (1 - w);
Ex = matrix.SourceMatrix.UMult(xnext, false);
Dx = Vector.Mult(di, xnext);
DEx = Vector.Division(Ex + Fx, di);
x = xnext;
xnext = (DEb - DEx) * w + x * (1 - w);
Dx = Vector.Mult(di, xnext);
Fx = matrix.SourceMatrix.LMult(xnext, false);
r = Dx + Fx + Ex - rightPart;
Residual = r.Norm() / rpnorm;
if (System.Double.IsInfinity(Residual))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by GaussSeidel .");
return(xnext);
}
if (System.Double.IsNaN(Residual))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by GaussSeidel.");
return(xnext);
}
solverLogger.AddIterationInfo(k, Residual);
}
return(xnext);
}
else
{
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GaussSeidel");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GaussSeidel");
}
}
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution,
ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
if (solverParametrs is LOSParametrs)
{
LOSParametrs LOSParametrs = solverParametrs as LOSParametrs;
Vector x, r, z, p;
double alpha, betta;
int size, k;
double Residual,pp,rightnorm;
Vector Ax;
Vector LAU, Ur;
size = initialSolution.Size;
x = new Vector(size);
r = new Vector(size);
z = new Vector(size);
p = new Vector(size);
Ax = new Vector(size);
LAU = new Vector(size);
Ur = new Vector(size);
x = initialSolution;
Ax = matrix.SourceMatrix.Multiply(x);
r = matrix.SSolve(rightPart - Ax);
z = matrix.QSolve(r);
p = matrix.SSolve(matrix.SourceMatrix.Multiply(z));
rightnorm = matrix.SSolve(rightPart).Norm();
Residual = r.Norm()/rightnorm;
solverLogger.AddIterationInfo(0, Residual);
for (k = 1; k <= LOSParametrs.MaxIterations && Residual > LOSParametrs.Epsilon; k++)
{
pp = p*p;
alpha = (p * r) / pp;
x = x + z * alpha;
r = r - p * alpha;
Ur = matrix.QSolve(r);
LAU = matrix.SourceMatrix.Multiply(Ur);
LAU = matrix.SSolve(LAU);
betta = - (p * LAU) / pp;
z = Ur + z * betta;
p = LAU + p * betta;
Residual = r.Norm() / rightnorm;
if (System.Double.IsInfinity(Residual))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by LOS.");
return x;
}
if (System.Double.IsNaN(Residual))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by LOS.");
return x;
}
solverLogger.AddIterationInfo(k, Residual);
}
return x;
}
else {
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GaussSeidel");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GaussSeidel");
}
}
private void InitializeSolver()
{
result = Vector<double>.Build.Dense(n * m);
Control.LinearAlgebraProvider = new OpenBlasLinearAlgebraProvider();
//Control.UseNativeMKL();
//Control.LinearAlgebraProvider = new MklLinearAlgebraProvider();
//Control.UseManaged();
var iterationCountStopCriterion = new IterationCountStopCriterion<double>(1000);
var residualStopCriterion = new ResidualStopCriterion<double>(1e-7);
monitor = new Iterator<double>(iterationCountStopCriterion, residualStopCriterion);
solver = new BiCgStab();
preconditioner = new MILU0Preconditioner();
}
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution,
ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
GMRESParameters GMRESParameters = solverParametrs as GMRESParameters;
if (GMRESParameters != null)
{
Vector[] V, H;
Vector residual, x, d, z, w, tmp;
bool continueCalculations;
double epsilon = GMRESParameters.Epsilon;
int m = GMRESParameters.M;
int maxIterations = GMRESParameters.MaxIterations;
int n = rightPart.Size;
x = initialSolution;
residual = rightPart - matrix.SourceMatrix.Multiply(x);
residual = matrix.SSolve(residual);
double rightPartNorm = matrix.SSolve(rightPart).Norm();
double residualNorm = residual.Norm();
if (System.Double.IsInfinity(residualNorm / rightPartNorm))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by GMRES.");
return x;
}
if (System.Double.IsNaN(residualNorm / rightPartNorm))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by GMRES.");
return x;
}
x = matrix.QMultiply(initialSolution);
V = new Vector[m];
for (int i = 0; i < m; i++)
V[i] = new Vector(n);
H = new Vector[m];
for (int i = 0; i < m; i++)
H[i] = new Vector(m + 1);
d = new Vector(m + 1);
for (int k = 1; k <= maxIterations && residualNorm / rightPartNorm > epsilon; k++)
{
d.Nullify();
V[0] = residual * (1.0 / residualNorm);
continueCalculations = true;
for (int j = 1; j <= m && continueCalculations; j++)
{
tmp = matrix.QSolve(V[j - 1]);
w = matrix.SSolve(matrix.SourceMatrix.Multiply(tmp));
for (int l = 1; l <= j; l++)
{
H[j - 1][l - 1] = V[l - 1] * w;
w = w - V[l - 1] * H[j - 1][l - 1];
}
H[j - 1][j] = w.Norm();
if (Math.Abs(H[j - 1][j]) < 1e-10)
{
m = j;
continueCalculations = false;
}
else
{
if(j != m)
V[j] = w * (1.0 / H[j - 1][j]);
}
}
d[0] = residualNorm;
z = solveMinSqrProblem(d, H, m);
x = x + multiplyMatrixVector(z, V);
tmp = rightPart - matrix.SourceMatrix.Multiply(matrix.QSolve(x));
residual = matrix.SSolve(tmp);
residualNorm = residual.Norm();
if (System.Double.IsInfinity(residualNorm / rightPartNorm))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by GMRES.");
return x;
}
if (System.Double.IsNaN(residualNorm / rightPartNorm))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by GMRES.");
return x;
}
solverLogger.AddIterationInfo(k, residualNorm / rightPartNorm);
}
return matrix.QSolve(x);
}
else {
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GMRES");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GMRES");
}
}
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution,
ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
if (solverParametrs is GaussSeidelParametrs)
{
GaussSeidelParametrs GZParametrs = solverParametrs as GaussSeidelParametrs;
Vector x,xnext, r, di;
int size, k;
double Residual, w;
Vector DEx,DEb, Dx, Fx, Ex;
size = initialSolution.Size;
x = new Vector(size);
xnext = new Vector(size);
r = new Vector(size);
di = new Vector(size);
x = initialSolution;
w = GZParametrs.Relaxation;
di = matrix.SourceMatrix.Diagonal;
Dx = Vector.Mult(di, x);
Fx = matrix.SourceMatrix.LMult(x, false);
Ex = matrix.SourceMatrix.UMult(x, false);
r = Dx+Fx+ Ex - rightPart;
var rpnorm = rightPart.Norm();
Residual = r.Norm() / rpnorm;
DEb = Vector.Division(rightPart,di);
solverLogger.AddIterationInfo(0, Residual);
for (k = 1; k <= GZParametrs.MaxIterations && Residual > GZParametrs.Epsilon; k++)
{
DEx = Vector.Division(Ex+Fx,di);
xnext = (DEb - DEx) * w + x * (1 - w);
Ex = matrix.SourceMatrix.UMult(xnext, false);
Dx = Vector.Mult(di, xnext);
DEx = Vector.Division(Ex + Fx, di);
x = xnext;
xnext = (DEb - DEx) * w + x * (1 - w);
Dx = Vector.Mult(di, xnext);
Fx = matrix.SourceMatrix.LMult(xnext, false);
r = Dx + Fx + Ex - rightPart;
Residual = r.Norm() / rpnorm;
if (System.Double.IsInfinity(Residual))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by GaussSeidel .");
return xnext;
}
if (System.Double.IsNaN(Residual))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by GaussSeidel.");
return xnext;
}
solverLogger.AddIterationInfo(k, Residual);
}
return xnext;
}
else {
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GaussSeidel");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GaussSeidel");
}
}
/// <summary>
/// Решение СЛАУ методом сопряженных градиентов
/// </summary>
/// <param name="A">Матрица СЛАУ</param>
/// <param name="b">Ветор правой части</param>
/// <param name="Initial">Ветор начального приближения</param>
/// <param name="Precision">Точность</param>
/// <param name="Maxiter">Максимальное число итераций</param>
/// <returns>Вектор x - решение СЛАУ Ax=b с заданной точностью</returns>
public IVector Solve(IPreconditioner Preconditioner, IMatrix A, IVector b, IVector Initial, double Precision, int Maxiter, ILogger Logger)
{
Logger.WriteNameSolution("MSG", Preconditioner.getName());
string start = DateTime.Now.ToString("dd.MM.yyyy hh:mm:ss:fff");
Logger.setMaxIter(Maxiter);
IVector x = Preconditioner.MultU(Initial);
double scalAzZ, scalRR, alpha, beta = 1.0;
IVector r = b.Add(A.Mult(Initial), 1, -1);
r = Preconditioner.T.SolveL(Preconditioner.SolveL(r));
IVector Az, Atz, z = A.Transpose.Mult(r);
z = Preconditioner.T.SolveU(z);
r = z.Clone() as IVector;
scalRR = r.ScalarMult(r);
double normR = Math.Sqrt(scalRR) / b.Norm;
for (int iter = 0; iter < Maxiter && normR > Precision && beta > 0; iter++)
{
Az = Preconditioner.SolveU(z);
Atz = A.Mult(Az);
Atz = Preconditioner.T.SolveL(Preconditioner.SolveL(Atz));
Az = A.Transpose.Mult(Atz);
Az = Preconditioner.T.SolveU(Az);
scalAzZ = Az.ScalarMult(z);
if (scalAzZ == 0)
{
Logger.WriteSolution(x, Maxiter, b.Add(A.Mult(x), -1, 1).Norm);
Logger.WriteTime(start, DateTime.Now.ToString("dd.MM.yyyy hh:mm:ss:fff"));
throw new DivideByZeroException("Division by 0");
}
alpha = scalRR / scalAzZ;
x.Add(z, 1, alpha, true);
r.Add(Az, 1, -alpha, true);
beta = scalRR;
if (scalRR == 0)
{
Logger.WriteSolution(x, Maxiter, b.Add(A.Mult(x), -1, 1).Norm);
Logger.WriteTime(start, DateTime.Now.ToString("dd.MM.yyyy hh:mm:ss:fff"));
throw new DivideByZeroException("Division by 0");
}
scalRR = r.ScalarMult(r);
beta = scalRR / beta;
z = r.Add(z, 1, beta);
normR = Math.Sqrt(scalRR) / b.Norm;
Factory.Residual.Add(normR);
Logger.WriteIteration(iter, normR);
if (double.IsNaN(normR) || double.IsInfinity(normR))
{
x = Preconditioner.SolveU(x);
Logger.WriteSolution(x, Maxiter, b.Add(A.Mult(x), -1, 1).Norm);
Logger.WriteTime(start, DateTime.Now.ToString("dd.MM.yyyy hh:mm:ss:fff"));
throw new CantSolveException();
}
}
;
x = Preconditioner.SolveU(x);
Logger.WriteSolution(x, Maxiter, b.Add(A.Mult(x), -1, 1).Norm);
Logger.WriteTime(start, DateTime.Now.ToString("dd.MM.yyyy hh:mm:ss:fff"));
return(x);
}
/// <summary>
/// Solves the matrix equation Ax = b, where A is the coefficient matrix, b is the
/// solution vector and x is the unknown vector.
/// </summary>
/// <param name="matrix">The coefficient <see cref="Matrix"/>, <c>A</c>.</param>
/// <param name="input">The solution <see cref="Vector"/>, <c>b</c>.</param>
/// <param name="result">The result <see cref="Vector"/>, <c>x</c>.</param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
public void Solve(Matrix <double> matrix, Vector <double> input, Vector <double> result,
Iterator <double> iterator, IPreconditioner <double> preconditioner)
{
var A = (SparseMatrix)matrix;
var M = preconditioner;
var b = (DenseVector)input;
var x = (DenseVector)result;
double atolf = 0.0;
double rtol_1 = 0.0;
int recompute_residual_p = 0;
var p = new DenseVector(b.Count);
var s = new DenseVector(b.Count);
var r = new DenseVector(b.Count);
double alpha, beta;
double gamma, gamma_old;
double bi_prod;
double sdotp;
bool recompute_true_residual = false;
int i = 0;
M.Initialize(A);
// Start pcg solve
// bi_prod = <C*b,b>
//VectorHelper.Clear(p);
M.Approximate(input, p);
bi_prod = VectorHelper.DotProduct(p, b);
if (bi_prod > 0.0)
{
if (atolf > 0) // mixed relative and absolute tolerance
{
bi_prod += atolf;
}
}
else // bi_prod==0.0: the rhs vector b is zero
{
// Set x equal to zero and return
VectorHelper.Copy(b, x);
return;
}
// r = b - Ax
VectorHelper.Copy(b, r);
A.Multiply(-1.0, result, 1.0, r);
// p = C*r
//VectorHelper.Clear(p);
M.Approximate(r, p);
// gamma = <r,p>
gamma = VectorHelper.DotProduct(r, p);
while (iterator.DetermineStatus(i, x, b, r) == IterationStatus.Continue)
{
// the core CG calculations...
i++;
// At user request, periodically recompute the residual from the formula
// r = b - A x (instead of using the recursive definition). Note that this
// is potentially expensive and can lead to degraded convergence (since it
// essentially a "restarted CG").
recompute_true_residual = (recompute_residual_p > 0) && !((i % recompute_residual_p) == 0);
// s = A*p
A.Multiply(1.0, p, 0.0, s);
// alpha = gamma / <s,p>
sdotp = VectorHelper.DotProduct(s, p);
if (sdotp == 0.0)
{
throw new NumericalBreakdownException();
}
alpha = gamma / sdotp;
gamma_old = gamma;
// x = x + alpha*p
VectorHelper.Add(alpha, p, x, x);
// r = r - alpha*s
if (recompute_true_residual)
{
//Recomputing the residual...
VectorHelper.Copy(b, r);
A.Multiply(-1.0, result, 1.0, r);
}
else
{
VectorHelper.Add(-alpha, s, r, r);
}
// s = C*r
VectorHelper.Clear(s);
M.Approximate(r, s);
// gamma = <r,s>
gamma = VectorHelper.DotProduct(r, s);
// residual-based stopping criteria: ||r_new-r_old||_C < rtol ||b||_C
if (rtol_1 > 0)
{
// use that ||r_new-r_old||_C^2 = (r_new ,C r_new) + (r_old, C r_old)
if ((gamma + gamma_old) / bi_prod < rtol_1 * rtol_1)
{
break;
}
}
// ... gamma should be >=0. IEEE subnormal numbers are < 2**(-1022)=2.2e-308
// (and >= 2**(-1074)=4.9e-324). So a gamma this small means we're getting
// dangerously close to subnormal or zero numbers (usually if gamma is small,
// so will be other variables). Thus further calculations risk a crash.
// Such small gamma generally means no hope of progress anyway.
if (Math.Abs(gamma) < TINY)
{
throw new NumericalBreakdownException();
}
// beta = gamma / gamma_old
beta = gamma / gamma_old;
// p = s + beta p
if (recompute_true_residual)
{
VectorHelper.Copy(s, p);
}
else
{
p.Scale(beta);
VectorHelper.Add(1.0, s, p, p);
}
}
}
/// <summary>
/// Решение СЛАУ методом локально-оптимальной схемы
/// </summary>
/// <param name="preconditioner">Матрица СЛАУ</param>
/// <param name="b">Ветор правой части</param>
/// <param name="Initial">Ветор начального приближения</param>
/// <param name="Precision">Точность</param>
/// <param name="Maxiter">Максимальное число итераций</param>
/// <returns>Вектор x - решение СЛАУ Ax=b с заданной точностью</returns>
public IVector Solve(IPreconditioner preconditioner, IMatrix A, IVector b, IVector Initial, double Precision, int Maxiter, ILogger Logger)
{
Logger.WriteNameSolution("LOS", preconditioner.getName());
string start = DateTime.Now.ToString("dd.MM.yyyy hh:mm:ss:fff");
Logger.setMaxIter(Maxiter);
IVector x = Initial.Clone() as IVector;
double alpha = 0.0, beta = 0.0;
IVector r = b.Add(A.Mult(Initial), 1, -1); //r_0 = f - Ax_0
r = preconditioner.SolveL(r); // r_0 = L^-1 * (f - Ax_0)
IVector Ar, z = preconditioner.SolveU(r); // z_0 = U^-1 * r_0
IVector p = preconditioner.SolveL(A.Mult(z)); // p_0 = L^-1 * Az_0
double p_r = 0.0, p_p = 0.0;
double scalRR = r.ScalarMult(r);
double normR = Math.Sqrt(scalRR) / b.Norm;
for (int iter = 0; iter < Maxiter && normR > Precision; iter++)
//for (int iter = 0; iter < Maxiter && ; iter++)
{
p_r = p.ScalarMult(r); //(p_k-1,r_k-1)
p_p = p.ScalarMult(p); //(p_k-1,p_k-1)
alpha = p_r / p_p;
x.Add(z, 1, alpha, true); // x_k = x_k-1 + alfa_k*z_k-1
r.Add(p, 1, -alpha, true); // r_k = r_k-1 - alfa_k*p_k-1
Ar = preconditioner.SolveL(A.Mult(preconditioner.SolveU(r))); //Ar_k = L^-1 * A * U^-1 * r_k
//Ar = A.SolveU(r);
//Ar = AA.Mult(Ar);
//Ar = A.SolveL(Ar);
beta = -(p.ScalarMult(Ar) / p_p);
z = preconditioner.SolveU(r).Add(z, 1, beta); //z_k = U^-1 * r_k + beta_k*z_k-1
p = Ar.Add(p, 1, beta); // p_k = L^-1 * A * U^-1 * r_k + beta_k*p_k-1
if (scalRR == 0)
{
Logger.WriteSolution(x, Maxiter, b.Add(A.Mult(x), -1, 1).Norm);
Logger.WriteTime(start, DateTime.Now.ToString("dd.MM.yyyy hh:mm:ss:fff"));
throw new DivideByZeroException("Division by 0");
}
scalRR = r.ScalarMult(r);
normR = Math.Sqrt(scalRR) / b.Norm;
Factory.Residual.Add(normR);
Logger.WriteIteration(iter, normR);
if (double.IsNaN(normR) || double.IsInfinity(normR))
{
Logger.WriteSolution(x, Maxiter, b.Add(A.Mult(x), -1, 1).Norm);
Logger.WriteTime(start, DateTime.Now.ToString("dd.MM.yyyy hh:mm:ss:fff"));
throw new CantSolveException();
}
}
Logger.WriteSolution(x, Maxiter, b.Add(A.Mult(x), -1, 1).Norm);
Logger.WriteTime(start, DateTime.Now.ToString("dd.MM.yyyy hh:mm:ss:fff"));
return(x);
}
/// <summary>
/// Solves the matrix equation AX = B, where A is the coefficient matrix (this matrix), B is the solution matrix and X is the unknown matrix.
/// </summary>
/// <param name="input">The solution matrix <c>B</c>.</param>
/// <param name="solver">The iterative solver to use.</param>
/// <param name="stopCriteria">Criteria to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
/// <returns>The result matrix <c>X</c>.</returns>
public Matrix <T> SolveIterative(Matrix <T> input, IIterativeSolver <T> solver, IPreconditioner <T> preconditioner, params IIterationStopCriterion <T>[] stopCriteria)
{
var result = Build.Dense(input.RowCount, input.ColumnCount);
TrySolveIterative(input, result, solver, preconditioner, stopCriteria);
return(result);
}
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution,
ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
JacobiParametrs JacParametrs = solverParametrs as JacobiParametrs;
if (JacParametrs != null)
{
Vector x, r, diagonal;
int size, k;
double relativeResidual, w, rightPartNorm;
Vector Db, Dx, Lx, Ux, DULx;
size = initialSolution.Size;
x = new Vector(size);
r = new Vector(size);
diagonal = new Vector(size);
x = initialSolution;
w = JacParametrs.Relaxation;
diagonal = matrix.SourceMatrix.Diagonal;
rightPartNorm = rightPart.Norm();
Dx = Vector.Mult(diagonal, x);
Lx = matrix.SourceMatrix.LMult(x, false);
Ux = matrix.SourceMatrix.UMult(x, false);
r = Lx + Dx + Ux - rightPart;
relativeResidual = r.Norm() / rightPartNorm;
if (System.Double.IsInfinity(relativeResidual))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by Jacobi.");
return(x);
}
if (System.Double.IsNaN(relativeResidual))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by Jacobi.");
return(x);
}
Db = Vector.Division(rightPart, diagonal);
solverLogger.AddIterationInfo(0, relativeResidual);
for (k = 1; k <= solverParametrs.MaxIterations && relativeResidual > solverParametrs.Epsilon; k++)
{
DULx = Vector.Division(Ux + Lx, diagonal);
x = (Db - DULx) * w + x * (1 - w);
Dx = Vector.Mult(diagonal, x);
Lx = matrix.SourceMatrix.LMult(x, false);
Ux = matrix.SourceMatrix.UMult(x, false);
r = Lx + Dx + Ux - rightPart;
relativeResidual = r.Norm() / rightPartNorm;
if (System.Double.IsInfinity(relativeResidual))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by Jacobi.");
return(x);
}
if (System.Double.IsNaN(relativeResidual))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by Jacobi.");
return(x);
}
solverLogger.AddIterationInfo(k, relativeResidual);
}
return(x);
}
else
{
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in Jacobi");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in Jacobi");
}
}
/// <summary>
/// Solves the matrix equation Ax = b, where A is the coefficient matrix, b is the
/// solution vector and x is the unknown vector.
/// </summary>
/// <param name="matrix">The coefficient matrix, <c>A</c>.</param>
/// <param name="input">The solution vector, <c>b</c></param>
/// <param name="result">The result vector, <c>x</c></param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
public void Solve(Matrix <Numerics.Complex32> matrix, Vector <Numerics.Complex32> input, Vector <Numerics.Complex32> result, Iterator <Numerics.Complex32> iterator, IPreconditioner <Numerics.Complex32> preconditioner)
{
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, nameof(matrix));
}
if (result.Count != input.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (input.Count != matrix.RowCount)
{
throw Matrix.DimensionsDontMatch <ArgumentException>(input, matrix);
}
if (iterator == null)
{
iterator = new Iterator <Numerics.Complex32>();
}
if (preconditioner == null)
{
preconditioner = new UnitPreconditioner <Numerics.Complex32>();
}
preconditioner.Initialize(matrix);
var d = new DenseVector(input.Count);
var r = DenseVector.OfVector(input);
var uodd = new DenseVector(input.Count);
var ueven = new DenseVector(input.Count);
var v = new DenseVector(input.Count);
var pseudoResiduals = DenseVector.OfVector(input);
var x = new DenseVector(input.Count);
var yodd = new DenseVector(input.Count);
var yeven = DenseVector.OfVector(input);
// Temp vectors
var temp = new DenseVector(input.Count);
var temp1 = new DenseVector(input.Count);
var temp2 = new DenseVector(input.Count);
// Define the scalars
Numerics.Complex32 alpha = 0;
Numerics.Complex32 eta = 0;
float theta = 0;
// Initialize
var tau = (float)input.L2Norm();
Numerics.Complex32 rho = tau * tau;
// Calculate the initial values for v
// M temp = yEven
preconditioner.Approximate(yeven, temp);
// v = A temp
matrix.Multiply(temp, v);
// Set uOdd
v.CopyTo(ueven);
// Start the iteration
var iterationNumber = 0;
while (iterator.DetermineStatus(iterationNumber, result, input, pseudoResiduals) == IterationStatus.Continue)
{
// First part of the step, the even bit
if (IsEven(iterationNumber))
{
// sigma = (v, r)
var sigma = r.ConjugateDotProduct(v);
if (sigma.Real.AlmostEqualNumbersBetween(0, 1) && sigma.Imaginary.AlmostEqualNumbersBetween(0, 1))
{
// FAIL HERE
iterator.Cancel();
break;
}
// alpha = rho / sigma
alpha = rho / sigma;
// yOdd = yEven - alpha * v
v.Multiply(-alpha, temp1);
yeven.Add(temp1, yodd);
// Solve M temp = yOdd
preconditioner.Approximate(yodd, temp);
// uOdd = A temp
matrix.Multiply(temp, uodd);
}
// The intermediate step which is equal for both even and
// odd iteration steps.
// Select the correct vector
var uinternal = IsEven(iterationNumber) ? ueven : uodd;
var yinternal = IsEven(iterationNumber) ? yeven : yodd;
// pseudoResiduals = pseudoResiduals - alpha * uOdd
uinternal.Multiply(-alpha, temp1);
pseudoResiduals.Add(temp1, temp2);
temp2.CopyTo(pseudoResiduals);
// d = yOdd + theta * theta * eta / alpha * d
d.Multiply(theta * theta * eta / alpha, temp);
yinternal.Add(temp, d);
// theta = ||pseudoResiduals||_2 / tau
theta = (float)pseudoResiduals.L2Norm() / tau;
var c = 1 / (float)Math.Sqrt(1 + (theta * theta));
// tau = tau * theta * c
tau *= theta * c;
// eta = c^2 * alpha
eta = c * c * alpha;
// x = x + eta * d
d.Multiply(eta, temp1);
x.Add(temp1, temp2);
temp2.CopyTo(x);
// Check convergence and see if we can bail
if (iterator.DetermineStatus(iterationNumber, result, input, pseudoResiduals) != IterationStatus.Continue)
{
// Calculate the real values
preconditioner.Approximate(x, result);
// Calculate the true residual. Use the temp vector for that
// so that we don't pollute the pseudoResidual vector for no
// good reason.
CalculateTrueResidual(matrix, temp, result, input);
// Now recheck the convergence
if (iterator.DetermineStatus(iterationNumber, result, input, temp) != IterationStatus.Continue)
{
// We're all good now.
return;
}
}
// The odd step
if (!IsEven(iterationNumber))
{
if (rho.Real.AlmostEqualNumbersBetween(0, 1) && rho.Imaginary.AlmostEqualNumbersBetween(0, 1))
{
// FAIL HERE
iterator.Cancel();
break;
}
var rhoNew = r.ConjugateDotProduct(pseudoResiduals);
var beta = rhoNew / rho;
// Update rho for the next loop
rho = rhoNew;
// yOdd = pseudoResiduals + beta * yOdd
yodd.Multiply(beta, temp1);
pseudoResiduals.Add(temp1, yeven);
// Solve M temp = yOdd
preconditioner.Approximate(yeven, temp);
// uOdd = A temp
matrix.Multiply(temp, ueven);
// v = uEven + beta * (uOdd + beta * v)
v.Multiply(beta, temp1);
uodd.Add(temp1, temp);
temp.Multiply(beta, temp1);
ueven.Add(temp1, v);
}
// Calculate the real values
preconditioner.Approximate(x, result);
iterationNumber++;
}
}
public PCG(IPreconditioner <double> M)
{
this.Preconditioner = M;
}
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution,
ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
JacobiParametrs JacParametrs = solverParametrs as JacobiParametrs;
if (JacParametrs != null)
{
Vector x, r, diagonal;
int size, k;
double relativeResidual, w, rightPartNorm;
Vector Db, Dx, Lx, Ux, DULx;
size = initialSolution.Size;
x = new Vector(size);
r = new Vector(size);
diagonal = new Vector(size);
x = initialSolution;
w = JacParametrs.Relaxation;
diagonal = matrix.SourceMatrix.Diagonal;
rightPartNorm = rightPart.Norm();
Dx = Vector.Mult(diagonal, x);
Lx = matrix.SourceMatrix.LMult(x, false);
Ux = matrix.SourceMatrix.UMult(x, false);
r = Lx + Dx + Ux - rightPart;
relativeResidual = r.Norm() / rightPartNorm;
if (System.Double.IsInfinity(relativeResidual))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by Jacobi.");
return x;
}
if (System.Double.IsNaN(relativeResidual))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by Jacobi.");
return x;
}
Db = Vector.Division(rightPart, diagonal);
solverLogger.AddIterationInfo(0, relativeResidual);
for (k = 1; k <= solverParametrs.MaxIterations && relativeResidual > solverParametrs.Epsilon; k++)
{
DULx = Vector.Division(Ux + Lx, diagonal);
x = (Db - DULx) * w + x * (1 - w);
Dx = Vector.Mult(diagonal, x);
Lx = matrix.SourceMatrix.LMult(x, false);
Ux = matrix.SourceMatrix.UMult(x, false);
r = Lx + Dx + Ux - rightPart;
relativeResidual = r.Norm() / rightPartNorm;
if (System.Double.IsInfinity(relativeResidual))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by Jacobi.");
return x;
}
if (System.Double.IsNaN(relativeResidual))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by Jacobi.");
return x;
}
solverLogger.AddIterationInfo(k, relativeResidual);
}
return x;
}
else {
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in Jacobi");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in Jacobi");
}
}
public abstract Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution,
ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs);
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution,
ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
GMRESParameters GMRESParameters = solverParametrs as GMRESParameters;
if (GMRESParameters != null)
{
Vector[] V, H;
Vector residual, x, d, z, w, tmp;
bool continueCalculations;
double epsilon = GMRESParameters.Epsilon;
int m = GMRESParameters.M;
int maxIterations = GMRESParameters.MaxIterations;
int n = rightPart.Size;
x = initialSolution;
residual = rightPart - matrix.SourceMatrix.Multiply(x);
residual = matrix.SSolve(residual);
double rightPartNorm = matrix.SSolve(rightPart).Norm();
double residualNorm = residual.Norm();
if (System.Double.IsInfinity(residualNorm / rightPartNorm))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by GMRES.");
return(x);
}
if (System.Double.IsNaN(residualNorm / rightPartNorm))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by GMRES.");
return(x);
}
x = matrix.QMultiply(initialSolution);
V = new Vector[m];
for (int i = 0; i < m; i++)
{
V[i] = new Vector(n);
}
H = new Vector[m];
for (int i = 0; i < m; i++)
{
H[i] = new Vector(m + 1);
}
d = new Vector(m + 1);
for (int k = 1; k <= maxIterations && residualNorm / rightPartNorm > epsilon; k++)
{
d.Nullify();
V[0] = residual * (1.0 / residualNorm);
continueCalculations = true;
for (int j = 1; j <= m && continueCalculations; j++)
{
tmp = matrix.QSolve(V[j - 1]);
w = matrix.SSolve(matrix.SourceMatrix.Multiply(tmp));
for (int l = 1; l <= j; l++)
{
H[j - 1][l - 1] = V[l - 1] * w;
w = w - V[l - 1] * H[j - 1][l - 1];
}
H[j - 1][j] = w.Norm();
if (Math.Abs(H[j - 1][j]) < 1e-10)
{
m = j;
continueCalculations = false;
}
else
{
if (j != m)
{
V[j] = w * (1.0 / H[j - 1][j]);
}
}
}
d[0] = residualNorm;
z = solveMinSqrProblem(d, H, m);
x = x + multiplyMatrixVector(z, V);
tmp = rightPart - matrix.SourceMatrix.Multiply(matrix.QSolve(x));
residual = matrix.SSolve(tmp);
residualNorm = residual.Norm();
if (System.Double.IsInfinity(residualNorm / rightPartNorm))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by GMRES.");
return(x);
}
if (System.Double.IsNaN(residualNorm / rightPartNorm))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by GMRES.");
return(x);
}
solverLogger.AddIterationInfo(k, residualNorm / rightPartNorm);
}
return(matrix.QSolve(x));
}
else
{
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GMRES");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in GMRES");
}
}
/// <summary> /// Check the result. /// </summary> /// <param name="preconditioner">Specific preconditioner.</param> /// <param name="matrix">Source matrix.</param> /// <param name="vector">Initial vector.</param> /// <param name="result">Result vector.</param> protected abstract void CheckResult(IPreconditioner<double> preconditioner, SparseMatrix matrix, Vector<double> vector, Vector<double> result);
/// <summary>
/// Solves the matrix equation Ax = b, where A is the coefficient matrix, b is the
/// solution vector and x is the unknown vector.
/// </summary>
/// <param name="matrix">The coefficient matrix, <c>A</c>.</param>
/// <param name="input">The solution vector, <c>b</c></param>
/// <param name="result">The result vector, <c>x</c></param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
public void Solve(Matrix <float> matrix, Vector <float> input, Vector <float> result, Iterator <float> iterator, IPreconditioner <float> preconditioner)
{
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resource.ArgumentMatrixSquare, "matrix");
}
if (result.Count != input.Count)
{
throw new ArgumentException(Resource.ArgumentVectorsSameLength);
}
if (iterator == null)
{
iterator = new Iterator <float>();
}
if (preconditioner == null)
{
preconditioner = new UnitPreconditioner <float>();
}
// Create a copy of the solution and result vectors so we can use them
// later on
var internalInput = input.Clone();
var internalResult = result.Clone();
foreach (var solver in _solvers)
{
// Store a reference to the solver so we can stop it.
IterationStatus status;
try
{
// Reset the iterator and pass it to the solver
iterator.Reset();
// Start the solver
solver.Item1.Solve(matrix, internalInput, internalResult, iterator, solver.Item2 ?? preconditioner);
status = iterator.Status;
}
catch (Exception)
{
// The solver broke down.
// Log a message about this
// Switch to the next preconditioner.
// Reset the solution vector to the previous solution
input.CopyTo(internalInput);
continue;
}
// There was no fatal breakdown so check the status
if (status == IterationStatus.Converged)
{
// We're done
internalResult.CopyTo(result);
break;
}
// We're not done
// Either:
// - calculation finished without convergence
if (status == IterationStatus.StoppedWithoutConvergence)
{
// Copy the internal result to the result vector and
// continue with the calculation.
internalResult.CopyTo(result);
}
else
{
// - calculation failed --> restart with the original vector
// - calculation diverged --> restart with the original vector
// - Some unknown status occurred --> To be safe restart.
input.CopyTo(internalInput);
}
}
}
// Iterative Solvers: Full
/// <summary>
/// Solves the matrix equation Ax = b, where A is the coefficient matrix (this matrix), b is the solution vector and x is the unknown vector.
/// </summary>
/// <param name="input">The solution vector <c>b</c>.</param>
/// <param name="result">The result vector <c>x</c>.</param>
/// <param name="solver">The iterative solver to use.</param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
public IterationStatus TrySolveIterative(Vector <T> input, Vector <T> result, IIterativeSolver <T> solver, Iterator <T> iterator = null, IPreconditioner <T> preconditioner = null)
{
if (iterator == null)
{
iterator = new Iterator <T>(Build.IterativeSolverStopCriteria());
}
if (preconditioner == null)
{
preconditioner = new UnitPreconditioner <T>();
}
solver.Solve(this, input, result, iterator, preconditioner);
return(iterator.Status);
}
/// <summary> /// Check the result. /// </summary> /// <param name="preconditioner">Specific preconditioner.</param> /// <param name="matrix">Source matrix.</param> /// <param name="vector">Initial vector.</param> /// <param name="result">Result vector.</param> protected abstract void CheckResult(IPreconditioner <Complex> preconditioner, SparseMatrix matrix, Vector <Complex> vector, Vector <Complex> result);
public override void HandleMatrixWillBeSet()
{
mustUpdatePreconditioner = true;
preconditioner = null;
}
/// <summary>
/// Solves the matrix equation Ax = b, where A is the coefficient matrix, b is the
/// solution vector and x is the unknown vector.
/// </summary>
/// <param name="matrix">The coefficient <see cref="Matrix"/>, <c>A</c>.</param>
/// <param name="input">The solution <see cref="Vector"/>, <c>b</c>.</param>
/// <param name="result">The result <see cref="Vector"/>, <c>x</c>.</param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
public void Solve(Matrix <Complex> matrix, Vector <Complex> input, Vector <Complex> result, Iterator <Complex> iterator, IPreconditioner <Complex> preconditioner)
{
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
if (result.Count != input.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (input.Count != matrix.RowCount)
{
throw Matrix.DimensionsDontMatch <ArgumentException>(input, result);
}
if (iterator == null)
{
iterator = new Iterator <Complex>();
}
if (preconditioner == null)
{
preconditioner = new UnitPreconditioner <Complex>();
}
preconditioner.Initialize(matrix);
// Compute r_0 = b - Ax_0 for some initial guess x_0
// In this case we take x_0 = vector
// This is basically a SAXPY so it could be made a lot faster
var residuals = new DenseVector(matrix.RowCount);
CalculateTrueResidual(matrix, residuals, result, input);
// Choose r~ (for example, r~ = r_0)
var tempResiduals = residuals.Clone();
// create seven temporary vectors needed to hold temporary
// coefficients. All vectors are mangled in each iteration.
// These are defined here to prevent stressing the garbage collector
var vecP = new DenseVector(residuals.Count);
var vecPdash = new DenseVector(residuals.Count);
var nu = new DenseVector(residuals.Count);
var vecS = new DenseVector(residuals.Count);
var vecSdash = new DenseVector(residuals.Count);
var temp = new DenseVector(residuals.Count);
var temp2 = new DenseVector(residuals.Count);
// create some temporary double variables that are needed
// to hold values in between iterations
Complex currentRho = 0;
Complex alpha = 0;
Complex omega = 0;
var iterationNumber = 0;
while (iterator.DetermineStatus(iterationNumber, result, input, residuals) == IterationStatus.Continue)
{
// rho_(i-1) = r~^T r_(i-1) // dotproduct r~ and r_(i-1)
var oldRho = currentRho;
currentRho = tempResiduals.ConjugateDotProduct(residuals);
// if (rho_(i-1) == 0) // METHOD FAILS
// If rho is only 1 ULP from zero then we fail.
if (currentRho.Real.AlmostEqualNumbersBetween(0, 1) && currentRho.Imaginary.AlmostEqualNumbersBetween(0, 1))
{
// Rho-type breakdown
throw new NumericalBreakdownException();
}
if (iterationNumber != 0)
{
// beta_(i-1) = (rho_(i-1)/rho_(i-2))(alpha_(i-1)/omega(i-1))
var beta = (currentRho / oldRho) * (alpha / omega);
// p_i = r_(i-1) + beta_(i-1)(p_(i-1) - omega_(i-1) * nu_(i-1))
nu.Multiply(-omega, temp);
vecP.Add(temp, temp2);
temp2.CopyTo(vecP);
vecP.Multiply(beta, vecP);
vecP.Add(residuals, temp2);
temp2.CopyTo(vecP);
}
else
{
// p_i = r_(i-1)
residuals.CopyTo(vecP);
}
// SOLVE Mp~ = p_i // M = preconditioner
preconditioner.Approximate(vecP, vecPdash);
// nu_i = Ap~
matrix.Multiply(vecPdash, nu);
// alpha_i = rho_(i-1)/ (r~^T nu_i) = rho / dotproduct(r~ and nu_i)
alpha = currentRho * 1 / tempResiduals.ConjugateDotProduct(nu);
// s = r_(i-1) - alpha_i nu_i
nu.Multiply(-alpha, temp);
residuals.Add(temp, vecS);
// Check if we're converged. If so then stop. Otherwise continue;
// Calculate the temporary result.
// Be careful not to change any of the temp vectors, except for
// temp. Others will be used in the calculation later on.
// x_i = x_(i-1) + alpha_i * p^_i + s^_i
vecPdash.Multiply(alpha, temp);
temp.Add(vecSdash, temp2);
temp2.CopyTo(temp);
temp.Add(result, temp2);
temp2.CopyTo(temp);
// Check convergence and stop if we are converged.
if (iterator.DetermineStatus(iterationNumber, temp, input, vecS) != IterationStatus.Continue)
{
temp.CopyTo(result);
// Calculate the true residual
CalculateTrueResidual(matrix, residuals, result, input);
// Now recheck the convergence
if (iterator.DetermineStatus(iterationNumber, result, input, residuals) != IterationStatus.Continue)
{
// We're all good now.
return;
}
// Continue the calculation
iterationNumber++;
continue;
}
// SOLVE Ms~ = s
preconditioner.Approximate(vecS, vecSdash);
// temp = As~
matrix.Multiply(vecSdash, temp);
// omega_i = temp^T s / temp^T temp
omega = temp.ConjugateDotProduct(vecS) / temp.ConjugateDotProduct(temp);
// x_i = x_(i-1) + alpha_i p^ + omega_i s^
temp.Multiply(-omega, residuals);
residuals.Add(vecS, temp2);
temp2.CopyTo(residuals);
vecSdash.Multiply(omega, temp);
result.Add(temp, temp2);
temp2.CopyTo(result);
vecPdash.Multiply(alpha, temp);
result.Add(temp, temp2);
temp2.CopyTo(result);
// for continuation it is necessary that omega_i != 0.0
// If omega is only 1 ULP from zero then we fail.
if (omega.Real.AlmostEqualNumbersBetween(0, 1) && omega.Imaginary.AlmostEqualNumbersBetween(0, 1))
{
// Omega-type breakdown
throw new NumericalBreakdownException();
}
if (iterator.DetermineStatus(iterationNumber, result, input, residuals) != IterationStatus.Continue)
{
// Recalculate the residuals and go round again. This is done to ensure that
// we have the proper residuals.
// The residual calculation based on omega_i * s can be off by a factor 10. So here
// we calculate the real residual (which can be expensive) but we only do it if we're
// sufficiently close to the finish.
CalculateTrueResidual(matrix, residuals, result, input);
}
iterationNumber++;
}
}
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution, ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
BCGStabParametrs ConGradParametrs = solverParametrs as BCGStabParametrs;
if (ConGradParametrs == null)
{
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in BSGStab");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in BSGSTab");
}
else
{
int oIter = 0;
double nPi, oPi, alpha, w, oNev, bNev, betta;
oPi = alpha = w = 1;
int size = rightPart.Size;
Vector x, r, rTab, s, t, z, sqt, y;
Vector v = new Vector(size);
Vector p = new Vector(size);
v.Nullify();
p.Nullify();
x = initialSolution;
r = rightPart - matrix.SourceMatrix.Multiply(x);
rTab = r;
bNev = rightPart.Norm();
oNev = r.Norm() / bNev;
solverLogger.AddIterationInfo(oIter, oNev);//logger
if (System.Double.IsInfinity(oNev))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by BSG Stab.");
return(x);
}
if (System.Double.IsNaN(oNev))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by BSG Stab.");
return(x);
}
while (oIter < ConGradParametrs.MaxIterations && oNev > ConGradParametrs.Epsilon)
{
nPi = rTab * r;
betta = (nPi / oPi) * (alpha / w);
p = r + (p - v * w) * betta;
y = matrix.QSolve(matrix.SSolve(p));
v = matrix.SourceMatrix.Multiply(y);
alpha = nPi / (rTab * v);
x = x + y * alpha;//УТОЧНИТЬ!!!!!!!!!!!!!!!
s = r - v * alpha;
if (s.Norm() / bNev < ConGradParametrs.Epsilon)
{
return(x);
}
z = matrix.QSolve(matrix.SSolve(s));
t = matrix.SourceMatrix.Multiply(z);
sqt = matrix.QSolve(matrix.SSolve(t));
w = (z * sqt) / (sqt * sqt);
x = x + z * w;//УТОЧНИТЬ!!!!!!!!!!!!!!!!!!!
r = s - t * w;
oPi = nPi;
oIter++;
oNev = r.Norm() / bNev;
if (System.Double.IsInfinity(oNev))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by BSG Stab.");
return(x);
}
if (System.Double.IsNaN(oNev))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by BSG Stab.");
return(x);
}
solverLogger.AddIterationInfo(oIter, oNev);//logger
}
return(x);
}
}
/// <summary>
/// Solves the matrix equation AX = B, where A is the coefficient matrix (this matrix), B is the solution matrix and X is the unknown matrix.
/// </summary>
/// <param name="input">The solution matrix <c>B</c>.</param>
/// <param name="result">The result matrix <c>X</c></param>
/// <param name="solver">The iterative solver to use.</param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
public IterationStatus TrySolveIterative(Matrix <T> input, Matrix <T> result, IIterativeSolver <T> solver, Iterator <T> iterator = null, IPreconditioner <T> preconditioner = null)
{
if (RowCount != input.RowCount || input.RowCount != result.RowCount || input.ColumnCount != result.ColumnCount)
{
throw DimensionsDontMatch <ArgumentException>(this, input, result);
}
if (iterator == null)
{
iterator = new Iterator <T>(Build.IterativeSolverStopCriteria());
}
if (preconditioner == null)
{
preconditioner = new UnitPreconditioner <T>();
}
for (var column = 0; column < input.ColumnCount; column++)
{
var solution = Vector <T> .Build.Dense(RowCount);
solver.Solve(this, input.Column(column), solution, iterator, preconditioner);
foreach (var element in solution.EnumerateIndexed(Zeros.AllowSkip))
{
result.At(element.Item1, column, element.Item2);
}
}
return(iterator.Status);
}
public abstract Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution,
ILogger logger,ISolverLogger solverLogger, ISolverParametrs solverParametrs);
// Iterative Solvers: Simple
/// <summary>
/// Solves the matrix equation Ax = b, where A is the coefficient matrix (this matrix), b is the solution vector and x is the unknown vector.
/// </summary>
/// <param name="input">The solution vector <c>b</c>.</param>
/// <param name="solver">The iterative solver to use.</param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
/// <returns>The result vector <c>x</c>.</returns>
public Vector <T> SolveIterative(Vector <T> input, IIterativeSolver <T> solver, Iterator <T> iterator = null, IPreconditioner <T> preconditioner = null)
{
var result = Vector <T> .Build.Dense(RowCount);
TrySolveIterative(input, result, solver, iterator, preconditioner);
return(result);
}
public IterativeStatistics Solve(IMatrixView matrix, IPreconditioner preconditioner, IVectorView rhs, IVector solution,
bool initialGuessIsZero, Func <IVector> zeroVectorInitializer)
{
return(Solve(new ExplicitMatrixTransformation(matrix), preconditioner, rhs, solution, initialGuessIsZero,
zeroVectorInitializer));
}
/// <summary>
/// Solves the matrix equation Ax = b, where A is the coefficient matrix (this matrix), b is the solution vector and x is the unknown vector.
/// </summary>
/// <param name="input">The solution vector <c>b</c>.</param>
/// <param name="solver">The iterative solver to use.</param>
/// <param name="stopCriteria">Criteria to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
/// <returns>The result vector <c>x</c>.</returns>
public Vector <T> SolveIterative(Vector <T> input, IIterativeSolver <T> solver, IPreconditioner <T> preconditioner, params IIterationStopCriterion <T>[] stopCriteria)
{
var result = Vector <T> .Build.Dense(RowCount);
TrySolveIterative(input, result, solver, preconditioner, stopCriteria);
return(result);
}
public IterativeStatistics Solve(ILinearTransformation matrix, IPreconditioner preconditioner, IVectorView rhs, IVector solution,
bool initialGuessIsZero, Func <IVector> zeroVectorInitializer)
{
Preconditions.CheckMultiplicationDimensions(matrix.NumColumns, solution.Length);
Preconditions.CheckSystemSolutionDimensions(matrix.NumRows, rhs.Length);
var innerIterations = innerIterationsProvider.GetMaxIterations(matrix.NumRows);
IVector[] v = new Vector[innerIterations + 1];
var y = Vector.CreateZero(innerIterations + 1);
var c = Vector.CreateZero(innerIterations + 1);
var s = Vector.CreateZero(innerIterations + 1);
var delta = 0.001;
double residualNorm = double.MaxValue;
var usedIterations = 0;
if (initialGuessIsZero)
{
residual = rhs.Copy();
}
else
{
residual = ExactResidual.Calculate(matrix, rhs, solution);
}
for (var iteration = 0; iteration < maximumIterations; iteration++)
{
preconditioner.SolveLinearSystem(residual, residual);
//var residual = ExactResidual.Calculate(matrix, rhs, solution);
residualNorm = residual.Norm2();
double residualTolerance;
if (iteration == 0)
{
residualTolerance = residualNorm * relativeTolerance;
}
v[0] = residual.Scale(1 / residualNorm);
var g = Vector.CreateZero(innerIterations + 1);
g[0] = residualNorm;
var hessenbergMatrix = Matrix.CreateZero(innerIterations + 1, innerIterations);
var indexIteration = 0;
for (int innerIteration = 0; innerIteration < innerIterations; innerIteration++)
{
indexIteration = innerIteration;
v[innerIteration + 1] = Vector.CreateZero(v[innerIteration].Length);
matrix.Multiply(v[innerIteration], v[innerIteration + 1]);
preconditioner.SolveLinearSystem(v[innerIteration + 1], v[innerIteration + 1]);
var av = v[innerIteration + 1].Norm2();
for (var j = 0; j <= innerIteration; j++)
{
hessenbergMatrix[j, innerIteration] = v[j].DotProduct(v[innerIteration + 1]);
v[innerIteration + 1] = v[innerIteration + 1].Subtract(v[j].Scale(hessenbergMatrix[j, innerIteration]));
}
hessenbergMatrix[innerIteration + 1, innerIteration] = v[innerIteration + 1].Norm2();
if (Math.Abs(av + delta * hessenbergMatrix[innerIteration + 1, innerIteration] - av) < 10e-9)
{
for (int j = 0; j <= innerIteration; j++)
{
var htmp = v[j].DotProduct(v[innerIteration + 1]);
hessenbergMatrix[j, innerIteration] += htmp;
v[innerIteration + 1].LinearCombinationIntoThis(1.0, v[j], -htmp);
}
hessenbergMatrix[innerIteration + 1, innerIteration] = v[innerIteration + 1].Norm2();
}
if (Math.Abs(hessenbergMatrix[innerIteration + 1, innerIteration]) > 10e-17)
{
v[innerIteration + 1].ScaleIntoThis(1 / hessenbergMatrix[innerIteration + 1, innerIteration]);
}
if (innerIteration > 0)
{
y = hessenbergMatrix.GetColumn(innerIteration).GetSubvector(0, innerIteration + 2);
for (int i = 0; i <= innerIteration - 1; i++)
{
y = CalculateGivensRotation(c[i], s[i], i, y);
}
hessenbergMatrix.SetSubcolumn(innerIteration, y);
}
var mu = Math.Sqrt(hessenbergMatrix[innerIteration, innerIteration] *
hessenbergMatrix[innerIteration, innerIteration] +
hessenbergMatrix[innerIteration + 1, innerIteration] *
hessenbergMatrix[innerIteration + 1, innerIteration]);
c[innerIteration] = hessenbergMatrix[innerIteration, innerIteration] / mu;
s[innerIteration] = -hessenbergMatrix[innerIteration + 1, innerIteration] / mu;
hessenbergMatrix[innerIteration, innerIteration] =
c[innerIteration] * hessenbergMatrix[innerIteration, innerIteration] -
s[innerIteration] * hessenbergMatrix[innerIteration + 1, innerIteration];
hessenbergMatrix[innerIteration + 1, innerIteration] = 0.0;
g = CalculateGivensRotation(c[innerIteration], s[innerIteration], innerIteration, g);
residualNorm = Math.Abs(g[innerIteration + 1]);
usedIterations++;
if (residualNorm <= relativeTolerance && residualNorm <= absoluteTolerance)
{
break;
}
}
indexIteration = indexIteration - 1;
y[indexIteration + 1] = g[indexIteration + 1] / hessenbergMatrix[indexIteration + 1, indexIteration + 1];
for (int i = indexIteration; i >= 0; i--)
{
y[i] = (g[i] - (hessenbergMatrix.GetRow(i).GetSubvector(i + 1, indexIteration + 2)
.DotProduct(y.GetSubvector(i + 1, indexIteration + 2)))) / hessenbergMatrix[i, i];
}
for (int i = 0; i < matrix.NumRows; i++)
{
var subV = Vector.CreateZero(indexIteration + 2);
for (int j = 0; j < indexIteration + 2; j++)
{
subV[j] = v[j][i];
}
solution.Set(i, solution[i] + subV.DotProduct(y.GetSubvector(0, indexIteration + 2)));
}
if (residualNorm <= relativeTolerance && residualNorm <= absoluteTolerance)
{
break;
}
}
return(new IterativeStatistics()
{
HasConverged = residualNorm <= relativeTolerance && residualNorm <= absoluteTolerance,
AlgorithmName = name,
NumIterationsRequired = usedIterations,
ResidualNormRatioEstimation = residualNorm
});
}
/// <summary>
/// Solves the matrix equation Ax = b, where A is the coefficient matrix, b is the
/// solution vector and x is the unknown vector.
/// </summary>
/// <param name="matrix">The coefficient matrix, <c>A</c>.</param>
/// <param name="input">The solution vector, <c>b</c></param>
/// <param name="result">The result vector, <c>x</c></param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
public void Solve(Matrix <Numerics.Complex32> matrix, Vector <Numerics.Complex32> input, Vector <Numerics.Complex32> result, Iterator <Numerics.Complex32> iterator, IPreconditioner <Numerics.Complex32> preconditioner)
{
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
if (result.Count != input.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (input.Count != matrix.RowCount)
{
throw Matrix.DimensionsDontMatch <ArgumentException>(input, matrix);
}
if (iterator == null)
{
iterator = new Iterator <Numerics.Complex32>();
}
if (preconditioner == null)
{
preconditioner = new UnitPreconditioner <Numerics.Complex32>();
}
preconditioner.Initialize(matrix);
// Choose an initial guess x_0
// Take x_0 = 0
var xtemp = new DenseVector(input.Count);
// Choose k vectors q_1, q_2, ..., q_k
// Build a new set if:
// a) the stored set doesn't exist (i.e. == null)
// b) Is of an incorrect length (i.e. too long)
// c) The vectors are of an incorrect length (i.e. too long or too short)
var useOld = false;
if (_startingVectors != null)
{
// We don't accept collections with zero starting vectors so ...
if (_startingVectors.Count <= NumberOfStartingVectorsToCreate(_numberOfStartingVectors, input.Count))
{
// Only check the first vector for sizing. If that matches we assume the
// other vectors match too. If they don't the process will crash
if (_startingVectors[0].Count == input.Count)
{
useOld = true;
}
}
}
_startingVectors = useOld ? _startingVectors : CreateStartingVectors(_numberOfStartingVectors, input.Count);
// Store the number of starting vectors. Not really necessary but easier to type :)
var k = _startingVectors.Count;
// r_0 = b - Ax_0
// This is basically a SAXPY so it could be made a lot faster
var residuals = new DenseVector(matrix.RowCount);
CalculateTrueResidual(matrix, residuals, xtemp, input);
// Define the temporary values
var c = new Numerics.Complex32[k];
// Define the temporary vectors
var gtemp = new DenseVector(residuals.Count);
var u = new DenseVector(residuals.Count);
var utemp = new DenseVector(residuals.Count);
var temp = new DenseVector(residuals.Count);
var temp1 = new DenseVector(residuals.Count);
var temp2 = new DenseVector(residuals.Count);
var zd = new DenseVector(residuals.Count);
var zg = new DenseVector(residuals.Count);
var zw = new DenseVector(residuals.Count);
var d = CreateVectorArray(_startingVectors.Count, residuals.Count);
// g_0 = r_0
var g = CreateVectorArray(_startingVectors.Count, residuals.Count);
residuals.CopyTo(g[k - 1]);
var w = CreateVectorArray(_startingVectors.Count, residuals.Count);
// FOR (j = 0, 1, 2 ....)
var iterationNumber = 0;
while (iterator.DetermineStatus(iterationNumber, xtemp, input, residuals) == IterationStatus.Continue)
{
// SOLVE M g~_((j-1)k+k) = g_((j-1)k+k)
preconditioner.Approximate(g[k - 1], gtemp);
// w_((j-1)k+k) = A g~_((j-1)k+k)
matrix.Multiply(gtemp, w[k - 1]);
// c_((j-1)k+k) = q^T_1 w_((j-1)k+k)
c[k - 1] = _startingVectors[0].ConjugateDotProduct(w[k - 1]);
if (c[k - 1].Real.AlmostEqualNumbersBetween(0, 1) && c[k - 1].Imaginary.AlmostEqualNumbersBetween(0, 1))
{
throw new Exception("Iterative solver experience a numerical break down");
}
// alpha_(jk+1) = q^T_1 r_((j-1)k+k) / c_((j-1)k+k)
var alpha = _startingVectors[0].ConjugateDotProduct(residuals) / c[k - 1];
// u_(jk+1) = r_((j-1)k+k) - alpha_(jk+1) w_((j-1)k+k)
w[k - 1].Multiply(-alpha, temp);
residuals.Add(temp, u);
// SOLVE M u~_(jk+1) = u_(jk+1)
preconditioner.Approximate(u, temp1);
temp1.CopyTo(utemp);
// rho_(j+1) = -u^t_(jk+1) A u~_(jk+1) / ||A u~_(jk+1)||^2
matrix.Multiply(temp1, temp);
var rho = temp.ConjugateDotProduct(temp);
// If rho is zero then temp is a zero vector and we're probably
// about to have zero residuals (i.e. an exact solution).
// So set rho to 1.0 because in the next step it will turn to zero.
if (rho.Real.AlmostEqualNumbersBetween(0, 1) && rho.Imaginary.AlmostEqualNumbersBetween(0, 1))
{
rho = 1.0f;
}
rho = -u.ConjugateDotProduct(temp) / rho;
// r_(jk+1) = rho_(j+1) A u~_(jk+1) + u_(jk+1)
u.CopyTo(residuals);
// Reuse temp
temp.Multiply(rho, temp);
residuals.Add(temp, temp2);
temp2.CopyTo(residuals);
// x_(jk+1) = x_((j-1)k_k) - rho_(j+1) u~_(jk+1) + alpha_(jk+1) g~_((j-1)k+k)
utemp.Multiply(-rho, temp);
xtemp.Add(temp, temp2);
temp2.CopyTo(xtemp);
gtemp.Multiply(alpha, gtemp);
xtemp.Add(gtemp, temp2);
temp2.CopyTo(xtemp);
// Check convergence and stop if we are converged.
if (iterator.DetermineStatus(iterationNumber, xtemp, input, residuals) != IterationStatus.Continue)
{
// Calculate the true residual
CalculateTrueResidual(matrix, residuals, xtemp, input);
// Now recheck the convergence
if (iterator.DetermineStatus(iterationNumber, xtemp, input, residuals) != IterationStatus.Continue)
{
// We're all good now.
// Exit from the while loop.
break;
}
}
// FOR (i = 1,2, ...., k)
for (var i = 0; i < k; i++)
{
// z_d = u_(jk+1)
u.CopyTo(zd);
// z_g = r_(jk+i)
residuals.CopyTo(zg);
// z_w = 0
zw.Clear();
// FOR (s = i, ...., k-1) AND j >= 1
Numerics.Complex32 beta;
if (iterationNumber >= 1)
{
for (var s = i; s < k - 1; s++)
{
// beta^(jk+i)_((j-1)k+s) = -q^t_(s+1) z_d / c_((j-1)k+s)
beta = -_startingVectors[s + 1].ConjugateDotProduct(zd) / c[s];
// z_d = z_d + beta^(jk+i)_((j-1)k+s) d_((j-1)k+s)
d[s].Multiply(beta, temp);
zd.Add(temp, temp2);
temp2.CopyTo(zd);
// z_g = z_g + beta^(jk+i)_((j-1)k+s) g_((j-1)k+s)
g[s].Multiply(beta, temp);
zg.Add(temp, temp2);
temp2.CopyTo(zg);
// z_w = z_w + beta^(jk+i)_((j-1)k+s) w_((j-1)k+s)
w[s].Multiply(beta, temp);
zw.Add(temp, temp2);
temp2.CopyTo(zw);
}
}
beta = rho * c[k - 1];
if (beta.Real.AlmostEqualNumbersBetween(0, 1) && beta.Imaginary.AlmostEqualNumbersBetween(0, 1))
{
throw new Exception("Iterative solver experience a numerical break down");
}
// beta^(jk+i)_((j-1)k+k) = -(q^T_1 (r_(jk+1) + rho_(j+1) z_w)) / (rho_(j+1) c_((j-1)k+k))
zw.Multiply(rho, temp2);
residuals.Add(temp2, temp);
beta = -_startingVectors[0].ConjugateDotProduct(temp) / beta;
// z_g = z_g + beta^(jk+i)_((j-1)k+k) g_((j-1)k+k)
g[k - 1].Multiply(beta, temp);
zg.Add(temp, temp2);
temp2.CopyTo(zg);
// z_w = rho_(j+1) (z_w + beta^(jk+i)_((j-1)k+k) w_((j-1)k+k))
w[k - 1].Multiply(beta, temp);
zw.Add(temp, temp2);
temp2.CopyTo(zw);
zw.Multiply(rho, zw);
// z_d = r_(jk+i) + z_w
residuals.Add(zw, zd);
// FOR (s = 1, ... i - 1)
for (var s = 0; s < i - 1; s++)
{
// beta^(jk+i)_(jk+s) = -q^T_s+1 z_d / c_(jk+s)
beta = -_startingVectors[s + 1].ConjugateDotProduct(zd) / c[s];
// z_d = z_d + beta^(jk+i)_(jk+s) * d_(jk+s)
d[s].Multiply(beta, temp);
zd.Add(temp, temp2);
temp2.CopyTo(zd);
// z_g = z_g + beta^(jk+i)_(jk+s) * g_(jk+s)
g[s].Multiply(beta, temp);
zg.Add(temp, temp2);
temp2.CopyTo(zg);
}
// d_(jk+i) = z_d - u_(jk+i)
zd.Subtract(u, d[i]);
// g_(jk+i) = z_g + z_w
zg.Add(zw, g[i]);
// IF (i < k - 1)
if (i < k - 1)
{
// c_(jk+1) = q^T_i+1 d_(jk+i)
c[i] = _startingVectors[i + 1].ConjugateDotProduct(d[i]);
if (c[i].Real.AlmostEqualNumbersBetween(0, 1) && c[i].Imaginary.AlmostEqualNumbersBetween(0, 1))
{
throw new Exception("Iterative solver experience a numerical break down");
}
// alpha_(jk+i+1) = q^T_(i+1) u_(jk+i) / c_(jk+i)
alpha = _startingVectors[i + 1].ConjugateDotProduct(u) / c[i];
// u_(jk+i+1) = u_(jk+i) - alpha_(jk+i+1) d_(jk+i)
d[i].Multiply(-alpha, temp);
u.Add(temp, temp2);
temp2.CopyTo(u);
// SOLVE M g~_(jk+i) = g_(jk+i)
preconditioner.Approximate(g[i], gtemp);
// x_(jk+i+1) = x_(jk+i) + rho_(j+1) alpha_(jk+i+1) g~_(jk+i)
gtemp.Multiply(rho * alpha, temp);
xtemp.Add(temp, temp2);
temp2.CopyTo(xtemp);
// w_(jk+i) = A g~_(jk+i)
matrix.Multiply(gtemp, w[i]);
// r_(jk+i+1) = r_(jk+i) - rho_(j+1) alpha_(jk+i+1) w_(jk+i)
w[i].Multiply(-rho * alpha, temp);
residuals.Add(temp, temp2);
temp2.CopyTo(residuals);
// We can check the residuals here if they're close
if (iterator.DetermineStatus(iterationNumber, xtemp, input, residuals) != IterationStatus.Continue)
{
// Recalculate the residuals and go round again. This is done to ensure that
// we have the proper residuals.
CalculateTrueResidual(matrix, residuals, xtemp, input);
}
}
} // END ITERATION OVER i
iterationNumber++;
}
// copy the temporary result to the real result vector
xtemp.CopyTo(result);
}
/// <summary> Sets left preconditioner</summary>
public virtual void setM1(IPreconditioner preconditioner)
{
M1 = preconditioner;
}
/// <summary>
/// Solves the matrix equation Ax = b, where A is the coefficient matrix, b is the
/// solution vector and x is the unknown vector.
/// </summary>
/// <param name="matrix">The coefficient matrix, <c>A</c>.</param>
/// <param name="input">The solution vector, <c>b</c></param>
/// <param name="result">The result vector, <c>x</c></param>
/// <param name="iterator">The iterator to use to control when to stop iterating.</param>
/// <param name="preconditioner">The preconditioner to use for approximations.</param>
public void Solve(Matrix <double> matrix, Vector <double> input, Vector <double> result, Iterator <double> iterator, IPreconditioner <double> preconditioner)
{
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, nameof(matrix));
}
if (result.Count != input.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (input.Count != matrix.RowCount)
{
throw Matrix.DimensionsDontMatch <ArgumentException>(input, matrix);
}
if (iterator == null)
{
iterator = new Iterator <double>();
}
if (preconditioner == null)
{
preconditioner = new UnitPreconditioner <double>();
}
preconditioner.Initialize(matrix);
// x_0 is initial guess
// Take x_0 = 0
var xtemp = new DenseVector(input.Count);
// r_0 = b - Ax_0
// This is basically a SAXPY so it could be made a lot faster
var residuals = new DenseVector(matrix.RowCount);
CalculateTrueResidual(matrix, residuals, xtemp, input);
// Define the temporary scalars
double beta = 0;
// Define the temporary vectors
// rDash_0 = r_0
var rdash = DenseVector.OfVector(residuals);
// t_-1 = 0
var t = new DenseVector(residuals.Count);
var t0 = new DenseVector(residuals.Count);
// w_-1 = 0
var w = new DenseVector(residuals.Count);
// Define the remaining temporary vectors
var c = new DenseVector(residuals.Count);
var p = new DenseVector(residuals.Count);
var s = new DenseVector(residuals.Count);
var u = new DenseVector(residuals.Count);
var y = new DenseVector(residuals.Count);
var z = new DenseVector(residuals.Count);
var temp = new DenseVector(residuals.Count);
var temp2 = new DenseVector(residuals.Count);
var temp3 = new DenseVector(residuals.Count);
// for (k = 0, 1, .... )
var iterationNumber = 0;
while (iterator.DetermineStatus(iterationNumber, xtemp, input, residuals) == IterationStatus.Continue)
{
// p_k = r_k + beta_(k-1) * (p_(k-1) - u_(k-1))
p.Subtract(u, temp);
temp.Multiply(beta, temp2);
residuals.Add(temp2, p);
// Solve M b_k = p_k
preconditioner.Approximate(p, temp);
// s_k = A b_k
matrix.Multiply(temp, s);
// alpha_k = (r*_0 * r_k) / (r*_0 * s_k)
var alpha = rdash.DotProduct(residuals) / rdash.DotProduct(s);
// y_k = t_(k-1) - r_k - alpha_k * w_(k-1) + alpha_k s_k
s.Subtract(w, temp);
t.Subtract(residuals, y);
temp.Multiply(alpha, temp2);
y.Add(temp2, temp3);
temp3.CopyTo(y);
// Store the old value of t in t0
t.CopyTo(t0);
// t_k = r_k - alpha_k s_k
s.Multiply(-alpha, temp2);
residuals.Add(temp2, t);
// Solve M d_k = t_k
preconditioner.Approximate(t, temp);
// c_k = A d_k
matrix.Multiply(temp, c);
var cdot = c.DotProduct(c);
// cDot can only be zero if c is a zero vector
// We'll set cDot to 1 if it is zero to prevent NaN's
// Note that the calculation should continue fine because
// c.DotProduct(t) will be zero and so will c.DotProduct(y)
if (cdot.AlmostEqualNumbersBetween(0, 1))
{
cdot = 1.0;
}
// Even if we don't want to do any BiCGStab steps we'll still have
// to do at least one at the start to initialize the
// system, but we'll only have to take special measures
// if we don't do any so ...
var ctdot = c.DotProduct(t);
double eta;
double sigma;
if (((_numberOfBiCgStabSteps == 0) && (iterationNumber == 0)) || ShouldRunBiCgStabSteps(iterationNumber))
{
// sigma_k = (c_k * t_k) / (c_k * c_k)
sigma = ctdot / cdot;
// eta_k = 0
eta = 0;
}
else
{
var ydot = y.DotProduct(y);
// yDot can only be zero if y is a zero vector
// We'll set yDot to 1 if it is zero to prevent NaN's
// Note that the calculation should continue fine because
// y.DotProduct(t) will be zero and so will c.DotProduct(y)
if (ydot.AlmostEqualNumbersBetween(0, 1))
{
ydot = 1.0;
}
var ytdot = y.DotProduct(t);
var cydot = c.DotProduct(y);
var denom = (cdot * ydot) - (cydot * cydot);
// sigma_k = ((y_k * y_k)(c_k * t_k) - (y_k * t_k)(c_k * y_k)) / ((c_k * c_k)(y_k * y_k) - (y_k * c_k)(c_k * y_k))
sigma = ((ydot * ctdot) - (ytdot * cydot)) / denom;
// eta_k = ((c_k * c_k)(y_k * t_k) - (y_k * c_k)(c_k * t_k)) / ((c_k * c_k)(y_k * y_k) - (y_k * c_k)(c_k * y_k))
eta = ((cdot * ytdot) - (cydot * ctdot)) / denom;
}
// u_k = sigma_k s_k + eta_k (t_(k-1) - r_k + beta_(k-1) u_(k-1))
u.Multiply(beta, temp2);
t0.Add(temp2, temp);
temp.Subtract(residuals, temp3);
temp3.CopyTo(temp);
temp.Multiply(eta, temp);
s.Multiply(sigma, temp2);
temp.Add(temp2, u);
// z_k = sigma_k r_k +_ eta_k z_(k-1) - alpha_k u_k
z.Multiply(eta, z);
u.Multiply(-alpha, temp2);
z.Add(temp2, temp3);
temp3.CopyTo(z);
residuals.Multiply(sigma, temp2);
z.Add(temp2, temp3);
temp3.CopyTo(z);
// x_(k+1) = x_k + alpha_k p_k + z_k
p.Multiply(alpha, temp2);
xtemp.Add(temp2, temp3);
temp3.CopyTo(xtemp);
xtemp.Add(z, temp3);
temp3.CopyTo(xtemp);
// r_(k+1) = t_k - eta_k y_k - sigma_k c_k
// Copy the old residuals to a temp vector because we'll
// need those in the next step
residuals.CopyTo(t0);
y.Multiply(-eta, temp2);
t.Add(temp2, residuals);
c.Multiply(-sigma, temp2);
residuals.Add(temp2, temp3);
temp3.CopyTo(residuals);
// beta_k = alpha_k / sigma_k * (r*_0 * r_(k+1)) / (r*_0 * r_k)
// But first we check if there is a possible NaN. If so just reset beta to zero.
beta = (!sigma.AlmostEqualNumbersBetween(0, 1)) ? alpha / sigma * rdash.DotProduct(residuals) / rdash.DotProduct(t0) : 0;
// w_k = c_k + beta_k s_k
s.Multiply(beta, temp2);
c.Add(temp2, w);
// Get the real value
preconditioner.Approximate(xtemp, result);
// Now check for convergence
if (iterator.DetermineStatus(iterationNumber, result, input, residuals) != IterationStatus.Continue)
{
// Recalculate the residuals and go round again. This is done to ensure that
// we have the proper residuals.
CalculateTrueResidual(matrix, residuals, result, input);
}
// Next iteration.
iterationNumber++;
}
}
/// <summary> Sets right preconditioner</summary>
public virtual void setM2(IPreconditioner preconditioner)
{
M2 = preconditioner;
}
/// <summary> /// Check the result. /// </summary> /// <param name="preconditioner">Specific preconditioner.</param> /// <param name="matrix">Source matrix.</param> /// <param name="vector">Initial vector.</param> /// <param name="result">Result vector.</param> protected abstract void CheckResult(IPreconditioner<Complex> preconditioner, SparseMatrix matrix, Vector<Complex> vector, Vector<Complex> result);
public QMRSolver() : base()
{
M1 = M;
M2 = M;
}
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution, ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
MSGParametrs ConGradParametrs = solverParametrs as MSGParametrs;
if (ConGradParametrs == null)
{
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in MSG");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in MSG");
}
else
{
//prestart
int oIter = 0;
double alpha, beta, oNev, bNev, scalRO, scaleRN;
Vector x = initialSolution, rNew, rOld, z, ap, p;
rOld = rightPart - matrix.SourceMatrix.Multiply(x);
z = matrix.QSolve(matrix.SSolve(rOld));
p = z;
ap = matrix.SourceMatrix.Multiply(p);
bNev = rightPart.Norm();
oNev = rOld.Norm() / bNev;
scalRO = z * rOld;
// x = matrix.QSolve(x);
solverLogger.AddIterationInfo(oIter, oNev);//logger
if (System.Double.IsInfinity(oNev))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by MSG.");
return x;
}
if (System.Double.IsNaN(oNev))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by MSG.");
return x;
}
while (oIter < ConGradParametrs.MaxIterations && oNev > ConGradParametrs.Epsilon)
{
alpha = scalRO / (ap * p);
x = x + p * alpha;
//if (oIter % 100 == 0)
//rNew = matrix.QMultiply(rightPart - matrix.SMultiply(matrix.SourceMatrix.Multiply(x)));
//else
rNew = rOld - ap * alpha;
z = matrix.QSolve(matrix.SSolve(rNew));
scaleRN = z * rNew;
beta = scaleRN / scalRO;
scalRO = scaleRN;
p = z + p * beta;
ap = matrix.SourceMatrix.Multiply(p);
rOld = rNew;
oIter++;
oNev = rNew.Norm() / bNev;
if (System.Double.IsInfinity(oNev))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by MSG.");
return x;
}
if (System.Double.IsNaN(oNev))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by MSG.");
return x;
}
solverLogger.AddIterationInfo(oIter, oNev);//logger
}
return x;
}
}
public override Vector Solve(IPreconditioner matrix, Vector rightPart, Vector initialSolution, ILogger logger, ISolverLogger solverLogger, ISolverParametrs solverParametrs)
{
MSGParametrs ConGradParametrs = solverParametrs as MSGParametrs;
if (ConGradParametrs == null)
{
logger.Error("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in MSG");
throw new Exception("Incorrect " + solverParametrs.GetType().Name.ToString() + " as a SolverParametrs in MSG");
}
else
{
//prestart
int oIter = 0;
double alpha, beta, oNev, bNev, scalRO, scaleRN;
Vector x = initialSolution, rNew, rOld, z, ap, p;
rOld = rightPart - matrix.SourceMatrix.Multiply(x);
z = matrix.QSolve(matrix.SSolve(rOld));
p = z;
ap = matrix.SourceMatrix.Multiply(p);
bNev = rightPart.Norm();
oNev = rOld.Norm() / bNev;
scalRO = z * rOld;
// x = matrix.QSolve(x);
solverLogger.AddIterationInfo(oIter, oNev);//logger
if (System.Double.IsInfinity(oNev))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by MSG.");
return(x);
}
if (System.Double.IsNaN(oNev))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by MSG.");
return(x);
}
while (oIter < ConGradParametrs.MaxIterations && oNev > ConGradParametrs.Epsilon)
{
alpha = scalRO / (ap * p);
x = x + p * alpha;
//if (oIter % 100 == 0)
//rNew = matrix.QMultiply(rightPart - matrix.SMultiply(matrix.SourceMatrix.Multiply(x)));
//else
rNew = rOld - ap * alpha;
z = matrix.QSolve(matrix.SSolve(rNew));
scaleRN = z * rNew;
beta = scaleRN / scalRO;
scalRO = scaleRN;
p = z + p * beta;
ap = matrix.SourceMatrix.Multiply(p);
rOld = rNew;
oIter++;
oNev = rNew.Norm() / bNev;
if (System.Double.IsInfinity(oNev))
{
logger.Error("Residual is infinity. It is impossible to solve this SLAE by MSG.");
return(x);
}
if (System.Double.IsNaN(oNev))
{
logger.Error("Residual is NaN. It is impossible to solve this SLAE by MSG.");
return(x);
}
solverLogger.AddIterationInfo(oIter, oNev);//logger
}
return(x);
}
}
