/// <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 <Complex> matrix, Vector <Complex> input, Vector <Complex> result, Iterator <Complex> iterator, IPreconditioner <Complex> preconditioner)
{
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, nameof(matrix));
}
if ((input.Count != matrix.RowCount) || (result.Count != input.Count))
{
throw Matrix.DimensionsDontMatch <ArgumentException>(matrix, input, result);
}
if (iterator == null)
{
iterator = new Iterator <Complex>();
}
if (preconditioner == null)
{
preconditioner = new UnitPreconditioner <Complex>();
}
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 Complex[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 NumericalBreakdownException();
}
// 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 = 1.0;
}
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
Complex 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 NumericalBreakdownException();
}
// 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 NumericalBreakdownException();
}
// 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>
/// 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>
public void Solve(Matrix matrix, Vector input, Vector result)
{
// If we were stopped before, we are no longer
// We're doing this at the start of the method to ensure
// that we can use these fields immediately.
_hasBeenStopped = false;
// Error checks
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
if (input == null)
{
throw new ArgumentNullException("input");
}
if (result == null)
{
throw new ArgumentNullException("result");
}
if (input.Count != matrix.RowCount || result.Count != input.Count)
{
throw Matrix.DimensionsDontMatch <ArgumentException>(matrix, input, result);
}
// Initialize the solver fields
// Set the convergence monitor
if (_iterator == null)
{
_iterator = Iterator.CreateDefault();
}
if (_preconditioner == null)
{
_preconditioner = new UnitPreconditioner();
}
_preconditioner.Initialize(matrix);
// x_0 is initial guess
// Take x_0 = 0
Vector xtemp = new DenseVector(input.Count);
// r_0 = b - Ax_0
// This is basically a SAXPY so it could be made a lot faster
Vector residuals = new DenseVector(matrix.RowCount);
CalculateTrueResidual(matrix, residuals, xtemp, input);
// Define the temporary scalars
Complex beta = 0;
// Define the temporary vectors
// rDash_0 = r_0
Vector rdash = DenseVector.OfVector(residuals);
// t_-1 = 0
Vector t = new DenseVector(residuals.Count);
Vector t0 = new DenseVector(residuals.Count);
// w_-1 = 0
Vector w = new DenseVector(residuals.Count);
// Define the remaining temporary vectors
Vector c = new DenseVector(residuals.Count);
Vector p = new DenseVector(residuals.Count);
Vector s = new DenseVector(residuals.Count);
Vector u = new DenseVector(residuals.Count);
Vector y = new DenseVector(residuals.Count);
Vector z = new DenseVector(residuals.Count);
Vector temp = new DenseVector(residuals.Count);
Vector temp2 = new DenseVector(residuals.Count);
Vector temp3 = new DenseVector(residuals.Count);
// for (k = 0, 1, .... )
var iterationNumber = 0;
while (ShouldContinue(iterationNumber, xtemp, input, residuals))
{
// 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.ConjugateDotProduct(residuals) / rdash.ConjugateDotProduct(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.ConjugateDotProduct(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.Real.AlmostEqual(0, 1) && cdot.Imaginary.AlmostEqual(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.ConjugateDotProduct(t);
Complex eta;
Complex 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.ConjugateDotProduct(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.Real.AlmostEqual(0, 1) && ydot.Imaginary.AlmostEqual(0, 1))
{
ydot = 1.0;
}
var ytdot = y.ConjugateDotProduct(t);
var cydot = c.ConjugateDotProduct(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.Real.AlmostEqual(0, 1) || !sigma.Imaginary.AlmostEqual(0, 1)) ? alpha / sigma * rdash.ConjugateDotProduct(residuals) / rdash.ConjugateDotProduct(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 (!ShouldContinue(iterationNumber, result, input, residuals))
{
// 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>
/// 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>
public void Solve(Matrix <Numerics.Complex32> matrix, Vector <Numerics.Complex32> input, Vector <Numerics.Complex32> result)
{
// If we were stopped before, we are no longer
// We're doing this at the start of the method to ensure
// that we can use these fields immediately.
_hasBeenStopped = false;
// Parameters checks
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
if (input == null)
{
throw new ArgumentNullException("input");
}
if (result == null)
{
throw new ArgumentNullException("result");
}
if (result.Count != input.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (input.Count != matrix.RowCount)
{
throw Matrix.DimensionsDontMatch <ArgumentException>(input, matrix);
}
// Initialize the solver fields
// Set the convergence monitor
if (_iterator == null)
{
_iterator = Iterator.CreateDefault();
}
if (_preconditioner == null)
{
_preconditioner = new UnitPreconditioner <Numerics.Complex32>();
}
_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 float variables that are needed
// to hold values in between iterations
Numerics.Complex32 currentRho = 0;
Numerics.Complex32 alpha = 0;
Numerics.Complex32 omega = 0;
var iterationNumber = 0;
while (ShouldContinue(iterationNumber, result, input, residuals))
{
// 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.AlmostEqual(0, 1) && currentRho.Imaginary.AlmostEqual(0, 1))
{
// Rho-type breakdown
throw new Exception("Iterative solver experience a numerical break down");
}
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 (!ShouldContinue(iterationNumber, temp, input, vecS))
{
temp.CopyTo(result);
// Calculate the true residual
CalculateTrueResidual(matrix, residuals, result, input);
// Now recheck the convergence
if (!ShouldContinue(iterationNumber, result, input, residuals))
{
// 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 != 0.0f
// If omega is only 1 ULP from zero then we fail.
if (omega.Real.AlmostEqual(0, 1) && omega.Imaginary.AlmostEqual(0, 1))
{
// Omega-type breakdown
throw new Exception("Iterative solver experience a numerical break down");
}
if (!ShouldContinue(iterationNumber, result, input, residuals))
{
// 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++;
}
}
/// <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>
public void Solve(Matrix matrix, Vector input, Vector result)
{
// If we were stopped before, we are no longer
// We're doing this at the start of the method to ensure
// that we can use these fields immediately.
_hasBeenStopped = false;
// Error checks
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
if (input == null)
{
throw new ArgumentNullException("input");
}
if (result == null)
{
throw new ArgumentNullException("result");
}
if (result.Count != input.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (input.Count != matrix.RowCount)
{
throw Matrix.DimensionsDontMatch<ArgumentException>(input, matrix);
}
// Initialize the solver fields
// Set the convergence monitor
if (_iterator == null)
{
_iterator = Iterator.CreateDefault();
}
if (_preconditioner == null)
{
_preconditioner = new UnitPreconditioner();
}
_preconditioner.Initialize(matrix);
// x_0 is initial guess
// Take x_0 = 0
Vector xtemp = new DenseVector(input.Count);
// r_0 = b - Ax_0
// This is basically a SAXPY so it could be made a lot faster
Vector residuals = new DenseVector(matrix.RowCount);
CalculateTrueResidual(matrix, residuals, xtemp, input);
// Define the temporary scalars
Complex32 beta = 0;
// Define the temporary vectors
// rDash_0 = r_0
Vector rdash = DenseVector.OfVector(residuals);
// t_-1 = 0
Vector t = new DenseVector(residuals.Count);
Vector t0 = new DenseVector(residuals.Count);
// w_-1 = 0
Vector w = new DenseVector(residuals.Count);
// Define the remaining temporary vectors
Vector c = new DenseVector(residuals.Count);
Vector p = new DenseVector(residuals.Count);
Vector s = new DenseVector(residuals.Count);
Vector u = new DenseVector(residuals.Count);
Vector y = new DenseVector(residuals.Count);
Vector z = new DenseVector(residuals.Count);
Vector temp = new DenseVector(residuals.Count);
Vector temp2 = new DenseVector(residuals.Count);
Vector temp3 = new DenseVector(residuals.Count);
// for (k = 0, 1, .... )
var iterationNumber = 0;
while (ShouldContinue(iterationNumber, xtemp, input, residuals))
{
// 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.ConjugateDotProduct(residuals)/rdash.ConjugateDotProduct(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.ConjugateDotProduct(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.Real.AlmostEqual(0, 1) && cdot.Imaginary.AlmostEqual(0, 1))
{
cdot = 1.0f;
}
// 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.ConjugateDotProduct(t);
Complex32 eta;
Complex32 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.ConjugateDotProduct(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.Real.AlmostEqual(0, 1) && ydot.Imaginary.AlmostEqual(0, 1))
{
ydot = 1.0f;
}
var ytdot = y.ConjugateDotProduct(t);
var cydot = c.ConjugateDotProduct(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.Real.AlmostEqual(0, 1) || !sigma.Imaginary.AlmostEqual(0, 1)) ? alpha/sigma*rdash.ConjugateDotProduct(residuals)/rdash.ConjugateDotProduct(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 (!ShouldContinue(iterationNumber, result, input, residuals))
{
// 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>
/// 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>
public void Solve(Matrix matrix, Vector input, Vector result)
{
// If we were stopped before, we are no longer
// We're doing this at the start of the method to ensure
// that we can use these fields immediately.
_hasBeenStopped = false;
// Error checks
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
if (input == null)
{
throw new ArgumentNullException("input");
}
if (result == null)
{
throw new ArgumentNullException("result");
}
if (input.Count != matrix.RowCount || result.Count != input.Count)
{
throw Matrix.DimensionsDontMatch<ArgumentException>(matrix, input, result);
}
// Initialize the solver fields
// Set the convergence monitor
if (_iterator == null)
{
_iterator = Iterator.CreateDefault();
}
if (_preconditioner == null)
{
_preconditioner = new UnitPreconditioner();
}
_preconditioner.Initialize(matrix);
// Choose an initial guess x_0
// Take x_0 = 0
Vector 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
Vector residuals = new DenseVector(matrix.RowCount);
CalculateTrueResidual(matrix, residuals, xtemp, input);
// Define the temporary values
var c = new Complex[k];
// Define the temporary vectors
Vector gtemp = new DenseVector(residuals.Count);
Vector u = new DenseVector(residuals.Count);
Vector utemp = new DenseVector(residuals.Count);
Vector temp = new DenseVector(residuals.Count);
Vector temp1 = new DenseVector(residuals.Count);
Vector temp2 = new DenseVector(residuals.Count);
Vector zd = new DenseVector(residuals.Count);
Vector zg = new DenseVector(residuals.Count);
Vector 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 (ShouldContinue(iterationNumber, xtemp, input, residuals))
{
// 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.AlmostEqual(0, 1) && c[k - 1].Imaginary.AlmostEqual(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.AlmostEqual(0, 1) && rho.Imaginary.AlmostEqual(0, 1))
{
rho = 1.0;
}
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 (!ShouldContinue(iterationNumber, xtemp, input, residuals))
{
// Calculate the true residual
CalculateTrueResidual(matrix, residuals, xtemp, input);
// Now recheck the convergence
if (!ShouldContinue(iterationNumber, xtemp, input, residuals))
{
// 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
Complex 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.AlmostEqual(0, 1) && beta.Imaginary.AlmostEqual(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.AlmostEqual(0, 1) && c[i].Imaginary.AlmostEqual(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 (!ShouldContinue(iterationNumber, xtemp, input, residuals))
{
// 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>
/// 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>
public void Solve(Matrix matrix, Vector input, Vector result)
{
// If we were stopped before, we are no longer
// We're doing this at the start of the method to ensure
// that we can use these fields immediately.
_hasBeenStopped = false;
// Parameters checks
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
if (input == null)
{
throw new ArgumentNullException("input");
}
if (result == null)
{
throw new ArgumentNullException("result");
}
if (result.Count != input.Count)
{
throw new ArgumentException(Resources.ArgumentVectorsSameLength);
}
if (input.Count != matrix.RowCount)
{
throw Matrix.DimensionsDontMatch<ArgumentException>(input, matrix);
}
// Initialize the solver fields
// Set the convergence monitor
if (_iterator == null)
{
_iterator = Iterator.CreateDefault();
}
if (_preconditioner == null)
{
_preconditioner = new UnitPreconditioner();
}
_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
Vector 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
Vector vecP = new DenseVector(residuals.Count);
Vector vecPdash = new DenseVector(residuals.Count);
Vector nu = new DenseVector(residuals.Count);
Vector vecS = new DenseVector(residuals.Count);
Vector vecSdash = new DenseVector(residuals.Count);
Vector temp = new DenseVector(residuals.Count);
Vector temp2 = new DenseVector(residuals.Count);
// create some temporary float variables that are needed
// to hold values in between iterations
Complex32 currentRho = 0;
Complex32 alpha = 0;
Complex32 omega = 0;
var iterationNumber = 0;
while (ShouldContinue(iterationNumber, result, input, residuals))
{
// 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.AlmostEqual(0, 1) && currentRho.Imaginary.AlmostEqual(0, 1))
{
// Rho-type breakdown
throw new Exception("Iterative solver experience a numerical break down");
}
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 (!ShouldContinue(iterationNumber, temp, input, vecS))
{
temp.CopyTo(result);
// Calculate the true residual
CalculateTrueResidual(matrix, residuals, result, input);
// Now recheck the convergence
if (!ShouldContinue(iterationNumber, result, input, residuals))
{
// 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 != 0.0f
// If omega is only 1 ULP from zero then we fail.
if (omega.Real.AlmostEqual(0, 1) && omega.Imaginary.AlmostEqual(0, 1))
{
// Omega-type breakdown
throw new Exception("Iterative solver experience a numerical break down");
}
if (!ShouldContinue(iterationNumber, result, input, residuals))
{
// 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++;
}
}
/// <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("Matrix must be square.", nameof(matrix));
}
if (result.Count != input.Count)
{
throw new ArgumentException("All vectors must have the same dimensionality.");
}
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++;
}
}
