/// <summary>
/// Initializes the preconditioner and loads the internal data structures.
/// </summary>
/// <param name="matrix">
/// The <see cref="Matrix"/> upon which this preconditioner is based. Note that the
/// method takes a general matrix type. However internally the data is stored
/// as a sparse matrix. Therefore it is not recommended to pass a dense matrix.
/// </param>
/// <exception cref="ArgumentNullException"> If <paramref name="matrix"/> is <see langword="null" />.</exception>
/// <exception cref="ArgumentException">If <paramref name="matrix"/> is not a square matrix.</exception>
public void Initialize(Matrix matrix)
{
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
var sparseMatrix = (matrix is SparseMatrix) ? matrix as SparseMatrix : new SparseMatrix(matrix);
// The creation of the preconditioner follows the following algorithm.
// spaceLeft = lfilNnz * nnz(A)
// for i = 1, .. , n
// {
// w = a(i,*)
// for j = 1, .. , i - 1
// {
// if (w(j) != 0)
// {
// w(j) = w(j) / a(j,j)
// if (w(j) < dropTol)
// {
// w(j) = 0;
// }
// if (w(j) != 0)
// {
// w = w - w(j) * U(j,*)
// }
// }
// }
//
// for j = i, .. ,n
// {
// if w(j) <= dropTol * ||A(i,*)||
// {
// w(j) = 0
// }
// }
//
// spaceRow = spaceLeft / (n - i + 1) // Determine the space for this row
// lfil = spaceRow / 2 // space for this row of L
// l(i,j) = w(j) for j = 1, .. , i -1 // only the largest lfil elements
//
// lfil = spaceRow - nnz(L(i,:)) // space for this row of U
// u(i,j) = w(j) for j = i, .. , n // only the largest lfil - 1 elements
// w = 0
//
// if max(U(i,i + 1: n)) > U(i,i) / pivTol then // pivot if necessary
// {
// pivot by swapping the max and the diagonal entries
// Update L, U
// Update P
// }
// spaceLeft = spaceLeft - nnz(L(i,:)) - nnz(U(i,:))
// }
// Create the lower triangular matrix
_lower = new SparseMatrix(sparseMatrix.RowCount);
// Create the upper triangular matrix and copy the values
_upper = new SparseMatrix(sparseMatrix.RowCount);
// Create the pivot array
_pivots = new int[sparseMatrix.RowCount];
for (var i = 0; i < _pivots.Length; i++)
{
_pivots[i] = i;
}
Vector workVector = new DenseVector(sparseMatrix.RowCount);
Vector rowVector = new DenseVector(sparseMatrix.ColumnCount);
var indexSorting = new int[sparseMatrix.RowCount];
// spaceLeft = lfilNnz * nnz(A)
var spaceLeft = (int)_fillLevel * sparseMatrix.NonZerosCount;
// for i = 1, .. , n
for (var i = 0; i < sparseMatrix.RowCount; i++)
{
// w = a(i,*)
sparseMatrix.Row(i, workVector);
// pivot the row
PivotRow(workVector);
var vectorNorm = workVector.Norm(Double.PositiveInfinity);
// for j = 1, .. , i - 1)
for (var j = 0; j < i; j++)
{
// if (w(j) != 0)
// {
// w(j) = w(j) / a(j,j)
// if (w(j) < dropTol)
// {
// w(j) = 0;
// }
// if (w(j) != 0)
// {
// w = w - w(j) * U(j,*)
// }
if (workVector[j] != 0.0)
{
// Calculate the multiplication factors that go into the L matrix
workVector[j] = workVector[j] / _upper[j, j];
if (workVector[j].Magnitude < _dropTolerance)
{
workVector[j] = 0.0;
}
// Calculate the addition factor
if (workVector[j] != 0.0)
{
// vector update all in one go
_upper.Row(j, rowVector);
// zero out columnVector[k] because we don't need that
// one anymore for k = 0 to k = j
for (var k = 0; k <= j; k++)
{
rowVector[k] = 0.0;
}
rowVector.Multiply(workVector[j], rowVector);
workVector.Subtract(rowVector, workVector);
}
}
}
// for j = i, .. ,n
for (var j = i; j < sparseMatrix.RowCount; j++)
{
// if w(j) <= dropTol * ||A(i,*)||
// {
// w(j) = 0
// }
if (workVector[j].Magnitude <= _dropTolerance * vectorNorm.Real)
{
workVector[j] = 0.0;
}
}
// spaceRow = spaceLeft / (n - i + 1) // Determine the space for this row
var spaceRow = spaceLeft / (sparseMatrix.RowCount - i + 1);
// lfil = spaceRow / 2 // space for this row of L
var fillLevel = spaceRow / 2;
FindLargestItems(0, i - 1, indexSorting, workVector);
// l(i,j) = w(j) for j = 1, .. , i -1 // only the largest lfil elements
var lowerNonZeroCount = 0;
var count = 0;
for (var j = 0; j < i; j++)
{
if ((count > fillLevel) || (indexSorting[j] == -1))
{
break;
}
_lower[i, indexSorting[j]] = workVector[indexSorting[j]];
count += 1;
lowerNonZeroCount += 1;
}
FindLargestItems(i + 1, sparseMatrix.RowCount - 1, indexSorting, workVector);
// lfil = spaceRow - nnz(L(i,:)) // space for this row of U
fillLevel = spaceRow - lowerNonZeroCount;
// u(i,j) = w(j) for j = i + 1, .. , n // only the largest lfil - 1 elements
var upperNonZeroCount = 0;
count = 0;
for (var j = 0; j < sparseMatrix.RowCount - i; j++)
{
if ((count > fillLevel - 1) || (indexSorting[j] == -1))
{
break;
}
_upper[i, indexSorting[j]] = workVector[indexSorting[j]];
count += 1;
upperNonZeroCount += 1;
}
// Simply copy the diagonal element. Next step is to see if we pivot
_upper[i, i] = workVector[i];
// if max(U(i,i + 1: n)) > U(i,i) / pivTol then // pivot if necessary
// {
// pivot by swapping the max and the diagonal entries
// Update L, U
// Update P
// }
// Check if we really need to pivot. If (i+1) >=(mCoefficientMatrix.Rows -1) then
// we are working on the last row. That means that there is only one number
// And pivoting is useless. Also the indexSorting array will only contain
// -1 values.
if ((i + 1) < (sparseMatrix.RowCount - 1))
{
if (workVector[i].Magnitude < _pivotTolerance * workVector[indexSorting[0]].Magnitude)
{
// swap columns of u (which holds the values of A in the
// sections that haven't been partitioned yet.
SwapColumns(_upper, i, indexSorting[0]);
// Update P
var temp = _pivots[i];
_pivots[i] = _pivots[indexSorting[0]];
_pivots[indexSorting[0]] = temp;
}
}
// spaceLeft = spaceLeft - nnz(L(i,:)) - nnz(U(i,:))
spaceLeft -= lowerNonZeroCount + upperNonZeroCount;
}
for (var i = 0; i < _lower.RowCount; i++)
{
_lower[i, i] = 1.0;
}
}
/// <summary>
/// Initializes the preconditioner and loads the internal data structures.
/// </summary>
/// <param name="matrix">
/// The <see cref="Matrix"/> upon which this preconditioner is based. Note that the
/// method takes a general matrix type. However internally the data is stored
/// as a sparse matrix. Therefore it is not recommended to pass a dense matrix.
/// </param>
/// <exception cref="ArgumentNullException"> If <paramref name="matrix"/> is <see langword="null" />.</exception>
/// <exception cref="ArgumentException">If <paramref name="matrix"/> is not a square matrix.</exception>
public void Initialize(Matrix matrix)
{
if (matrix == null)
{
throw new ArgumentNullException("matrix");
}
if (matrix.RowCount != matrix.ColumnCount)
{
throw new ArgumentException(Resources.ArgumentMatrixSquare, "matrix");
}
var sparseMatrix = (matrix is SparseMatrix) ? matrix as SparseMatrix : new SparseMatrix(matrix);
// The creation of the preconditioner follows the following algorithm.
// spaceLeft = lfilNnz * nnz(A)
// for i = 1, .. , n
// {
// w = a(i,*)
// for j = 1, .. , i - 1
// {
// if (w(j) != 0)
// {
// w(j) = w(j) / a(j,j)
// if (w(j) < dropTol)
// {
// w(j) = 0;
// }
// if (w(j) != 0)
// {
// w = w - w(j) * U(j,*)
// }
// }
// }
//
// for j = i, .. ,n
// {
// if w(j) <= dropTol * ||A(i,*)||
// {
// w(j) = 0
// }
// }
//
// spaceRow = spaceLeft / (n - i + 1) // Determine the space for this row
// lfil = spaceRow / 2 // space for this row of L
// l(i,j) = w(j) for j = 1, .. , i -1 // only the largest lfil elements
//
// lfil = spaceRow - nnz(L(i,:)) // space for this row of U
// u(i,j) = w(j) for j = i, .. , n // only the largest lfil - 1 elements
// w = 0
//
// if max(U(i,i + 1: n)) > U(i,i) / pivTol then // pivot if necessary
// {
// pivot by swapping the max and the diagonal entries
// Update L, U
// Update P
// }
// spaceLeft = spaceLeft - nnz(L(i,:)) - nnz(U(i,:))
// }
// Create the lower triangular matrix
_lower = new SparseMatrix(sparseMatrix.RowCount);
// Create the upper triangular matrix and copy the values
_upper = new SparseMatrix(sparseMatrix.RowCount);
// Create the pivot array
_pivots = new int[sparseMatrix.RowCount];
for (var i = 0; i < _pivots.Length; i++)
{
_pivots[i] = i;
}
Vector workVector = new DenseVector(sparseMatrix.RowCount);
Vector rowVector = new DenseVector(sparseMatrix.ColumnCount);
var indexSorting = new int[sparseMatrix.RowCount];
// spaceLeft = lfilNnz * nnz(A)
var spaceLeft = (int)_fillLevel * sparseMatrix.NonZerosCount;
// for i = 1, .. , n
for (var i = 0; i < sparseMatrix.RowCount; i++)
{
// w = a(i,*)
sparseMatrix.Row(i, workVector);
// pivot the row
PivotRow(workVector);
var vectorNorm = workVector.Norm(Double.PositiveInfinity);
// for j = 1, .. , i - 1)
for (var j = 0; j < i; j++)
{
// if (w(j) != 0)
// {
// w(j) = w(j) / a(j,j)
// if (w(j) < dropTol)
// {
// w(j) = 0;
// }
// if (w(j) != 0)
// {
// w = w - w(j) * U(j,*)
// }
if (workVector[j] != 0.0)
{
// Calculate the multiplication factors that go into the L matrix
workVector[j] = workVector[j] / _upper[j, j];
if (workVector[j].Magnitude < _dropTolerance)
{
workVector[j] = 0.0;
}
// Calculate the addition factor
if (workVector[j] != 0.0)
{
// vector update all in one go
_upper.Row(j, rowVector);
// zero out columnVector[k] because we don't need that
// one anymore for k = 0 to k = j
for (var k = 0; k <= j; k++)
{
rowVector[k] = 0.0;
}
rowVector.Multiply(workVector[j], rowVector);
workVector.Subtract(rowVector, workVector);
}
}
}
// for j = i, .. ,n
for (var j = i; j < sparseMatrix.RowCount; j++)
{
// if w(j) <= dropTol * ||A(i,*)||
// {
// w(j) = 0
// }
if (workVector[j].Magnitude <= _dropTolerance * vectorNorm.Real)
{
workVector[j] = 0.0;
}
}
// spaceRow = spaceLeft / (n - i + 1) // Determine the space for this row
var spaceRow = spaceLeft / (sparseMatrix.RowCount - i + 1);
// lfil = spaceRow / 2 // space for this row of L
var fillLevel = spaceRow / 2;
FindLargestItems(0, i - 1, indexSorting, workVector);
// l(i,j) = w(j) for j = 1, .. , i -1 // only the largest lfil elements
var lowerNonZeroCount = 0;
var count = 0;
for (var j = 0; j < i; j++)
{
if ((count > fillLevel) || (indexSorting[j] == -1))
{
break;
}
_lower[i, indexSorting[j]] = workVector[indexSorting[j]];
count += 1;
lowerNonZeroCount += 1;
}
FindLargestItems(i + 1, sparseMatrix.RowCount - 1, indexSorting, workVector);
// lfil = spaceRow - nnz(L(i,:)) // space for this row of U
fillLevel = spaceRow - lowerNonZeroCount;
// u(i,j) = w(j) for j = i + 1, .. , n // only the largest lfil - 1 elements
var upperNonZeroCount = 0;
count = 0;
for (var j = 0; j < sparseMatrix.RowCount - i; j++)
{
if ((count > fillLevel - 1) || (indexSorting[j] == -1))
{
break;
}
_upper[i, indexSorting[j]] = workVector[indexSorting[j]];
count += 1;
upperNonZeroCount += 1;
}
// Simply copy the diagonal element. Next step is to see if we pivot
_upper[i, i] = workVector[i];
// if max(U(i,i + 1: n)) > U(i,i) / pivTol then // pivot if necessary
// {
// pivot by swapping the max and the diagonal entries
// Update L, U
// Update P
// }
// Check if we really need to pivot. If (i+1) >=(mCoefficientMatrix.Rows -1) then
// we are working on the last row. That means that there is only one number
// And pivoting is useless. Also the indexSorting array will only contain
// -1 values.
if ((i + 1) < (sparseMatrix.RowCount - 1))
{
if (workVector[i].Magnitude < _pivotTolerance * workVector[indexSorting[0]].Magnitude)
{
// swap columns of u (which holds the values of A in the
// sections that haven't been partitioned yet.
SwapColumns(_upper, i, indexSorting[0]);
// Update P
var temp = _pivots[i];
_pivots[i] = _pivots[indexSorting[0]];
_pivots[indexSorting[0]] = temp;
}
}
// spaceLeft = spaceLeft - nnz(L(i,:)) - nnz(U(i,:))
spaceLeft -= lowerNonZeroCount + upperNonZeroCount;
}
for (var i = 0; i < _lower.RowCount; i++)
{
_lower[i, i] = 1.0;
}
}
/// <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 new ArgumentException(Resources.ArgumentMatrixDimensions);
}
// Initialize the solver fields
// Set the convergence monitor
if (_iterator == null)
{
_iterator = Iterator.CreateDefault();
}
if (_preconditioner == null)
{
_preconditioner = new UnitPreconditioner();
}
_preconditioner.Initialize(matrix);
var d = new DenseVector(input.Count);
var r = new DenseVector(input);
var uodd = new DenseVector(input.Count);
var ueven = new DenseVector(input.Count);
var v = new DenseVector(input.Count);
var pseudoResiduals = new DenseVector(input);
var x = new DenseVector(input.Count);
var yodd = new DenseVector(input.Count);
var yeven = new DenseVector(input);
// Temp vectors
var temp = new DenseVector(input.Count);
var temp1 = new DenseVector(input.Count);
var temp2 = new DenseVector(input.Count);
// Initialize
var startNorm = input.Norm(2);
// Define the scalars
Complex alpha = 0;
Complex eta = 0;
double theta = 0;
var tau = startNorm.Real;
Complex 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 (ShouldContinue(iterationNumber, result, input, pseudoResiduals))
{
// First part of the step, the even bit
if (IsEven(iterationNumber))
{
// sigma = (v, r)
var sigma = v.DotProduct(r.Conjugate());
if (sigma.Real.AlmostEqual(0, 1) && sigma.Imaginary.AlmostEqual(0, 1))
{
// FAIL HERE
_iterator.IterationCancelled();
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 = pseudoResiduals.Norm(2).Real / tau;
var c = 1 / 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 (!ShouldContinue(iterationNumber, result, input, pseudoResiduals))
{
// 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 (!ShouldContinue(iterationNumber, result, input, temp))
{
// We're all good now.
return;
}
}
// The odd step
if (!IsEven(iterationNumber))
{
if (rho.Real.AlmostEqual(0, 1) && rho.Imaginary.AlmostEqual(0, 1))
{
// FAIL HERE
_iterator.IterationCancelled();
break;
}
var rhoNew = pseudoResiduals.DotProduct(r.Conjugate());
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++;
}
}
/// <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);
var d = new DenseVector(input.Count);
var r = new DenseVector(input);
var uodd = new DenseVector(input.Count);
var ueven = new DenseVector(input.Count);
var v = new DenseVector(input.Count);
var pseudoResiduals = new DenseVector(input);
var x = new DenseVector(input.Count);
var yodd = new DenseVector(input.Count);
var yeven = new DenseVector(input);
// Temp vectors
var temp = new DenseVector(input.Count);
var temp1 = new DenseVector(input.Count);
var temp2 = new DenseVector(input.Count);
// Initialize
var startNorm = input.Norm(2);
// Define the scalars
double alpha = 0;
double eta = 0;
double theta = 0;
var tau = startNorm;
var 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 (ShouldContinue(iterationNumber, result, input, pseudoResiduals))
{
// First part of the step, the even bit
if (IsEven(iterationNumber))
{
// sigma = (v, r)
var sigma = v.DotProduct(r);
if (sigma.AlmostEqual(0, 1))
{
// FAIL HERE
_iterator.IterationCancelled();
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 = pseudoResiduals.Norm(2) / tau;
var c = 1 / 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 (!ShouldContinue(iterationNumber, result, input, pseudoResiduals))
{
// 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 (!ShouldContinue(iterationNumber, result, input, temp))
{
// We're all good now.
return;
}
}
// The odd step
if (!IsEven(iterationNumber))
{
if (rho.AlmostEqual(0, 1))
{
// FAIL HERE
_iterator.IterationCancelled();
break;
}
var rhoNew = pseudoResiduals.DotProduct(r);
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++;
}
}
