public static void PreconditioningPosDefSparse()
{
Console.WriteLine("Assessing correctness and efficiency of preconditioned MINRES:\n");
var A = Matrix.CreateFromArray(SparsePosDef10by10.Matrix);
var b = Vector.CreateFromArray(SparsePosDef10by10.Rhs);
var xExpected = Vector.CreateFromArray(SparsePosDef10by10.Lhs);
var M = new JacobiPreconditioner(A.GetDiagonalAsArray());
var minres = new MinRes(A.NumRows, 1e-10, 0, true, false);
// Without preconditioning
Console.WriteLine("Sparse pos-def system WITHOUT preconditioning:");
(IVector xSimple, MinresStatistics statsSimple) = minres.Solve(A, b);
Console.Write(statsSimple);
if (xSimple != null)
{
CheckSolution(xExpected, xSimple);
}
Console.WriteLine();
// With preconditioning
Console.WriteLine("Sparse pos-def system WITH preconditioning:");
(IVector xPrec, MinresStatistics statsPrec) = minres.Solve(A, b, M);
Console.Write(statsPrec);
if (xPrec != null)
{
CheckSolution(xExpected, xPrec);
}
Console.WriteLine();
}
private static void TestPosDefSparseSystem()
{
var A = Matrix.CreateFromArray(SparsePosDef10by10.Matrix);
var b = Vector.CreateFromArray(SparsePosDef10by10.Rhs);
var xExpected = Vector.CreateFromArray(SparsePosDef10by10.Lhs);
var builder = new ReorthogonalizedPcg.Builder();
builder.ResidualTolerance = 1E-7;
builder.MaxIterationsProvider = new PercentageMaxIterationsProvider(1.0);
var pcg = builder.Build();
var M = new JacobiPreconditioner(A.GetDiagonalAsArray());
Vector xComputed = Vector.CreateZero(A.NumRows);
IterativeStatistics stats = pcg.Solve(A, M, b, xComputed, true, () => Vector.CreateZero(b.Length));
comparer.AssertEqual(xExpected, xComputed);
}
private static void TestPosDefSparseSystem(LinearAlgebraProviderChoice providers)
{
TestSettings.RunMultiproviderTest(providers, delegate()
{
var A = Matrix.CreateFromArray(SparsePosDef10by10.Matrix);
var b = Vector.CreateFromArray(SparsePosDef10by10.Rhs);
var xExpected = Vector.CreateFromArray(SparsePosDef10by10.Lhs);
var builder = new PcgAlgorithm.Builder();
builder.ResidualTolerance = 1E-7;
builder.MaxIterationsProvider = new PercentageMaxIterationsProvider(1.0);
var pcg = builder.Build();
var M = new JacobiPreconditioner(A.GetDiagonalAsArray());
Vector xComputed = Vector.CreateZero(A.NumRows);
IterativeStatistics stats = pcg.Solve(A, M, b, xComputed, true, () => Vector.CreateZero(b.Length));
comparer.AssertEqual(xExpected, xComputed);
});
}
private static void TestNearbyProblems(double noiseWidth, int maxIterations, int numRhsVectors)
{
int order = SymmPosDef10by10.Order;
var A = Matrix.CreateFromArray(SymmPosDef10by10.Matrix);
var builder = new ReorthogonalizedPcg.Builder();
builder.ResidualTolerance = 1E-6;
builder.MaxIterationsProvider = new PercentageMaxIterationsProvider(1.0);
builder.Convergence = new RhsNormalizedConvergence();
var pcg = builder.Build();
var M = new JacobiPreconditioner(A.GetDiagonalAsArray());
// Initial run
Vector x0 = Vector.CreateWithValue(order, 1);
Vector x0Expected = x0.Copy();
Vector b0 = A * x0Expected;
Vector x0Computed = Vector.CreateZero(A.NumRows);
IterativeStatistics stats0 = pcg.Solve(A, M, b0, x0Computed, true, () => Vector.CreateZero(order));
Debug.WriteLine($"Initial run: iterations = {stats0.NumIterationsRequired}");
comparer.AssertEqual(x0Expected, x0Computed);
// Subsequent runs
int seed = 345;
for (int i = 0; i < numRhsVectors; ++i)
{
Vector dx = Vector.CreateFromArray(RandomMatrices.CreateRandomVector(order, seed));
Vector xExpected = x0 + noiseWidth * dx;
Vector b = A * xExpected;
pcg.Clear(); //TODO: preferably do not call this.
//pcg.ReorthoCache.Clear();
Vector xComputed = Vector.CreateZero(A.NumRows);
IterativeStatistics stats = pcg.Solve(A, M, b, xComputed, true, () => Vector.CreateZero(b.Length));
Debug.WriteLine($"Subsequent run: iterations = {stats.NumIterationsRequired}");
comparer.AssertEqual(xExpected, xComputed);
Assert.InRange(stats.NumIterationsRequired, 1, maxIterations);
}
}
/// <summary>
/// </summary>
/// <param name="evenOrder">The number of rows/columns. It must be even.</param>
internal static (Matrix A, Vector b, Vector x, IPreconditioner M) BuildIndefiniteSystem(int evenOrder)
{
// Example taken from Matlab docs. Creates a diagonal matrix with half its entries being negative.
int n = evenOrder;
var A = Matrix.CreateZero(n, n);
for (int i = 0; i < n / 2; ++i)
{
A[i, i] = n / 2 - i;
}
for (int i = 0; i < n / 2; ++i)
{
A[n / 2 + i, n / 2 + i] = -1 - i;
}
// x = [1, 1, ... 1]
var x = Vector.CreateWithValue(n, 1.0);
// b[i] = A[i, i]
var b = Vector.CreateZero(n);
for (int i = 0; i < n; ++i)
{
b[i] = A[i, i];
}
// The usual Jacobi preconditioner worsens convergence. Modifications are needed.
var positiveDiagonal = new double[n];
for (int i = 0; i < n; ++i)
{
positiveDiagonal[i] = Math.Abs(A[i, i]);
}
var M = new JacobiPreconditioner(positiveDiagonal);
return(A, b, x, M);
}
