// return true if it converges. Output: solution matrix, errors, loops it took
public static Boolean solve(Matrix A, Matrix b, out Matrix x, out Matrix err, out int loops, Intracommunicator comm)
{
// check sanity. rank 0 only
if (comm.Rank == 0 && (!A.isSquare || !b.isColumn || (A.Height != b.Height)))
{
Exception e = new Exception("Matrix A must be square! Matrix b must be a column matrix with the same height as matrix A!");
throw e;
}
// follow samples in Wikipedia step by step https://en.wikipedia.org/wiki/Gauss%E2%80%93Seidel_method
benchmark bm = new benchmark(), bm2 = new benchmark(), bm3 = new benchmark();
double sequential = 0, parallel = 0, communication = 0;
bm.start();
bm2.start();
// decompose A into the sum of a lower triangular component L* and a strict upper triangular component U
int size = 0; Matrix L = null, U = null, L_1;
if (comm.Rank == 0)
{
size = A.Height;
Matrix.Decompose(A, out L, out U);
}
bm2.pause();
sequential += bm2.getElapsedSeconds();
bm2.start();
comm.Broadcast(ref size, 0);
comm.Broadcast(ref U, 0);
comm.Broadcast(ref b, 0);
bm2.pause();
communication += bm2.getElapsedSeconds();
// Inverse matrix L*
comm.Barrier();
L_1 = MatrixParallel.Inverse(L, comm, ref sequential, ref parallel, ref communication);
// Main iteration: x (at step k+1) = T * x (at step k) + C
// where T = - (inverse of L*) * U, and C = (inverse of L*) * b
// split T & C into groups of rows, each for one slave, according to the nature of this algorithm
// each slave will have one piece of T & one piece of C stored locally. the rest of T & C is not needed
// there might be cases where jobs > slaves, so some might get no job at all
// Changes: only split L_1. Slaves will calculate T & C (pieces) themselves
bm2.start();
Matrix jobDistro = Utils.splitJob(size, comm.Size);
int startRow = 0, endRow = 0, myJobSize = (int)jobDistro[0, comm.Rank];
for (int p = 0; p < comm.Size; p++)
{
if (p != comm.Rank)
{
startRow += (int)jobDistro[0, p];
}
else
{
endRow = startRow + (int)jobDistro[0, p] - 1;
break;
}
}
Matrix[] L_1Ps = new Matrix[comm.Size];
if (comm.Rank == 0)
{
int slaveStart = 0;
for (int p = 0; p < comm.Size; p++)
{
L_1Ps[p] = Matrix.extractRows(L_1, slaveStart, slaveStart + (int)jobDistro[0, p] - 1);
slaveStart += (int)jobDistro[0, p];
}
}
bm2.pause();
sequential += bm2.getElapsedSeconds();
bm2.start();
Matrix L_1P = comm.Scatter(L_1Ps, 0);
bm2.pause();
communication += bm2.getElapsedSeconds();
bm2.start();
Matrix T = -L_1P * U; Matrix C = L_1P * b;
bm2.pause();
parallel += bm2.getElapsedSeconds();
// the actual iteration
// if it still doesn't converge after this many loops, assume it won't converge and give up
Boolean converge = false;
int loopLimit = 100;
x = Matrix.zeroLike(b); // at step k
for (loops = 0; loops < loopLimit; loops++)
{
bm3.start();
// (re-)distributing x vector. Must be done every single loop
// this loop needs x from the previous loop
comm.Broadcast(ref x, 0);
bm3.pause();
communication += bm3.getElapsedSeconds();
// calculation step
bm3.start();
comm.Barrier();
Matrix new_x = T * x + C;
// check convergence
converge = Matrix.SomeClose(new_x, x, 1e-15, startRow);
// collect result x
comm.Barrier();
x = comm.Reduce(new_x, Matrix.Concatenate, 0);
// collect convergence. consider converged if ALL slaves claim so
converge = comm.Reduce(converge, bothTrue, 0);
comm.Broadcast(ref converge, 0); // make sure EVERYONE breaks/coninues
bm3.pause();
parallel += bm3.getElapsedSeconds();
if (converge)
{
loops++;
break;
}
}
bm2.start();
// round the result slightly
err = null;
if (comm.Rank == 0)
{
x.Round(1e-14);
err = A * x - b;
err.Round(1e-14);
}
bm2.pause();
sequential += bm2.getElapsedSeconds();
bm.pause();
if (showBenchmark)
{
Console.WriteLine("Sequential part took " + sequential + " secs.");
Console.WriteLine("Parallel part took " + parallel + " secs.");
Console.WriteLine("Communication took " + communication + " secs.");
Console.WriteLine("Total: " + bm.getResult() + " (" + bm.getElapsedSeconds() + " secs). Seq + Parallel: " + (sequential + parallel));
}
return(converge);
}
public static Matrix Inverse(Matrix matrix, Intracommunicator comm, ref double timeS, ref double timeP, ref double timeC)
{
if (comm.Rank == 0 && !matrix.isSquare)
{
Exception e = new Exception("Matrix must be square!");
throw e;
}
benchmark bm = new benchmark(), bm2 = new benchmark();
bm.start();
int n = 0;
int[] perm = new int[10]; int toggle = 0; Matrix lum = null;
if (comm.Rank == 0)
{
n = matrix.dim1;
lum = LUPDecompose(matrix, out perm, out toggle);
}
bm.pause();
timeS += bm.getElapsedSeconds();
bm.start();
comm.Broadcast(ref n, 0);
comm.Broadcast(ref lum, 0);
if (comm.Rank != 0)
{
perm = new int[n];
}
comm.Broadcast(ref perm, 0);
comm.Broadcast(ref toggle, 0);
comm.Barrier();
bm.pause();
timeC += bm.getElapsedSeconds();
if (lum == null)
{
return(zeroLike(matrix));
}
bm.start();
Double det = 0;
if (comm.Rank == 0)
{
det = Determinant(lum, perm, toggle);
}
bm.pause();
timeS += bm.getElapsedSeconds();
bm.start();
comm.Broadcast(ref det, 0);
comm.Barrier();
bm.pause();
timeC += bm.getElapsedSeconds();
if (det == 0) // not invertible
{
// still return for the sake of simplicity
// Zero matrix * any matrix = zero matrix
// so it's never a valid answer
return(zeroLike(matrix));
}
bm.pause();
int slaves = comm.Size;
Matrix jobDistro = Utils.splitJob(n, slaves);
int startCol = 0, endCol = 0, size = (int)jobDistro[0, comm.Rank];
for (int p = 0; p < slaves; p++)
{
if (p != comm.Rank)
{
startCol += (int)jobDistro[0, p];
}
else
{
endCol = startCol + (int)jobDistro[0, p] - 1;
break;
}
}
bm.pause();
timeP += bm.getElapsedSeconds();
bm.start();
Matrix result = new Matrix(n, size);
for (int i = startCol; i < startCol + size; ++i)
{
double[] b = new double[n];
for (int j = 0; j < n; ++j)
{
if (i == perm[j])
{
b[j] = 1.0;
}
else
{
b[j] = 0.0;
}
}
double[] x = HelperSolve(lum, b);
for (int j = 0; j < n; ++j)
{
result[j, i - startCol] = x[j];
}
}
bm.pause();
timeP += bm.getElapsedSeconds();
bm.start();
// collect result
result = comm.Reduce(result, ConcatenateColumn, 0);
bm.pause();
timeP += bm.getElapsedSeconds();
return(result);
}
