internal Vector SolveInternal(Matrix A, Vector b, Vector c, StartingBasis B, Matrix AB, Vector bB)
{
//Init simplex
var m = A.RowCount;
var ABi = AB.Inverse();
//X is a starting vector
var x = AB.LU().Solve(bB);
while (true)
{
iteration++;
//Compute lambda (cT*AB.inv())T
var lambda = (c.ToRowMatrix()*ABi).Transpose().ToRowWiseArray();
//check for optimality
if (lambda.All(l => l >= 0)) return (Vector) x;
//Find leaving index r (first index where component < 0)
var r = lambda.Select((i, index) => new {i, index})
.Where((i, index) => i.i < 0)
.First().index;
//compute direction to move int - take r-th column
var d = ABi.Column(r)*(-1);
//Determine the set K (all indexes of positive values of lambda)
//all k that a(k).T*d>0, 1 <= i <=m
var K = new List<int>();
for (int k = 0; k < m; k++)
{
var val = A.Row(k)*d;
if (val > 0 && !val.FloatEquals(0))
K.Add(k);
}
if (K.Count == 0)
throw new SimplexException("Problem is unbounded") {Iteration = iteration};
//Find entering index e
int e = 0;
var v = double.MaxValue;
foreach (var k in K)
{
var w = (b[k] - A.Row(k)*x)/(A.Row(k)*d);
if (!(w < v)) continue;
v = w;
e = k;
}
//Update basis
B.InequalityIndexes[r] = e;
AB.SetRow(r, A.Row(e));
bB[r] = b[e];
//Trick - lets update inverse AB in a smart way - sinse there is only one new inequality we only need to
//compute new inversed row (should drop complexity of whole algo to n*n)
var f = AB.Row(r)*ABi;
var g = -f;
g[r] = 1;
g /= f[r];
g[r] -= 1;
ABi = ABi.Add(ABi.Column(r).ToColumnMatrix()*g.ToRowMatrix());
//Compute new x
x = x + v*d;
}
}