/// <summary>
/// Solves simultaneous linear equations.
/// </summary>
/// <typeparam name="T"></typeparam>
/// <param name="a"> [<c>a.Rank == 2 && a.Shape[0] == a.Shape[1]</c>] </param>
/// <param name="b"> [<c>(b.Rank == 1 || b.Rank == 2) && b.Shape[0] == a.Shape[0]</c>] </param>
/// <returns></returns>
/// <exception cref="ShapeMismatchException">
/// <para> When <c>n := a.Shape[0]</c>, </para>
/// <para> - If <c>b.Shape == {n}</c>, then <c>$ReturnValue.Shape == {n}</c>. </para>
/// <para>
/// - If <c>b.Shape == {n, p}</c>, then <c>$ReturnValue.Shape == {n, p}</c>.
/// Each column of return value is the solution against corresponding column of b.
/// </para>
/// </exception>
public static NdArray <T> Solve <T>(this INdArray <T> a, INdArray <T> b)
{
Guard.AssertShapeMatch(a.Rank == 2 && a.Shape[0] == a.Shape[1], "a.Rank == 2 && a.Shape[0] == a.Shape[1]");
if (b.Rank == 1 && b.Shape[0] == a.Shape[0])
{
var(l, u, perms) = a.LUWithPermutationsLegacy();
var x = NdArray.CreateMutable(new T[b.Shape[0]]);
SolveCore(l, u, perms, b, x);
return(x.MoveToImmutable());
}
if (b.Rank == 2 && b.Shape[0] == a.Shape[0])
{
var(l, u, perms) = a.LUWithPermutationsLegacy();
var x = NdArray.CreateMutable(new T[b.Shape[0], b.Shape[1]]);
var col = b.Shape[1];
for (var j = 0; j < col; ++j)
{
var bj = b[Range.Whole, new Index(j, false)];
var xj = x[Range.Whole, new Index(j, false)];
SolveCore(l, u, perms, bj, xj);
}
return(x.MoveToImmutable());
}
Guard.ThrowShapeMismatch("(b.Rank == 1 || b.Rank == 2) && b.Shape[0] == a.Shape[0]");
throw new NotSupportedException();
}
/// <summary>
/// Calculates a matrix determinant.
/// </summary>
/// <typeparam name="T"></typeparam>
/// <param name="ndArray"></param>
/// <param name="strategy"></param>
/// <returns></returns>
public static T Determinant <T>(this INdArray <T> ndArray, IIterationStrategy?strategy = default)
{
Guard.AssertShapeMatch(ndArray.Shape.Length == 2 && ndArray.Shape[0] == ndArray.Shape[1],
"x must be a square matrix.");
var(_, u, permutations) = ndArray.LUWithPermutationsLegacy(strategy);
var n = ndArray.Shape[0];
var x = One <T>();
for (var i = 0; i < n; ++i)
{
x = Multiply(x, u[i, i]);
}
if (permutations.Count % 2 == 1)
{
x = UnaryNegate(x);
}
return(x);
}
