private bool CoreBuildSpecial()
{
if (m_X.Count > 4)
{
return(false);
}
if (m_X.Count == 0)
{
m_A = new BigRational[] { BigRational.NaN };
m_B = new BigRational[] { 0 };
m_C = new BigRational[] { 0 };
m_D = new BigRational[] { 0 };
}
else if (m_X.Count == 1)
{
m_A = new BigRational[] { m_Y[0] };
m_B = new BigRational[] { 0 };
m_C = new BigRational[] { 0 };
m_D = new BigRational[] { 0 };
}
else if (m_X.Count == 2)
{
m_A = new BigRational[] { (m_Y[0] * m_X[1] - m_Y[1] * m_X[0]) / (m_X[1] - m_X[0]) };
m_B = new BigRational[] { (m_Y[1] - m_Y[0]) / (m_X[1] - m_X[0]) };
m_C = new BigRational[] { 0 };
m_D = new BigRational[] { 0 };
}
else if (m_X.Count == 3)
{
BigRational[] s = MatrixLowLevel.Solve(new BigRational[][] {
new BigRational[] { 1, m_X[0], m_X[0] * m_X[0], m_Y[0] },
new BigRational[] { 1, m_X[1], m_X[1] * m_X[1], m_Y[1] },
new BigRational[] { 1, m_X[2], m_X[2] * m_X[2], m_Y[2] },
});
m_A = new BigRational[] { s[0] };
m_B = new BigRational[] { s[1] };
m_C = new BigRational[] { s[2] };
m_D = new BigRational[] { 0 };
}
else if (m_X.Count == 4)
{
BigRational[] s = MatrixLowLevel.Solve(new BigRational[][] {
new BigRational[] { 1, m_X[0], m_X[0] * m_X[0], m_X[0] * m_X[0] * m_X[0], m_Y[0] },
new BigRational[] { 1, m_X[1], m_X[1] * m_X[1], m_X[1] * m_X[1] * m_X[1], m_Y[1] },
new BigRational[] { 1, m_X[2], m_X[2] * m_X[2], m_X[2] * m_X[2] * m_X[2], m_Y[2] },
new BigRational[] { 1, m_X[3], m_X[3] * m_X[3], m_X[3] * m_X[3] * m_X[3], m_Y[3] },
});
m_A = new BigRational[] { s[0] };
m_B = new BigRational[] { s[1] };
m_C = new BigRational[] { s[2] };
m_D = new BigRational[] { s[3] };
}
return(true);
}
// http://web.snauka.ru/issues/2015/05/53846
// https://en.wikipedia.org/wiki/Akima_spline
private void CoreBuild()
{
if (CoreBuildSpecial())
{
return;
}
int N = m_X.Count - 1;
Dictionary <int, BigRational> m = new();
for (int i = 0; i < N; ++i)
{
m.Add(i, (m_Y[i + 1] - m_Y[i]) / (m_X[i + 1] - m_X[i]));
}
m.Add(-2, 3 * m[0] - 2 * m[1]);
m.Add(-1, 2 * m[0] - 2 * m[1]);
m.Add(N, 2 * m[N - 1] - 2 * m[N - 2]);
m.Add(N + 1, 3 * m[N - 1] - 2 * m[N - 2]);
BigRational[] s = new BigRational[N + 1];
for (int i = 0; i <= N; ++i)
{
s[i] = ((m[i + 1] - m[i]).Abs() * m[i - 1] +
(m[i - 1] - m[i - 2]).Abs() * m[i]) /
((m[i + 1] - m[i]).Abs() + (m[i - 1] - m[i - 2]).Abs());
}
for (int i = 0; i < N; ++i)
{
BigRational[][] mt = new BigRational[][] {
new BigRational[] { 1, m_X[i], m_X[i] * m_X[i], m_X[i] * m_X[i] * m_X[i], m_Y[i] },
new BigRational[] { 1, m_X[i + 1], m_X[i + 1] * m_X[i + 1], m_X[i + 1] * m_X[i + 1] * m_X[i + 1], m_Y[i + 1] },
new BigRational[] { 0, 1, 2 * m_X[i], 3 * m_X[i] * m_X[i], s[i] },
new BigRational[] { 0, 1, 2 * m_X[i + 1], 3 * m_X[i + 1] * m_X[i + 1], s[i + 1] },
};
BigRational[] z = MatrixLowLevel.Solve(mt);
m_A[i] = z[0];
m_B[i] = z[1];
m_C[i] = z[2];
m_D[i] = z[3];
}
}
private void CoreBuild()
{
if (m_X.Count == 0)
{
m_A = new BigRational[] { BigRational.NaN };
m_Polynom = RationalPolynom.NaN;
return;
}
else if (m_X.Count == 1)
{
m_A = new BigRational[] { m_Y[0] };
m_Polynom = new RationalPolynom(m_A);
return;
}
BigRational[][] M = new BigRational[m_Y.Count][];
for (int r = 0; r < M.Length; ++r)
{
BigRational[] row = new BigRational[M.Length + 1];
row[M.Length] = m_Y[r];
BigRational x = m_X[r];
BigRational v = 1;
for (int i = 0; i < M.Length; ++i)
{
row[i] = v;
v *= x;
}
M[r] = row;
}
m_A = MatrixLowLevel.Solve(M);
m_Polynom = new RationalPolynom(m_A);
}
/// <summary>
/// Reconstruct polynom from its values
/// for P(0) = 1, P(1) = ?, P(2) = 9, P(3) = 16, P(4) = 25, P(5) = ?
/// var result = Reconstruct(0, new BigRational[] {1, BigRational.NaN, 9, 16, 25, BigRational.NaN});
/// </summary>
/// <param name="startAt">starting point</param>
/// <param name="values">values (put BigRational.NaN for abscent points)</param>
/// <returns></returns>
public static RationalPolynom Reconstruct(int startAt, IEnumerable <BigRational> values)
{
if (values is null)
{
throw new ArgumentNullException(nameof(values));
}
var points = values
.Select((v, i) => (x: (BigRational)i + startAt, y: v))
.Where(item => !item.y.IsNaN)
.ToArray();
if (points.Length == 0)
{
return(NaN);
}
BigRational[][] M = new BigRational[points.Length][];
for (int r = 0; r < M.Length; ++r)
{
BigRational[] row = new BigRational[M.Length + 1];
M[r] = row;
row[M.Length] = points[r].y;
BigRational x = points[r].x;
BigRational v = 1;
for (int c = 0; c < M.Length; ++c)
{
row[c] = v;
v *= x;
}
}
return(new RationalPolynom(MatrixLowLevel.Solve(M)));
}