/// <summary>
/// Solves a system of linear equations, <c>A'x = b</c>.
/// </summary>
/// <param name="input">The right hand side vector, <c>b</c>.</param>
/// <param name="result">The left hand side vector, <c>x</c>.</param>
public void SolveTranspose(Complex[] input, Complex[] result)
{
if (input == null)
{
throw new ArgumentNullException(nameof(input));
}
if (result == null)
{
throw new ArgumentNullException(nameof(result));
}
int m2 = S.m2;
var x = new Complex[m2];
if (m >= n)
{
// x(q(0:m-1)) = b(0:m-1)
Permutation.Apply(S.q, input, x, n);
SolverHelper.SolveUpperTranspose(R, x); // x = R'\x
// Apply Householder reflection to x.
for (int k = n - 1; k >= 0; k--)
{
ApplyHouseholder(Q, k, beta[k], x);
}
// b(0:n-1) = x(p(0:n-1))
Permutation.Apply(S.pinv, x, result, m);
}
else
{
// x(0:m-1) = b(p(0:m-1)
Permutation.ApplyInverse(S.pinv, input, x, n);
// Apply Householder reflection to x.
for (int k = 0; k < m; k++)
{
ApplyHouseholder(Q, k, beta[k], x);
}
SolverHelper.SolveUpper(R, x); // x = R\x
// b(q(0:n-1)) = x(0:n-1)
Permutation.ApplyInverse(S.q, x, result, m);
}
}
/// <summary>
/// Solves a system of linear equations, <c>Ax = b</c>.
/// </summary>
/// <param name="input">The right hand side vector, <c>b</c>.</param>
/// <param name="result">The left hand side vector, <c>x</c>.</param>
/// <remarks>
/// Let A be a m-by-n matrix. If m >= n a least-squares problem (min |Ax-b|)
/// is solved. If m < n the underdetermined system is solved.
/// </remarks>
public override void Solve(double[] input, double[] result)
{
if (input == null)
{
throw new ArgumentNullException("input");
}
if (result == null)
{
throw new ArgumentNullException("result");
}
var x = new double[S.m2];
if (m >= n)
{
// x(0:m-1) = b(p(0:m-1)
Permutation.ApplyInverse(S.pinv, input, x, m);
// Apply Householder reflection to x.
for (int k = 0; k < n; k++)
{
ApplyHouseholder(Q, k, beta[k], x);
}
SolverHelper.SolveUpper(R, x); // x = R\x
// b(q(0:n-1)) = x(0:n-1)
Permutation.ApplyInverse(S.q, x, result, n);
}
else
{
// x(q(0:m-1)) = b(0:m-1)
Permutation.Apply(S.q, input, x, m);
SolverHelper.SolveUpperTranspose(R, x); // x = R'\x
// Apply Householder reflection to x.
for (int k = m - 1; k >= 0; k--)
{
ApplyHouseholder(Q, k, beta[k], x);
}
// b(0:n-1) = x(p(0:n-1))
Permutation.Apply(S.pinv, x, result, n);
}
}
/// <summary>
/// Solves a system of linear equations, <c>A'x = b</c>.
/// </summary>
/// <param name="input">The right hand side vector, <c>b</c>.</param>
/// <param name="result">The left hand side vector, <c>x</c>.</param>
public void SolveTranspose(Complex[] input, Complex[] result)
{
if (input == null)
{
throw new ArgumentNullException(nameof(input));
}
if (result == null)
{
throw new ArgumentNullException(nameof(result));
}
var x = this.temp;
Permutation.Apply(S.q, input, x, n); // x = Q'*b
SolverHelper.SolveUpperTranspose(U, x); // x = U'\x.
SolverHelper.SolveLowerTranspose(L, x); // x = L'\x.
Permutation.Apply(pinv, x, result, n); // b = P'*x
}