/// <summary>
/// QR decomposition with pivoting, possibly economical sized
/// </summary>
/// <param name="A">general input matrix A of size [m x n]</param>
/// <param name="R">output parameter. Upper triangular matrix R as
/// result of decomposition. Size [m x n] or [min(m,n) x n] depending
/// on <paramref name="economySize"/> (see remarks). </param>
/// <param name="economySize"><para>if true, <list type="bullet">
/// <item>the size of Q and R will be [m x m] and [m x n] respectively.
/// However, if m < n, the economySize parameter has no effect on
/// those sizes.</item>
/// <item>the output parameter E will be returned as row permutation
/// vector rather than as permutation matrix</item></list></para>
/// <para>if false, this function acts exactly as its overload
/// <see cref="ILNumerics.BuiltInFunctions.ILMath.qr(ILArray<double>,ref ILArray<double>,ref ILArray<double>)"/></para>
/// </param>
/// <param name="E">permutation matrix from pivoting. Size [m x m].
/// If this is not null, the permutation matrix/ vector E will be returned.
/// <para>E is of size [n x n], if <paramref name="economySize"/> is
/// true, a row vector of length n otherwise</para></param>
/// <returns>Orthonormal / unitary matrix Q as result of decomposition.
/// Size [m x m] or [m x min(m,n)], depending on <paramref name="economySize"/>
/// (see remarks below)</returns>
/// <remarks><para> If <paramref name="economySize"/> is false, the function
/// returns Q, R and E such that the equation A * E = Q * R holds except
/// roundoff errors. </para>
/// <para>If <paramref name="economySize"/> is true, E will be a permutation
/// vector and the equation A[null,E] == Q * R holds (except roundoff).</para>
/// <para>E reflects the pivoting of A done inside LAPACK in order to give R
/// increasingly diagonal elements.</para>
/// <para>Q, R and E will be solid ILArray's.</para></remarks>
public static ILArray <complex> qr(
ILArray <complex> A,
ref ILArray <complex> R,
ref ILArray <complex> E,
bool economySize)
{
if (Object.Equals(R, null))
{
return(qr(A));
}
int m = A.Dimensions[0];
int n = A.Dimensions[1];
if (m < n && economySize)
{
return(qr(A, ref R, false));
}
ILArray <complex> ret;
if (m == 0 || n == 0)
{
R = new ILArray <complex> (new complex [0], m, n);
E = new ILArray <complex> (new complex [0], 1, 0);
return(ILArray <complex> .empty(A.Dimensions));
}
// prepare IPVT
if (object.Equals(E, null))
{
return(qr(A, ref R, economySize));
}
if (!economySize)
{
E = new ILArray <complex> (new complex [n * n], n, n);
}
else
{
E = new ILArray <complex> (new complex [n], 1, n);
}
int [] ipvt = new int[n];
int minMN = (m < n)?m:n;
int info = 0;
complex [] tau = new complex [minMN];
complex [] QArr;
if (m >= n)
{
ret = new ILArray <complex> (
new complex [(economySize)? minMN * m : m * m],
m, (economySize)? minMN: m);
}
else
{
// economySize is always false ... !
// a temporary array is needed for extraction of the compact lapack Q (?geqrf)
ret = new ILArray <complex> (
new complex [m * n], m, n);
}
QArr = ret.m_data;
for (int i = m * n; i-- > 0;)
{
QArr[i] = A.GetValue(i);
}
Lapack.zgeqp3(m, n, QArr, m, ipvt, tau, ref info);
if (info != 0)
{
throw new ILArgumentException("qr: error inside lapack library (?geqrf). info=" + info.ToString());
}
// extract R, Q
if (economySize)
{
R = copyUpperTriangle(QArr, m, n, minMN);
Lapack.zungqr(m, minMN, tau.Length, QArr, m, tau, ref info);
// transform E into out typed vector
for (int i = 0; i < n; i++)
{
E.m_data[i] = ipvt[i] - 1;
}
}
else
{
R = copyUpperTriangle(QArr, m, n, m);
Lapack.zungqr(m, m, tau.Length, QArr, m, tau, ref info);
if (m < n)
{
ret = ret[":;0:" + (m - 1)];
}
// transform E into matrix
for (int i = 0; i < n; i++)
{
E.m_data[(ipvt[i] - 1) + n * i] = new complex(1.0, 0.0);
}
}
if (info != 0)
{
throw new ILArgumentException("qr: error in lapack library (???gqr). info=" + info.ToString());
}
return(ret);
}