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Factorizes a row-major matrix by Householder reflections.

public static QrDecomposition Householder(ReadOnlySpan<double> matrix, int rowCount, int columnCount)

Parameters — matrix is the matrix, row-major: columnCount values per row. rowCount and columnCount are its shape, and there must be at least as many rows as columns.

Returns — the thin factorization.

Exceptions — ArgumentOutOfRangeException when either dimension is not positive, or when there are more columns than rows. ArgumentException when matrix does not hold exactly rowCount × columnCount values.

Example — the factors multiply back to the matrix.

using Lodestar.Decomposition;

double[] matrix = [1.0, 2.0, 3.0, 4.0, 5.0, 7.0];
QrDecomposition qr = QrDecomposition.Householder(matrix, rowCount: 3, columnCount: 2);

// Q · R reproduces the first entry.
double rebuilt = (qr.Q[0] * qr.R[0]) + (qr.Q[1] * qr.R[2]);   // => 0.9999999999999997

// Q's columns are orthonormal, so the first has unit length.
double lengthSquared = (qr.Q[0] * qr.Q[0]) + (qr.Q[2] * qr.Q[2]) + (qr.Q[4] * qr.Q[4]);

Remarks — the reflections are LAPACK's, dgeqr2 and dorg2r, so the signs are numpy.linalg.qr's and, on a matrix of full rank, the two agree entry for entry, to rounding. So do their infinities and NaNs: a matrix holding ±∞ or NaN comes back with the same entries infinite, NaN or finite as numpy's. On a rank-deficient matrix, past the pivot that vanishes, a column of Q is built from rounding noise on both sides, and the two agree only in Q · R.

A wide matrix is refused rather than padded: there is no thin QR of one, and answering with a full factorization under a method that promises a thin one would be worse than saying so.

Applies to — net10.0, netstandard2.0.

See also — QrDecomposition, TruncatedSvd.Fit.