F64Decompositions

The vector kernels this half's inherited routines run on; the installed ones by default.

Apply Q, or Qᵀ when transpose, from qr to a length-m y, into a fresh result (LAPACK dormqr restricted to a single column).

Apply Q, or Qᵀ when transpose, from qr to y into out, which is returned. out may be y.

Cholesky factorization A = L·Lᵀ of a symmetric positive-definite a (dpotrf with uplo = 'L'). Reads only the lower triangle of a, so an upper-only matrix factors to silent nonsense.

Throws

LU factorization with partial pivoting of a square a (LAPACK dgetrf). a is not modified.

Refactorize a into out's existing buffers, returning out. out must have a's dimension and its previous contents are discarded.

Invert a general matrix from its LU factorization, returning A⁻¹ given P·A = L·U (LAPACK dgetri). Prefer solve to apply A⁻¹, which costs less and is more accurate.

Throws

Invert an SPD matrix from its Cholesky factorization, returning A⁻¹ given chol (LAPACK dpotri).

Symmetric indefinite factorization A = L·D·Lᵀ with Bunch-Kaufman pivoting (LAPACK dsytrf, lower). Reads only the lower triangle of a, so an upper-only matrix factors to silent nonsense.

QR factorization A = Q·R of an m×n a via Householder reflections (LAPACK dgeqrf). a is not modified, any shape is accepted, and rank deficiency is not detected.

QR with column pivoting, A·P = Q·R (LAPACK dgeqp3), reporting F64PivotedQrDecomposition.rank as the count of leading diagonal entries with |R_kk| > tolerance · |R₀₀|. tolerance is a fraction of |R₀₀|; AUTOMATIC_RANK_TOLERANCE derives one from the shape, max(m, n) · ε, and a negative value is rejected.

Throws

Order-of-magnitude estimate of 1 / (anorm · est(‖A⁻¹‖₁)) (LAPACK dgecon), where anorm is the 1-norm of the unfactored matrix (see norm1). Returns 1.0 when n == 0 and 0.0 when singular.

Solve A · x = b, or Aᵀ · x = b when transpose, for the factorization lu (LAPACK dgetrs).

Solve A · X = B, or Aᵀ · X = B when transpose, for all right-hand-side columns of b at once (LAPACK dgetrs with nrhs).

Solve A · x = b for a symmetric indefinite factorization ldl (LAPACK dsytrs).

Solve A · X = B for all right-hand-side columns of b at once against a symmetric indefinite factorization (LAPACK dsytrs with nrhs).

Least-squares solve from a pivoted factorization, with the column permutation undone. A rank-deficient factorization returns the basic solution, not the minimum-norm one: zero outside the pivoted rank.

Solve from a QR factorization. By default, finds the least-squares solution min ‖A·x − b‖₂ for a tall or square A; it requires full column rank and returns R⁻¹·(Qᵀb). With minimumNorm, finds the minimum-norm solution of a consistent wide system from qr(Aᵀ); it requires full row rank.

Solve A · x = b for the Cholesky factorization chol (LAPACK dpotrs). b is not modified.

Solve A · x = b, or Aᵀ · x = b when transpose, into out, which is returned. out may be b, and a workspace lends the transposed direction's staging buffer.

Solve A · X = B, or Aᵀ · X = B when transpose, into out, which is returned. out may be b, and a workspace lends the transposed direction's n·nrhs staging block.

Solve A · x = b into out, which is returned. out may be b.

Solve A · X = B into out, which is returned. out may be b.

solve into out, which is returned.

solve into out, which is returned. Its length is n by default and m with minimumNorm. A workspace lends the intermediate for applying Q or Qᵀ.

Invert a triangular matrix into a fresh result (LAPACK dtrtri), returning T⁻¹ for the lower or upper triangle of the square a, taking the diagonal as 1 when unitDiag.

Throws