F64ReferenceBackend

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.

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.

gemm with alpha = 1, beta = 0, into a fresh matrix. A.cols must equal B.rows.

C = alpha · op(A) · op(B) + beta · C (BLAS dgemm), with shapes op(A): m×k, op(B): k×n, C: m×n. beta == 0.0 overwrites c without reading it.

gemv with alpha = 1, beta = 0, into a fresh result.

y = alpha · op(A) · x + beta · y (BLAS dgemv), with op(A) being Aᵀ when transpose. beta == 0.0 overwrites y without reading it.

A = A + alpha · x · yᵀ (BLAS dger), the dense form a backend can dispatch. The free ger accepts F64VectorView operands and takes a sparse fast path.

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

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.

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.

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 for the Cholesky factorization chol (LAPACK dpotrs). b is not modified.

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

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

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

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 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 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, 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 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.

C = alpha · A · B + beta · C, or C = alpha · B · A + beta · C when right (BLAS dsymm). Only the lower triangle of a is read; beta == 0.0 overwrites c without reading it.

y = alpha · A · x + beta · y for a symmetric a (BLAS dsymv). Only the lower triangle is read, diagonal included; beta == 0.0 overwrites y without reading it.

A += alpha · x · xᵀ (BLAS dsyr), writing the triangles uplo selects. syrk is the rank-k form.

A += alpha · (x · yᵀ + y · xᵀ) (BLAS dsyr2), writing the triangles uplo selects.

C = alpha · (op(A) · op(B)ᵀ + op(B) · op(A)ᵀ) + beta · C (BLAS dsyr2k), where op transposes when transpose. Writes the triangles uplo selects.

C = alpha · A·Aᵀ + beta · C, or alpha · Aᵀ·A + beta · C when transpose (BLAS dsyrk). Uplo.FULL writes both triangles, unlike standard dsyrk; beta == 0.0 overwrites without reading.

B = alpha · op(T) · B, or B = alpha · B · op(T) when right (BLAS dtrmm), the counterpart of trsm.

x = op(T) · x in place (BLAS dtrmv), the product counterpart of trsv.

B = alpha · op(T)⁻¹ · B in place, or B = alpha · B · op(T)⁻¹ when right (BLAS dtrsm). Flags follow trsv; the right-hand sides are the columns of b from the left and its rows from the right.

Solve op(T) · x = b in place (BLAS dtrsv) for the lower or upper triangle of the square a, op transposing when transpose and unitDiag taking the diagonal as 1. x carries b in and x out.

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.