Module linalg 

Dense linear solves and general complex eigensystems with analytic derivatives, and restarted GMRES. treams-rs extension.

Two libraries share the linear algebra, by one rule:

• nalgebra’s DMatrix stores every dense matrix.
• faer factors and multiplies dense complex matrices of any size, through this module: LU solves (solve, Lu), singular values (svdvals), eigensystems (eig) and products (product and its variants). view and view_mut lend a DMatrix to faer without a copy.
• Real symmetric eigenproblems use nalgebra’s SymmetricEigen: special::wigner diagonalizes the tridiagonal generator of the small-d matrix, of size 2l + 1.
• Fixed-size matrices of at most 4 x 4 stay in nalgebra: the 3 x 3 inverse of lattice::reciprocal, the closed-form 2 x 2 inverse of coeffs::mie, the 4 x 4 and 2 x 2 inverses of coeffs::mie_cyl and the 4 x 4 inverse of smatrix::interface.

Modules

gmres 🔒

Restarted GMRES (generalized minimal residual method) for complex linear systems given only through matrix-vector products. treams-rs extension.

saved 🔒

Fixed numerical layouts for invocation-owned JAX state. Dimensions are stored as u64; all other data are IEEE f64 components in column-major order. treams-rs extension.

Structs

Convergence

Proof of convergence for one right-hand side: the residual b - A x recomputed from the returned answer x, not the estimate of the Arnoldi iteration.

EigResidual

What eig saves for its derivatives: the right eigensystem, with unit vectors whose largest component is real and positive.

Equilibration 🔒

Positive diagonal scales for the equivalent operator R A C.

GmresOptions

Accuracy and memory budget of restarted GMRES.

Lu 🔒

Packed pivoted LU: overwrite the operator and share its triangular storage.

SolveGradient

Cotangents of the inputs of solve and solve_owned, in the order of their arguments.

SolveResidual

What solve saves for its derivatives: the LU factors and the solution. Both derivative directions solve with the same factors.

SvdvalsResidual

What svdvals saves for its derivatives: the thin singular vectors and the singular values.

ThinSvd 🔒

Thin singular vectors and values. faer’s Svd reads faer’s global parallelism, so the decomposition is called with DECOMPOSITION instead.

Constants

BALANCE_SWEEPS 🔒

The most sweeps of balance; dense operators take at most three.

DECOMPOSITION 🔒

Eigen- and singular-value decompositions run on the calling thread. faer’s parallel Hessenberg and bidiagonal reductions split inner products by worker count, so their last bits, and the order of eigenvalues, would follow the budget. On four workers they gained at most 1.3× (eigenvalues) and 1.7× (singular values) at n = 768, and nothing below n = 384.

Functions

balance 🔒

Even out the row and column sums of a max-equilibrated operator in place with power-of-two factors, folded into row and column.

eig

General complex eigendecomposition. Repeated eigenvalues are allowed in the primal.

eigen 🔒

Eigenvalues and right eigenvectors, called with an explicit Par as thin_svd.

eigvals

Eigenvalues ordered by real part, then imaginary part. The same eigensystem residual supplies derivatives without imposing an eigenvector phase constraint.

equilibrate 🔒

Equilibrate severely unbalanced finite square operators in place.

faer_zeros 🔒

Reserve decomposition vectors before initializing their entries.

lu_threads 🔒

The faer workers of an LU factorization or solve with rows rows and columns right-hand sides, on a pool of budget threads.

lu_workers 🔒

The faer workers of an LU call on the treams-rs budget.

product 🔒

left right.

product_adjoint_right 🔒

left rightᴴ.

product_adjoint_right_into 🔒

out = left rightᴴ, reusing the allocation of out.

product_into 🔒

out = left right on the treams-rs pool (crate::threads::product). Pass .adjoint() views for conjugate-transposed factors; faer conjugates them inside the product kernel.

product_views 🔒

left right in a new matrix; see product_into.

scale_rows 🔒

matrix[(i, j)] scales[i] 2^shifts[j], rounded once unless subnormal. A shifted column splits each scale into a mantissa and a power of two, and applies the power first when it scales up, so no intermediate product overflows or flushes a subnormal entry.

scratch 🔒

A faer workspace of req, or Error::OutOfMemory when the system refuses it or the requirement overflowed.

solve

Solve A X = B with pivoted LU; B is a matrix of right-hand sides.

solve_owned

Solve while reusing the owned operator buffer for its LU factors.

svdvals

Singular values of a finite nonempty rectangular complex matrix.

thin_svd 🔒

view 🔒

Column-major faer view of a nalgebra matrix.

view_mut 🔒

Mutable column-major faer view of a nalgebra matrix.