Module krylov 

Krylov subspace methods for solving linear equations with abstract tensors.

This module provides iterative solvers that work with any type implementing TensorVectorSpace, enabling their use in tensor network algorithms without requiring dense vector representations.

Solvers

• gmres: Generalized Minimal Residual Method (GMRES) for non-symmetric systems
• hermitian_lanczos_lowest_eigenpair: Matrix-free lowest eigenpair solver for Hermitian operators

Future Extensions

• CG (Conjugate Gradient) for symmetric positive definite systems
• BiCGSTAB for non-symmetric systems with better convergence properties

Example

use tensor4all_core::{
    krylov::{gmres, GmresOptions},
    DynIndex, TensorDynLen, TensorVectorSpace,
};

let i = DynIndex::new_dyn(2);
let rhs = TensorDynLen::from_dense(vec![i.clone()], vec![1.0, -1.0])?;
let initial_guess = TensorDynLen::from_dense(vec![i.clone()], vec![0.0, 0.0])?;

let apply_operator = |x: &TensorDynLen| Ok(x.clone());
let result = gmres(apply_operator, &rhs, &initial_guess, &GmresOptions::default())?;

assert!(result.converged);
assert!(result.solution.sub(&rhs)?.maxabs() < 1e-12);

Structs

GmresOptions

Options for GMRES solver.

GmresResult

Result of GMRES solver.

HermitianKrylovExpmOptions

Options for hermitian_krylov_expm_multiply.

HermitianKrylovExpmResult

Result of hermitian_krylov_expm_multiply.

HermitianLanczosOptions

Options for hermitian_lanczos_lowest_eigenpair.

HermitianLanczosResult

Result of hermitian_lanczos_lowest_eigenpair.

RestartGmresOptions

Options for restarted GMRES with truncation.

RestartGmresResult

Result of restarted GMRES solver.

Functions

gmres

Solve A x = b using GMRES (Generalized Minimal Residual Method).

gmres_affine

Solve (a0 I + a1 A) x = b using GMRES with relative residual tolerance.

gmres_affine_with_absolute_tolerance

Solve (a0 I + a1 A) x = b using GMRES with an absolute residual tolerance.

gmres_with_absolute_tolerance

Solve A x = b using GMRES with an absolute residual tolerance.

gmres_with_total_iteration_limit

Solve A x = b using GMRES while enforcing a total iteration limit.

gmres_with_truncation

Solve A x = b using GMRES with optional truncation after each iteration.

hermitian_krylov_expm_multiply

Apply exp(exponent * A) to a vector using a matrix-free Hermitian Krylov method.

hermitian_lanczos_lowest_eigenpair

Compute the lowest eigenpair of a Hermitian matrix-free operator.

restart_gmres_with_truncation

Solve A x = b using restarted GMRES with truncation.