Krylov Newton Algorithm

The Krylov Newton algorithm is a memory-efficient variation of the Newton-Raphson method that uses a Krylov subspace to approximate the solution. It is particularly useful for large-scale problems where memory usage is a concern.

Description

The Krylov Newton algorithm: - Uses a Krylov subspace to approximate the solution - Updates the geometry and internal forces based on the displacement increment - Repeats until convergence is achieved - Can control the tangent matrix used for iterations and increments

Parameters

Parameter	Type	Description
tang_iter	str	Tangent matrix to use for iterations (“current” or “initial”) (default: “current”)
tang_incr	str	Tangent matrix to use for increments (“current” or “initial”) (default: “current”)
max_dim	int	Maximum dimension of the Krylov subspace (default: 3)

Usage Example

# Create a Femora instance


# Create a Krylov Newton algorithm with default settings
fm.analysis.algorithm.create_algorithm("KrylovNewton")

# Create a Krylov Newton algorithm with custom settings
fm.analysis.algorithm.create_algorithm(
    "KrylovNewton",
    tang_iter="initial",
    tang_incr="current",
    max_dim=5
)

Notes

• The Krylov Newton algorithm is memory-efficient compared to standard Newton

• It is particularly useful for large-scale problems

• The convergence rate may be slower than standard Newton

• The max_dim parameter controls the trade-off between memory usage and convergence rate