Generic interface

Krylov.jl provides a generic interface for solving linear systems using (block) Krylov methods. The interface is designed to be common for all methods and contains three routines krylov_workspace, krylov_solve and krylov_solve!. They allow to build Krylov workspaces and call both in-place and out-of-place variants of the solvers with a unified API.

Krylov.krylov_workspace — Function

workspace = krylov_workspace(Val(method), args...; kwargs...)

Generic function that dispatches to the appropriate workspace constructor for each subtype of KrylovWorkspace and BlockKrylovWorkspace. The first argument Val(method), where method is a symbol (such as :cg, :gmres or :block_minres), specifies the (block) Krylov method for which a workspace is desired. The returned workspace can later be used by krylov_solve! to execute the (block) Krylov method in-place.

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Krylov.krylov_solve — Function

krylov_solve(Val(method), args...; kwargs...)

Generic function that dispatches to the appropriate out-of-place (block) Krylov method specified by symbol method (such as :cg, :gmres or :block_minres).

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Krylov.krylov_solve! — Function

krylov_solve!(workspace, args...; kwargs...)

Generic function that dispatches to the appropriate in-place (block) Krylov method based on the type of workspace. The argument workspace must be a subtype of KrylovWorkspace or BlockKrylovWorkspace (such as CgWorkspace, GmresWorkspace or BlockMinresWorkspace).

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The section on workspace accessors describes how to retrieve the solution, statistics, and other results from a workspace after calling krylov_solve!.

Examples

using Krylov, SparseArrays, LinearAlgebra

# Define a symmetric positive definite matrix A and a right-hand side vector b
n = 1000
A = sprandn(n, n, 0.005)
A = A * A' + I
b = randn(n)

# Out-of-place interface
for method in (:cg, :cr, :car)
    x, stats = krylov_solve(Val(method), A, b)
    r = b - A * x
    println("Residual norm for $(method): ", norm(r))
end

Residual norm for cg: 3.406895148541806e-7
Residual norm for cr: 3.720577865400298e-7
Residual norm for car: 4.3843116777622553e-7

using Krylov, SparseArrays, LinearAlgebra

# Define a square nonsymmetric matrix A and a right-hand side vector b
n = 100
A = sprand(n, n, 0.05) + I
b = rand(n)

# In-place interface
for method in (:bicgstab, :gmres, :qmr)
    # Create a workspace for the Krylov method
    workspace = krylov_workspace(Val(method), A, b)

    # Solve the system in-place
    krylov_solve!(workspace, A, b)

    # Get the statistics
    stats = Krylov.statistics(workspace)

    # Retrieve the solution
    x = Krylov.solution(workspace)

    # Check if the solver converged
    solved = Krylov.issolved(workspace)
    println("Convergence of $method: ", solved)

    # Display the number of iterations
    niter = Krylov.iteration_count(workspace)
    println("Number of iterations for $method: ", niter)

    # Display the elapsed timer
    timer = Krylov.elapsed_time(workspace)
    println("Elapsed time for $method: ", timer, " seconds")

    # Display the number of operator-vector products with A and A'
    nAprod = Krylov.Aprod_count(workspace)
    nAtprod = Krylov.Atprod_count(workspace)
    println("Number of operator-vector products with A and A' for $method: ", (nAprod, nAtprod))

    println()
end

Convergence of bicgstab: true
Number of iterations for bicgstab: 194
Elapsed time for bicgstab: 0.000255769 seconds
Number of operator-vector products with A and A' for bicgstab: (388, 0)

Convergence of gmres: true
Number of iterations for gmres: 85
Elapsed time for gmres: 0.000277671 seconds
Number of operator-vector products with A and A' for gmres: (85, 0)

Convergence of qmr: true
Number of iterations for qmr: 128
Elapsed time for qmr: 0.000179046 seconds
Number of operator-vector products with A and A' for qmr: (128, 128)