gko::solver::Cgs

Conjugate gradient squared. Replaces Bicg’s Aᵀ apply with another apply of A, doubling the per-iteration residual reduction but amplifying numerical noise. Cheap per iteration but less robust than Bicgstab; useful when A is well conditioned.

template<typename ValueType = default_precision>
class Cgs

Inherits from

• public gko::LinOp

• public gko::solver::EnablePreconditionedIterativeSolver<default_precision, Cgs<default_precision>>

• public gko::Transposable

CGS or the conjugate gradient square method is an iterative type Krylov subspace method which is suitable for general systems.

CGS rests on the identity that BiCG produces a residual of the form \( r_k^{\mathrm{BiCG}} = \psi_k(A) r_0 \), where \( \psi_k \) is the residual polynomial. Squaring this polynomial yields the CGS residual

\[ r_k^{\mathrm{CGS}} = \psi_k(A)^2 r_0, \]

so the iteration avoids the explicit \( A^H \) apply that BiCG requires, and it contracts the residual twice as fast per step when BiCG converges.

Starting from \( r_0 = b - A x_0 \), a shadow residual \( \tilde r_0 \) (Ginkgo uses \( \tilde r_0 = r_0 \)), \( q_{-1} = p_{-1} = 0 \) and \( \rho_{-1} = 1 \), one iteration with the preconditioner \( M \approx A^{-1} \) reads

\[\begin{split} \begin{aligned} \rho_k &= \langle \tilde r_0, r_k \rangle, &\qquad \beta_k &= \rho_k / \rho_{k-1}, \\ u_k &= r_k + \beta_k q_{k-1}, &\qquad p_k &= u_k + \beta_k (q_{k-1} + \beta_k p_{k-1}), \\ \hat v_k &= A M p_k, &\qquad \alpha_k &= \rho_k / \langle \tilde r_0, \hat v_k \rangle, \\ q_k &= u_k - \alpha_k \hat v_k, &\qquad \hat u_k &= M (u_k + q_k), \\ x_{k+1} &= x_k + \alpha_k \hat u_k, &\qquad r_{k+1} &= r_k - \alpha_k A \hat u_k. \end{aligned} \end{split}\]

The pair \( u_k, q_k \) takes the place of the single BiCG search direction, which is why the update of \( x \) uses their sum. The trade-off for dropping \( A^H \) is that the squared polynomial also squares the error: the residuals lose monotonicity and can oscillate strongly on poorly conditioned systems. When this matters in practice, BiCGSTAB replaces the second application of \( \psi_k \) by a locally minimizing polynomial for the same number of operator applies and one extra inner product.

The implementation in Ginkgo makes use of the merged kernel to make the best use of data locality. The inner operations in one iteration of CGS are merged into 3 separate steps.

Template Parameters:

ValueType – precision of matrix elements

Public Functions

virtual std::unique_ptr<LinOp> transpose() const override

Returns a LinOp representing the transpose of the Transposable object.

Returns:

a pointer to the new transposed object

virtual std::unique_ptr<LinOp> conj_transpose() const override

Returns a LinOp representing the conjugate transpose of the Transposable object.

Returns:

a pointer to the new conjugate transposed object

inline bool apply_uses_initial_guess() const override

Return true as iterative solvers use the data in x as an initial guess.

Returns:

true as iterative solvers use the data in x as an initial guess.

Public Static Functions

static parameters_type parse(

const config::pnode &config,

const config::registry &context,

const config::type_descriptor &td_for_child = config::make_type_descriptor<ValueType>(),

)

Create the parameters from the property_tree. Because this is directly tied to the specific type, the value/index type settings within config are ignored and type_descriptor is only used for children configs.

Parameters:

• config – the property tree for setting

• context – the registry

• td_for_child – the type descriptor for children configs. The default uses the value type of this class.

Returns:

parameters

struct parameters_type

Inherits from

• public gko::solver::enable_preconditioned_iterative_solver_factory_parameters<parameters_type, Factory>