ssy Interface

public interface ssy

Description

Given a linear operator $A$ with full column rank and the associated saddle-point problem

$$\begin{bmatrix} I & A \\ A^\top & \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} b \\ c \end{bmatrix},$$

the Saunders-Simon-Yip factorization computes orthonormal bases $U$ and $V$ for the column-span and of $AA^\top$ and $A^\top A$, respectively, and a tridiagonal matrix $T$ such that

$$U^\top A V = T.$$

with $\beta u_1 = b$ and $\gamma v_1 = c$.

Algorithmic Features

• The operator $A$ only needs to be accessed through matrix-vector products.
• Constructs an orthonormal basis for the Krylov subspace $\mathcal{K} = \left( b, AA^\top b, \cdots \right)$.
• Constructs an orthonormal basis for the Krylov subspace $\mathcal{K} = \left( c, A^\top A c, \cdots, \right)$.
• Constructs a tridiagonal matrix $T$ such that $U^\top A V = T$.
• Checks for convergence and invariant subspaces.

References

• A. Buttari, D. Orban, D. Ruiz, and D. Titly-Peloquin. "A tridiagonalization method for symmetric saddle-point systems". SIAM Journal on Scientific Computing, 41(5), 2019. (PDF)

### Syntax

fortran call ssy(A, U, V, T, info [, kstart] [, kend] [, tol])

### Arguments

• A : Linear operator derived from one the base types provided by the AbstractLinops module. It is an intent(inout) argument.

• U : Arrays of types derived from one the base types provided by the AbstractVectors module. It needs to be consistent with the type of A. On entry, U(1) = b. On exit, it contains an orthonormal basis for $\mathcal{K} = \left(b, A A^\top b, \cdots \right)$. It is an intent(inout) argument.

• V : Arrays of types derived from one the base types provided by the AbstractVectors module. It needs to be consistent with the type o fA. On entry, V(1) = c. On exit, it contains an orthonormal basis for $\mathcal{K} = \left(c, A^\top A c, \cdots \right)$. It is an intent(inout) argument.

• T : real or complex rank-2 array. On exit, it contains the $(k+1) \times (k+1)$ tridiagonal matrix computed from the SSY factorization. It is an intent(inout) argument.

• info : integer variable. It is the LightKrylov information flag. On exit, if info > 0 the SSY process experienced a lucky breakdown.

• kstart (optional) : integer value determining the index of the first SSY step to be computed. It is an optional intent(in) argument. By default, kstart = 1.

• kend (optional) : integer value determining the index of the last SSY step to be computed. It is an optional intent(in) argument. By default, kend = size(U) - 1.

• tol (optional) : Numerical tolerance below which a subspace is considered to be $A$-invariant. It is an optional intent(in) argument. By default tol = atol_sp or tol = atol_dp depending on the kind of A.

Subroutines