bidiagonalization Interface

public interface bidiagonalization

Description

Given a general linear operator $A$, find matrices $U$, $V$ and $B$ such that

$$\begin{aligned} AV_k & = U_k T_k + t_{k+1, k} v_{k+1} e_k^T, \\ A^H U_k & = V_k T_k^T + t_{k+1, k} u_{k+1} e_k^T \end{aligned}$$

where $U$ and $V$ are orthogonal bases for the column span and row span of $A$, respectively, and $B$ is a bidiagonal matrix.

Algorithmic Features

• The operator $A$ only needs to be accessed through matrix-vector products.
• Constructs an orthonormal Krylov basis $U$ for the column span of $A$.
• Constructs an orthonormal Krylov basis $V$ for the row span of $A$.
• Constructs a bidiagonal matrix $B$ whose singular values approximates those of $A$.
• Checks for convergence and invariant subspaces.

References

• R. M. Larsen. "Lanczos bidiagonalization with partial reorthogonalization." Technical Report, 1998. (PDF)

Syntax

    call bidiagonalization(A, U, V, B, 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 : Array of types derived from one the base types provided by the AbstractVectors module. It needs to be consistent with the type of A. On exit, it contains the the computed Krylov vectors for the column span of A. The first entry U(1) is the starting vector for the Lanczos factorization. Additionally, the maximum number of Lanczos steps is equal to size(X) - 1. It is an intent(inout) argument.

• V : Array of types derived from one the base types provided by the AbstractVectors module. It needs to be consistent with the type of A. On exit, it contains the the computed Krylov vectors for the row span of A. It is an intent(inout) argument.

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

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

• kstart (optional) : integer value determining the index of the first Lanczos 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 Lanczos step to be computed. It is an optional intent(in) argument. By default, kend = size(X) - 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_rp depending on the kind of A.

Subroutines