BVMatArnoldi — SLEPc 3.25.1 documentation

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# BVMatArnoldi

Computes an Arnoldi factorization associated with a matrix.

## Synopsis

    #include "slepcbv.h"
    PetscErrorCode BVMatArnoldi(BV V,Mat A,Mat H,PetscInt k,PetscInt *m,PetscReal *beta,PetscBool *breakdown)

Collective

## Input Parameters

- V - basis vectors context

- A - the matrix

- H - (optional) the upper Hessenberg matrix

- k - number of locked columns

- m - dimension of the Arnoldi basis, may be modified

## Output Parameters

- beta - (optional) norm of last vector before normalization

- breakdown - (optional) flag indicating that breakdown occurred

## Notes

Computes an `m`-step Arnoldi factorization for matrix `A`. The first `k` columns are assumed to be locked and therefore they are not modified. On exit, the following relation is satisfied

\[ A V - V H = \beta v_m e_m^* \]

where the columns of \(V\) are the Arnoldi vectors (which are orthonormal), \(H\) is an upper Hessenberg matrix, \(e_m\) is the \(m\)-th vector of the canonical basis. On exit, `beta` contains the norm of \(v_m\) before normalization.

The breakdown flag indicates that orthogonalization failed, see `BVOrthonormalizeColumn()`. In that case, on exit `m` contains the index of the column that failed.

The values of `k` and `m` are not restricted to the active columns of `V`.

To create an Arnoldi factorization from scratch, set `k=0` and make sure the first column contains the normalized initial vector.

## See Also

BV: Basis Vectors, `BVMatLanczos()`, `BVSetActiveColumns()`, `BVOrthonormalizeColumn()`

## Level

advanced

## Location

src/sys/classes/bv/interface/bvkrylov.c

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