Arnoldi Iteration

arnoldi(A,b[;m,tol,opnorm,iop]) -> Ks

Perform Arnoldi iterations to obtain the Krylov subspace $K_m(A, b)$.

Arguments

• A: a square matrix or a matrix-free operator satisfying the Matrix-Free Operator Interface page in the manual.
• b: starting vector.

Keywords

• m: requested Krylov dimension, default min(30, size(A, 1)).
• ishermitian: select Lanczos for a Hermitian operator; defaults to LinearAlgebra.ishermitian(A).
• Additional keywords are forwarded to arnoldi!, including tol, iop, and opnorm.

Returns

A populated KrylovSubspace. Its m can be smaller than requested after a happy breakdown.

Examples

A = [-2.0 1.0; 0.0 -1.0]
b = [1.0, 0.0]
Ks = arnoldi(A, b; m = 2)

The first m + 1 columns of Ks.V and the leading (m + 1) by m block of Ks.H are related by the recurrence formula

v_1=b,\quad Av_j = \sum_{i=1}^{j+1}h_{ij}v_i\quad(j = 1,2,\ldots,m)

iop determines the length of the incomplete orthogonalization procedure ^[1]. The default value of 0 indicates full Arnoldi. For symmetric/Hermitian A, iop will be ignored and the Lanczos algorithm will be used instead.

Refer to KrylovSubspace for more information regarding the output.

Happy-breakdown occurs whenever norm(v_j) < tol * opnorm, in this case, the dimension of Ks is smaller than m.

arnoldi!(Ks,A,b[;tol,m,opnorm,iop,init]) -> Ks

Populate an existing KrylovSubspace without allocating its basis.

Arguments

• Ks: reusable Krylov storage whose element and storage types must support A and b.
• A: square matrix or matrix-free operator.
• b: starting vector.

Keywords

• tol: happy-breakdown threshold, default 1e-7.
• m: requested Krylov dimension, bounded by Ks.maxiter unless the cache is resized.
• ishermitian: select lanczos! for Hermitian operators.
• iop: incomplete-orthogonalization length; 0 performs full Arnoldi.
• init, t, mu, l: advanced controls for continuing or augmented Krylov constructions.

Returns

The mutated Ks.

Examples

A = [-2.0 1.0; 0.0 -1.0]
b = [1.0, 0.0]
Ks = KrylovSubspace{Float64}(length(b), 2)
arnoldi!(Ks, A, b; m = 2)

lanczos!(Ks,A,b[;tol,m,opnorm]) -> Ks

Populate an existing KrylovSubspace with the Lanczos recurrence for a Hermitian operator.

Arguments

• Ks: reusable Krylov storage with real-valued Hessenberg coefficients.
• A: Hermitian matrix or matrix-free operator.
• b: starting vector.

Keywords

tol, m, opnorm, init, t, mu, and l have the same role as for arnoldi!. Call this only when A is Hermitian for the scalar product implemented by mul!.

Returns

The mutated Ks.

Examples

using LinearAlgebra

A = Hermitian([-2.0 1.0; 1.0 -2.0])
b = [1.0, 0.0]
Ks = KrylovSubspace{Float64, Float64}(length(b), 2)
lanczos!(Ks, A, b; m = 2)

KrylovSubspace{T}(n, [maxiter = 30]) -> Ks
KrylovSubspace{T, U, VType}(n, [maxiter = 30]) -> Ks

Construct an uninitialized Krylov subspace to be filled by arnoldi! or lanczos!.

Arguments

• T: element type of the Krylov basis.
• U: element type of the Hessenberg matrix. Use real(T) for a Hermitian problem.
• VType: matrix storage type for the basis. It must support VType(undef, rows, columns); choose a GPU-backed matrix type for GPU vectors.
• n: dimension of the operator domain.
• maxiter: maximum Krylov dimension, defaulting to 30.
• augmented: number of rows reserved for an augmented operator, defaulting to 0.

Fields

• m: current Krylov dimension. It can be smaller than maxiter after a happy breakdown.
• maxiter: allocated maximum Krylov dimension.
• augmented: number of augmented rows.
• beta: norm of the starting vector from the last factorization.
• wasbreakdown: whether the last factorization ended in a happy breakdown.
• V: orthonormal Krylov basis storage.
• H: Hessenberg or tridiagonal recurrence-coefficient storage.

Returns

An uninitialized KrylovSubspace. Call arnoldi! or lanczos! before using its basis or coefficients.

Examples

A = [-2.0 1.0; 1.0 -2.0]
b = [1.0, 0.0]
Ks = KrylovSubspace{Float64}(length(b), 2)
arnoldi!(Ks, A, b)