# Golub-Kahan-Lanczos (SVDL)

The SVDL method computes a partial, approximate SVD decomposition of a general linear operator $A$.

## Usage

IterativeSolvers.svdlFunction
svdl(A) -> Σ, L, [history]

Compute some singular values (and optionally vectors) using Golub-Kahan-Lanczos bidiagonalization [Golub1965] with thick restarting [Wu2000].

If log is set to true is given, method will output a tuple X, L, ch. Where ch is a ConvergenceHistory object. Otherwise it will only return X, L.

Arguments

• A : The matrix or matrix-like object whose singular values are desired.

Keywords

• nsv::Int = 6: number of singular values requested;
• v0 = random unit vector: starting guess vector in the domain of A. The length of q should be the number of columns in A;
• k::Int = 2nsv: maximum number of Lanczos vectors to compute before restarting;
• j::Int = nsv: number of vectors to keep at the end of the restart. We don't recommend j < nsv;
• maxiter::Int = minimum(size(A)): maximum number of iterations to run;
• verbose::Bool = false: print information at each iteration;
• tol::Real = √eps(): maximum absolute error in each desired singular value;
• reltol::Real=√eps(): maximum error in each desired singular value relative to the estimated norm of the input matrix;
• method::Symbol=:ritz: restarting algorithm to use. Valid choices are:
1. :ritz: Thick restart with Ritz values [Wu2000].
2. :harmonic: Restart with harmonic Ritz values [Baglama2005].
• vecs::Symbol = :none: singular vectors to return.
1. :both: Both left and right singular vectors are returned.
2. :left: Only the left singular vectors are returned.
3. :right: Only the right singular vectors are returned.
4. :none: No singular vectors are returned.
• dolock::Bool=false: If true, locks converged Ritz values, removing them from the Krylov subspace being searched in the next macroiteration;
• log::Bool = false: output an extra element of type ConvergenceHistory containing extra information of the method execution.

Return values

if log is false

• Σ: list of the desired singular values if vecs == :none (the default), otherwise returns an SVD object with the desired singular vectors filled in;
• L: computed partial factorizations of A.

if log is true

• Σ: list of the desired singular values if vecs == :none (the default),

otherwise returns an SVD object with the desired singular vectors filled in;

• L: computed partial factorizations of A;
• history: convergence history.

ConvergenceHistory keys

• :betas => betas: The history of the computed betas.
• :Bs => Bs: The history of the computed projected matrices.
• :ritz => ritzvalhist: Ritz values computed at each iteration.
• :conv => convhist: Convergence data.
source

## Implementation details

The implementation of thick restarting follows closely that of SLEPc as described in [Hernandez2008]. Thick restarting can be turned off by setting k = maxiter, but most of the time this is not desirable.

The singular vectors are computed directly by forming the Ritz vectors from the product of the Lanczos vectors L.P/L.Q and the singular vectors of L.B. Additional accuracy in the singular triples can be obtained using inverse iteration.

• Golub1965Golub, Gene, and William Kahan. "Calculating the singular values and pseudo-inverse of a matrix." Journal of the Society for Industrial and Applied Mathematics, Series B: Numerical Analysis 2.2 (1965): 205-224.
• Wu2000Wu, Kesheng, and Horst Simon. "Thick-restart Lanczos method for large symmetric eigenvalue problems." SIAM Journal on Matrix Analysis and Applications 22.2 (2000): 602-616.
• Hernandez2008Vicente Hernández, José E. Román, and Andrés Tomás. "A robust and efficient parallel SVD solver based on restarted Lanczos bidiagonalization." Electronic Transactions on Numerical Analysis 31 (2008): 68-85.