# irlba

##### Find a few approximate singular values and corresponding singular vectors of a matrix.

The augmented implicitly restarted Lanczos bidiagonalization algorithm (IRLBA) finds a few approximate largest (or, optionally, smallest) singular values and corresponding singular vectors of a sparse or dense matrix using a method of Baglama and Reichel. It is a fast and memory-efficient way to compute a partial SVD.

##### Usage

```
irlba(A, nv = 5, nu = nv, maxit = 1000, work = nv + 7, reorth = TRUE,
tol = 1e-05, v = NULL, right_only = FALSE, verbose = FALSE,
scale = NULL, center = NULL, shift = NULL, mult = NULL,
fastpath = TRUE, svtol = tol, smallest = FALSE, ...)
```

##### Arguments

- A
numeric real- or complex-valued matrix or real-valued sparse matrix.

- nv
number of right singular vectors to estimate.

- nu
number of left singular vectors to estimate (defaults to

`nv`

).- maxit
maximum number of iterations.

- work
working subspace dimension, larger values can speed convergence at the cost of more memory use.

- reorth
if

`TRUE`

, apply full reorthogonalization to both SVD bases, otherwise only apply reorthogonalization to the right SVD basis vectors; the latter case is cheaper per iteration but, overall, may require more iterations for convergence. Automatically`TRUE`

when`fastpath=TRUE`

(see below).- tol
convergence is determined when \(\|A^TU - VS\| < tol\|A\|\), and when the maximum relative change in estimated singular values from one iteration to the next is less than

`svtol = tol`

(see`svtol`

below), where the spectral norm ||A|| is approximated by the largest estimated singular value, and U, V, S are the matrices corresponding to the estimated left and right singular vectors, and diagonal matrix of estimated singular values, respectively.- v
optional starting vector or output from a previous run of

`irlba`

used to restart the algorithm from where it left off (see the notes).- right_only
logical value indicating return only the right singular vectors (

`TRUE`

) or both sets of vectors (`FALSE`

). The right_only option can be cheaper to compute and use much less memory when`nrow(A) >> ncol(A)`

but note that obtained solutions typically lose accuracy due to lack of re-orthogonalization in the algorithm and that`right_only = TRUE`

sets`fastpath = FALSE`

(only use this option for really large problems that run out of memory and when`nrow(A) >> ncol(A)`

). Consider increasing the`work`

option to improve accuracy with`right_only=TRUE`

.- verbose
logical value that when

`TRUE`

prints status messages during the computation.- scale
optional column scaling vector whose values divide each column of

`A`

; must be as long as the number of columns of`A`

(see notes).- center
optional column centering vector whose values are subtracted from each column of

`A`

; must be as long as the number of columns of`A`

and may not be used together with the deflation options below (see notes).- shift
optional shift value (square matrices only, see notes).

- mult
DEPRECATED optional custom matrix multiplication function (default is

`%*%`

, see notes).- fastpath
try a fast C algorithm implementation if possible; set

`fastpath=FALSE`

to use the reference R implementation. See the notes for more details.- svtol
additional stopping tolerance on maximum allowed absolute relative change across each estimated singular value between iterations. The default value of this parameter is to set it to

`tol`

. You can set`svtol=Inf`

to effectively disable this stopping criterion. Setting`svtol=Inf`

allows the method to terminate on the first Lanczos iteration if it finds an invariant subspace, but with less certainty that the converged subspace is the desired one. (It may, for instance, miss some of the largest singular values in difficult problems.)- smallest
set

`smallest=TRUE`

to estimate the smallest singular values and associated singular vectors. WARNING: this option is somewhat experimental, and may produce poor estimates for ill-conditioned matrices.- ...
optional additional arguments used to support experimental and deprecated features.

##### Value

Returns a list with entries:

- d:
max(nu, nv) approximate singular values

- u:
nu approximate left singular vectors (only when right_only=FALSE)

- v:
nv approximate right singular vectors

- iter:
The number of Lanczos iterations carried out

- mprod:
The total number of matrix vector products carried out

##### Note

The syntax of `irlba`

partially follows `svd`

, with an important
exception. The usual R `svd`

function always returns a complete set of
singular values, even if the number of singular vectors `nu`

or `nv`

is set less than the maximum. The `irlba`

function returns a number of
estimated singular values equal to the maximum of the number of specified
singular vectors `nu`

and `nv`

.

Use the optional `scale`

parameter to implicitly scale each column of
the matrix `A`

by the values in the `scale`

vector, computing the
truncated SVD of the column-scaled `sweep(A, 2, scale, FUN=`/`)`

, or
equivalently, `A %*% diag(1 / scale)`

, without explicitly forming the
scaled matrix. `scale`

must be a non-zero vector of length equal
to the number of columns of `A`

.

Use the optional `center`

parameter to implicitly subtract the values
in the `center`

vector from each column of `A`

, computing the
truncated SVD of `sweep(A, 2, center, FUN=`-`)`

,
without explicitly forming the centered matrix. `center`

must be a vector of length equal to the number of columns of `A`

.
This option may be used to efficiently compute principal components without
explicitly forming the centered matrix (which can, importantly, preserve
sparsity in the matrix). See the examples.

The optional `shift`

scalar valued argument applies only to square matrices; use it
to estimate the partial svd of `A + diag(shift, nrow(A), nrow(A))`

(without explicitly forming the shifted matrix).

(Deprecated) Specify an optional alternative matrix multiplication operator in the
`mult`

parameter. `mult`

must be a function of two arguments,
and must handle both cases where one argument is a vector and the other
a matrix. This option is deprecated and will be removed in a future version.
The new preferred method simply uses R itself to define a custom matrix class
with your user-defined matrix multiplication operator. See the examples.

Use the `v`

option to supply a starting vector for the iterative
method. A random vector is used by default (precede with `set.seed()`

for reproducibility). Optionally set `v`

to
the output of a previous run of `irlba`

to restart the method, adding
additional singular values/vectors without recomputing the solution
subspace. See the examples.

The function may generate the following warnings:

"did not converge--results might be invalid!; try increasing work or maxit" means that the algorithm didn't converge -- this is potentially a serious problem and the returned results may not be valid.

`irlba`

reports a warning here instead of an error so that you can inspect whatever is returned. If this happens, carefully heed the warning and inspect the result. You may also try setting`fastpath=FALSE`

."You're computing a large percentage of total singular values, standard svd might work better!"

`irlba`

is designed to efficiently compute a few of the largest singular values and associated singular vectors of a matrix. The standard`svd`

function will be more efficient for computing large numbers of singular values than`irlba`

."convergence criterion below machine epsilon" means that the product of

`tol`

and the largest estimated singular value is really small and the normal convergence criterion is only met up to round off error.

The function might return an error for several reasons including a situation when the starting
vector `v`

is near the null space of the matrix. In that case, try a different `v`

.

The `fastpath=TRUE`

option only supports real-valued matrices and sparse matrices
of type `dgCMatrix`

(for now). Other problems fall back to the reference
R implementation.

##### References

Baglama, James, and Lothar Reichel. "Augmented implicitly restarted Lanczos bidiagonalization methods." SIAM Journal on Scientific Computing 27.1 (2005): 19-42.

##### See Also

##### Examples

```
# NOT RUN {
set.seed(1)
A <- matrix(runif(400), nrow=20)
S <- irlba(A, 3)
S$d
# Compare with svd
svd(A)$d[1:3]
# Restart the algorithm to compute more singular values
# (starting with an existing solution S)
S1 <- irlba(A, 5, v=S)
# Estimate smallest singular values
irlba(A, 3, smallest=TRUE)$d
#Compare with
tail(svd(A)$d, 3)
# Principal components (see also prcomp_irlba)
P <- irlba(A, nv=1, center=colMeans(A))
# Compare with prcomp and prcomp_irlba (might vary up to sign)
cbind(P$v,
prcomp(A)$rotation[, 1],
prcomp_irlba(A)$rotation[, 1])
# A custom matrix multiplication function that scales the columns of A
# (cf the scale option). This function scales the columns of A to unit norm.
col_scale <- sqrt(apply(A, 2, crossprod))
setClass("scaled_matrix", contains="matrix", slots=c(scale="numeric"))
setMethod("%*%", signature(x="scaled_matrix", y="numeric"),
function(x ,y) x@.Data %*% (y / x@scale))
setMethod("%*%", signature(x="numeric", y="scaled_matrix"),
function(x ,y) (x %*% y@.Data) / y@scale)
a <- new("scaled_matrix", A, scale=col_scale)
irlba(a, 3)$d
# Compare with:
svd(sweep(A, 2, col_scale, FUN=`/`))$d[1:3]
# }
```

*Documentation reproduced from package irlba, version 2.3.3, License: GPL-3*