**RSpectra** is an R interface to the
Spectra library.
It is typically used to compute a few eigenvalues/vectors of an `n`

by `n`

matrix, e.g., the `k`

largest eigen values, which
is usually more efficient than `eigen()`

if `k << n`

.

Currently this package provides the function `eigs()`

for eigenvalue/eigenvector
problems, and `svds()`

for truncated SVD. Different matrix types in R,
including sparse matrices, are supported. Below is a list of implemented ones:

`matrix`

(defined in base R)`dgeMatrix`

(defined in**Matrix**package, for general matrices)`dgCMatrix`

(defined in**Matrix**package, for column oriented sparse matrices)`dgRMatrix`

(defined in**Matrix**package, for row oriented sparse matrices)`dsyMatrix`

(defined in**Matrix**package, for symmetric matrices)`dsCMatrix`

(defined in**Matrix**package, for symmetric column oriented sparse matrices)`dsRMatrix`

(defined in**Matrix**package, for symmetric row oriented sparse matrices)`function`

(implicitly specify the matrix by providing a function that calculates matrix product`A %*% x`

)

We first generate some matrices:

```
library(Matrix)
n = 20
k = 5
set.seed(111)
A1 = matrix(rnorm(n^2), n) ## class "matrix"
A2 = Matrix(A1) ## class "dgeMatrix"
```

General matrices have complex eigenvalues:

```
eigs(A1, k)
eigs(A2, k, opts = list(retvec = FALSE)) ## eigenvalues only
```

**RSpectra** also works on sparse matrices:

```
A1[sample(n^2, n^2 / 2)] = 0
A3 = as(A1, "dgCMatrix")
A4 = as(A1, "dgRMatrix")
eigs(A3, k)
eigs(A4, k)
```

Function interface is also supported:

```
f = function(x, args)
{
as.numeric(args %*% x)
}
eigs(f, k, n = n, args = A3)
```

Symmetric matrices have real eigenvalues.

```
A5 = crossprod(A1)
eigs_sym(A5, k)
```

To find the smallest (in absolute value) `k`

eigenvalues of `A5`

,
we have two approaches:

```
eigs_sym(A5, k, which = "SM")
eigs_sym(A5, k, sigma = 0)
```

The results should be the same, but the latter method is far more stable on large matrices.

For SVD problems, you can specify the number of singular values
(`k`

), number of left singular vectors (`nu`

) and number of right
singular vectors(`nv`

).

```
m = 100
n = 20
k = 5
set.seed(111)
A = matrix(rnorm(m * n), m)
svds(A, k)
svds(t(A), k, nu = 0, nv = 3)
```

Similar to `eigs()`

, `svds()`

supports sparse matrices:

```
A[sample(m * n, m * n / 2)] = 0
Asp1 = as(A, "dgCMatrix")
Asp2 = as(A, "dgRMatrix")
svds(Asp1, k)
svds(Asp2, k, nu = 0, nv = 0)
```

and function interface

```
f = function(x, args)
{
as.numeric(args %*% x)
}
g = function(x, args)
{
as.numeric(crossprod(args, x))
}
svds(f, k, Atrans = g, dim = c(m, n), args = Asp1)
```

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