`123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100101102103104105106107108109110111112113114115116117118119120121122` ``````## Solvers for Large Scale Eigenvalue and SVD Problems ### Introduction **RSpectra** is an R interface to the [Spectra library](https://spectralib.org/). 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`) ### Examples We first generate some matrices: ```r 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: ```r eigs(A1, k) eigs(A2, k, opts = list(retvec = FALSE)) ## eigenvalues only ``` **RSpectra** also works on sparse matrices: ```r 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: ```r f = function(x, args) { as.numeric(args %*% x) } eigs(f, k, n = n, args = A3) ``` Symmetric matrices have real eigenvalues. ```r A5 = crossprod(A1) eigs_sym(A5, k) ``` To find the smallest (in absolute value) `k` eigenvalues of `A5`, we have two approaches: ```r 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`). ```r 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: ```r 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 ```r 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) ``` ``````