kaczmarz: Solve a linear system defined by factors

In lfe: Linear Group Fixed Effects

Solve a linear system defined by factors

Description

Uses the Kaczmarz method to solve a system of the type Dx = R, where D is the matrix of dummies created from a list of factors.

Usage

kaczmarz(
  fl,
  R,
  eps = getOption("lfe.eps"),
  init = NULL,
  threads = getOption("lfe.threads")
)

Arguments

fl	A list of arbitrary factors of the same length
R	numeric. A vector, matrix or list of such of the same length as the factors
eps	a tolerance for the method
init	numeric. A vector to use as initial value for the Kaczmarz iterations. The algorithm converges to the solution closest to this
threads	integer. The number of threads to use when R is more than one vector

Value

A vector x of length equal to the sum of the number of levels of the factors in fl, which solves the system Dx=R. If the system is inconsistent, the algorithm may not converge, it will give a warning and return something which may or may not be close to a solution. By setting eps=0, maximum accuracy (with convergence warning) will be achieved.

Note

This function is used by getfe(), it's quite specialized, but it might be useful for other purposes too.

In case of convergence problems, setting options(lfe.usecg=TRUE) will cause the kaczmarz() function to dispatch to the more general conjugate gradient method of cgsolve(). This may or may not be faster.

See Also

cgsolve()

Examples

## create factors
f1 <- factor(sample(24000, 100000, replace = TRUE))
f2 <- factor(sample(20000, length(f1), replace = TRUE))
f3 <- factor(sample(10000, length(f1), replace = TRUE))
f4 <- factor(sample(8000, length(f1), replace = TRUE))
## the matrix of dummies
D <- makeDmatrix(list(f1, f2, f3, f4))
dim(D)
## an x
truex <- runif(ncol(D))
## and the right hand side
R <- as.vector(D %*% truex)
## solve it
sol <- kaczmarz(list(f1, f2, f3, f4), R)
## verify that the solution solves the system Dx = R
sqrt(sum((D %*% sol - R)^2))
## but the solution is not equal to the true x, because the system is
## underdetermined
sqrt(sum((sol - truex)^2))
## moreover, the solution from kaczmarz has smaller norm
sqrt(sum(sol^2)) < sqrt(sum(truex^2))

lfe documentation built on Feb. 16, 2023, 7:32 p.m.