[R] Solving sparse, singular systems of equations

[Berend Hasselman]

> On 20 Apr 2016, at 13:22, A A via R-help <r-help at r-project.org> wrote:
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> I have a situation in R where I would like to find any x (if one exists) that solves the linear system of equations Ax = b, where A is square, sparse, and singular, and b is a vector. Here is some code that mimics my issue with a relatively simple A and b, along with three other methods of solving this system that I found online, two of which give me an error and one of which succeeds on the simplified problem, but fails on my data set(attached). Is there a solver in R that I can use in order to get x without any errors given the structure of A? Thanks for your time.
> #CODE STARTS HEREA = as(matrix(c(1.5,-1.5,0,-1.5,2.5,-1,0,-1,1),nrow=3,ncol=3),"sparseMatrix")b = matrix(c(-30,40,-10),nrow=3,ncol=1)
> #solve for x, Error in LU.dgC(a) : cs_lu(A) failed: near-singular A (or out of memory)solve(A,b,sparse=TRUE,tol=.Machine$double.eps)
> #one x that happens to solve Ax = bx = matrix(c(-10,10,0),nrow=3,ncol=1)A %*% x
> #Error in lsfit(A, b) : only 3 cases, but 4 variableslsfit(A,b)#solves the system, but fails belowsolve(qr(A, LAPACK=TRUE),b)#Error in qr.solve(A, b) : singular matrix 'a' in solveqr.solve(A,b)
> #matrices used in my actual problem (see attached files)A = readMM("A.txt")b = readMM("b.txt")
> #Error in as(x, "matrix")[i, , drop = drop] : subscript out of boundssolve(qr(A, LAPACK=TRUE),b)

Your code is a mess. 

A singular square system of linear equations has an infinity of solutions if a solution exists at all.
How that works you can find here: https://en.wikipedia.org/wiki/System_of_linear_equations
in the section "Matrix solutions".

For your simple example you can do it like this:

library(MASS)
Ag <- ginv(A)	# pseudoinverse

xb <- Ag %*% b # minimum norm solution

Aw <- diag(nrow=nrow(Ag)) - Ag %*% A  # see the Wikipedia page
w <- runif(3)
z <- xb + Aw %*% w
A %*% z - b

N <- Null(t(A))	 # null space of A;  see the help for Null in package MASS
A %*% N
A %*% (xb + 2 * N) - b

For sparse systems you will have to approach this differently; I have no experience with that.

Berend