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Solving sparse, singular systems of equations

Berend Hasselman 
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

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a a Solving sparse, singular systems of equations Apr 20 William Dunlap Solving sparse, singular systems of equations Apr 20 Jeff Newmiller Solving sparse, singular systems of equations Apr 20 Berend Hasselman Solving sparse, singular systems of equations Apr 20 a a Solving sparse, singular systems of equations Apr 20 a a Solving sparse, singular systems of equations Apr 20 a a Solving sparse, singular systems of equations Apr 20 Jeff Newmiller Solving sparse, singular systems of equations Apr 20