solve() only works for nonsingular systems of equations.
Use a generalized inverse for singular systems:
> A<- matrix(c(1,2,1,1, 3,0,0,4, 1,-4,-2,-2, 0,0,0,0), ncol=4, byrow=TRUE)
> A
[,1] [,2] [,3] [,4]
[1,] 1 2 1 1
[2,] 3 0 0 4
[3,] 1 -4 -2 -2
[4,] 0 0 0 0
> b<- c(0,2,2,0) #rhs
> b
[1] 0 2 2 0
>
> require('MASS')
> giA<- ginv(A) #M-P generalized inverse
> giA
[,1] [,2] [,3] [,4]
[1,] 0.6666667 1.431553e-16 0.33333333 0
[2,] 0.3333333 -1.000000e-01 -0.03333333 0
[3,] 0.1666667 -5.000000e-02 -0.01666667 0
[4,] -0.5000000 2.500000e-01 -0.25000000 0
>
> require('Matrix')
> I<- as.matrix(Diagonal(4)) #order 4 identity matrix
> I
[,1] [,2] [,3] [,4]
[1,] 1 0 0 0
[2,] 0 1 0 0
[3,] 0 0 1 0
[4,] 0 0 0 1
>
> giA%*%b #particular solution
[,1]
[1,] 6.666667e-01
[2,] -2.666667e-01
[3,] -1.333333e-01
[4,] -2.220446e-16
> giA%*%A - I #matrix for parametric homogeneous solution
[,1] [,2] [,3] [,4]
[1,] 0.000000e+00 0.0 0.0 5.551115e-16
[2,] 3.469447e-17 -0.2 0.4 4.024558e-16
[3,] 4.510281e-17 0.4 -0.8 2.706169e-16
[4,] -3.330669e-16 0.0 0.0 -7.771561e-16
At 09:34 PM 5/21/2011, dslowik wrote:
I have a simple system of linear equations to solve for X, aX=b:
> a
[,1] [,2] [,3] [,4]
[1,] 1 2 1 1
[2,] 3 0 0 4
[3,] 1 -4 -2 -2
[4,] 0 0 0 0
> b
[,1]
[1,] 0
[2,] 2
[3,] 2
[4,] 0
(This is ex Ch1, 2.2 of Artin, Algebra).
So, 3 eqs in 4 unknowns. One can easily use row-reductions to find a
homogeneous solution(b=0) of:
X_1 = 0, X_2 = -c/2, X_3 = c, X_4 = 0
and solutions of the above system are:
X_1 = 2/3, X_2 = -1/3-c/2, X_3 = c, X_4 = 0.
So the Kernel is 1-D spanned by X_2 = -X_3 /2, (nulliity=1), rank is 3.
In R I use solve():
> solve(a,b)
Error in solve.default(a, b) :
Lapack routine dgesv: system is exactly singular
and it gives the error that the system is exactly singular, since it seems
to be trying to invert a.
So my question is:
Can R only solve non-singular linear systems? If not, what routine should I
be using? If so, why? It seems that it would be simple and useful enough to
have a routine which, given a system as above, returns the null-space
(kernel) and the particular solution.
--
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