How can I solve this system of linear equations?

Question

I cannot find a complete answer to this question. I am attempting to solve a similar system of equations to this:

r_Aus <- 8.7 + r_Fra + r_Ser + r_USA
r_Fra <- 2.7 + r_Aus + r_Chi + r_Ser
r_USA <- 37 + r_Chi + r_Ven + r_Aus
r_Chi <- -29.7 + r_USA + r_Fra + r_Ven
r_Ser <- 2.7 + r_Ven + r_Aus + r_Fra
r_Ven <- -21.3 + r_Ser + r_USA + r_Chi

How could I solve for each country variable??

Answer 1

Preparation

We first express your linear system in matrix form A * x = b . In case you are unclear of how to do this, have a read on General forms . For your example, you can express it as:

## x = r_Aus, r_Chi, r_Fra, r_Ser, r_USA, r_Ven
  r_Aus         - r_Fra - r_Ser - r_USA         =  8.7
- r_Aus - r_Chi + r_Fra - r_Ser                 =  2.7
- r_Aus - r_Chi                 + r_USA - r_Ven =  37
        + r_Chi - r_Fra         - r_USA - r_Ven = -29.7
- r_Aus         - r_Fra + r_Ser         - r_Ven =  2.7
        - r_Chi         - r_Ser - r_USA + r_Ven = -21.3

then define coefficient matrix A and RHS vector b :

A <- matrix(c( 1,  0, -1, -1, -1,  0,
              -1, -1,  1, -1,  0,  0,
              -1, -1,  0,  0,  1, -1,
               0,  1, -1,  0, -1, -1,
              -1,  0, -1,  1,  0, -1,
               0, -1,  0, -1, -1,  1),
            nrow = 6, ncol = 6, byrow = TRUE)

b <- as.matrix(c(8.7, 2.7, 37, -29.7, 2.7, -21.3))

---

Trying solve()

Almost always, we think about solve first. But solve() is based on LU factorization, and requires a full rank coefficient matrix A ; when A is found rank-deficient, LU factorization meets a 0 diagonal element and fails. Let's try your A and b :

solve(A, b)
#Error in solve.default(A, b) : 
#  Lapack routine dgesv: system is exactly singular: U[6,6] = 0

U[0,0] = 0 tells you, your A has only a rank of 5.

---

A stable method: QR factorization

QR factorization is known to be a very stable method. We can make use of .lm.fit() to do this:

x <- .lm.fit(A, b)
x$coef
# [1]   4.783333  -5.600000 -21.450000 -18.650000  40.866667   0.000000
x$rank
# [1] 5

Your system is of rank-5, so least square fitting is performed. The 6th value is r_Ven is constrained at 0, and none of your equations are exactly satisfied. x$resi gives you residuals, ie, b - A %*% x$beta .

---

Gaussian elimination

To complete the picture, I have to mention Gaussian elimination. In theory this is the best approach, as you can determine whether:

1. there is a unique solution;
2. there is no solution;
3. there are infinite number of solutions

as well as solving the linear system.

There is a small R package optR around, but as I found out, it is not doing a perfect job.

#install.packages("optR")
library(optR)

?optR gives a full rank linear system as an example which works certainly fine (where simply using solve(A, b) would also work!). But for your system with rank-5, it gives:

optR(A, b, method="gauss")

call: 
optR.default(x = A, y = b, method = "gauss")

 Coefficients: 
           [,1]
[1,]   9.466667
[2,] -24.333333
[3,] -16.766667
[4,]  -4.600000
[5,]  22.133333
[6,]   0.000000
Warning messages:
1: In opt.matrix.reorder(A, tol) : Singular Matrix
2: In opt.matrix.reorder(A, tol) : Singular Matrix

Note the warning message that your linear system is rank-deficient. To understand what optR does in such case, compare b with

A %*% x$beta
#      [,1]
#[1,]   8.7
#[2,]   2.7
#[3,]  37.0
#[4,] -29.7
#[5,]   2.7
#[6,]   6.8

The first 5 equations are satisfied, except the 6th. So, optR abandoned your last equation to address rank-deficiency, instead of doing least square fitting.