[R] Solve for x in Ax=B with vectors, not matrices

[(Ted Harding)]

On 22-Jan-06 Lapointe, Pierre wrote:
> Hello R-helpers,
> 
> What I have: I am working with vectors not matrice:
> 
>#Basic equations
> A <-c(-20,-9,-2)
> x <-c(0.17,0.22,0.61)
> B <- crossprod(A,x)
> 
># R matrix multiplication works with vectors
> A%*%x==B      # Is true...
> 
> Question: If x is unknown and A and B are known,
> how do I solve for x in R?
> solve(A,B) won't work because A is not a square matrix 
> 
> solve(A,B)
> Error in solve.default(A, B) : 'A' (3 x 1) must be square
> 
> I understand that I might have many solutions but note
> that the sum of x is 1 and all x are positive (x are
> weightings in % of the total).

For the example you have given, in "classical" vector
algebra notation the equation is

  A.x = B   [ = -6.6 in this case ]

where A and x are two vectors.

Note the explanataion resulting from ?"%*%"

     Multiplies two matrices, if they are conformable.
     If one argument is a vector, it will be coerced
     to a either a row or column matrix to make the
     two arguments conformable. If both are vectors it
     will return the inner product.

If this is the interpretation you intend, and if the above
is a typical problem of yours, then if you divide by the
"lengths" of A and x you will get an equation

  V.y = cos(u)

where V (corresponding to A) and y (corresponding to x)
are unit vectors, and cos(u) corresponds to B.

Now you want to find solutions y from this equation.
You are in the first instance looking for all vectors
y which are at a fixed angle u in (0,pi) to the vector
V (the "elevation", if you like), which you can find by
choosing another angle v (the "azimuth", say) arbitrarily
in (0,2*pi). All such angles give a solution of V.y = cos(u),
and the endpoints of the vectors describe a circle.

You can get back to x by re-scaling, and now you want
the solutions such that the sum of the elements is 1
(which defines a plane). There are either two of these
(where the circle cuts the plane, or infinitely many
(if the circle in question lies in the plane).

Does this outline point in s useful direction?

Best wishes,
Ted.

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E-Mail: (Ted Harding) <Ted.Harding at nessie.mcc.ac.uk>
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Date: 22-Jan-06                                       Time: 18:27:18
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