LEFTDIVIDE Matrix Equation Solver/Divide Operator
Section: Mathematical Operators
Usage
The divide operator\
is really a combination of three
operators, all of which have the same general syntax:
Y = A \ B
where A
and B
are arrays of numerical type. The result Y
depends
on which of the following three situations applies to the arguments
A
and B
:

A
is a scalar,B
is an arbitraryn
dimensional numerical array, in which case the output is each element ofB
divided by the scalarA
. 
A,B
are matrices with the same number of rows, i.e.,A
is of sizeM x K
, andB
is of sizeM x L
, in which case the output is of sizeK x L
.
A
and B
are integers, the output is an integer also, while in the third case if A
and B
are integers, the output is of type double
.
A few additional words about the third version, in which A
and B
are matrices. Very loosely speaking, Y
is the matrix that satisfies A * Y = B
. In cases where such a matrix exists. If such a matrix does not exist, then a matrix Y
is returned that approximates A * Y \approx B
.
Function Internals
There are three formulae for the times operator. For the first form
In the second form, the calculation of the output depends on the size of A
. Because each column of B
is treated independantly, we can rewrite the equation A Y = B
as
where y_i
are the columns of Y
, and b_i
are the columns of the matrix B
. If A
is a square matrix, then the LAPACK routine *gesvx
(where the *
is replaced with sdcz
depending on the type of the arguments) is used, which uses an LU decomposition of A
to solve the sequence of equations sequentially. If A
is singular, then a warning is emitted.
On the other hand, if A
is rectangular, then the LAPACK routine *gelsy
is used. Note that these routines are designed to work with matrices A
that are full rank  either full column rank or full row rank. If A
fails to satisfy this assumption, a warning is emitted. If A
has full column rank (and thus necessarily has more rows than columns), then theoretically, this operator finds the columns y_i
that satisfy:
and each column is thus the Least Squares solution of A y = b_i
. On the other hand, if A
has full row rank (and thus necessarily has more columns than rows), then theoretically, this operator finds the columns y_i
that satisfy
and each column is thus the Minimum Norm vector y_i
that satisfies A y_i = b_i
.
In the event that the matrix A
is neither full row rank nor full column rank, a solution is returned, that is the minimum norm least squares solution. The solution is computed using an orthogonal factorization technique that is documented in the LAPACK User's Guide (see the References section for details).
Examples
Here are some simple examples of the divide operator. We start with a simple example of a full rank, square matrix:> A = [1,1;0,1] A = 1 1 0 1
Suppose we wish to solve
(which by inspection has the solution y_1 = 1
, y_2 = 2
). Thus we compute:
> B = [3;2] B = 3 2 > Y = A\B Y = 1 2
Suppose we wish to solve a trivial Least Squares (LS) problem. We want to find a simple scaling of the vector [1;1]
that is closest to the point [2,1]
. This is equivalent to solving
in a least squares sense. For fun, we can calculate the solution using calculus by hand. The error we wish to minimize is
Taking a derivative with respect to y
, and setting to zero (which we must have for an extrema when y
is unconstrained)
which we can simplify to 4y = 6
or y = 3/2
(we must, technically, check to make sure this is a minimum, and not a maximum or an inflection point). Here is the same calculation performed using FreeMat:
> A = [1;1] A = 1 1 > B = [2;1] B = 2 1 > A\B ans = 1.5000
which is the same solution.