## Solve Ax=b in Maxima, part 2

In a previous post, I included my little coding project to implement a general backsolve() function to use with the built-in maxima matrix function echelon(), producing an easy-to-call matrix solver matsolve(A,b).  The result is meant to solve a general matrix vector equation $Ax=b$ , including cases when $A$ is non-square and/or non-invertible.

Here’s a quicker approach — convert the matrix into an explicit system of equations using a vector of dummy variables, feed the result into the built-in Maxima function linsolve(), and then extract the right hand sides of the resulting solutions and put them into a column vector.

The two methods often behave identically, but here’s an example that breaks the linsolve() method, where the backsolve() method gives a correct solution:

*Note, I’ve found that the symbol rhs is a very popular thing for users to call their problem-specific vectors or functions.  Maxima’s  “all symbols are global” bug/feature generally wouldn’t cause a problem with a function call to rhs(), but the function map(rhs, list of equations)  ignores that rhs() is a function and uses user-defined rhs.  For that reason I protect that name in the block declarations so that rhs() works as expected in the map() line at the bottom.  I think I could have done the same thing with a quote: map(‘rhs, list of equations).

```matsolve2(A,b):=block(
[rhs,inp,sol,Ax,m,n,vars],
[m,n]:[length(A),length(transpose(A))],
vars:makelist(xx[i],i,1,n,1),
Ax:A.vars,
inp:makelist(part(Ax,i,1)=b[i],i,1,n,1),
sol:linsolve(inp,vars),
expand(transpose(matrix(map(rhs,sol))))
);```