Linear solving

Overview of the functionality

The module AbstractAlgebra.Solve provides the following four functions for solving linear systems:

solve
can_solve
can_solve_with_solution
can_solve_with_solution_and_kernel

All of these take the same set of arguments, namely:

a matrix $A$ of type MatElem;
a vector or matrix $B$ of type Vector or MatElem;
a keyword argument side which can be either :left (default) or :right.

If side is :left, the system $xA = B$ is solved, otherwise the system $Ax = B$ is solved.

The functionality of the functions can be summarized as follows.

solve: return a solution, if it exists, otherwise throw an error.
can_solve: return true, if a solution exists, false otherwise.
can_solve_with_solution: return true and a solution, if this exists, and false and an empty vector or matrix otherwise.
can_solve_with_solution_and_kernel: like can_solve_with_solution and additionally return a matrix whose rows (respectively columns) give a basis of the kernel of $A$.

Solving with several right hand sides

Systems $xA = b_1,\dots, xA = b_k$ with the same matrix $A$, but several right hand sides $b_i$ can be solved more efficiently, by first initializing a "context object" C.

AbstractAlgebra.Solve.solve_init — Function

solve_init(A::MatElem)

Return a context object C that allows to efficiently solve linear systems $Ax = b$ or $xA = b$ for different $b$.

Example

julia> A = QQ[1 2 3; 0 3 0; 5 0 0];

julia> C = solve_init(A)
Linear solving context of matrix
  [1//1   2//1   3//1]
  [0//1   3//1   0//1]
  [5//1   0//1   0//1]

julia> solve(C, [QQ(1), QQ(1), QQ(1)], side = :left)
3-element Vector{Rational{BigInt}}:
 1//3
 1//9
 2//15

julia> solve(C, [QQ(1), QQ(1), QQ(1)], side = :right)
3-element Vector{Rational{BigInt}}:
 1//5
 1//3
 2//45

source

Now the functions solve, can_solve, etc. can be used with C in place of $A$. This way the time-consuming part of the solving (i.e. computing a reduced form of $A$) is only done once and the result cached in C to be reused.

Detailed documentation

AbstractAlgebra.Solve.solve — Function

solve(A::MatElem{T}, b::Vector{T}; side::Symbol = :left) where T
solve(A::MatElem{T}, b::MatElem{T}; side::Symbol = :left) where T
solve(C::SolveCtx{T}, b::Vector{T}; side::Symbol = :left) where T
solve(C::SolveCtx{T}, b::MatElem{T}; side::Symbol = :left) where T

Return $x$ of same type as $b$ solving the linear system $xA = b$, if side == :left (default), or $Ax = b$, if side == :right.

If no solution exists, an error is raised.

If a context object C is supplied, then the above applies for A = matrix(C).