linear_solving.md

CurrentModule = AbstractAlgebra.Solve

[Linear solving](@id solving_chapter)

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.

solve_init

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

solve
can_solve
can_solve_with_solution
can_solve_with_solution_and_kernel
kernel