# ApCoCoA-1:LinBox.Solve

## LinBox.Solve

Solves a system of linear equations.

### Syntax

LinBox.Solve(M:MAT, B:MAT):MAT LinBox.Solve(M:MAT, B:MAT, METHOD:STRING):MAT

### Description

*Please note:* The function(s) explained on this page is/are using the *ApCoCoAServer*. You will have to start the ApCoCoAServer in order to use it/them.

Let `M` and `B` be matrices defined over the ring of integers, a finite field or the field of rationals. This function tries to solve the linear equation system `M*X = B` by using the ApCoCoAServer supported by the LinBox library. If your base ring is the ring of integers or a finite field, you can pass `Wiedemann` as `METHOD` to let the ApCoCoAServer compute the solution by using the LinBox Wiedemann implementation. If you pass `BlasElim` instead in this case, the solution will be computed by using the LinBox BLAS elimination implementation. If you omit the parameter `METHOD`, `BlasElim` will be used as default value where applicable.

*Please note:* The parameter `METHOD` will be ignored if your base ring is the field of rationals, i.e. in this case it is always the LinBox rational solver implementation that will be used for computing a solution.

The return value will be a solution vector of the linear equation system or an empty matrix if no solution has been found.

@param

*M*A matrix with components either of type`INT`,`ZMOD`or`RAT`.@param

*B*A matrix with components either of type`INT`,`ZMOD`or`RAT`.@return A matrix

`X`representing a solution vector of the linear equation system`M*X = B`if a solution exists or the empty matrix otherwise.

The following parameter is optional.

@param

*METHOD*A string specifying the solution method to use. Available methods are "Wiedemann" and "BlasElim". Please read the detailed description about this parameter, too.

#### Example

Use ZZ/(19)[x]; M := BringIn(Mat([[1,3,4], [0,2,1]])); B := BringIn(Mat([[1], [2]])); LinBox.Solve(M, B); ------------------------------- Mat([ [-2 % 19], [1 % 19], [0 % 19] ]) -------------------------------

#### Example

Use QQ[x]; M := Mat([ [1,3,4], [0,2,1], [1,3,4] ]); B := Mat([ [1], [2], [0] ]); LinBox.Solve(M, B); ------------------------------- Mat([ [ ] ]) -------------------------------