### The Result of a computation

The line of commands

```
using HomotopyContinuation
@polyvar x y;
f = [x^2+2y, y^2-2])
result = solve(f)
```

gives the output.

```
Result with 4 solutions
==================================
• 4 non-singular solutions (2 real)
• 0 singular solutions (0 real)
• 4 paths tracked
• random seed: 902575
```

Let us look at this output in more detail: 4 non-singular solutions have been found, and 2 of them are real solutions.

The line “4 paths tracked” means that the algorithm tracked 4 paths with homotopy continuation. In the guide on How does it work? we explain in detail the meaning of this. If you are only interested in the solutions, however, you can skip this part.

### The entries of Result: PathResult

`result`

is an array of the data structure `PathResult`

). For instance, the first entry of `result`

is

```
julia> result[1]
PathResult
=================
• return_code: success
• solution: Complex{Float64}[5.11415e-16+1.68179im, 1.41421-4.09132e-16im]
• accuracy: 3.434e-11
• residual: 2.894e-15
• condition_jacobian: 4.655e+00
• path_number: 1
```

The meaning of those entries is as follows:

`return_code: success`

means that the computation was successful.`solution`

is the solution that was computed.`accuracy`

is an approximation of $\Vert x-x^* \Vert$ where $x$ is the computed solution and $x^* $ is the true solution.`residual`

is the value of the euclidean norm of $f(x)$, where $x$ is the computed solution.`condition_jacobian`

is the condition number of the Jacobian of $f$ at the solution. A small value of this indicates that the numerical values are trustworthy.`path_number`

the number of the path in the enumeration of all paths.