# Solve Examples

## A First Example (no start system)

We can solve the system $F(x,y) = (x^2+y^2+1, 2x+3y-1)$ in the following way

```
@var x y
F = System([x^2+y^2+1, 2x+3y-1], variables = [x, y])
solve(F)
```

Result with 2 solutions ======================= • 2 paths tracked • 2 non-singular solutions (0 real) • random_seed: 0xfa2a29a4 • start_system: :polyhedral

Here, the call

`F = System([x^2+y^2+1, 2x+3y-1], variables = [x, y])`

also determines the ordering of the variables in the solution vectors. By default, variables are ordered *lexciographically*. If this is okay, you can also call `solve`

without first constructing a system, i.e.,

`solve([x^2+y^2+1, 2x+3y-1])`

## Parameter Homotopy

Using the syntax

`solve(F, startsolutions; start_parameters, target_parameters)`

We can track the given start solutions alogn the parameter homotopy

\[H(x, t) = F(x, tp₁+(1-t)p₀),\]

where $p₁$ (=`start_parameters`

) and $p₀$ (=`target_parameters`

) are vectors of parameter values for $F$ where $F$ is a `System`

depending on parameters.

Assume we want to perform a parameter homotopy $H(x,t) := F(x; t[1, 0]+(1-t)[2, 4])$ where

\[F(x; a) := (x₁^2-a₁, x₁x₂-a₁+a₂)\]

and let's say we are only interested in tracking the solution $[1,1]$. This can be accomplished as follows

```
@var x[1:2] a[1:2]
F = System([x[1]^2-a[1], x[1]*x[2]-a[1]+a[2]], parameters = a)
start_solutions = [[1, 1]]
p₁ = [1, 0]
p₀ = [2, 4]
solve(F, start_solutions; start_parameters=p₁, target_parameters=p₀)
```

Result with 1 solution ====================== • 1 path tracked • 1 non-singular solution (1 real) • random_seed: 0x73594df8

## Start Target Homotopy

`solve(G, F, start_solutions; options...)`

This constructs the homotopy $H(x,t) = tG(x)+(1-t)F(x)$ to compute solutions of the system `F`

. `start_solutions`

is a list of solutions of `G`

which are tracked to solutions of `F`

.

```
@var x y
G = System([x^2+1,y+1])
F = System([x^2+y^2+1, 2x+3y-1])
solve(G, F, [[im, -1], [-im, -1]])
```

Result with 2 solutions ======================= • 2 paths tracked • 2 non-singular solutions (0 real) • random_seed: 0x57a56561