# Optimization and iterative solves


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``` python
import rustymath as rm
```

## Minimization and scalar roots

`rm.optimize` takes SciPy’s names. `minimize` runs Brent’s method,
Nelder–Mead, Powell’s method, L-BFGS or differential evolution. Without
a `method`, it picks one from the number of variables, whether `jac`
gives a gradient and whether there are bounds. `root_scalar` finds a
zero of a function of one variable by Brent’s method, within a bracket
or by stepping outward from a start. Here Nelder–Mead finds the minimum
of Rosenbrock’s function, at (1, 1), and Brent’s method finds where cos
x = x:

``` python
rosen = lambda x: (1-x[0])**2 + 100*(x[1]-x[0]**2)**2
rm.optimize.minimize(rosen, [-1.2, 1]).x, rm.optimize.root_scalar(lambda x: rm.cos(x) - x, [0, 1]).root
```

    (Array([1.0, 1.0], dtype=float64), 0.7390851350713886)

## Fitting and linear systems

`least_squares` fits by Levenberg–Marquardt. `cg`, `minres`, `gmres` and
`lsmr` solve linear systems through the matrix’s products alone. They
take the matrix as an array or as anything with a `shape` and a `matvec`
method, such as a SciPy `LinearOperator`. An array’s products run in
Rust, with no Python call for each product. Here `least_squares`
recovers a = 2 and b = 0.3 from five points on y = a e^(bx), and `lsmr`
gives the least-squares line through five points that no line fits:

``` python
t = rm.array([0., 0.5, 1, 1.5, 2])
fit = rm.optimize.least_squares(lambda p: p[0]*rm.exp(p[1]*t) - 2*rm.exp(0.3*t), [1, 1])
line = rm.optimize.lsmr([[1., 0], [1, 1], [1, 2], [1, 3], [1, 4]], [1, 2.9, 5.2, 6.8, 9.1])
fit.x, line.x
```

    (Array([2.0, 0.3], dtype=float64), Array([0.98, 2.01], dtype=float64))