Module solve 

Root finding, polynomial roots and minimization.

The numerical substrate every intersection, projection and extrema algorithm in the kernel sits on. Nothing here is geometric; it is deliberately kept separate so those algorithms are about geometry rather than about convergence.

What to reach for

• A root inside a known bracket: brent. Guaranteed to converge, and nearly as fast as Newton in practice.
• A root with a known derivative and a good starting point: newton, which falls back to bisection whenever a step would leave the bracket. Unsafeguarded Newton diverges on the configurations that matter: a tangential intersection is exactly where the derivative vanishes.
• Roots of a polynomial up to quartic: roots. Closed form, and the quadratic is written to avoid the cancellation the schoolbook formula suffers.
• A system of equations: newton_system. Surface projection is two equations in two unknowns; intersection marching is much the same.
• A minimum without derivatives: minimize.

Structs

Criteria

Stopping criteria for an iterative solver.

Solution

A solver result: the estimate, the residual there, and how it finished.

SystemSolution

The outcome of solving a system of equations.

Enums

Convergence

How a solver finished.

Functions

brent

Find a root of f in [a, b] by Brent’s method.

cubic_roots

Real roots of a x^3 + b x^2 + c x + d.

minimize

Minimize a scalar function on [a, b] without derivatives.

newton

Find a root of f near start, using its derivative, safeguarded by a bracket.

newton_system

Solve f(x) = 0 for a vector x, by damped Newton.

newton_system_2

A two-unknown newton_system, allocation-free.

quadratic_roots

Real roots of a x^2 + b x + c.

roots

The real roots of a polynomial, in increasing order.