# Predictors - Correctors

The continuation method works with the following pattern (see ^{[Allgower1990]}):

- compute
**tangent** - call
**predictor**(based on tangent, mostly) - call
**corrector**

There are several couples predictor-tangent/corrector which can be used in **BifurcationKit.jl** as we now explain. The tangent computation is formally included in the predictor whereas it is a distinct function in the code.

Note that setting the predictor also sets the corresponding corrector: it selects the couple predictor-corrector. You don't have (in fact cannot) set them independently.

## 1. Natural, zeroth order predictor

This is the dumbest predictor based on the formula $(x_1,p_1) = (x_0, p_0 + ds)$ with Newton corrector ; it fails at Turning points. This is set by the algorithm `Natural()`

in `continuation`

. For matrix based jacobian, it is not faster than the pseudo-arclength predictor because the factorisation of the jacobian is cached. For Matrix-free methods, this predictor can be faster than the following ones until it hits a Turning point.

## 2. First order predictor

This predictor is based on a computation of the tangent $\tau = (dx,dp)$ to the curve of solutions, it is given by $(x_1,p_1) = (x_0,p_0) + ds\cdot \tau$. This predictor passes Turning points when used with PALC Newton corrector. **BifurcationKit.jl** provides two ways to compute the tangent $(dx, dp)$.

### 2a. Secant predictor

This predictor is called **secant** and is parametrized by the algorithm `PALC(tangent = Secant())`

in `continuation`

with `Secant`

. It is computed by $(dx, dp) = (z_1, p_1) - (z_0, p_0)$ and normalized by the norm $\|(x, p)\|^2_\theta = \frac{\theta}{length(x)} \langle x,x\rangle + (1 - \theta)\cdot p^2$.

The parameter `θ`

in the struct `ContinuationPar`

is very important. It should be tuned for the continuation to work properly especially in the case of large problems where the $\langle x - x_0, dx_0\rangle$ component in the constraint might be favored too much. Also, large `θ`

s favour `p`

as the corresponding term in the constraint $N$ involves the term $1-θ$.

### 2b. Bordered predictor

This predictor departs from the previous one in the way the tangent is estimated. It computes $(dx, dp)$ by solving solving the bordered linear system $\begin{bmatrix} F_x & F_p \\ \frac{\theta}{length(x)}dx_0 & (1-\theta)dp_0\end{bmatrix}\begin{bmatrix}dx \\ dp\end{bmatrix} =\begin{bmatrix}0 \\ 1\end{bmatrix}$.

It is set by the algorithm `PALC(tangent = Bordered())`

in `continuation`

with `Bordered`

. The linear solver for the linear problem in $(dx, dp)$ is set by the option `bls`

in `PALC`

: it is one of Bordered linear solvers (BLS).

## 3. Polynomial predictor

The polynomial predictor is based on a fit (least square regression) of an $n$th-order polynomial $P$ on the last $k$ solution vectors, where $n < k$. The arclength $s$ is used for the polynomial which then fits the solution $(x_i,p_i,s_i)$ as $P(s_i)\approx (x_i,p_i)$. To keep $s$ in suitable range (see ^{[Waugh]}), we rescale it as $s\to \frac{s-\bar s}{\sigma}$ where $\sigma$ is the standard deviation of the $s_i$.

This algorithm is parametrized by `alg = Polynomial(Fred, n, k, v0)`

where `pred::AbstractTangentComputation`

is the tangent predictor used only for the first $k$ solutions before the polynomial predictor is operational and `v0`

is an example of guess. More information is available in `Polynomial`

.

## 4. Multiple predictor (aka `pmcont`

in `pde2path`

)

The predictor is designed ^{[Uecker2014]} to avoid spurious branch switching and pass singular points especially in PDE where branch point density can be quite high. It is based on the use of many predictors with increasing "jumps" $(x_i,p_i) = (x_0,p_0) + i\cdot ds\cdot \tau,\ i\leq nb$ and use a corrector (PALC Newton) with the following twist. The criterion is that in each Newton step, the residual has to decrease by a factor $0<\alpha<1$:

\[\| F(u_n,p_n)\|\leq \alpha \| F(u_{n-1},p_{n-1}) \|\]

otherwise the corrector fails. The solution that is returned is the one for the highest $i$. We refer to ^{[Uecker2014]} for an exposition of the step size adaption strategy.

This algorithm is parametrized by `alg = Multiple(pred, x0, α, nb)`

where `τ`

is an initial tangent vector (used to set the types) and `pred::PALC`

is a predictor. The default value is `pred = PALC()`

. More information is available in `Multiple`

.

## References

- Allgower1990
Allgower and Georg, Numerical Continuation Methods, 1990

- Uecker2014
1.Uecker, H. pde2path - A Matlab Package for Continuation and Bifurcation in 2D Elliptic Systems. NMTMA 7, 58–106 (2014).

- Waugh
Waugh, Illingworth, and Juniper, “Matrix-Free Continuation of Limit Cycles for Bifurcation Analysis of Large Thermoacoustic Systems.”