ptcsolsc: pseudo-transient continuation

ptcsolsc(f, x0, fp=difffp; rtol=1.e-6, atol=1.e-12, maxit=100, delta0=1.e-6, dx=1.e-7, pdata=nothing, printerr = true, keepsolhist=true)

C. T. Kelley, 2022

Scalar pseudo-transient continuation solver. PTC is designed to find stable steady state solutions of

dx/dt = - f(x)

The scalar code is a simple wrapper around a call to ptcsol.jl, the PTC solver for systems.

–> PTC is ABSOLUTELY NOT a general purpose nonlinear solver.

Input:

f: function

x: initial iterate/data

fp: derivative. If your derivative function is fp, you give me its name. For example fp=foobar tells me that foobar is your function for the derivative. The default is a forward difference Jacobian that I provide.

Keyword Arguments:

rtol, atol: real and absolute error tolerances

maxit: upper bound on number of nonlinear iterations. This is coupled to delta0. If your choice of delta0 is too small (conservative) then you'll need many iterations to converge and will need a larger value of maxit.

delta0: initial pseudo time step. The default value of 1.e-3 is a bit conservative and is one option you really should play with. Look at the example where I set it to 1.0!

dx: default = 1.e-7

difference increment in finite-difference derivatives h=dx*norm(x)+1.e-6

pdata:

precomputed data for the function/derivative. Things will go better if you use this rather than hide the data in global variables within the module for your function/derivative If you use this option your function and derivative must take pdata as a second argument. eg f(x,pdata) and fp(x,pdata)

printerr: default = true

I print a helpful message when the solver fails. To suppress that message set printerr to false.

keepsolhist: if true you get the history of the iteration in the output tuple. This is on by default for scalar equations and off for systems. Only turn it on if you have use for the data, which can get REALLY LARGE.

Output: A tuple (solution, functionval, history, idid, errcode, solhist) where history is the array of absolute function values |f(x)| of residual norms and time steps. Unless something has gone badly wrong, delta approx |f(x_0)|/|f(x)|.

idid=true if the iteration succeeded and false if not.

errcode = 0 if if the iteration succeeded = -1 if the initial iterate satisfies the termination criteria = 10 if no convergence after maxit iterations

solhist=entire history of the iteration if keepsolhist=true

ptcsolsc builds solhist with a function from the Tools directory. For systems, solhist is an N x K array where N is the length of x and K is the number of iteration + 1. So, for scalar equations (N=1), solhist is a row vector. Hence I use [ptcout.solhist' ptcout.history] in the example below.

If the iteration fails it's time to play with the tolerances, delta0, and maxit. You are certain to fail if there is no stable solution to the equation.

Examples for ptcsolsc

julia> ptcout=ptcsolsc(sptest,.2;delta0=2.0,rtol=1.e-3,atol=1.e-3);

julia> [ptcout.solhist' ptcout.history]
7×2 Array{Float64,2}:
 2.00000e-01  9.20000e-02
 9.66666e-01  4.19962e-01
 8.75086e-01  2.32577e-01
 7.99114e-01  1.10743e-01
 7.44225e-01  4.00926e-02
 7.15163e-01  8.19395e-03
 7.07568e-01  4.61523e-04