step.SW: Stepleman-Winarsky automatic step selection

In pnd: Parallel Numerical Derivatives, Gradients, Jacobians, and Hessians of Arbitrary Accuracy Order

Stepleman–Winarsky automatic step selection

Description

Stepleman–Winarsky automatic step selection

Usage

step.SW(
  FUN,
  x,
  h0 = 1e-05 * (abs(x) + (x == 0)),
  shrink.factor = 0.5,
  range = h0/c(1e+12, 1e-08),
  seq.tol = 1e-04,
  max.rel.error = .Machine$double.eps/2,
  maxit = 40L,
  cores = 1,
  preschedule = getOption("pnd.preschedule", TRUE),
  cl = NULL,
  ...
)

Arguments

FUN	Function for which the optimal numerical derivative step size is needed.
x	Numeric scalar: the point at which the derivative is computed and the optimal step size is estimated.
h0	Numeric scalar: initial step size, defaulting to a relative step of slightly greater than .Machine$double.eps^(1/3) (or absolute step if x == 0).
shrink.factor	A scalar less than 1 that is used to multiply the step size during the search. The authors recommend 0.25, but this may be result in earlier termination at slightly sub-optimal steps. Change to 0.5 for a more thorough search.
range	Numeric vector of length 2 defining the valid search range for the step size.
seq.tol	Numeric scalar: maximum relative difference between old and new step sizes for declaring convergence.
max.rel.error	Positive numeric scalar > 0 indicating the maximum relative error of function evaluation. For highly accurate functions with all accurate bits is equal to half of machine epsilon. For noisy functions (derivatives, integrals, output of optimisation routines etc.), it is higher.
maxit	Maximum number of algorithm iterations to avoid infinite loops. Consider trying some smaller or larger initial step size h0 if this limit is reached.
cores	Integer specifying the number of CPU cores used for parallel computation. Recommended to be set to the number of physical cores on the machine minus one.
preschedule	Logical: if TRUE, disables pre-scheduling for mclapply() or enables load balancing with parLapplyLB(). Recommended for functions that take less than 0.1 s per evaluation.
cl	An optional user-supplied cluster object (created by makeCluster or similar functions). If not NULL, the code uses parLapply() (if preschedule is TRUE) or parLapplyLB() on that cluster on Windows, and mclapply (fork cluster) on everything else.
...	Passed to FUN.

Details

This function computes the optimal step size for central differences using the \insertCitestepleman1979adaptivepnd algorithm.

Value

A list similar to the one returned by optim():

• par – the optimal step size found.

• value – the estimated numerical first derivative (using central differences).

• counts – the number of iterations (each iteration includes four function evaluations).

• abs.error – an estimate of the truncation and rounding errors.

• exitcode – an integer code indicating the termination status:

  • 0 – Optimal termination within tolerance.

  • 2 – No change in step size within tolerance.

  • 3 – Solution lies at the boundary of the allowed value range.

  • 4 – Step trimmed to 0.1|x| when |x| is not tiny and within range.

  • 5 – Maximum number of iterations reached.

• message – A summary message of the exit status.

• iterations – A list including the full step size search path, argument grids, function values on those grids, estimated derivative values, estimated error values, and monotonicity check results.

References

\insertAllCited

Examples

f <- function(x) x^4  # The derivative at 1 is 4
step.SW(x = 1, f)
step.SW(x = 1, f, h0 = 1e-9) # Starting too low
# Starting somewhat high leads to too many preliminary iterations
step.SW(x = 1, f, h0 = 10)
step.SW(x = 1, f, h0 = 1000) # Starting absurdly high

f <- sin  # The derivative at pi/4 is sqrt(2)/2
step.SW(x = pi/4, f)
step.SW(x = pi/4, f, h0 = 1e-9) # Starting too low
step.SW(x = pi/4, f, h0 = 0.1) # Starting slightly high
# The following two example fail because the truncation error estimate is invalid
step.SW(x = pi/4, f, h0 = 10)   # Warning
step.SW(x = pi/4, f, h0 = 1000) # Warning

pnd documentation built on Sept. 9, 2025, 5:44 p.m.