sp1.poly: Shape-preserving polynomial approximation

In squipbar/cheby: Computes Chebychev approximations to 1- and 2-dimensional functions

Description

Approximates the function fn using shape-preserving polynomials evaluated at the points in grid.

Usage

sp1.poly(fn, range, iOrder, iPts, fn.opts = NULL, fn.vals = NULL,
  grid = NULL, n.shape = 0, sign.deriv = NULL, x0 = NULL,
  solver = "NLOPT_LD_SLSQP", tol = 1e-06, details = FALSE,
  quiet = FALSE)

Arguments

fn	a function f(x) or f(x,β) for β a list of function parameters. If the latter, must be coded with second argument a list names opts, i.e. fn <- function( x, opts )
range	the range of the approximation.
iOrder	the order of the polynomial approximation.
iPts	the number of points at which the approximation is computed. Must be at least as large as iOrder.
fn.opts	(optional) options passed to fn
fn.vals	the values of fn on grid. Useful if fn is very slow to evaluate.
grid	(optional) the grid on which the function is to be approximated.
n.shape	a vector of the number of shape-preserving points for each order of differentiation. For example, to specify the slope at 5 Chebychev points and the concavity at 10, use n.shape=c( 5, 10 )
sign.deriv	a vector of signs +1, 0, -1 defining the sign of each derivative. For example, for a concave approximation with positive slope sign.deriv=c(1,-1).
x0	initial guess of the polynomial approximation. Default is the standard Chebychev approximation.
solver	the nlopt solver to use in computing the best fit. Default is NLOPT_LD_SLSQP.
tol	tolerance for solver convergence. Default is 1e-06
details	If TRUE, returns extra details about the approximation.
quiet	Supresses output about success of least-error fitting. Failure will always be reported

Value

A function which approximates fn. If details=TRUE, return is a list with entries fn, poly, fn.deriv, poly.deriv, residuals, which are, respectively, the approximating function, the polynomial desciption over [-1,1], the derivative of the approximation, the polynomial desciption of the derivative, and the approximation errors.

References

nloptr documentation

See Also

d1.poly

Examples

base <- d1.poly( log, c(0,4), 6, 10, details=TRUE )
sp.compare <- sp1.poly(  log, c(0,4), 6, 10, details=TRUE )
sp.flat.x0 <- sp1.poly(  log, c(0,4), 6, 10, x0=c(1,1,1,1,1,1,1), details=TRUE )
print( base$poly - sp.compare$poly )
print( base$poly - sp.flat.x0$poly )
   # Comparison without using shape-preserving methods
sp.concave <- sp1.poly( log, c(0,4), 6, 10, n.shape=c(5,10), sign.deriv=c(1,-1) )
pp <-  seq( 0, 4, length.out=100 )
plot( pp, sapply(pp, base$fn), lwd=2, col=2, type='l' )
lines( pp, sapply(pp, log), lwd=2, col=1 )
lines( pp, sapply(pp, sp.concave), lwd=2, col=4 )
legend( 'bottomright', c( 'log', 'Order 6 polynomial approx',
     'Order 6 shape-preserving polynomial approx' ), lwd=2, 
     col=c(1,2,4), bty='n' )
   # Compare the Chebychev and shape-preserving approximations

squipbar/cheby documentation built on May 30, 2019, 8 a.m.