capn (version 1.0.0)

aproxdef: Defining Approximation Space

Description

The function defines an approximation space for all three approximation apporoaches (V, P, and Pdot).

Usage

aproxdef(deg, lb, ub, delta)

Arguments

deg

An array of degrees of approximation function: degrees of Chebyshev polynomials

lb

An array of lower bounds

ub

An array of upper bounds

delta

discount rate

Value

A list containing the approximation space

Details

For the \(i\)-th dimension of \(i = 1, 2, \cdots, d\), suppose a polynomial approximant \(s_{i}\) over a bounded interval \([a_{i},b_{i}]\) is defined by Chebysev nodes. Then, a \(d\)-dimensional Chebyshev grids can be defined as:

\(\mathbf{S} = \left\{ (s_{1},s_{2},\cdots,s_{d}) \vert a_{i} \leq s_{1} \leq b_{i}, i = 1, 2, \cdots, d \right\} \).

Suppose we impletement \(n_{i}\) numbers of polynomials (i.e., \((n_{i}-1)\)-th order) for the \(i\)-th dimension. The approximation space is defined as:

deg = c(\(n_{1},n_{2},\cdots,n_{d}\)), lb = c(\(a_{1},a_{2},\cdots,a_{d}\)), and ub = c(\(b_{1},b_{2},\cdots,b_{d}\)).

delta is the given constant discount rate.

References

Fenichel, Eli P. and Joshua K. Abbott. (2014) "Natural Capital: From Metaphor to Measurement." Journal of the Association of Environmental Economists. 1(1/2):1-27.

See Also

vaprox, vsim, paprox, psim, pdotaprox, pdotsim

Examples

## Reef-fish example: see Fenichel and Abbott (2014)
delta <- 0.02
upper <- 359016000     # upper bound on approximation space
lower <- 5*10^6        # lower bound on approximation space
myspace <- aproxdef(50,lower,upper,delta)
## Two dimensional example
ub <- c(1.5,1.5)
lb <- c(0.1,0.1)
deg <- c(20,20)
delta <- 0.03
myspace <- aproxdef(deg,lb,ub,delta)