aproxdef: Defining Approximation Space
In capn: Capital Asset Pricing for Nature

Description Usage Arguments Details Value References See Also Examples

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

1	aproxdef(deg, lb, ub, delta)

`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

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} = ≤ft\{ (s_{1},s_{2},\cdots,s_{d}) \vert a_{i} ≤q s_{1} ≤q 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.

A list containing the approximation space

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.

vaprox, vsim, paprox, psim, pdotaprox, pdotsim

## 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)