Get the order of canonical polynomial chaos expansion

Description

For a d-dimensional polynomial chaos expansion up to order p, there are a total of M polynomials, determine from get(d,p). Each polynomial $phi_m(x)$ is expressed as the product of the polynomial from each random variables, i.e. $phi_m(x) = phi_m,1(x_1)phi_m,2(x_2)...phi_m,d(x_d)$, each with a different polynomial order. We can thus very succintly denote phi_m(x) with a n-tuple vector containing the polynomial order from each dimension and the entire canonical PCE expansion can be express as a (m x n) matrix.

Usage

indexCardinal(d,p,m,index)

Arguments

Value

Author(s)

Jordan Ko

References

R. Ghanem and P. Spanos, 1991, Stochastic Finite Elements: A Spectral Approach. Berlin: Springer.

J. Ko, 2009, Applications of the generalized polynomial chaos to the numerical simulationof stochastic shear flows, Doctoral thesis, Universit'e Paris VI.

See Also

getM

Examples

d <- 3
p <- 5
m <- getM(d,p)
index <- indexCardinal(d,p)
print(t(index))

Results

R version 3.3.1 (2016-06-21) -- "Bug in Your Hair"
Copyright (C) 2016 The R Foundation for Statistical Computing
Platform: x86_64-pc-linux-gnu (64-bit)

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> library(GPC)
Loading required package: randtoolbox
Loading required package: rngWELL
This is randtoolbox. For overview, type 'help("randtoolbox")'.
Loading required package: orthopolynom
Loading required package: polynom
Loading required package: ks
Loading required package: KernSmooth
KernSmooth 2.23 loaded
Copyright M. P. Wand 1997-2009
Loading required package: misc3d
Loading required package: mvtnorm
Loading required package: rgl
Loading required package: lars
Loaded lars 1.2

> png(filename="/home/ddbj/snapshot/RGM3/R_CC/result/GPC/indexCardinal.Rd_%03d_medium.png", width=480, height=480)
> ### Name: indexCardinal
> ### Title: Get the order of canonical polynomial chaos expansion
> ### Aliases: indexCardinal
> 
> ### ** Examples
> 
> d <- 3
> p <- 5
> m <- getM(d,p)
> index <- indexCardinal(d,p)
> print(t(index))
      [,1] [,2] [,3]
 [1,]    0    0    0
 [2,]    1    0    0
 [3,]    0    1    0
 [4,]    0    0    1
 [5,]    2    0    0
 [6,]    1    1    0
 [7,]    1    0    1
 [8,]    0    2    0
 [9,]    0    1    1
[10,]    0    0    2
[11,]    3    0    0
[12,]    2    1    0
[13,]    2    0    1
[14,]    1    2    0
[15,]    1    1    1
[16,]    1    0    2
[17,]    0    3    0
[18,]    0    2    1
[19,]    0    1    2
[20,]    0    0    3
[21,]    4    0    0
[22,]    3    1    0
[23,]    3    0    1
[24,]    2    2    0
[25,]    2    1    1
[26,]    2    0    2
[27,]    1    3    0
[28,]    1    2    1
[29,]    1    1    2
[30,]    1    0    3
[31,]    0    4    0
[32,]    0    3    1
[33,]    0    2    2
[34,]    0    1    3
[35,]    0    0    4
[36,]    5    0    0
[37,]    4    1    0
[38,]    4    0    1
[39,]    3    2    0
[40,]    3    1    1
[41,]    3    0    2
[42,]    2    3    0
[43,]    2    2    1
[44,]    2    1    2
[45,]    2    0    3
[46,]    1    4    0
[47,]    1    3    1
[48,]    1    2    2
[49,]    1    1    3
[50,]    1    0    4
[51,]    0    5    0
[52,]    0    4    1
[53,]    0    3    2
[54,]    0    2    3
[55,]    0    1    4
[56,]    0    0    5
> 
> 
> 
> 
> 
> dev.off()
null device 
          1 
>