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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_rand_field_2d_predef_setup (g05zr)

## Purpose

nag_rand_field_2d_predef_setup (g05zr) performs the setup required in order to simulate stationary Gaussian random fields in two dimensions, for a preset variogram, using the circulant embedding method. Specifically, the eigenvalues of the extended covariance matrix (or embedding matrix) are calculated, and their square roots output, for use by nag_rand_field_2d_generate (g05zs), which simulates the random field.

## Syntax

[lam, xx, yy, m, approx, rho, icount, eig, ifail] = g05zr(ns, xmin, xmax, ymin, ymax, maxm, var, icov2, params, 'norm_p', norm_p, 'np', np, 'pad', pad, 'icorr', icorr)
[lam, xx, yy, m, approx, rho, icount, eig, ifail] = nag_rand_field_2d_predef_setup(ns, xmin, xmax, ymin, ymax, maxm, var, icov2, params, 'norm_p', norm_p, 'np', np, 'pad', pad, 'icorr', icorr)

## Description

A two-dimensional random field Z(x)$Z\left(\mathbf{x}\right)$ in 2${ℝ}^{2}$ is a function which is random at every point x2$\mathbf{x}\in {ℝ}^{2}$, so Z(x)$Z\left(\mathbf{x}\right)$ is a random variable for each x$\mathbf{x}$. The random field has a mean function μ(x) = 𝔼[Z(x)]$\mu \left(\mathbf{x}\right)=𝔼\left[Z\left(\mathbf{x}\right)\right]$ and a symmetric positive semidefinite covariance function C(x,y) = 𝔼[(Z(x)μ(x))(Z(y)μ(y))]$C\left(\mathbf{x},\mathbf{y}\right)=𝔼\left[\left(Z\left(\mathbf{x}\right)-\mu \left(\mathbf{x}\right)\right)\left(Z\left(\mathbf{y}\right)-\mu \left(\mathbf{y}\right)\right)\right]$. Z(x)$Z\left(\mathbf{x}\right)$ is a Gaussian random field if for any choice of n$n\in ℕ$ and x1,,xn2${\mathbf{x}}_{1},\dots ,{\mathbf{x}}_{n}\in {ℝ}^{2}$, the random vector [Z(x1),,Z(xn)]T${\left[Z\left({\mathbf{x}}_{1}\right),\dots ,Z\left({\mathbf{x}}_{n}\right)\right]}^{\mathrm{T}}$ follows a multivariate Gaussian distribution, which would have a mean vector μ̃$\stackrel{~}{\mathbf{\mu }}$ with entries μ̃i = μ(xi)${\stackrel{~}{\mu }}_{i}=\mu \left({\mathbf{x}}_{i}\right)$ and a covariance matrix $\stackrel{~}{C}$ with entries ij = C(xi,xj)${\stackrel{~}{C}}_{ij}=C\left({\mathbf{x}}_{i},{\mathbf{x}}_{j}\right)$. A Gaussian random field Z(x)$Z\left(\mathbf{x}\right)$ is stationary if μ(x)$\mu \left(\mathbf{x}\right)$ is constant for all x2$\mathbf{x}\in {ℝ}^{2}$ and C(x,y) = C(x + a,y + a)$C\left(\mathbf{x},\mathbf{y}\right)=C\left(\mathbf{x}+\mathbf{a},\mathbf{y}+\mathbf{a}\right)$ for all x,y,a2$\mathbf{x},\mathbf{y},\mathbf{a}\in {ℝ}^{2}$ and hence we can express the covariance function C(x,y)$C\left(\mathbf{x},\mathbf{y}\right)$ as a function γ$\gamma$ of one variable: C(x,y) = γ(xy)$C\left(\mathbf{x},\mathbf{y}\right)=\gamma \left(\mathbf{x}-\mathbf{y}\right)$. γ$\gamma$ is known as a variogram (or more correctly, a semivariogram) and includes the multiplicative factor σ2${\sigma }^{2}$ representing the variance such that γ(0) = σ2$\gamma \left(0\right)={\sigma }^{2}$.
The functions nag_rand_field_2d_predef_setup (g05zr) and nag_rand_field_2d_generate (g05zs) are used to simulate a two-dimensional stationary Gaussian random field, with mean function zero and variogram γ(x)$\gamma \left(\mathbf{x}\right)$, over a domain [xmin,xmax] × [ymin,ymax]$\left[{x}_{\mathrm{min}},{x}_{\mathrm{max}}\right]×\left[{y}_{\mathrm{min}},{y}_{\mathrm{max}}\right]$, using an equally spaced set of N1 × N2${N}_{1}×{N}_{2}$ gridpoints; N1${N}_{1}$ gridpoints in the x$x$-direction and N2${N}_{2}$ gridpoints in the y$y$-direction. The problem reduces to sampling a Gaussian random vector X$\mathbf{X}$ of size N1 × N2${N}_{1}×{N}_{2}$, with mean vector zero and a symmetric covariance matrix A$A$, which is an N2${N}_{2}$ by N2${N}_{2}$ block Toeplitz matrix with Toeplitz blocks of size N1${N}_{1}$ by N1${N}_{1}$. Since A$A$ is in general expensive to factorize, a technique known as the circulant embedding method is used. A$A$ is embedded into a larger, symmetric matrix B$B$, which is an M2${M}_{2}$ by M2${M}_{2}$ block circulant matrix with circulant blocks of size M1${M}_{1}$ by M1${M}_{1}$, where M12(N11)${M}_{1}\ge 2\left({N}_{1}-1\right)$ and M22(N21)${M}_{2}\ge 2\left({N}_{2}-1\right)$. B$B$ can now be factorized as B = WΛW* = R*R$B=W\Lambda {W}^{*}={R}^{*}R$, where W$W$ is the two-dimensional Fourier matrix (W*${W}^{*}$ is the complex conjugate of W$W$), Λ$\Lambda$ is the diagonal matrix containing the eigenvalues of B$B$ and R = Λ(1/2)W*$R={\Lambda }^{\frac{1}{2}}{W}^{*}$. B$B$ is known as the embedding matrix. The eigenvalues can be calculated by performing a discrete Fourier transform of the first row (or column) of B$B$ and multiplying by M1 × M2${M}_{1}×{M}_{2}$, and so only the first row (or column) of B$B$ is needed – the whole matrix does not need to be formed.
As long as all of the values of Λ$\Lambda$ are non-negative (i.e., B$B$ is positive semidefinite), B$B$ is a covariance matrix for a random vector Y$\mathbf{Y}$ which has M2${M}_{2}$ blocks of size M1${M}_{1}$. Two samples of Y$\mathbf{Y}$ can now be simulated from the real and imaginary parts of R*(U + iV)${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)$, where U$\mathbf{U}$ and V$\mathbf{V}$ have elements from the standard Normal distribution. Since R*(U + iV) = WΛ(1/2)(U + iV)${R}^{*}\left(\mathbf{U}+i\mathbf{V}\right)=W{\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$, this calculation can be done using a discrete Fourier transform of the vector Λ(1/2)(U + iV)${\Lambda }^{\frac{1}{2}}\left(\mathbf{U}+i\mathbf{V}\right)$. Two samples of the random vector X$\mathbf{X}$ can now be recovered by taking the first N1${N}_{1}$ elements of the first N2${N}_{2}$ blocks of each sample of Y$\mathbf{Y}$ – because the original covariance matrix A$A$ is embedded in B$B$, X$\mathbf{X}$ will have the correct distribution.
If B$B$ is not positive semidefinite, larger embedding matrices B$B$ can be tried; however if the size of the matrix would have to be larger than maxm, an approximation procedure is used. We write Λ = Λ+ + Λ$\Lambda ={\Lambda }_{+}+{\Lambda }_{-}$, where Λ+${\Lambda }_{+}$ and Λ${\Lambda }_{-}$ contain the non-negative and negative eigenvalues of B$B$ respectively. Then B$B$ is replaced by ρB+$\rho {B}_{+}$ where B+ = WΛ+W*${B}_{+}=W{\Lambda }_{+}{W}^{*}$ and ρ(0,1]$\rho \in \left(0,1\right]$ is a scaling factor. The error ε$\epsilon$ in approximating the distribution of the random field is given by
 ε = sqrt( ( (1 − ρ)2 traceΛ + ρ2 traceΛ− )/M ) . $ε= (1-ρ) 2 trace⁡Λ + ρ2 trace⁡Λ- M .$
Three choices for ρ$\rho$ are available, and are determined by the input parameter icorr:
• setting icorr = 0${\mathbf{icorr}}=0$ sets
 ρ = (traceΛ)/(traceΛ+) , $ρ= trace⁡Λ trace⁡Λ+ ,$
• setting icorr = 1${\mathbf{icorr}}=1$ sets
 ρ = sqrt( (traceΛ)/(traceΛ+) ) , $ρ= trace⁡Λ trace⁡Λ+ ,$
• setting icorr = 2${\mathbf{icorr}}=2$ sets ρ = 1$\rho =1$.
nag_rand_field_2d_predef_setup (g05zr) finds a suitable positive semidefinite embedding matrix B$B$ and outputs its sizes in the vector m and the square roots of its eigenvalues in lam. If approximation is used, information regarding the accuracy of the approximation is output. Note that only the first row (or column) of B$B$ is actually formed and stored.

## References

Dietrich C R and Newsam G N (1997) Fast and exact simulation of stationary Gaussian processes through circulant embedding of the covariance matrix SIAM J. Sci. Comput. 18 1088–1107
Schlather M (1999) Introduction to positive definite functions and to unconditional simulation of random fields Technical Report ST 99–10 Lancaster University
Wood A T A and Chan G (1997) Algorithm AS 312: An Algorithm for Simulating Stationary Gaussian Random Fields Journal of the Royal Statistical Society, Series C (Applied Statistics) (Volume 46) 1 171–181

## Parameters

### Compulsory Input Parameters

1:     ns(2$2$) – int64int32nag_int array
The number of sample points (gridpoints) to use in each direction, with ns(1)${\mathbf{ns}}\left(1\right)$ sample points in the x$x$-direction, N1${N}_{1}$ and ns(2)${\mathbf{ns}}\left(2\right)$ sample points in the y$y$-direction, N2${N}_{2}$. The total number of sample points on the grid is therefore ns(1) × ns(2) ${\mathbf{ns}}\left(1\right)×{\mathbf{ns}}\left(2\right)$.
Constraints:
• ns(1)1${\mathbf{ns}}\left(1\right)\ge 1$;
• ns(2)1${\mathbf{ns}}\left(2\right)\ge 1$.
2:     xmin – double scalar
The lower bound for the x$x$-coordinate, for the region in which the random field is to be simulated.
Constraint: ${\mathbf{xmin}}<{\mathbf{xmax}}$.
3:     xmax – double scalar
The upper bound for the x$x$-coordinate, for the region in which the random field is to be simulated.
Constraint: ${\mathbf{xmin}}<{\mathbf{xmax}}$.
4:     ymin – double scalar
The lower bound for the y$y$-coordinate, for the region in which the random field is to be simulated.
Constraint: ${\mathbf{ymin}}<{\mathbf{ymax}}$.
5:     ymax – double scalar
The upper bound for the y$y$-coordinate, for the region in which the random field is to be simulated.
Constraint: ${\mathbf{ymin}}<{\mathbf{ymax}}$.
6:     maxm(2$2$) – int64int32nag_int array
Determines the maximum size of the circulant matrix to use – a maximum of maxm(1)${\mathbf{maxm}}\left(1\right)$ elements in the x$x$-direction, and a maximum of maxm(2)${\mathbf{maxm}}\left(2\right)$ elements in the y$y$-direction. The maximum size of the circulant matrix is thus maxm(1)${\mathbf{maxm}}\left(1\right)$ × $×$maxm(2)${\mathbf{maxm}}\left(2\right)$.
Constraint: maxm(i) 2k ${\mathbf{maxm}}\left(i\right)\ge {2}^{k}$, where k$k$ is the smallest integer satisfying 2k 2 (ns(i)1) ${2}^{k}\ge 2\left({\mathbf{ns}}\left(i\right)-1\right)$, for i = 1,2$i=1,2$.
7:     var – double scalar
The multiplicative factor σ2${\sigma }^{2}$ of the variogram γ(x)$\gamma \left(\mathbf{x}\right)$.
Constraint: var0.0${\mathbf{var}}\ge 0.0$.
8:     icov2 – int64int32nag_int scalar
Determines which of the preset variograms to use. The choices are given below. Note that x = x/(1),y/(2) ${x}^{\prime }=‖\frac{x}{{\ell }_{1}},\frac{y}{{\ell }_{2}}‖$, where 1${\ell }_{1}$ and 2${\ell }_{2}$ are correlation lengths in the x$x$ and y$y$ directions respectively and are parameters for most of the variograms, and σ2${\sigma }^{2}$ is the variance specified by var.
icov2 = 1${\mathbf{icov2}}=1$
Symmetric stable variogram
 γ(x) = σ2 exp( − (x′)ν) , $γ(x) = σ2 exp( - (x′) ν ) ,$
where
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$,
• ν = params(3)$\nu ={\mathbf{params}}\left(3\right)$, 0 < ν2$0<\nu \le 2$.
icov2 = 2${\mathbf{icov2}}=2$
Cauchy variogram
 γ(x) = σ2 (1 + (x′)2) − ν , $γ(x) = σ2 ( 1 + (x′) 2 ) -ν ,$
where
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$,
• ν = params(3)$\nu ={\mathbf{params}}\left(3\right)$, ν > 0$\nu >0$.
icov2 = 3${\mathbf{icov2}}=3$
Differential variogram with compact support
γ(x) =
 { σ2 (1 + 8x′ + 25(x′)2 + 32(x′)3) (1 − x′)8 , x′ < 1 , 0 , x′ ≥ 1 ,
$γ(x) = { σ2 ( 1 + 8 x′ + 25 (x′) 2 + 32 (x′) 3 ) ( 1 - x′ ) 8 , x′<1 , 0 , x′ ≥ 1 ,$
where
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$.
icov2 = 4${\mathbf{icov2}}=4$
Exponential variogram
 γ(x) = σ2 exp( − x′) , $γ(x) = σ2 exp(-x′) ,$
where
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$.
icov2 = 5${\mathbf{icov2}}=5$
Gaussian variogram
 γ(x) = σ2 exp( − (x′)2) , $γ(x) = σ2 exp( -(x′) 2 ) ,$
where
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$.
icov2 = 6${\mathbf{icov2}}=6$
Nugget variogram
γ(x) =
 { σ2, x = 0, 0, x ≠ 0.
$γ(x) = { σ2, x=0, 0, x≠0.$
No parameters need be set for this value of icov2.
icov2 = 7${\mathbf{icov2}}=7$
Spherical variogram
γ(x) =
 { σ2 (1 − 1.5x′ + 0.5(x′)3) , x′ < 1 , 0, x′ ≥ 1 ,
$γ(x) = { σ2 ( 1 - 1.5x′ + 0.5 (x′) 3 ) , x′ < 1 , 0, x′ ≥ 1 ,$
where
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$.
icov2 = 8${\mathbf{icov2}}=8$
Bessel variogram
 γ(x) = σ2 ( 2ν Γ (ν + 1) Jν (x′) )/((x′)ν) , $γ(x) = σ2 2ν Γ (ν+1) Jν (x′) (x′) ν ,$
where
• Jν( · )${J}_{\nu }\left(·\right)$ is the Bessel function of the first kind,
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$,
• ν = params(3)$\nu ={\mathbf{params}}\left(3\right)$, ν0$\nu \ge 0$.
icov2 = 9${\mathbf{icov2}}=9$
Hole effect variogram
 γ(x) = σ2 (sin(x′))/(x′) , $γ(x) = σ2 sin(x′) x′ ,$
where
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$.
icov2 = 10${\mathbf{icov2}}=10$
Whittle-Matérn variogram
 γ(x) = σ2 ( 21 − ν (x′)ν Kν (x′) )/(Γ(ν)) , $γ(x) = σ2 21-ν (x′) ν Kν (x′) Γ(ν) ,$
where
• Kν( · )${K}_{\nu }\left(·\right)$ is the modified Bessel function of the second kind,
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$,
• ν = params(3)$\nu ={\mathbf{params}}\left(3\right)$, ν > 0$\nu >0$.
icov2 = 11${\mathbf{icov2}}=11$
Continuously parameterised variogram with compact support
γ(x) =
 { σ2 ( 21 − ν (x′)ν Kν (x′) )/(Γ(ν)) (1 + 8x′′ + 25(x′′)2 + 32(x′′)3)(1 − x′′)8, x′′ < 1, 0, x′′ ≥ 1,
$γ(x) = { σ2 21-ν (x′)ν Kν (x′) Γ(ν) (1+8x′′+25(x′′)2+32(x′′)3)(1-x′′)8, x′′<1, 0, x′′≥1,$
where
• x′′ = (x)/(1s1),(y)/(2s2) ${x}^{\mathrm{\prime \prime }}=‖\frac{{x}^{\prime }}{{\ell }_{1}{s}_{1}},\frac{{y}^{\prime }}{{\ell }_{2}{s}_{2}}‖$,
• Kν( · )${K}_{\nu }\left(·\right)$ is the modified Bessel function of the second kind,
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$,
• s1 = params(3)${s}_{1}={\mathbf{params}}\left(3\right)$, s1 > 0${s}_{1}>0$,
• s2 = params(4)${s}_{2}={\mathbf{params}}\left(4\right)$, s2 > 0${s}_{2}>0$,
• ν = params(5)$\nu ={\mathbf{params}}\left(5\right)$, ν > 0$\nu >0$.
icov2 = 12${\mathbf{icov2}}=12$
Generalized hyperbolic distribution variogram
 γ(x) = σ2((δ2 + (x′)2)λ/2)/(δλKλ(κδ))Kλ(κ(δ2 + (x′)2)(1/2)), $γ(x)=σ2(δ2+(x′)2)λ2δλKλ(κδ)Kλ(κ(δ2+(x′)2)12),$
where
• Kλ( · )${K}_{\lambda }\left(·\right)$ is the modified Bessel function of the second kind,
• 1 = params(1)${\ell }_{1}={\mathbf{params}}\left(1\right)$, 1 > 0${\ell }_{1}>0$,
• 2 = params(2)${\ell }_{2}={\mathbf{params}}\left(2\right)$, 2 > 0${\ell }_{2}>0$,
• λ = params(3)$\lambda ={\mathbf{params}}\left(3\right)$, no constraint on λ$\lambda$,
• δ = params(4)$\delta ={\mathbf{params}}\left(4\right)$, δ > 0$\delta >0$,
• κ = params(5)$\kappa ={\mathbf{params}}\left(5\right)$, κ > 0$\kappa >0$.
9:     params(np) – double array
The parameters for the variogram as detailed in the description of icov2.
Constraint: see icov2 for a description of the individual parameter constraints.

### Optional Input Parameters

1:     norm_p – int64int32nag_int scalar
Determines which norm to use when calculating the variogram.
norm = 1${\mathbf{norm}}=1$
The 1-norm is used, i.e., x,y = |x| + |y|$‖x,y‖=|x|+|y|$.
norm = 2${\mathbf{norm}}=2$
The 2-norm (Euclidean norm) is used, i.e., x,y = sqrt(x2 + y2)$‖x,y‖=\sqrt{{x}^{2}+{y}^{2}}$.
Default: norm = 2${\mathbf{norm}}=2$
Constraint: norm = 1${\mathbf{norm}}=1$ or 2$2$.
2:     np – int64int32nag_int scalar
Default: The dimension of the array params.
The number of parameters to be set. Different covariance functions need a different number of parameters.
icov2 = 6${\mathbf{icov2}}=6$
np must be set to 0$0$.
icov2 = 3${\mathbf{icov2}}=3$, 4$4$, 5$5$, 7$7$ or 9$9$
np must be set to 2$2$.
icov2 = 1${\mathbf{icov2}}=1$, 2$2$, 8$8$ or 10$10$
np must be set to 3$3$.
icov2 = 11${\mathbf{icov2}}=11$ or 12$12$
np must be set to 5$5$.
Determines whether the embedding matrix is padded with zeros, or padded with values of the variogram. The choice of padding may affect how big the embedding matrix must be in order to be positive semidefinite.
pad = 0${\mathbf{pad}}=0$
The embedding matrix is padded with zeros.
pad = 1${\mathbf{pad}}=1$
The embedding matrix is padded with values of the variogram.
Default: pad = 1${\mathbf{pad}}=1$
Constraint: pad = 0${\mathbf{pad}}=0$ or 1$1$.
4:     icorr – int64int32nag_int scalar
Determines which approximation to implement if required, as described in Section [Description].
Default: icorr = 0${\mathbf{icorr}}=0$
Constraint: icorr = 0${\mathbf{icorr}}=0$, 1$1$ or 2$2$.

None.

### Output Parameters

1:     lam(maxm(1) × maxm(2)${\mathbf{maxm}}\left(1\right)×{\mathbf{maxm}}\left(2\right)$) – double array
Contains the square roots of the eigenvalues of the embedding matrix.
2:     xx(ns(1)${\mathbf{ns}}\left(1\right)$) – double array
The gridpoints of the x$x$-coordinates at which values of the random field will be output.
3:     yy(ns(2)${\mathbf{ns}}\left(2\right)$) – double array
The gridpoints of the y$y$-coordinates at which values of the random field will be output.
4:     m(2$2$) – int64int32nag_int array
m(1)${\mathbf{m}}\left(1\right)$ contains M1${M}_{1}$, the size of the circulant blocks and m(2)${\mathbf{m}}\left(2\right)$ contains M2${M}_{2}$, the number of blocks, resulting in a final square matrix of size M1 × M2${M}_{1}×{M}_{2}$.
5:     approx – int64int32nag_int scalar
Indicates whether approximation was used.
approx = 0${\mathbf{approx}}=0$
No approximation was used.
approx = 1${\mathbf{approx}}=1$
Approximation was used.
6:     rho – double scalar
Indicates the scaling of the covariance matrix. rho = 1.0${\mathbf{rho}}=1.0$ unless approximation was used with icorr = 0${\mathbf{icorr}}=0$ or 1$1$.
7:     icount – int64int32nag_int scalar
Indicates the number of negative eigenvalues in the embedding matrix which have had to be set to zero.
8:     eig(3$3$) – double array
Indicates information about the negative eigenvalues in the embedding matrix which have had to be set to zero. eig(1)${\mathbf{eig}}\left(1\right)$ contains the smallest eigenvalue, eig(2)${\mathbf{eig}}\left(2\right)$ contains the sum of the squares of the negative eigenvalues, and eig(3)${\mathbf{eig}}\left(3\right)$ contains the sum of the absolute values of the negative eigenvalues.
9:     ifail – int64int32nag_int scalar
${\mathrm{ifail}}={\mathbf{0}}$ unless the function detects an error (see [Error Indicators and Warnings]).

## Error Indicators and Warnings

Errors or warnings detected by the function:
ifail = 1${\mathbf{ifail}}=1$
Constraint: ns(1)1${\mathbf{ns}}\left(1\right)\ge 1$, ns(2)1${\mathbf{ns}}\left(2\right)\ge 1$.
ifail = 2${\mathbf{ifail}}=2$
Constraint: ${\mathbf{xmin}}<{\mathbf{xmax}}$.
ifail = 4${\mathbf{ifail}}=4$
Constraint: ${\mathbf{ymin}}<{\mathbf{ymax}}$.
ifail = 6${\mathbf{ifail}}=6$
Constraint: the calculated minimum value for maxm are [_,_]$\left[_,_\right]$.
Where the minimum calculated value of maxm(i)${\mathbf{maxm}}\left(i\right)$ is given by 2k ${2}^{k}$, where k$k$ is the smallest integer satisfying 2k 2 (ns(i)1) ${2}^{k}\ge 2\left({\mathbf{ns}}\left(i\right)-1\right)$.
ifail = 7${\mathbf{ifail}}=7$
Constraint: var0.0${\mathbf{var}}\ge 0.0$.
ifail = 8${\mathbf{ifail}}=8$
Constraint: icov21${\mathbf{icov2}}\ge 1$ and icov212${\mathbf{icov2}}\le 12$.
ifail = 9${\mathbf{ifail}}=9$
Constraint: norm = 1${\mathbf{norm}}=1$ or 2$2$.
ifail = 10${\mathbf{ifail}}=10$
Constraint: for icov2 = _${\mathbf{icov2}}=_$.
On entry, np is not the correct number of parameters for the specified variogram.
ifail = 11${\mathbf{ifail}}=11$
Constraint: dependent on icov2, see documentation.
ifail = 12${\mathbf{ifail}}=12$
Constraint: pad = 0${\mathbf{pad}}=0$ or 1$1$.
ifail = 13${\mathbf{ifail}}=13$
Constraint: icorr = 0${\mathbf{icorr}}=0$, 1$1$ or 2$2$.

Not applicable.

None.

## Example

```function nag_rand_field_2d_predef_setup_example
icov2 = int64(1); % symmetric stable
params = [0.1; 0.15; 1.2];
var = 0.5;
xmin = -1;
xmax = 1;
ymin = -0.5;
ymax = 0.5;
ns = [int64(5), 5];
maxm = [int64(64), 64];
icorr = int64(2);

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, yy, m, approx, rho, icount, eig, ifail] = ...
nag_rand_field_2d_predef_setup(ns, xmin, xmax, ymin, ymax, ...
maxm, var, icov2, params, 'icorr', icorr);

fprintf('\nSize of embedding matrix = %d\n\n', m(1)*m(2));

% Display approximation information if approximation used
if approx == 1
fprintf('Approximation required\n\n');
fprintf('rho = %10.5f\n', rho);
fprintf('eig = %10.5f%10.5f%10.5f\n', eig(1:3));
fprintf('icount = %d\n', icount);
else
fprintf('Approximation not required\n\n');
end

% Display square roots of the eigenvalues of the embedding matrix
fprintf('Square roots of eigenvalues of embedding matrix:\n');
disp(reshape(lam(1:m(1)*m(2)), m(1), m(2)));
```
```

Size of embedding matrix = 64

Approximation not required

Square roots of eigenvalues of embedding matrix:
0.8966    0.8234    0.6810    0.5757    0.5391    0.5757    0.6810    0.8234
0.8940    0.8217    0.6804    0.5756    0.5391    0.5756    0.6804    0.8217
0.8877    0.8175    0.6792    0.5754    0.5391    0.5754    0.6792    0.8175
0.8813    0.8133    0.6780    0.5751    0.5390    0.5751    0.6780    0.8133
0.8787    0.8116    0.6774    0.5750    0.5390    0.5750    0.6774    0.8116
0.8813    0.8133    0.6780    0.5751    0.5390    0.5751    0.6780    0.8133
0.8877    0.8175    0.6792    0.5754    0.5391    0.5754    0.6792    0.8175
0.8940    0.8217    0.6804    0.5756    0.5391    0.5756    0.6804    0.8217

```
```function g05zr_example
icov2 = int64(1); % symmetric stable
params = [0.1; 0.15; 1.2];
var = 0.5;
xmin = -1;
xmax = 1;
ymin = -0.5;
ymax = 0.5;
ns = [int64(5), 5];
maxm = [int64(64), 64];
icorr = int64(2);

% Get square roots of the eigenvalues of the embedding matrix
[lam, xx, yy, m, approx, rho, icount, eig, ifail] = ...
g05zr(ns, xmin, xmax, ymin, ymax, maxm, var, ...
icov2, params, 'icorr', icorr);

fprintf('\nSize of embedding matrix = %d\n\n', m(1)*m(2));

% Display approximation information if approximation used
if approx == 1
fprintf('Approximation required\n\n');
fprintf('rho = %10.5f\n', rho);
fprintf('eig = %10.5f%10.5f%10.5f\n', eig(1:3));
fprintf('icount = %d\n', icount);
else
fprintf('Approximation not required\n\n');
end

% Display square roots of the eigenvalues of the embedding matrix
fprintf('Square roots of eigenvalues of embedding matrix:\n');
disp(reshape(lam(1:m(1)*m(2)), m(1), m(2)));
```
```

Size of embedding matrix = 64

Approximation not required

Square roots of eigenvalues of embedding matrix:
0.8966    0.8234    0.6810    0.5757    0.5391    0.5757    0.6810    0.8234
0.8940    0.8217    0.6804    0.5756    0.5391    0.5756    0.6804    0.8217
0.8877    0.8175    0.6792    0.5754    0.5391    0.5754    0.6792    0.8175
0.8813    0.8133    0.6780    0.5751    0.5390    0.5751    0.6780    0.8133
0.8787    0.8116    0.6774    0.5750    0.5390    0.5750    0.6774    0.8116
0.8813    0.8133    0.6780    0.5751    0.5390    0.5751    0.6780    0.8133
0.8877    0.8175    0.6792    0.5754    0.5391    0.5754    0.6792    0.8175
0.8940    0.8217    0.6804    0.5756    0.5391    0.5756    0.6804    0.8217

```