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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_tsa_multi_spectrum_lag (g13cc)

## Purpose

nag_tsa_multi_spectrum_lag (g13cc) calculates the smoothed sample cross spectrum of a bivariate time series using one of four lag windows: rectangular, Bartlett, Tukey or Parzen.

## Syntax

[cxy, cyx, xg, yg, ng, ifail] = g13cc(nxy, mtxy, pxy, iw, mw, ish, ic, cxy, cyx, kc, l, 'nc', nc, 'xg', xg, 'yg', yg)
[cxy, cyx, xg, yg, ng, ifail] = nag_tsa_multi_spectrum_lag(nxy, mtxy, pxy, iw, mw, ish, ic, cxy, cyx, kc, l, 'nc', nc, 'xg', xg, 'yg', yg)

## Description

The smoothed sample cross spectrum is a complex valued function of frequency $\omega$, ${f}_{xy}\left(\omega \right)=cf\left(\omega \right)+iqf\left(\omega \right)$, defined by its real part or co-spectrum
 $cfω=12π ∑k=-M+1 M-1wkCxyk+Scosωk$
and imaginary part or quadrature spectrum
 $qfω=12π ∑k=-M+ 1 M- 1wkCxyk+Ssinω k$
where ${w}_{\mathit{k}}={w}_{-\mathit{k}}$, for $\mathit{k}=0,1,\dots ,M-1$, is the smoothing lag window as defined in the description of nag_tsa_uni_spectrum_lag (g13ca). The alignment shift $S$ is recommended to be chosen as the lag $k$ at which the cross-covariances ${c}_{xy}\left(k\right)$ peak, so as to minimize bias.
The results are calculated for frequency values
 $ωj=2πjL, j=0,1,…,L/2,$
where $\left[\right]$ denotes the integer part.
The cross-covariances ${c}_{xy}\left(k\right)$ may be supplied by you, or constructed from supplied series ${x}_{1},{x}_{2},\dots ,{x}_{n}$; ${y}_{1},{y}_{2},\dots ,{y}_{n}$ as
 $cxyk=∑t=1 n-kxtyt+kn, k≥0$
 $cxyk=∑t= 1-knxtyt+kn=cyx-k, k< 0$
this convolution being carried out using the finite Fourier transform.
The supplied series may be mean and trend corrected and tapered before calculation of the cross-covariances, in exactly the manner described in nag_tsa_uni_spectrum_lag (g13ca) for univariate spectrum estimation. The results are corrected for any bias due to tapering.
The bandwidth associated with the estimates is not returned. It will normally already have been calculated in previous calls of nag_tsa_uni_spectrum_lag (g13ca) for estimating the univariate spectra of ${y}_{t}$ and ${x}_{t}$.

## References

Bloomfield P (1976) Fourier Analysis of Time Series: An Introduction Wiley
Jenkins G M and Watts D G (1968) Spectral Analysis and its Applications Holden–Day

## Parameters

### Compulsory Input Parameters

1:     $\mathrm{nxy}$int64int32nag_int scalar
$n$, the length of the time series $x$ and $y$.
Constraint: ${\mathbf{nxy}}\ge 1$.
2:     $\mathrm{mtxy}$int64int32nag_int scalar
If cross-covariances are to be calculated by the function (${\mathbf{ic}}=0$), mtxy must specify whether the data is to be initially mean or trend corrected.
${\mathbf{mtxy}}=0$
For no correction.
${\mathbf{mtxy}}=1$
For mean correction.
${\mathbf{mtxy}}=2$
For trend correction.
If cross-covariances are supplied $\left({\mathbf{ic}}\ne 0\right)$, mtxy is not used.
Constraint: if ${\mathbf{ic}}=0$, ${\mathbf{mtxy}}=0$, $1$ or $2$.
3:     $\mathrm{pxy}$ – double scalar
If cross-covariances are to be calculated by the function (${\mathbf{ic}}=0$), pxy must specify the proportion of the data (totalled over both ends) to be initially tapered by the split cosine bell taper. A value of $0.0$ implies no tapering.
If cross-covariances are supplied $\left({\mathbf{ic}}\ne 0\right)$, pxy is not used.
Constraint: if ${\mathbf{ic}}=0$, $0.0\le {\mathbf{pxy}}\le 1.0$.
4:     $\mathrm{iw}$int64int32nag_int scalar
The choice of lag window.
${\mathbf{iw}}=1$
Rectangular.
${\mathbf{iw}}=2$
Bartlett.
${\mathbf{iw}}=3$
Tukey.
${\mathbf{iw}}=4$
Parzen.
Constraint: $1\le {\mathbf{iw}}\le 4$.
5:     $\mathrm{mw}$int64int32nag_int scalar
$M$, the ‘cut-off’ point of the lag window, relative to any alignment shift that has been applied. Windowed cross-covariances at lags $\left(-{\mathbf{mw}}+{\mathbf{ish}}\right)$ or less, and at lags $\left({\mathbf{mw}}+{\mathbf{ish}}\right)$ or greater are zero.
Constraints:
• ${\mathbf{mw}}\ge 1$;
• ${\mathbf{mw}}+\left|{\mathbf{ish}}\right|\le {\mathbf{nxy}}$.
6:     $\mathrm{ish}$int64int32nag_int scalar
$S$, the alignment shift between the $x$ and $y$ series. If $x$ leads $y$, the shift is positive.
Constraint: $-{\mathbf{mw}}<{\mathbf{ish}}<{\mathbf{mw}}$.
7:     $\mathrm{ic}$int64int32nag_int scalar
Indicates whether cross-covariances are to be calculated in the function or supplied in the call to the function.
${\mathbf{ic}}=0$
Cross-covariances are to be calculated.
${\mathbf{ic}}\ne 0$
Cross-covariances are to be supplied.
8:     $\mathrm{cxy}\left({\mathbf{nc}}\right)$ – double array
If ${\mathbf{ic}}\ne 0$, cxy must contain the nc cross-covariances between values in the $y$ series and earlier values in time in the $x$ series, for lags from $0$ to $\left({\mathbf{nc}}-1\right)$.
If ${\mathbf{ic}}=0$, cxy need not be set.
9:     $\mathrm{cyx}\left({\mathbf{nc}}\right)$ – double array
If ${\mathbf{ic}}\ne 0$, cyx must contain the nc cross-covariances between values in the $y$ series and later values in time in the $x$ series, for lags from $0$ to $\left({\mathbf{nc}}-1\right)$.
If ${\mathbf{ic}}=0$, cyx need not be set.
10:   $\mathrm{kc}$int64int32nag_int scalar
If ${\mathbf{ic}}=0$, kc must specify the order of the fast Fourier transform (FFT) used to calculate the cross-covariances. kc should be a product of small primes such as ${2}^{m}$ where $m$ is the smallest integer such that ${2}^{m}\ge n+{\mathbf{nc}}$.
If ${\mathbf{ic}}\ne 0$, that is if covariances are supplied, kc is not used.
Constraint: ${\mathbf{kc}}\ge {\mathbf{nxy}}+{\mathbf{nc}}$. The largest prime factor of kc must not exceed $19$, and the total number of prime factors of kc, counting repetitions, must not exceed $20$. These two restrictions are imposed by the internal FFT algorithm used.
11:   $\mathrm{l}$int64int32nag_int scalar
$L$, the frequency division of the spectral estimates as $\frac{2\pi }{L}$. Therefore it is also the order of the FFT used to construct the sample spectrum from the cross-covariances. l should be a product of small primes such as ${2}^{m}$ where $m$ is the smallest integer such that ${2}^{m}\ge 2M-1$.
Constraint: ${\mathbf{l}}\ge 2×{\mathbf{mw}}-1$. The largest prime factor of l must not exceed $19$, and the total number of prime factors of l, counting repetitions, must not exceed $20$. These two restrictions are imposed by the internal FFT algorithm used.

### Optional Input Parameters

1:     $\mathrm{nc}$int64int32nag_int scalar
Default: the dimension of the arrays cxy, cyx. (An error is raised if these dimensions are not equal.)
The number of cross-covariances to be calculated in the function or supplied in the call to the function.
Constraint: ${\mathbf{mw}}+\left|{\mathbf{ish}}\right|\le {\mathbf{nc}}\le {\mathbf{nxy}}$.
2:     $\mathrm{xg}\left(\mathit{nxyg}\right)$ – double array
If the cross-covariances are to be calculated, then xg must contain the nxy data points of the $x$ series. If covariances are supplied, xg need not be set.
3:     $\mathrm{yg}\left(\mathit{nxyg}\right)$ – double array
If cross-covariances are to be calculated, yg must contain the nxy data points of the $y$ series. If covariances are supplied, yg need not be set.

### Output Parameters

1:     $\mathrm{cxy}\left({\mathbf{nc}}\right)$ – double array
If ${\mathbf{ic}}=0$, cxy will contain the nc calculated cross-covariances.
If ${\mathbf{ic}}\ne 0$, the contents of cxy will be unchanged.
2:     $\mathrm{cyx}\left({\mathbf{nc}}\right)$ – double array
If ${\mathbf{ic}}=0$, cyx will contain the nc calculated cross-covariances.
If ${\mathbf{ic}}\ne 0$, the contents of cyx will be unchanged.
3:     $\mathrm{xg}\left(\mathit{nxyg}\right)$ – double array
Contains the real parts of the ng complex spectral estimates in elements ${\mathbf{xg}}\left(1\right)$ to ${\mathbf{xg}}\left({\mathbf{ng}}\right)$, and ${\mathbf{xg}}\left({\mathbf{ng}}+1\right)$ to ${\mathbf{xg}}\left(\mathit{nxyg}\right)$ contain $0.0$. The $y$ series leads the $x$ series.
4:     $\mathrm{yg}\left(\mathit{nxyg}\right)$ – double array
Contains the imaginary parts of the ng complex spectral estimates in elements ${\mathbf{yg}}\left(1\right)$ to ${\mathbf{yg}}\left({\mathbf{ng}}\right)$, and ${\mathbf{yg}}\left({\mathbf{ng}}+1\right)$ to ${\mathbf{yg}}\left(\mathit{nxyg}\right)$ contain $0.0$. The $y$ series leads the $x$ series.
5:     $\mathrm{ng}$int64int32nag_int scalar
The number, $\left[{\mathbf{l}}/2\right]+1$, of complex spectral estimates, whose separate parts are held in xg and yg.
6:     $\mathrm{ifail}$int64int32nag_int scalar
${\mathbf{ifail}}={\mathbf{0}}$ unless the function detects an error (see Error Indicators and Warnings).

## Error Indicators and Warnings

Errors or warnings detected by the function:
${\mathbf{ifail}}=1$
 On entry, ${\mathbf{nxy}}<1$, or ${\mathbf{mtxy}}<0$ and ${\mathbf{ic}}=0$, or ${\mathbf{mtxy}}>2$ and ${\mathbf{ic}}=0$, or ${\mathbf{pxy}}<0.0$ and ${\mathbf{ic}}=0$, or ${\mathbf{pxy}}>1.0$ and ${\mathbf{ic}}=0$, or ${\mathbf{iw}}\le 0$, or ${\mathbf{iw}}>4$, or ${\mathbf{mw}}<1$, or ${\mathbf{mw}}+\left|{\mathbf{ish}}\right|>{\mathbf{nxy}}$, or $\left|{\mathbf{ish}}\right|\ge {\mathbf{mw}}$, or ${\mathbf{nc}}<{\mathbf{mw}}+\left|{\mathbf{ish}}\right|$, or ${\mathbf{nc}}>{\mathbf{nxy}}$, or $\mathit{nxyg}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{kc}},{\mathbf{l}}\right)$ and ${\mathbf{ic}}=0$, or $\mathit{nxyg}<{\mathbf{l}}$ and ${\mathbf{ic}}\ne 0$.
${\mathbf{ifail}}=2$
 On entry, ${\mathbf{kc}}<{\mathbf{nxy}}+{\mathbf{nc}}$, or kc has a prime factor exceeding $19$, or kc has more than $20$ prime factors, counting repetitions.
This error only occurs when ${\mathbf{ic}}=0$.
${\mathbf{ifail}}=3$
 On entry, ${\mathbf{l}}<2×{\mathbf{mw}}-1$, or l has a prime factor exceeding $19$, or l has more than $20$ prime factors, counting repetitions.
${\mathbf{ifail}}=-99$
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.

## Accuracy

The FFT is a numerically stable process, and any errors introduced during the computation will normally be insignificant compared with uncertainty in the data.

nag_tsa_multi_spectrum_lag (g13cc) carries out two FFTs of length kc to calculate the sample cross-covariances and one FFT of length $L$ to calculate the sample spectrum. The timing of nag_tsa_multi_spectrum_lag (g13cc) is therefore dependent on the choice of these values. The time taken for an FFT of length $n$ is approximately proportional to $n\mathrm{log}\left(n\right)$ (but see Further Comments in nag_sum_fft_realherm_1d (c06pa) for further details).

## Example

This example reads two time series of length $296$. It then selects mean correction, a 10% tapering proportion, the Parzen smoothing window and a cut-off point of $35$ for the lag window. The alignment shift is set to $3$ and $50$ cross-covariances are chosen to be calculated. The program then calls nag_tsa_multi_spectrum_lag (g13cc) to calculate the cross spectrum and then prints the cross-covariances and cross spectrum.
```function g13cc_example

fprintf('g13cc example results\n\n');

% Problem size
nxy  = int64(296);
ic   = int64(0);
nc   = 50;

% Control parameters
mtxy = int64(1);
pxy  = 0.1;
iw   = int64(4);
mw   = int64(35);
ish  = int64(3);
kc   = int64(350);
l    = int64(80);

cxy  = zeros(nc, 1);
cyx  = zeros(nc, 1);

% Series
xg = zeros(kc, 1);
xg(1:nxy) = ...
[-0.109; 0.000; 0.178; 0.339; 0.373; 0.441; 0.461; 0.348; 0.127;-0.180;
-0.588;-1.055;-1.421;-1.520;-1.302;-0.814;-0.475;-0.193; 0.088; 0.435;
0.771; 0.866; 0.875; 0.891; 0.987; 1.263; 1.775; 1.976; 1.934; 1.866;
1.832; 1.767; 1.608; 1.265; 0.790; 0.360; 0.115; 0.088; 0.331; 0.645;
0.960; 1.409; 2.670; 2.834; 2.812; 2.483; 1.929; 1.485; 1.214; 1.239;
1.608; 1.905; 2.023; 1.815; 0.535; 0.122; 0.009; 0.164; 0.671; 1.019;
1.146; 1.155; 1.112; 1.121; 1.223; 1.257; 1.157; 0.913; 0.620; 0.255;
-0.280;-1.080;-1.551;-1.799;-1.825;-1.456;-0.944;-0.570;-0.431;-0.577;
-0.960;-1.616;-1.875;-1.891;-1.746;-1.474;-1.201;-0.927;-0.524; 0.040;
0.788; 0.943; 0.930; 1.006; 1.137; 1.198; 1.054; 0.595;-0.080;-0.314;
-0.288;-0.153;-0.109;-0.187;-0.255;-0.299;-0.007; 0.254; 0.330; 0.102;
-0.423;-1.139;-2.275;-2.594;-2.716;-2.510;-1.790;-1.346;-1.081;-0.910;
-0.876;-0.885;-0.800;-0.544;-0.416;-0.271; 0.000; 0.403; 0.841; 1.285;
1.607; 1.746; 1.683; 1.485; 0.993; 0.648; 0.577; 0.577; 0.632; 0.747;
0.999; 0.993; 0.968; 0.790; 0.399;-0.161;-0.553;-0.603;-0.424;-0.194;
-0.049; 0.060; 0.161; 0.301; 0.517; 0.566; 0.560; 0.573; 0.592; 0.671;
0.933; 1.337; 1.460; 1.353; 0.772; 0.218;-0.237;-0.714;-1.099;-1.269;
-1.175;-0.676; 0.033; 0.556; 0.643; 0.484; 0.109;-0.310;-0.697;-1.047;
-1.218;-1.183;-0.873;-0.336; 0.063; 0.084; 0.000; 0.001; 0.209; 0.556;
0.782; 0.858; 0.918; 0.862; 0.416;-0.336;-0.959;-1.813;-2.378;-2.499;
-2.473;-2.330;-2.053;-1.739;-1.261;-0.569;-0.137;-0.024;-0.050;-0.135;
-0.276;-0.534;-0.871;-1.243;-1.439;-1.422;-1.175;-0.813;-0.634;-0.582;
-0.625;-0.713;-0.848;-1.039;-1.346;-1.628;-1.619;-1.149;-0.488;-0.160;
-0.007;-0.092;-0.620;-1.086;-1.525;-1.858;-2.029;-2.024;-1.961;-1.952;
-1.794;-1.302;-1.030;-0.918;-0.798;-0.867;-1.047;-1.123;-0.876;-0.395;
0.185; 0.662; 0.709; 0.605; 0.501; 0.603; 0.943; 1.223; 1.249; 0.824;
0.102; 0.025; 0.382; 0.922; 1.032; 0.866; 0.527; 0.093;-0.458;-0.748;
-0.947;-1.029;-0.928;-0.645;-0.424;-0.276;-0.158;-0.033; 0.102; 0.251;
0.280; 0.000;-0.493;-0.759;-0.824;-0.740;-0.528;-0.204; 0.034; 0.204;
0.253; 0.195; 0.131; 0.017;-0.182;-0.262];
yg = zeros(kc, 1);
yg(1:nxy) = ...
[53.8; 53.6; 53.5; 53.5; 53.4; 53.1; 52.7; 52.4; 52.2; 52.0; 52.0; 52.4;
53.0; 54.0; 54.9; 56.0; 56.8; 56.8; 56.4; 55.7; 55.0; 54.3; 53.2; 52.3;
51.6; 51.2; 50.8; 50.5; 50.0; 49.2; 48.4; 47.9; 47.6; 47.5; 47.5; 47.6;
48.1; 49.0; 50.0; 51.1; 51.8; 51.9; 51.7; 51.2; 50.0; 48.3; 47.0; 45.8;
45.6; 46.0; 46.9; 47.8; 48.2; 48.3; 47.9; 47.2; 47.2; 48.1; 49.4; 50.6;
51.5; 51.6; 51.2; 50.5; 50.1; 49.8; 49.6; 49.4; 49.3; 49.2; 49.3; 49.7;
50.3; 51.3; 52.8; 54.4; 56.0; 56.9; 57.5; 57.3; 56.6; 56.0; 55.4; 55.4;
56.4; 57.2; 58.0; 58.4; 58.4; 58.1; 57.7; 57.0; 56.0; 54.7; 53.2; 52.1;
51.6; 51.0; 50.5; 50.4; 51.0; 51.8; 52.4; 53.0; 53.4; 53.6; 53.7; 53.8;
53.8; 53.8; 53.3; 53.0; 52.9; 53.4; 54.6; 56.4; 58.0; 59.4; 60.2; 60.0;
59.4; 58.4; 57.6; 56.9; 56.4; 56.0; 55.7; 55.3; 55.0; 54.4; 53.7; 52.8;
51.6; 50.6; 49.4; 48.8; 48.5; 48.7; 49.2; 49.8; 50.4; 50.7; 50.9; 50.7;
50.5; 50.4; 50.2; 50.4; 51.2; 52.3; 53.2; 53.9; 54.1; 54.0; 53.6; 53.2;
53.0; 52.8; 52.3; 51.9; 51.6; 51.6; 51.4; 51.2; 50.7; 50.0; 49.4; 49.3;
49.7; 50.6; 51.8; 53.0; 54.0; 55.3; 55.9; 55.9; 54.6; 53.5; 52.4; 52.1;
52.3; 53.0; 53.8; 54.6; 55.4; 55.9; 55.9; 55.2; 54.4; 53.7; 53.6; 53.6;
53.2; 52.5; 52.0; 51.4; 51.0; 50.9; 52.4; 53.5; 55.6; 58.0; 59.5; 60.0;
60.4; 60.5; 60.2; 59.7; 59.0; 57.6; 56.4; 55.2; 54.5; 54.1; 54.1; 54.4;
55.5; 56.2; 57.0; 57.3; 57.4; 57.0; 56.4; 55.9; 55.5; 55.3; 55.2; 55.4;
56.0; 56.5; 57.1; 57.3; 56.8; 55.6; 55.0; 54.1; 54.3; 55.3; 56.4; 57.2;
57.8; 58.3; 58.6; 58.8; 58.8; 58.6; 58.0; 57.4; 57.0; 56.4; 56.3; 56.4;
56.4; 56.0; 55.2; 54.0; 53.0; 52.0; 51.6; 51.6; 51.1; 50.4; 50.0; 50.0;
52.0; 54.0; 55.1; 54.5; 52.8; 51.4; 50.8; 51.2; 52.0; 52.8; 53.8; 54.5;
54.9; 54.9; 54.8; 54.4; 53.7; 53.3; 52.8; 52.6; 52.6; 53.0; 54.3; 56.0;
57.0; 58.0; 58.6; 58.5; 58.3; 57.8; 57.3; 57];

[cxy, cyx, xg, yg, ng, ifail] = ...
g13cc( ...
nxy, mtxy, pxy, iw, mw, ish, ic, cxy, cyx, kc, l, 'xg', xg, 'yg', yg);

% Display results
fprintf('                  Returned cross covariances\n\n');
fprintf(' Lag     XY       YX   Lag     XY       YX   Lag     XY       YX\n');
result = [double([0:nc-1]); cxy'; cyx'];
for j = 1:3:nc
fprintf('%4d%9.4f%9.4f', result(:,j:min(j+2,nc)));
fprintf('\n');
end
fprintf('\n                      Returned sample spectrum\n\n');
fprintf('%23s%22s%22s\n', 'Real  Imaginary', 'Real  Imaginary', ...
'Real  Imaginary');
fprintf('%21s%22s%22s\n', 'Lag    part     part', '  Lag    part     part', ...
'  Lag    part     part');
result = [double([0:ng-1]); xg(1:ng)'; yg(1:ng)'];
for j = 1:3:ng
fprintf('%4d%9.4f%9.4f', result(:,j:min(j+2,ng)));
fprintf('\n');
end

```
```g13cc example results

Returned cross covariances

Lag     XY       YX   Lag     XY       YX   Lag     XY       YX
0  -1.6700  -1.6700   1  -2.0581  -1.3606   2  -2.4859  -1.1383
3  -2.8793  -0.9926   4  -3.1473  -0.9009   5  -3.2239  -0.8382
6  -3.0929  -0.7804   7  -2.7974  -0.7074   8  -2.4145  -0.6147
9  -2.0237  -0.5080  10  -1.6802  -0.4032  11  -1.4065  -0.3159
12  -1.2049  -0.2554  13  -1.0655  -0.2250  14  -0.9726  -0.2238
15  -0.9117  -0.2454  16  -0.8658  -0.2784  17  -0.8180  -0.3081
18  -0.7563  -0.3257  19  -0.6750  -0.3315  20  -0.5754  -0.3321
21  -0.4701  -0.3308  22  -0.3738  -0.3312  23  -0.3023  -0.3332
24  -0.2665  -0.3384  25  -0.2645  -0.3506  26  -0.2847  -0.3727
27  -0.3103  -0.3992  28  -0.3263  -0.4152  29  -0.3271  -0.4044
30  -0.3119  -0.3621  31  -0.2837  -0.2919  32  -0.2568  -0.2054
33  -0.2427  -0.1185  34  -0.2490  -0.0414  35  -0.2774   0.0227
36  -0.3218   0.0697  37  -0.3705   0.1039  38  -0.4083   0.1356
39  -0.4197   0.1805  40  -0.3920   0.2460  41  -0.3241   0.3319
42  -0.2273   0.4325  43  -0.1216   0.5331  44  -0.0245   0.6199
45   0.0528   0.6875  46   0.1074   0.7329  47   0.1448   0.7550
48   0.1713   0.7544  49   0.1943   0.7349

Returned sample spectrum

Real  Imaginary       Real  Imaginary       Real  Imaginary
Lag    part     part  Lag    part     part  Lag    part     part
0  -6.5500   0.0000   1  -5.4267  -1.9842   2  -3.1323  -2.7307
3  -1.2649  -2.3998   4  -0.2102  -1.7520   5   0.3411  -1.1903
6   0.6063  -0.7420   7   0.6178  -0.3586   8   0.4391  -0.1008
9   0.2422   0.0061  10   0.1233   0.0409  11   0.0574   0.0529
12   0.0174   0.0452  13  -0.0008   0.0289  14  -0.0058   0.0161
15  -0.0051   0.0084  16  -0.0027   0.0040  17  -0.0010   0.0015
18  -0.0006   0.0006  19  -0.0005   0.0003  20  -0.0003   0.0003
21  -0.0003   0.0004  22  -0.0003   0.0003  23  -0.0003   0.0002
24  -0.0004   0.0001  25  -0.0004  -0.0000  26  -0.0003  -0.0001
27  -0.0002  -0.0001  28  -0.0001   0.0001  29  -0.0002   0.0003
30  -0.0003   0.0002  31  -0.0002   0.0001  32  -0.0001   0.0000
33  -0.0000  -0.0000  34   0.0001  -0.0001  35   0.0001  -0.0002
36   0.0001  -0.0001  37   0.0001  -0.0001  38   0.0001  -0.0001
39   0.0001  -0.0001  40   0.0001   0.0000
```

Chapter Contents
Chapter Introduction
NAG Toolbox

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