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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 ω$\omega$, fxy(ω) = cf(ω) + iqf(ω)${f}_{xy}\left(\omega \right)=cf\left(\omega \right)+iqf\left(\omega \right)$, defined by its real part or co-spectrum
 M − 1 cf(ω) = 1/(2π) ∑ wkCxy(k + S)cos(ωk) k = − M + 1
$cf(ω)=12π ∑k=-M+1 M-1wkCxy(k+S)cos(ωk)$
and imaginary part or quadrature spectrum
 M − 1 qf(ω) = 1/(2π ) ∑ wkCxy(k + S)sin(ωk) k = − M + 1
$qf(ω)=12π ∑k=-M+ 1 M- 1wkCxy(k+S)sin(ω k)$
where wk = wk${w}_{\mathit{k}}={w}_{-\mathit{k}}$, for k = 0,1,,M1$\mathit{k}=0,1,\dots ,M-1$, is the smoothing lag window as defined in the description of nag_tsa_uni_spectrum_lag (g13ca). The alignment shift S$S$ is recommended to be chosen as the lag k$k$ at which the cross-covariances cxy(k)${c}_{xy}\left(k\right)$ peak, so as to minimize bias.
The results are calculated for frequency values
 ωj = (2πj)/L,  j = 0,1, … ,[L / 2], $ωj=2πjL, j=0,1,…,[L/2],$
where []$\left[\right]$ denotes the integer part.
The cross-covariances cxy(k)${c}_{xy}\left(k\right)$ may be supplied by you, or constructed from supplied series x1,x2,,xn${x}_{1},{x}_{2},\dots ,{x}_{n}$; y1,y2,,yn${y}_{1},{y}_{2},\dots ,{y}_{n}$ as
 cxy(k) = ( ∑ t = 1n − kxtyt + k)/n,  k ≥ 0 $cxy(k)=∑t=1 n-kxtyt+kn, k≥0$
 cxy(k) = ( ∑ t = 1 − knxtyt + k)/n = cyx( − k),   k < 0 $cxy(k)=∑t= 1-knxtyt+kn=cyx(-k), k< 0$
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 yt${y}_{t}$ and xt${x}_{t}$.

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:     nxy – int64int32nag_int scalar
n$n$, the length of the time series x$x$ and y$y$.
Constraint: nxy1${\mathbf{nxy}}\ge 1$.
2:     mtxy – int64int32nag_int scalar
If cross-covariances are to be calculated by the function (ic = 0${\mathbf{ic}}=0$), mtxy must specify whether the data is to be initially mean or trend corrected.
mtxy = 0${\mathbf{mtxy}}=0$
For no correction.
mtxy = 1${\mathbf{mtxy}}=1$
For mean correction.
mtxy = 2${\mathbf{mtxy}}=2$
For trend correction.
If cross-covariances are supplied (ic0)$\left({\mathbf{ic}}\ne 0\right)$, mtxy is not used.
Constraint: if ic = 0${\mathbf{ic}}=0$, mtxy = 0${\mathbf{mtxy}}=0$, 1$1$ or 2$2$.
3:     pxy – double scalar
If cross-covariances are to be calculated by the function (ic = 0${\mathbf{ic}}=0$), 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$0.0$ implies no tapering.
If cross-covariances are supplied (ic0)$\left({\mathbf{ic}}\ne 0\right)$, pxy is not used.
Constraint: if ic = 0${\mathbf{ic}}=0$, 0.0pxy1.0$0.0\le {\mathbf{pxy}}\le 1.0$.
4:     iw – int64int32nag_int scalar
The choice of lag window.
iw = 1${\mathbf{iw}}=1$
Rectangular.
iw = 2${\mathbf{iw}}=2$
Bartlett.
iw = 3${\mathbf{iw}}=3$
Tukey.
iw = 4${\mathbf{iw}}=4$
Parzen.
Constraint: 1iw4$1\le {\mathbf{iw}}\le 4$.
5:     mw – int64int32nag_int scalar
M$M$, the ‘cut-off’ point of the lag window, relative to any alignment shift that has been applied. Windowed cross-covariances at lags (mw + ish)$\left(-{\mathbf{mw}}+{\mathbf{ish}}\right)$ or less, and at lags (mw + ish)$\left({\mathbf{mw}}+{\mathbf{ish}}\right)$ or greater are zero.
Constraints:
• mw1${\mathbf{mw}}\ge 1$;
• mw + |ish|nxy${\mathbf{mw}}+|{\mathbf{ish}}|\le {\mathbf{nxy}}$.
6:     ish – int64int32nag_int scalar
S$S$, the alignment shift between the x$x$ and y$y$ series. If x$x$ leads y$y$, the shift is positive.
Constraint: mw < ish < mw$-{\mathbf{mw}}<{\mathbf{ish}}<{\mathbf{mw}}$.
7:     ic – int64int32nag_int scalar
Indicates whether cross-covariances are to be calculated in the function or supplied in the call to the function.
ic = 0${\mathbf{ic}}=0$
Cross-covariances are to be calculated.
ic0${\mathbf{ic}}\ne 0$
Cross-covariances are to be supplied.
8:     cxy(nc) – double array
nc, the dimension of the array, must satisfy the constraint mw + |ish|ncnxy${\mathbf{mw}}+|{\mathbf{ish}}|\le {\mathbf{nc}}\le {\mathbf{nxy}}$.
If ic0${\mathbf{ic}}\ne 0$, cxy must contain the nc cross-covariances between values in the y$y$ series and earlier values in time in the x$x$ series, for lags from 0$0$ to (nc1)$\left({\mathbf{nc}}-1\right)$.
If ic = 0${\mathbf{ic}}=0$, cxy need not be set.
9:     cyx(nc) – double array
nc, the dimension of the array, must satisfy the constraint mw + |ish|ncnxy${\mathbf{mw}}+|{\mathbf{ish}}|\le {\mathbf{nc}}\le {\mathbf{nxy}}$.
If ic0${\mathbf{ic}}\ne 0$, cyx must contain the nc cross-covariances between values in the y$y$ series and later values in time in the x$x$ series, for lags from 0$0$ to (nc1)$\left({\mathbf{nc}}-1\right)$.
If ic = 0${\mathbf{ic}}=0$, cyx need not be set.
10:   kc – int64int32nag_int scalar
If ic = 0${\mathbf{ic}}=0$, 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 2m${2}^{m}$ where m$m$ is the smallest integer such that 2mn + nc${2}^{m}\ge n+{\mathbf{nc}}$.
If ic0${\mathbf{ic}}\ne 0$, that is if covariances are supplied, kc is not used.
Constraint: kcnxy + nc${\mathbf{kc}}\ge {\mathbf{nxy}}+{\mathbf{nc}}$. The largest prime factor of kc must not exceed 19$19$, and the total number of prime factors of kc, counting repetitions, must not exceed 20$20$. These two restrictions are imposed by the internal FFT algorithm used.
11:   l – int64int32nag_int scalar
L$L$, the frequency division of the spectral estimates as (2π)/L $\frac{2\pi }{L}$. 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 2m${2}^{m}$ where m$m$ is the smallest integer such that 2m2M1${2}^{m}\ge 2M-1$.
Constraint: l2 × mw1${\mathbf{l}}\ge 2×{\mathbf{mw}}-1$. The largest prime factor of l must not exceed 19$19$, and the total number of prime factors of l, counting repetitions, must not exceed 20$20$. These two restrictions are imposed by the internal FFT algorithm used.

Optional Input Parameters

1:     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: mw + |ish|ncnxy${\mathbf{mw}}+|{\mathbf{ish}}|\le {\mathbf{nc}}\le {\mathbf{nxy}}$.
2:     xg(nxyg) – double array
If the cross-covariances are to be calculated, then xg must contain the nxy data points of the x$x$ series. If covariances are supplied, xg need not be set.
3:     yg(nxyg) – double array
If cross-covariances are to be calculated, yg must contain the nxy data points of the y$y$ series. If covariances are supplied, yg need not be set.

nxyg

Output Parameters

1:     cxy(nc) – double array
If ic = 0${\mathbf{ic}}=0$, cxy will contain the nc calculated cross-covariances.
If ic0${\mathbf{ic}}\ne 0$, the contents of cxy will be unchanged.
2:     cyx(nc) – double array
If ic = 0${\mathbf{ic}}=0$, cyx will contain the nc calculated cross-covariances.
If ic0${\mathbf{ic}}\ne 0$, the contents of cyx will be unchanged.
3:     xg(nxyg) – double array
Contains the real parts of the ng complex spectral estimates in elements xg(1)${\mathbf{xg}}\left(1\right)$ to xg(ng)${\mathbf{xg}}\left({\mathbf{ng}}\right)$, and xg(ng + 1)${\mathbf{xg}}\left({\mathbf{ng}}+1\right)$ to xg(nxyg)${\mathbf{xg}}\left(\mathit{nxyg}\right)$ contain 0.0$0.0$. The y$y$ series leads the x$x$ series.
4:     yg(nxyg) – double array
Contains the imaginary parts of the ng complex spectral estimates in elements yg(1)${\mathbf{yg}}\left(1\right)$ to yg(ng)${\mathbf{yg}}\left({\mathbf{ng}}\right)$, and yg(ng + 1)${\mathbf{yg}}\left({\mathbf{ng}}+1\right)$ to yg(nxyg)${\mathbf{yg}}\left(\mathit{nxyg}\right)$ contain 0.0$0.0$. The y$y$ series leads the x$x$ series.
5:     ng – int64int32nag_int scalar
The number, [l / 2] + 1$\left[{\mathbf{l}}/2\right]+1$, of complex spectral estimates, whose separate parts are held in xg and yg.
6:     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${\mathbf{ifail}}=1$
 On entry, nxy < 1${\mathbf{nxy}}<1$, or mtxy < 0${\mathbf{mtxy}}<0$ and ic = 0${\mathbf{ic}}=0$, or mtxy > 2${\mathbf{mtxy}}>2$ and ic = 0${\mathbf{ic}}=0$, or pxy < 0.0${\mathbf{pxy}}<0.0$ and ic = 0${\mathbf{ic}}=0$, or pxy > 1.0${\mathbf{pxy}}>1.0$ and ic = 0${\mathbf{ic}}=0$, or iw ≤ 0${\mathbf{iw}}\le 0$, or iw > 4${\mathbf{iw}}>4$, or mw < 1${\mathbf{mw}}<1$, or mw + |ish| > nxy${\mathbf{mw}}+|{\mathbf{ish}}|>{\mathbf{nxy}}$, or |ish| ≥ mw$|{\mathbf{ish}}|\ge {\mathbf{mw}}$, or nc < mw + |ish|${\mathbf{nc}}<{\mathbf{mw}}+|{\mathbf{ish}}|$, or ${\mathbf{nc}}>{\mathbf{nxy}}$, or nxyg < max (kc,l)$\mathit{nxyg}<\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{kc}},{\mathbf{l}}\right)$ and ic = 0${\mathbf{ic}}=0$, or nxyg < l$\mathit{nxyg}<{\mathbf{l}}$ and ic ≠ 0${\mathbf{ic}}\ne 0$.
ifail = 2${\mathbf{ifail}}=2$
 On entry, kc < nxy + nc${\mathbf{kc}}<{\mathbf{nxy}}+{\mathbf{nc}}$, or kc has a prime factor exceeding 19$19$, or kc has more than 20$20$ prime factors, counting repetitions.
This error only occurs when ic = 0${\mathbf{ic}}=0$.
ifail = 3${\mathbf{ifail}}=3$
 On entry, l < 2 × mw − 1${\mathbf{l}}<2×{\mathbf{mw}}-1$, or l has a prime factor exceeding 19$19$, or l has more than 20$20$ prime factors, counting repetitions.

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$L$ 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$n$ is approximately proportional to nlog(n)$n\mathrm{log}\left(n\right)$ (but see Section [Further Comments] in (c06pa) for further details).

Example

```function nag_tsa_multi_spectrum_lag_example
nxy = int64(296);
mtxy = int64(1);
pxy = 0.1;
iw = int64(4);
mw = int64(35);
ish = int64(3);
ic = int64(0);
cxy = zeros(50, 1);
cyx = zeros(50, 1);
kc = int64(350);
l = int64(80);
xg = zeros(350, 1);
xg(1:296) = ...
[-0.109;0;0.178;0.339;0.373;0.441;0.461;0.348;0.127;-0.18;-0.588;-1.055;
-1.421;-1.52;-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;0.79;
0.36;0.115;0.088;0.331;0.645;0.96;1.409;2.67;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;0.62;0.255;-0.28;
-1.08;-1.551;-1.799;-1.825;-1.456;-0.944;-0.57;-0.431;-0.577;-0.96;-1.616;
-1.875;-1.891;-1.746;-1.474;-1.201;-0.927;-0.524;0.04;0.788;0.943;0.93;
1.006;1.137;1.198;1.054;0.595;-0.08;-0.314;-0.288;-0.153;-0.109;-0.187;
-0.255;-0.299;-0.007;0.254;0.33;0.102;-0.423;-1.139;-2.275;-2.594;-2.716;
-2.51;-1.79;-1.346;-1.081;-0.91;-0.876;-0.885;-0.8;-0.544;-0.416;-0.271;
0;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;0.79;0.399;-0.161;-0.553;-0.603;-0.424;-0.194;
-0.049;0.06;0.161;0.301;0.517;0.566;0.56;0.573;0.592;0.671;0.933;1.337;
1.46;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;-0.31;-0.697;-1.047;-1.218;-1.183;-0.873;-0.336;
0.063;0.084;0;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;-2.33;-2.053;-1.739;-1.261;-0.569;
-0.137;-0.024;-0.05;-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;-0.16;-0.007;-0.092;-0.62;-1.086;-1.525;-1.858;
-2.029;-2.024;-1.961;-1.952;-1.794;-1.302;-1.03;-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;0.28;0;-0.493;-0.759;-0.824;-0.74;-0.528;-0.204;0.034;0.204;0.253;
0.195;0.131;0.017;-0.182;-0.262];
yg = zeros(350, 1);
yg(1:296) = ...
[53.8;53.6;53.5;53.5;53.4;53.1;52.7;52.4;52.2;52;52;52.4;53;54;54.9;56;
56.8;56.8;56.4;55.7;55;54.3;53.2;52.3;51.6;51.2;50.8;50.5;50;49.2;48.4;
47.9;47.6;47.5;47.5;47.6;48.1;49;50;51.1;51.8;51.9;51.7;51.2;50;48.3;47;
45.8;45.6;46;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;
56.9;57.5;57.3;56.6;56;55.4;55.4;56.4;57.2;58;58.4;58.4;58.1;57.7;57;56;
54.7;53.2;52.1;51.6;51;50.5;50.4;51;51.8;52.4;53;53.4;53.6;53.7;53.8;53.8;
53.8;53.3;53;52.9;53.4;54.6;56.4;58;59.4;60.2;60;59.4;58.4;57.6;56.9;56.4;
56;55.7;55.3;55;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;53.6;
53.2;53;52.8;52.3;51.9;51.6;51.6;51.4;51.2;50.7;50;49.4;49.3;49.7;50.6;
51.8;53;54;55.3;55.9;55.9;54.6;53.5;52.4;52.1;52.3;53;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;51.4;51;50.9;52.4;53.5;
55.6;58;59.5;60;60.4;60.5;60.2;59.7;59;57.6;56.4;55.2;54.5;54.1;54.1;54.4;
55.5;56.2;57;57.3;57.4;57;56.4;55.9;55.5;55.3;55.2;55.4;56;56.5;57.1;57.3;
56.8;55.6;55;54.1;54.3;55.3;56.4;57.2;57.8;58.3;58.6;58.8;58.8;58.6;58;
57.4;57;56.4;56.3;56.4;56.4;56;55.2;54;53;52;51.6;51.6;51.1;50.4;50;50;52;
54;55.1;54.5;52.8;51.4;50.8;51.2;52;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;54.3;56;57;58;58.6;58.5;58.3;57.8;57.3;57];
[cxyOut, cyxOut, xg, yg, ng, ifail] = ...
nag_tsa_multi_spectrum_lag(nxy, mtxy, pxy, iw, mw, ish, ic, cxy, cyx, kc, l, 'xg', xg, 'yg', yg);
cxyOut, cyxOut, ng, ifail
```
```

cxyOut =

-1.6700
-2.0581
-2.4859
-2.8793
-3.1473
-3.2239
-3.0929
-2.7974
-2.4145
-2.0237
-1.6802
-1.4065
-1.2049
-1.0655
-0.9726
-0.9117
-0.8658
-0.8180
-0.7563
-0.6750
-0.5754
-0.4701
-0.3738
-0.3023
-0.2665
-0.2645
-0.2847
-0.3103
-0.3263
-0.3271
-0.3119
-0.2837
-0.2568
-0.2427
-0.2490
-0.2774
-0.3218
-0.3705
-0.4083
-0.4197
-0.3920
-0.3241
-0.2273
-0.1216
-0.0245
0.0528
0.1074
0.1448
0.1713
0.1943

cyxOut =

-1.6700
-1.3606
-1.1383
-0.9926
-0.9009
-0.8382
-0.7804
-0.7074
-0.6147
-0.5080
-0.4032
-0.3159
-0.2554
-0.2250
-0.2238
-0.2454
-0.2784
-0.3081
-0.3257
-0.3315
-0.3321
-0.3308
-0.3312
-0.3332
-0.3384
-0.3506
-0.3727
-0.3992
-0.4152
-0.4044
-0.3621
-0.2919
-0.2054
-0.1185
-0.0414
0.0227
0.0697
0.1039
0.1356
0.1805
0.2460
0.3319
0.4325
0.5331
0.6199
0.6875
0.7329
0.7550
0.7544
0.7349

ng =

41

ifail =

0

```
```function g13cc_example
nxy = int64(296);
mtxy = int64(1);
pxy = 0.1;
iw = int64(4);
mw = int64(35);
ish = int64(3);
ic = int64(0);
cxy = zeros(50, 1);
cyx = zeros(50, 1);
kc = int64(350);
l = int64(80);
xg = zeros(350, 1);
xg(1:296) = ...
[-0.109;0;0.178;0.339;0.373;0.441;0.461;0.348;0.127;-0.18;-0.588;-1.055;
-1.421;-1.52;-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;0.79;
0.36;0.115;0.088;0.331;0.645;0.96;1.409;2.67;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;0.62;0.255;-0.28;
-1.08;-1.551;-1.799;-1.825;-1.456;-0.944;-0.57;-0.431;-0.577;-0.96;-1.616;
-1.875;-1.891;-1.746;-1.474;-1.201;-0.927;-0.524;0.04;0.788;0.943;0.93;
1.006;1.137;1.198;1.054;0.595;-0.08;-0.314;-0.288;-0.153;-0.109;-0.187;
-0.255;-0.299;-0.007;0.254;0.33;0.102;-0.423;-1.139;-2.275;-2.594;-2.716;
-2.51;-1.79;-1.346;-1.081;-0.91;-0.876;-0.885;-0.8;-0.544;-0.416;-0.271;
0;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;0.79;0.399;-0.161;-0.553;-0.603;-0.424;-0.194;
-0.049;0.06;0.161;0.301;0.517;0.566;0.56;0.573;0.592;0.671;0.933;1.337;
1.46;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;-0.31;-0.697;-1.047;-1.218;-1.183;-0.873;-0.336;
0.063;0.084;0;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;-2.33;-2.053;-1.739;-1.261;-0.569;
-0.137;-0.024;-0.05;-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;-0.16;-0.007;-0.092;-0.62;-1.086;-1.525;-1.858;
-2.029;-2.024;-1.961;-1.952;-1.794;-1.302;-1.03;-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;0.28;0;-0.493;-0.759;-0.824;-0.74;-0.528;-0.204;0.034;0.204;0.253;
0.195;0.131;0.017;-0.182;-0.262];
yg = zeros(350, 1);
yg(1:296) = ...
[53.8;53.6;53.5;53.5;53.4;53.1;52.7;52.4;52.2;52;52;52.4;53;54;54.9;56;
56.8;56.8;56.4;55.7;55;54.3;53.2;52.3;51.6;51.2;50.8;50.5;50;49.2;48.4;
47.9;47.6;47.5;47.5;47.6;48.1;49;50;51.1;51.8;51.9;51.7;51.2;50;48.3;47;
45.8;45.6;46;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;
56.9;57.5;57.3;56.6;56;55.4;55.4;56.4;57.2;58;58.4;58.4;58.1;57.7;57;56;
54.7;53.2;52.1;51.6;51;50.5;50.4;51;51.8;52.4;53;53.4;53.6;53.7;53.8;53.8;
53.8;53.3;53;52.9;53.4;54.6;56.4;58;59.4;60.2;60;59.4;58.4;57.6;56.9;56.4;
56;55.7;55.3;55;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;53.6;
53.2;53;52.8;52.3;51.9;51.6;51.6;51.4;51.2;50.7;50;49.4;49.3;49.7;50.6;
51.8;53;54;55.3;55.9;55.9;54.6;53.5;52.4;52.1;52.3;53;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;51.4;51;50.9;52.4;53.5;
55.6;58;59.5;60;60.4;60.5;60.2;59.7;59;57.6;56.4;55.2;54.5;54.1;54.1;54.4;
55.5;56.2;57;57.3;57.4;57;56.4;55.9;55.5;55.3;55.2;55.4;56;56.5;57.1;57.3;
56.8;55.6;55;54.1;54.3;55.3;56.4;57.2;57.8;58.3;58.6;58.8;58.8;58.6;58;
57.4;57;56.4;56.3;56.4;56.4;56;55.2;54;53;52;51.6;51.6;51.1;50.4;50;50;52;
54;55.1;54.5;52.8;51.4;50.8;51.2;52;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;54.3;56;57;58;58.6;58.5;58.3;57.8;57.3;57];
[cxyOut, cyxOut, xg, yg, ng, ifail] = ...
g13cc(nxy, mtxy, pxy, iw, mw, ish, ic, cxy, cyx, kc, l, 'xg', xg, 'yg', yg);
cxyOut, cyxOut, ng, ifail
```
```

cxyOut =

-1.6700
-2.0581
-2.4859
-2.8793
-3.1473
-3.2239
-3.0929
-2.7974
-2.4145
-2.0237
-1.6802
-1.4065
-1.2049
-1.0655
-0.9726
-0.9117
-0.8658
-0.8180
-0.7563
-0.6750
-0.5754
-0.4701
-0.3738
-0.3023
-0.2665
-0.2645
-0.2847
-0.3103
-0.3263
-0.3271
-0.3119
-0.2837
-0.2568
-0.2427
-0.2490
-0.2774
-0.3218
-0.3705
-0.4083
-0.4197
-0.3920
-0.3241
-0.2273
-0.1216
-0.0245
0.0528
0.1074
0.1448
0.1713
0.1943

cyxOut =

-1.6700
-1.3606
-1.1383
-0.9926
-0.9009
-0.8382
-0.7804
-0.7074
-0.6147
-0.5080
-0.4032
-0.3159
-0.2554
-0.2250
-0.2238
-0.2454
-0.2784
-0.3081
-0.3257
-0.3315
-0.3321
-0.3308
-0.3312
-0.3332
-0.3384
-0.3506
-0.3727
-0.3992
-0.4152
-0.4044
-0.3621
-0.2919
-0.2054
-0.1185
-0.0414
0.0227
0.0697
0.1039
0.1356
0.1805
0.2460
0.3319
0.4325
0.5331
0.6199
0.6875
0.7329
0.7550
0.7544
0.7349

ng =

41

ifail =

0

```