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Chapter Contents
Chapter Introduction
NAG Toolbox

# NAG Toolbox: nag_correg_ridge (g02kb)

## Purpose

nag_correg_ridge (g02kb) calculates a ridge regression, with ridge parameters supplied by you.

## Syntax

[nep, b, vf, pe, ifail] = g02kb(x, isx, ip, y, h, wantb, wantvf, pec, 'n', n, 'm', m, 'lh', lh, 'lpec', lpec)
[nep, b, vf, pe, ifail] = nag_correg_ridge(x, isx, ip, y, h, wantb, wantvf, pec, 'n', n, 'm', m, 'lh', lh, 'lpec', lpec)

## Description

A linear model has the form:
 y = c + Xβ + ε , $y = c+Xβ+ε ,$
where
• y$y$ is an n$n$ by 1$1$ matrix of values of a dependent variable;
• c$c$ is a scalar intercept term;
• X$X$ is an n$n$ by m$m$ matrix of values of independent variables;
• β$\beta$ is a m$m$ by 1$1$ matrix of unknown values of parameters;
• ε$\epsilon$ is an n$n$ by 1$1$ matrix of unknown random errors such that variance of ε = σ2I${\epsilon =\sigma }^{2}I$.
Let $\stackrel{~}{X}$ be the mean-centred X$X$ and $\stackrel{~}{y}$ the mean-centred y$y$. Furthermore, $\stackrel{~}{X}$ is scaled such that the diagonal elements of the cross product matrix T${\stackrel{~}{X}}^{\mathrm{T}}\stackrel{~}{X}$ are one. The linear model now takes the form:
 ỹ = X̃ β̃ + ε . $y~ = X~ β~ + ε .$
Ridge regression estimates the parameters β̃$\stackrel{~}{\beta }$ in a penalised least squares sense by finding the $\stackrel{~}{b}$ that minimizes
 ‖X̃b̃ − ỹ‖2 + h ‖b̃‖2 ,   h > 0 , $‖ X~ b~ - y~ ‖ 2 + h ‖b~‖ 2 , h>0 ,$
where ·$‖·‖$ denotes the 2${\ell }_{2}$-norm and h$h$ is a scalar regularization or ridge parameter. For a given value of h$h$, the parameters estimates $\stackrel{~}{b}$ are found by evaluating
 b̃ = (X̃TX̃ + hI) − 1 X̃T ỹ . $b~ = ( X~T X~+hI )-1 X~T y~ .$
Note that if h = 0$h=0$ the ridge regression solution is equivalent to the ordinary least squares solution.
Rather than calculate the inverse of (T + hI${\stackrel{~}{X}}^{\mathrm{T}}\stackrel{~}{X}+hI$) directly, nag_correg_ridge (g02kb) uses the singular value decomposition (SVD) of $\stackrel{~}{X}$. After decomposing $\stackrel{~}{X}$ into UDVT$UD{V}^{\mathrm{T}}$ where U$U$ and V$V$ are orthogonal matrices and D$D$ is a diagonal matrix, the parameter estimates become
 b̃ = V (DTD + hI) − 1 DUT ỹ . $b~ = V ( DTD+hI )-1 DUT y~ .$
A consequence of introducing the ridge parameter is that the effective number of parameters, γ$\gamma$, in the model is given by the sum of diagonal elements of
 DT D (DTD + hI) − 1 , $DT D ( DT D+hI )-1 ,$
see Moody (1992) for details.
Any multi-collinearity in the design matrix X$X$ may be highlighted by calculating the variance inflation factors for the fitted model. The j$j$th variance inflation factor, vj${v}_{j}$, is a scaled version of the multiple correlation coefficient between independent variable j$j$ and the other independent variables, Rj${R}_{j}$, and is given by
 vj = 1/(1 − Rj) ,   j = 1,2, … ,m . $vj = 1 1-Rj , j=1,2,…,m .$
The m$m$ variance inflation factors are calculated as the diagonal elements of the matrix:
 (X̃TX̃ + hI) − 1 X̃T X̃ (X̃TX̃ + hI) − 1 , $( X~T X~+hI )-1 X~T X~ (X~T X~+hI)-1 ,$
which, using the SVD of $\stackrel{~}{X}$, is equivalent to the diagonal elements of the matrix:
 V (DTD + hI) − 1 DT D (DTD + hI) − 1 VT . $V ( DT D+hI )-1 DT D ( DT D+hI )-1 VT .$
Given a value of h$h$, any or all of the following prediction criteria are available:
(a) Generalized cross-validation (GCV):
 (ns)/((n − γ)2) ; $ns (n-γ) 2 ;$
(b) Unbiased estimate of variance (UEV):
 s/(n − γ) ; $s n-γ ;$
(c) Future prediction error (FPE):
 1/n (s + (2γs)/(n − γ)) ; $1n ( s+ 2γs n-γ ) ;$
(d) Bayesian information criterion (BIC):
 1/n (s + (log(n)γs)/(n − γ)) ; $1n ( s+ log(n)γs n-γ ) ;$
(e) Leave-one-out cross-validation (LOOCV),
where s$s$ is the sum of squares of residuals.
Although parameter estimates $\stackrel{~}{b}$ are calculated by using $\stackrel{~}{X}$, it is usual to report the parameter estimates b$b$ associated with X$X$. These are calculated from $\stackrel{~}{b}$, and the means and scalings of X$X$. Optionally, either $\stackrel{~}{b}$ or b$b$ may be calculated.

## References

Hastie T, Tibshirani R and Friedman J (2003) The Elements of Statistical Learning: Data Mining, Inference and Prediction Springer Series in Statistics
Moody J.E. (1992) The effective number of parameters: An analysis of generalisation and regularisation in nonlinear learning systems In: Neural Information Processing Systems (eds J E Moody, S J Hanson, and R P Lippmann) 4 847–854 Morgan Kaufmann San Mateo CA

## Parameters

### Compulsory Input Parameters

1:     x(ldx,m) – double array
ldx, the first dimension of the array, must satisfy the constraint ldxn$\mathit{ldx}\ge {\mathbf{n}}$.
The values of independent variables in the data matrix X$X$.
2:     isx(m) – int64int32nag_int array
m, the dimension of the array, must satisfy the constraint mn${\mathbf{m}}\le {\mathbf{n}}$.
Indicates which m$m$ independent variables are included in the model.
isx(j) = 1${\mathbf{isx}}\left(j\right)=1$
The j$j$th variable in x will be included in the model.
isx(j) = 0${\mathbf{isx}}\left(j\right)=0$
Variable j$j$ is excluded.
Constraint: isx(j) = 0 ​ or ​ 1${\mathbf{isx}}\left(\mathit{j}\right)=0\text{​ or ​}1$, for j = 1,2,,m$\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
3:     ip – int64int32nag_int scalar
m$m$, the number of independent variables in the model.
Constraints:
• 1ipm$1\le {\mathbf{ip}}\le {\mathbf{m}}$;
• Exactly ip elements of isx must be equal to 1$1$.
4:     y(n) – double array
n, the dimension of the array, must satisfy the constraint n1${\mathbf{n}}\ge 1$.
The n$n$ values of the dependent variable y$y$.
5:     h(lh) – double array
lh, the dimension of the array, must satisfy the constraint lh > 0${\mathbf{lh}}>0$.
h(j)${\mathbf{h}}\left(j\right)$ is the value of the j$j$th ridge parameter h$h$.
Constraint: h(j)0.0${\mathbf{h}}\left(\mathit{j}\right)\ge 0.0$, for j = 1,2,,lh$\mathit{j}=1,2,\dots ,{\mathbf{lh}}$.
6:     wantb – int64int32nag_int scalar
Defines the options for parameter estimates.
wantb = 0${\mathbf{wantb}}=0$
Parameter estimates are not calculated and b is not referenced.
wantb = 1${\mathbf{wantb}}=1$
Parameter estimates b$b$ are calculated for the original data.
wantb = 2${\mathbf{wantb}}=2$
Parameter estimates $\stackrel{~}{b}$ are calculated for the standardized data.
Constraint: wantb = 0${\mathbf{wantb}}=0$, 1$1$ or 2$2$.
7:     wantvf – int64int32nag_int scalar
Defines the options for variance inflation factors.
wantvf = 0${\mathbf{wantvf}}=0$
Variance inflation factors are not calculated and the array vf is not referenced.
wantvf = 1${\mathbf{wantvf}}=1$
Variance inflation factors are calculated.
Constraints:
• wantvf = 0${\mathbf{wantvf}}=0$ or 1$1$;
• if wantb = 0${\mathbf{wantb}}=0$, wantvf = 1${\mathbf{wantvf}}=1$.
8:     pec(lpec) – cell array of strings
If lpec > 0${\mathbf{lpec}}>0$, pec(j)${\mathbf{pec}}\left(\mathit{j}\right)$ defines the j$\mathit{j}$th prediction error, for j = 1,2,,lpec$\mathit{j}=1,2,\dots ,{\mathbf{lpec}}$; otherwise pec is not referenced.
pec(j) = 'B'${\mathbf{pec}}\left(j\right)=\text{'B'}$
Bayesian information criterion (BIC).
pec(j) = 'F'${\mathbf{pec}}\left(j\right)=\text{'F'}$
Future prediction error (FPE).
pec(j) = 'G'${\mathbf{pec}}\left(j\right)=\text{'G'}$
Generalized cross-validation (GCV).
pec(j) = 'L'${\mathbf{pec}}\left(j\right)=\text{'L'}$
Leave-one-out cross-validation (LOOCV).
pec(j) = 'U'${\mathbf{pec}}\left(j\right)=\text{'U'}$
Unbiased estimate of variance (UEV).
Constraint: if lpec > 0${\mathbf{lpec}}>0$, pec(j) = 'B'${\mathbf{pec}}\left(\mathit{j}\right)=\text{'B'}$, 'F'$\text{'F'}$, 'G'$\text{'G'}$, 'L'$\text{'L'}$ or 'U'$\text{'U'}$, for j = 1,2,,lpec$\mathit{j}=1,2,\dots ,{\mathbf{lpec}}$.

### Optional Input Parameters

1:     n – int64int32nag_int scalar
Default: The dimension of the array y and the first dimension of the array x. (An error is raised if these dimensions are not equal.)
n$n$, the number of observations.
Constraint: n1${\mathbf{n}}\ge 1$.
2:     m – int64int32nag_int scalar
Default: The dimension of the array isx and the second dimension of the array x. (An error is raised if these dimensions are not equal.)
The number of independent variables available in the data matrix X$X$.
Constraint: mn${\mathbf{m}}\le {\mathbf{n}}$.
3:     lh – int64int32nag_int scalar
Default: The dimension of the array h.
The number of supplied ridge parameters.
Constraint: lh > 0${\mathbf{lh}}>0$.
4:     lpec – int64int32nag_int scalar
Default: The dimension of the array pec.
The number of prediction error statistics to return; set lpec0${\mathbf{lpec}}\le 0$ for no prediction error estimates.

### Input Parameters Omitted from the MATLAB Interface

ldx ldb ldvf ldpe

### Output Parameters

1:     nep(lh) – double array
nep(j)${\mathbf{nep}}\left(\mathit{j}\right)$ is the number of effective parameters, γ$\gamma$, in the j$\mathit{j}$th model, for j = 1,2,,lh$\mathit{j}=1,2,\dots ,{\mathbf{lh}}$.
2:     b(ldb, : $:$) – double array
The first dimension, ldb, of the array b will be
• if wantb0${\mathbf{wantb}}\ne 0$, ldbip + 1$\mathit{ldb}\ge {\mathbf{ip}}+1$;
• otherwise ldb1$\mathit{ldb}\ge 1$.
The second dimension of the array will be lh${\mathbf{lh}}$ if wantb0${\mathbf{wantb}}\ne 0$, and at least 1$1$ otherwise
If wantb0${\mathbf{wantb}}\ne 0$, b contains the intercept and parameter estimates for the fitted ridge regression model in the order indicated by isx. b(1,j)${\mathbf{b}}\left(1,\mathit{j}\right)$, for j = 1,2,,lh$\mathit{j}=1,2,\dots ,{\mathbf{lh}}$, contains the estimate for the intercept; b(i + 1,j)${\mathbf{b}}\left(\mathit{i}+1,j\right)$ contains the parameter estimate for the i$\mathit{i}$th independent variable in the model fitted with ridge parameter h(j)${\mathbf{h}}\left(j\right)$, for i = 1,2,,ip$\mathit{i}=1,2,\dots ,{\mathbf{ip}}$.
3:     vf(ldvf, : $:$) – double array
The first dimension, ldvf, of the array vf will be
• if wantvf0${\mathbf{wantvf}}\ne 0$, ldvfip$\mathit{ldvf}\ge {\mathbf{ip}}$;
• otherwise ldvf1$\mathit{ldvf}\ge 1$.
The second dimension of the array will be lh${\mathbf{lh}}$ if wantvf0${\mathbf{wantvf}}\ne 0$, and at least 1$1$ otherwise
If wantvf = 1${\mathbf{wantvf}}=1$, the variance inflation factors. For the i$\mathit{i}$th independent variable in a model fitted with ridge parameter h(j)${\mathbf{h}}\left(j\right)$, vf(i,j)${\mathbf{vf}}\left(\mathit{i},j\right)$ is the value of vi${v}_{\mathit{i}}$, for i = 1,2,,ip$\mathit{i}=1,2,\dots ,{\mathbf{ip}}$.
4:     pe(ldpe, : $:$) – double array
The first dimension, ldpe, of the array pe will be
• if lpec > 0${\mathbf{lpec}}>0$, ldpelpec$\mathit{ldpe}\ge {\mathbf{lpec}}$;
• otherwise ldpe1$\mathit{ldpe}\ge 1$.
The second dimension of the array will be lh${\mathbf{lh}}$ if lpec > 0${\mathbf{lpec}}>0$, and at least 1$1$ otherwise
If lpec0${\mathbf{lpec}}\le 0$, pe is not referenced; otherwise pe(i,j)${\mathbf{pe}}\left(\mathit{i},\mathit{j}\right)$ contains the prediction error of criterion pec(i)${\mathbf{pec}}\left(\mathit{i}\right)$ for the model fitted with ridge parameter h(j)${\mathbf{h}}\left(\mathit{j}\right)$, for i = 1,2,,lpec$\mathit{i}=1,2,\dots ,{\mathbf{lpec}}$ and j = 1,2,,lh$\mathit{j}=1,2,\dots ,{\mathbf{lh}}$.
5:     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, n < 1${\mathbf{n}}<1$, or h(j) < 0.0${\mathbf{h}}\left(j\right)<0.0$, or lh ≤ 0${\mathbf{lh}}\le 0$, or wantb ≠ 0${\mathbf{wantb}}\ne 0$, 1$1$ or 2$2$, or wantb ≠ 0${\mathbf{wantb}}\ne 0$ and ldb < ip + 1$\mathit{ldb}<{\mathbf{ip}}+1$, or wantvf ≠ 0${\mathbf{wantvf}}\ne 0$ or 1$1$, or an element of pec is not defined.
ifail = 2${\mathbf{ifail}}=2$
 On entry, m > n${\mathbf{m}}>{\mathbf{n}}$, or ldx < n$\mathit{ldx}<{\mathbf{n}}$, or ip < 1${\mathbf{ip}}<1$ or ip > m${\mathbf{ip}}>{\mathbf{m}}$, or an element of isx ≠ 0${\mathbf{isx}}\ne 0$ or 1$1$, or ip does not equal the sum of elements in isx, or wantvf ≠ 0${\mathbf{wantvf}}\ne 0$ and ldvf < ip$\mathit{ldvf}<{\mathbf{ip}}$, or ldpe < lpec$\mathit{ldpe}<{\mathbf{lpec}}$.
ifail = 3${\mathbf{ifail}}=3$
Both wantb and wantvf are zero.
ifail = 4${\mathbf{ifail}}=4$
ifail = 999${\mathbf{ifail}}=-999$
Internal memory allocation failed.

## Accuracy

The accuracy of nag_correg_ridge (g02kb) is closely related to that of the singular value decomposition.

nag_correg_ridge (g02kb) allocates internally max (5 × (n1),2 × ip × ip) + (n + 3) × ip + n$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(5×\left({\mathbf{n}}-1\right),2×{\mathbf{ip}}×{\mathbf{ip}}\right)+\left({\mathbf{n}}+3\right)×{\mathbf{ip}}+{\mathbf{n}}$ elements of double precision storage.

## Example

```function nag_correg_ridge_example
x = [19.5, 43.1, 29.1;
24.7, 49.8, 28.2;
30.7, 51.9, 37;
29.8, 54.3, 31.1;
19.1, 42.2, 30.9;
25.6, 53.9, 23.7;
31.4, 58.5, 27.6;
27.9, 52.1, 30.6;
22.1, 49.9, 23.2;
25.5, 53.5, 24.8;
31.1, 56.6, 30;
30.4, 56.7, 28.3;
18.7, 46.5, 23;
19.7, 44.2, 28.6;
14.6, 42.7, 21.3;
29.5, 54.4, 30.1;
27.7, 55.3, 25.7;
30.2, 58.6, 24.6;
22.7, 48.2, 27.1;
25.2, 51, 27.5];
isx = [int64(1);1;1];
ip = int64(3);
y = [11.9;
22.8;
18.7;
20.1;
12.9;
21.7;
27.1;
25.4;
21.3;
19.3;
25.4;
27.2;
11.7;
17.8;
12.8;
23.9;
22.6;
25.4;
14.8;
21.1];
h = [0;
0.002;
0.004;
0.006;
0.008;
0.01;
0.012;
0.014;
0.016;
0.018;
0.02;
0.022;
0.024;
0.026;
0.028;
0.03];
wantb = int64(1);
wantvf = int64(1);
pec = {'L'; 'G'; 'U'; 'F'; 'B'};
[nep, b, vf, pe, ifail] = nag_correg_ridge(x, isx, ip, y, h, wantb, wantvf, pec)
```
```

nep =

4.0000
3.2634
3.1475
3.0987
3.0709
3.0523
3.0386
3.0278
3.0189
3.0112
3.0045
2.9984
2.9928
2.9876
2.9828
2.9782

b =

Columns 1 through 9

117.0847   22.2748    7.7209    1.8363   -1.3396   -3.3219   -4.6734   -5.6511   -6.3891
4.3341    1.4644    1.0229    0.8437    0.7465    0.6853    0.6432    0.6125    0.5890
-2.8568   -0.4012   -0.0242    0.1282    0.2105    0.2618    0.2968    0.3222    0.3413
-2.1861   -0.6738   -0.4408   -0.3460   -0.2944   -0.2619   -0.2393   -0.2228   -0.2100

Columns 10 through 16

-6.9642   -7.4236   -7.7978   -8.1075   -8.3673   -8.5874   -8.7758
0.5704    0.5554    0.5429    0.5323    0.5233    0.5155    0.5086
0.3562    0.3681    0.3779    0.3859    0.3926    0.3984    0.4033
-0.1999   -0.1916   -0.1847   -0.1788   -0.1737   -0.1693   -0.1653

vf =

Columns 1 through 9

708.8429   50.5592   16.9816    8.5033    5.1472    3.4855    2.5434    1.9581    1.5698
564.3434   40.4483   13.7247    6.9764    4.3046    2.9813    2.2306    1.7640    1.4541
104.6060    8.2797    3.3628    2.1185    1.6238    1.3770    1.2356    1.1463    1.0859

Columns 10 through 16

1.2990    1.1026    0.9556    0.8427    0.7541    0.6832    0.6257
1.2377    1.0805    0.9627    0.8721    0.8007    0.7435    0.6969
1.0428    1.0105    0.9855    0.9655    0.9491    0.9353    0.9235

pe =

Columns 1 through 9

8.0368    7.5464    7.5575    7.5656    7.5701    7.5723    7.5732    7.5734    7.5731
7.6879    7.4238    7.4520    7.4668    7.4749    7.4796    7.4823    7.4838    7.4845
6.1503    6.2124    6.2793    6.3100    6.3272    6.3381    6.3455    6.3508    6.3548
7.3804    7.2261    7.2675    7.2876    7.2987    7.3053    7.3095    7.3122    7.3140
8.6052    8.2355    8.2515    8.2611    8.2661    8.2685    8.2695    8.2696    8.2691

Columns 10 through 16

7.5724    7.5715    7.5705    7.5694    7.5682    7.5669    7.5657
7.4848    7.4847    7.4843    7.4838    7.4832    7.4825    7.4818
6.3578    6.3603    6.3623    6.3639    6.3654    6.3666    6.3677
7.3151    7.3158    7.3161    7.3162    7.3162    7.3161    7.3159
8.2683    8.2671    8.2659    8.2645    8.2630    8.2615    8.2600

ifail =

0

```
```function g02kb_example
x = [19.5, 43.1, 29.1;
24.7, 49.8, 28.2;
30.7, 51.9, 37;
29.8, 54.3, 31.1;
19.1, 42.2, 30.9;
25.6, 53.9, 23.7;
31.4, 58.5, 27.6;
27.9, 52.1, 30.6;
22.1, 49.9, 23.2;
25.5, 53.5, 24.8;
31.1, 56.6, 30;
30.4, 56.7, 28.3;
18.7, 46.5, 23;
19.7, 44.2, 28.6;
14.6, 42.7, 21.3;
29.5, 54.4, 30.1;
27.7, 55.3, 25.7;
30.2, 58.6, 24.6;
22.7, 48.2, 27.1;
25.2, 51, 27.5];
isx = [int64(1);1;1];
ip = int64(3);
y = [11.9;
22.8;
18.7;
20.1;
12.9;
21.7;
27.1;
25.4;
21.3;
19.3;
25.4;
27.2;
11.7;
17.8;
12.8;
23.9;
22.6;
25.4;
14.8;
21.1];
h = [0;
0.002;
0.004;
0.006;
0.008;
0.01;
0.012;
0.014;
0.016;
0.018;
0.02;
0.022;
0.024;
0.026;
0.028;
0.03];
wantb = int64(1);
wantvf = int64(1);
pec = {'L'; 'G'; 'U'; 'F'; 'B'};
[nep, b, vf, pe, ifail] = g02kb(x, isx, ip, y, h, wantb, wantvf, pec)
```
```

nep =

4.0000
3.2634
3.1475
3.0987
3.0709
3.0523
3.0386
3.0278
3.0189
3.0112
3.0045
2.9984
2.9928
2.9876
2.9828
2.9782

b =

Columns 1 through 9

117.0847   22.2748    7.7209    1.8363   -1.3396   -3.3219   -4.6734   -5.6511   -6.3891
4.3341    1.4644    1.0229    0.8437    0.7465    0.6853    0.6432    0.6125    0.5890
-2.8568   -0.4012   -0.0242    0.1282    0.2105    0.2618    0.2968    0.3222    0.3413
-2.1861   -0.6738   -0.4408   -0.3460   -0.2944   -0.2619   -0.2393   -0.2228   -0.2100

Columns 10 through 16

-6.9642   -7.4236   -7.7978   -8.1075   -8.3673   -8.5874   -8.7758
0.5704    0.5554    0.5429    0.5323    0.5233    0.5155    0.5086
0.3562    0.3681    0.3779    0.3859    0.3926    0.3984    0.4033
-0.1999   -0.1916   -0.1847   -0.1788   -0.1737   -0.1693   -0.1653

vf =

Columns 1 through 9

708.8429   50.5592   16.9816    8.5033    5.1472    3.4855    2.5434    1.9581    1.5698
564.3434   40.4483   13.7247    6.9764    4.3046    2.9813    2.2306    1.7640    1.4541
104.6060    8.2797    3.3628    2.1185    1.6238    1.3770    1.2356    1.1463    1.0859

Columns 10 through 16

1.2990    1.1026    0.9556    0.8427    0.7541    0.6832    0.6257
1.2377    1.0805    0.9627    0.8721    0.8007    0.7435    0.6969
1.0428    1.0105    0.9855    0.9655    0.9491    0.9353    0.9235

pe =

Columns 1 through 9

8.0368    7.5464    7.5575    7.5656    7.5701    7.5723    7.5732    7.5734    7.5731
7.6879    7.4238    7.4520    7.4668    7.4749    7.4796    7.4823    7.4838    7.4845
6.1503    6.2124    6.2793    6.3100    6.3272    6.3381    6.3455    6.3508    6.3548
7.3804    7.2261    7.2675    7.2876    7.2987    7.3053    7.3095    7.3122    7.3140
8.6052    8.2355    8.2515    8.2611    8.2661    8.2685    8.2695    8.2696    8.2691

Columns 10 through 16

7.5724    7.5715    7.5705    7.5694    7.5682    7.5669    7.5657
7.4848    7.4847    7.4843    7.4838    7.4832    7.4825    7.4818
6.3578    6.3603    6.3623    6.3639    6.3654    6.3666    6.3677
7.3151    7.3158    7.3161    7.3162    7.3162    7.3161    7.3159
8.2683    8.2671    8.2659    8.2645    8.2630    8.2615    8.2600

ifail =

0

```