# NAG CL Interfaceg02kbc (ridge)

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## 1Purpose

g02kbc calculates a ridge regression, with ridge parameters supplied by you.

## 2Specification

 #include
 void g02kbc (Nag_OrderType order, Integer n, Integer m, const double x[], Integer pdx, const Integer isx[], Integer ip, const double y[], Integer lh, const double h[], double nep[], Nag_ParaOption wantb, double b[], Integer pdb, Nag_VIFOption wantvf, double vf[], Integer pdvf, Integer lpec, const Nag_PredictError pec[], double pe[], Integer pdpe, NagError *fail)
The function may be called by the names: g02kbc, nag_correg_ridge or nag_regsn_ridge.

## 3Description

A linear model has the form:
 $y = c+Xβ+ε ,$
where
• $y$ is an $n×1$ matrix of values of a dependent variable;
• $c$ is a scalar intercept term;
• $X$ is an $n×m$ matrix of values of independent variables;
• $\beta$ is an $m×1$ matrix of unknown values of parameters;
• $\epsilon$ is an $n×1$ matrix of unknown random errors such that variance of ${\epsilon =\sigma }^{2}I$.
Let $\stackrel{~}{X}$ be the mean-centred $X$ and $\stackrel{~}{y}$ the mean-centred $y$. Furthermore, $\stackrel{~}{X}$ is scaled such that the diagonal elements of the cross product matrix ${\stackrel{~}{X}}^{\mathrm{T}}\stackrel{~}{X}$ are one. The linear model now takes the form:
 $y~ = X~ β~ + ε .$
Ridge regression estimates the parameters $\stackrel{~}{\beta }$ in a penalised least squares sense by finding the $\stackrel{~}{b}$ that minimizes
 $‖X~b~-y~‖ 2 + h ‖b~‖ 2 , h>0 ,$
where $‖·‖$ denotes the ${\ell }_{2}$-norm and $h$ is a scalar regularization or ridge parameter. For a given value of $h$, the parameters estimates $\stackrel{~}{b}$ are found by evaluating
 $b~ = (X~TX~+hI)-1 X~T y~ .$
Note that if $h=0$ the ridge regression solution is equivalent to the ordinary least squares solution.
Rather than calculate the inverse of (${\stackrel{~}{X}}^{\mathrm{T}}\stackrel{~}{X}+hI$) directly, g02kbc uses the singular value decomposition (SVD) of $\stackrel{~}{X}$. After decomposing $\stackrel{~}{X}$ into $UD{V}^{\mathrm{T}}$ where $U$ and $V$ are orthogonal matrices and $D$ is a diagonal matrix, the parameter estimates become
 $b~ = V (DTD+hI)-1 DUT y~ .$
A consequence of introducing the ridge parameter is that the effective number of parameters, $\gamma$, in the model is given by the sum of diagonal elements of
 $DT D (DTD+hI)-1 ,$
see Moody (1992) for details.
Any multi-collinearity in the design matrix $X$ may be highlighted by calculating the variance inflation factors for the fitted model. The $j$th variance inflation factor, ${v}_{j}$, is a scaled version of the multiple correlation coefficient between independent variable $j$ and the other independent variables, ${R}_{j}$, and is given by
 $vj = 1 1-Rj , j=1,2,…,m .$
The $m$ variance inflation factors are calculated as the diagonal elements of the matrix:
 $(X~TX~+hI)-1 X~T X~ (X~TX~+hI)-1 ,$
which, using the SVD of $\stackrel{~}{X}$, is equivalent to the diagonal elements of the matrix:
 $V (DTD+hI)-1 DT D (DTD+hI)-1 VT .$
Given a value of $h$, any or all of the following prediction criteria are available:
1. (a)Generalized cross-validation (GCV):
 $ns (n-γ) 2 ;$
2. (b)Unbiased estimate of variance (UEV):
 $s n-γ ;$
3. (c)Future prediction error (FPE):
 $1n (s+ 2γs n-γ ) ;$
4. (d)Bayesian information criterion (BIC):
 $1n (s+ log(n)γs n-γ ) ;$
5. (e)Leave-one-out cross-validation (LOOCV),
where $s$ 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 $b$ associated with $X$. These are calculated from $\stackrel{~}{b}$, and the means and scalings of $X$. Optionally, either $\stackrel{~}{b}$ or $b$ may be calculated.
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

## 5Arguments

1: $\mathbf{order}$Nag_OrderType Input
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2: $\mathbf{n}$Integer Input
On entry: $n$, the number of observations.
Constraint: ${\mathbf{n}}\ge 1$.
3: $\mathbf{m}$Integer Input
On entry: the number of independent variables available in the data matrix $X$.
Constraint: ${\mathbf{m}}\le {\mathbf{n}}$.
4: $\mathbf{x}\left[\mathit{dim}\right]$const double Input
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{m}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdx}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
the $\left(i,j\right)$th element of the matrix $X$ is stored in
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: the values of independent variables in the data matrix $X$.
5: $\mathbf{pdx}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{m}}$.
6: $\mathbf{isx}\left[{\mathbf{m}}\right]$const Integer Input
On entry: indicates which $m$ independent variables are included in the model.
${\mathbf{isx}}\left[j-1\right]=1$
The $j$th variable in x will be included in the model.
${\mathbf{isx}}\left[j-1\right]=0$
Variable $j$ is excluded.
Constraint: ${\mathbf{isx}}\left[\mathit{j}-1\right]=0$ or $1$, for $\mathit{j}=1,2,\dots ,{\mathbf{m}}$.
7: $\mathbf{ip}$Integer Input
On entry: $m$, the number of independent variables in the model.
Constraints:
• $1\le {\mathbf{ip}}\le {\mathbf{m}}$;
• Exactly ip elements of isx must be equal to $1$.
8: $\mathbf{y}\left[{\mathbf{n}}\right]$const double Input
On entry: the $n$ values of the dependent variable $y$.
9: $\mathbf{lh}$Integer Input
On entry: the number of supplied ridge parameters.
Constraint: ${\mathbf{lh}}>0$.
10: $\mathbf{h}\left[{\mathbf{lh}}\right]$const double Input
On entry: ${\mathbf{h}}\left[j-1\right]$ is the value of the $j$th ridge parameter $h$.
Constraint: ${\mathbf{h}}\left[\mathit{j}-1\right]\ge 0.0$, for $\mathit{j}=1,2,\dots ,{\mathbf{lh}}$.
11: $\mathbf{nep}\left[{\mathbf{lh}}\right]$double Output
On exit: ${\mathbf{nep}}\left[\mathit{j}-1\right]$ is the number of effective parameters, $\gamma$, in the $\mathit{j}$th model, for $\mathit{j}=1,2,\dots ,{\mathbf{lh}}$.
12: $\mathbf{wantb}$Nag_ParaOption Input
On entry: defines the options for parameter estimates.
${\mathbf{wantb}}=\mathrm{Nag_NoPara}$
Parameter estimates are not calculated and b is not referenced.
${\mathbf{wantb}}=\mathrm{Nag_OrigPara}$
Parameter estimates $b$ are calculated for the original data.
${\mathbf{wantb}}=\mathrm{Nag_StandPara}$
Parameter estimates $\stackrel{~}{b}$ are calculated for the standardized data.
Constraint: ${\mathbf{wantb}}=\mathrm{Nag_NoPara}$, $\mathrm{Nag_OrigPara}$ or $\mathrm{Nag_StandPara}$.
13: $\mathbf{b}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array b must be at least
• ${\mathbf{pdb}}×{\mathbf{lh}}$ when ${\mathbf{wantb}}\ne \mathrm{Nag_NoPara}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,\left({\mathbf{ip}}+1\right)×{\mathbf{pdb}}\right)$ when ${\mathbf{wantb}}\ne \mathrm{Nag_NoPara}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $1$ otherwise.
where ${\mathbf{B}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{b}}\left[\left(j-1\right)×{\mathbf{pdb}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{b}}\left[\left(i-1\right)×{\mathbf{pdb}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{wantb}}\ne \mathrm{Nag_NoPara}$, b contains the intercept and parameter estimates for the fitted ridge regression model in the order indicated by isx. ${\mathbf{B}}\left(1,\mathit{j}\right)$, for $\mathit{j}=1,2,\dots ,{\mathbf{lh}}$, contains the estimate for the intercept; ${\mathbf{B}}\left(\mathit{i}+1,j\right)$ contains the parameter estimate for the $\mathit{i}$th independent variable in the model fitted with ridge parameter ${\mathbf{h}}\left[j-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{ip}}$.
14: $\mathbf{pdb}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array b.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{wantb}}\ne \mathrm{Nag_NoPara}$, ${\mathbf{pdb}}\ge {\mathbf{ip}}+1$;
• otherwise ${\mathbf{pdb}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{wantb}}\ne \mathrm{Nag_NoPara}$, ${\mathbf{pdb}}\ge {\mathbf{lh}}$;
• otherwise ${\mathbf{pdb}}\ge 1$.
15: $\mathbf{wantvf}$Nag_VIFOption Input
On entry: defines the options for variance inflation factors.
${\mathbf{wantvf}}=\mathrm{Nag_NoVIF}$
Variance inflation factors are not calculated and the array vf is not referenced.
${\mathbf{wantvf}}=\mathrm{Nag_WantVIF}$
Variance inflation factors are calculated.
Constraints:
• ${\mathbf{wantvf}}=\mathrm{Nag_NoVIF}$ or $\mathrm{Nag_WantVIF}$;
• if ${\mathbf{wantb}}=\mathrm{Nag_NoPara}$, ${\mathbf{wantvf}}=\mathrm{Nag_WantVIF}$.
16: $\mathbf{vf}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array vf must be at least
• ${\mathbf{pdvf}}×{\mathbf{lh}}$ when ${\mathbf{wantvf}}\ne \mathrm{Nag_NoVIF}$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{ip}}×{\mathbf{pdvf}}\right)$ when ${\mathbf{wantvf}}\ne \mathrm{Nag_NoVIF}$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• $1$ otherwise.
where ${\mathbf{VF}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{vf}}\left[\left(j-1\right)×{\mathbf{pdvf}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{vf}}\left[\left(i-1\right)×{\mathbf{pdvf}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{wantvf}}=\mathrm{Nag_WantVIF}$, the variance inflation factors. For the $\mathit{i}$th independent variable in a model fitted with ridge parameter ${\mathbf{h}}\left[j-1\right]$, ${\mathbf{VF}}\left(\mathit{i},j\right)$ is the value of ${v}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{ip}}$.
17: $\mathbf{pdvf}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array vf.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{wantvf}}\ne \mathrm{Nag_NoVIF}$, ${\mathbf{pdvf}}\ge {\mathbf{ip}}$;
• otherwise ${\mathbf{pdvf}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{wantvf}}\ne \mathrm{Nag_NoVIF}$, ${\mathbf{pdvf}}\ge {\mathbf{lh}}$;
• otherwise ${\mathbf{pdvf}}\ge 1$.
18: $\mathbf{lpec}$Integer Input
On entry: the number of prediction error statistics to return; set ${\mathbf{lpec}}\le 0$ for no prediction error estimates.
19: $\mathbf{pec}\left[{\mathbf{lpec}}\right]$const Nag_PredictError Input
On entry: if ${\mathbf{lpec}}>0$, ${\mathbf{pec}}\left[\mathit{j}-1\right]$ defines the $\mathit{j}$th prediction error, for $\mathit{j}=1,2,\dots ,{\mathbf{lpec}}$; otherwise pec is not referenced and may be NULL.
${\mathbf{pec}}\left[j-1\right]=\mathrm{Nag_BIC}$
Bayesian information criterion (BIC).
${\mathbf{pec}}\left[j-1\right]=\mathrm{Nag_FPE}$
Future prediction error (FPE).
${\mathbf{pec}}\left[j-1\right]=\mathrm{Nag_GCV}$
Generalized cross-validation (GCV).
${\mathbf{pec}}\left[j-1\right]=\mathrm{Nag_LOOCV}$
Leave-one-out cross-validation (LOOCV).
${\mathbf{pec}}\left[j-1\right]=\mathrm{Nag_EUV}$
Unbiased estimate of variance (UEV).
Constraint: if ${\mathbf{lpec}}>0$, ${\mathbf{pec}}\left[\mathit{j}-1\right]=\mathrm{Nag_BIC}$, $\mathrm{Nag_FPE}$, $\mathrm{Nag_GCV}$, $\mathrm{Nag_LOOCV}$ or $\mathrm{Nag_EUV}$, for $\mathit{j}=1,2,\dots ,{\mathbf{lpec}}$.
20: $\mathbf{pe}\left[\mathit{dim}\right]$double Output
Note: the dimension, dim, of the array pe must be at least
• ${\mathbf{pdpe}}×{\mathbf{lh}}$ when ${\mathbf{lpec}}>0$ and ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{lpec}}×{\mathbf{pdpe}}$ when ${\mathbf{lpec}}>0$ and ${\mathbf{order}}=\mathrm{Nag_RowMajor}$;
• otherwise pe may be NULL.
where ${\mathbf{PE}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{pe}}\left[\left(j-1\right)×{\mathbf{pdpe}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{pe}}\left[\left(i-1\right)×{\mathbf{pdpe}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On exit: if ${\mathbf{lpec}}\le 0$, pe is not referenced and may be NULL; otherwise ${\mathbf{PE}}\left(\mathit{i},\mathit{j}\right)$ contains the prediction error of criterion ${\mathbf{pec}}\left[\mathit{i}-1\right]$ for the model fitted with ridge parameter ${\mathbf{h}}\left[\mathit{j}-1\right]$, for $\mathit{i}=1,2,\dots ,{\mathbf{lpec}}$ and $\mathit{j}=1,2,\dots ,{\mathbf{lh}}$.
21: $\mathbf{pdpe}$Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array pe.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$,
• if ${\mathbf{lpec}}>0$, ${\mathbf{pdpe}}\ge {\mathbf{lpec}}$;
• otherwise ${\mathbf{pdpe}}\ge 1$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$,
• if ${\mathbf{lpec}}>0$, ${\mathbf{pdpe}}\ge {\mathbf{lh}}$;
• otherwise pe may be NULL.
22: $\mathbf{fail}$NagError * Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument $⟨\mathit{\text{value}}⟩$ had an illegal value.
NE_CONSTRAINT
On entry, ${\mathbf{wantb}}=\mathrm{Nag_NoPara}$ and ${\mathbf{wantvf}}=\mathrm{Nag_NoVIF}$.
Constraint: ${\mathbf{wantb}}=\mathrm{Nag_NoPara}$, ${\mathbf{wantvf}}=\mathrm{Nag_WantVIF}$.
NE_ENUM_INT_2
On entry, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ip}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{wantb}}\ne \mathrm{Nag_NoPara}$, ${\mathbf{pdb}}\ge {\mathbf{ip}}+1$.
On entry, ${\mathbf{pdvf}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{ip}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{wantvf}}\ne \mathrm{Nag_NoVIF}$, ${\mathbf{pdvf}}\ge {\mathbf{ip}}$.
On entry, ${\mathbf{wantb}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdb}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{lh}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{wantb}}\ne \mathrm{Nag_NoPara}$, ${\mathbf{pdb}}\ge {\mathbf{lh}}$;
otherwise ${\mathbf{pdb}}\ge 1$.
On entry, ${\mathbf{wantvf}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{pdvf}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{lh}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{wantvf}}\ne \mathrm{Nag_NoVIF}$, ${\mathbf{pdvf}}\ge {\mathbf{lh}}$;
otherwise ${\mathbf{pdvf}}\ge 1$.
NE_INT
On entry, ${\mathbf{lh}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{lh}}>0$.
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 1$.
NE_INT_2
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\le {\mathbf{n}}$.
On entry, ${\mathbf{pdpe}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{lpec}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdpe}}\ge {\mathbf{lpec}}$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{n}}$.
NE_INT_3
On entry, ${\mathbf{pdpe}}=⟨\mathit{\text{value}}⟩$, ${\mathbf{lpec}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{lh}}=⟨\mathit{\text{value}}⟩$.
Constraint: if ${\mathbf{lpec}}>0$, ${\mathbf{pdpe}}\ge {\mathbf{lh}}$.
NE_INT_ARG_CONS
On entry, ip is not equal to the sum of elements in isx.
Constraint: exactly ip elements of isx must be equal to $1$.
NE_INT_ARRAY_VAL_1_OR_2
On entry, ${\mathbf{isx}}\left[j-1\right]\ne 0$ or $1$ for at least one $j$.
Constraint: ${\mathbf{isx}}\left[j-1\right]=0$ or $1$, for all $j$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_REAL_ARRAY_CONS
On entry, ${\mathbf{h}}\left[j-1\right]<0$ for at least one $j$.
Constraint: ${\mathbf{h}}\left[j-1\right]\le 0.0$, for all $j$.

## 7Accuracy

The accuracy of g02kbc is closely related to that of the singular value decomposition.

## 8Parallelism and Performance

g02kbc is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02kbc makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

g02kbc allocates internally $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(5×\left({\mathbf{n}}-1\right),2×{\mathbf{ip}}×{\mathbf{ip}}\right)+\left({\mathbf{n}}+3\right)×{\mathbf{ip}}+{\mathbf{n}}$ elements of double precision storage.

## 10Example

This example reads in data from an experiment to model body fat, and a selection of ridge regression models are calculated.

### 10.1Program Text

Program Text (g02kbce.c)

### 10.2Program Data

Program Data (g02kbce.d)

### 10.3Program Results

Program Results (g02kbce.r)