# NAG Library Routine Document

## 1Purpose

g02dkf calculates the estimates of the parameters of a general linear regression model for given constraints from the singular value decomposition results.

## 2Specification

Fortran Interface
 Subroutine g02dkf ( ip, p, c, ldc, b, rss, idf, se, cov, wk,
 Integer, Intent (In) :: ip, iconst, ldc, idf Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: p(ip*ip+2*ip), c(ldc,iconst), rss Real (Kind=nag_wp), Intent (Inout) :: b(ip) Real (Kind=nag_wp), Intent (Out) :: se(ip), cov(ip*(ip+1)/2), wk(2*ip*ip+ip*iconst+2*iconst*iconst+4*iconst)
#include <nagmk26.h>
 void g02dkf_ (const Integer *ip, const Integer *iconst, const double p[], const double c[], const Integer *ldc, double b[], const double *rss, const Integer *idf, double se[], double cov[], double wk[], Integer *ifail)

## 3Description

g02dkf computes the estimates given a set of linear constraints for a general linear regression model which is not of full rank. It is intended for use after a call to g02daf or g02ddf.
In the case of a model not of full rank the routines use a singular value decomposition (SVD) to find the parameter estimates, ${\stackrel{^}{\beta }}_{\text{svd}}$, and their variance-covariance matrix. Details of the SVD are made available in the form of the matrix ${P}^{*}$:
 $P*= D-1 P1T P0T ,$
as described by g02daf and g02ddf.
Alternative solutions can be formed by imposing constraints on the parameters. If there are $p$ parameters and the rank of the model is $k$, then ${n}_{c}=p-k$ constraints will have to be imposed to obtain a unique solution.
Let $C$ be a $p$ by ${n}_{c}$ matrix of constraints, such that
 $CTβ=0$
then the new parameter estimates ${\stackrel{^}{\beta }}_{c}$ are given by
 $β^c =Aβ^svd; =I-P0CTP0-1β^svd,$
where $I$ is the identity matrix, and the variance-covariance matrix is given by
 $AP1D-2P1TAT,$
provided ${\left({C}^{\mathrm{T}}{P}_{0}\right)}^{-1}$ exists.

## 4References

Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore
Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25
Searle S R (1971) Linear Models Wiley

## 5Arguments

1:     $\mathbf{ip}$ – IntegerInput
On entry: $p$, the number of terms in the linear model.
Constraint: ${\mathbf{ip}}\ge 1$.
2:     $\mathbf{iconst}$ – IntegerInput
On entry: the number of constraints to be imposed on the parameters, ${n}_{\mathrm{c}}$.
Constraint: $0<{\mathbf{iconst}}<{\mathbf{ip}}$.
3:     $\mathbf{p}\left({\mathbf{ip}}×{\mathbf{ip}}+2×{\mathbf{ip}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: as returned by g02daf and g02ddf.
4:     $\mathbf{c}\left({\mathbf{ldc}},{\mathbf{iconst}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the iconst constraints stored by column, i.e., the $i$th constraint is stored in the $i$th column of c.
5:     $\mathbf{ldc}$ – IntegerInput
On entry: the first dimension of the array c as declared in the (sub)program from which g02dkf is called.
Constraint: ${\mathbf{ldc}}\ge {\mathbf{ip}}$.
6:     $\mathbf{b}\left({\mathbf{ip}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: the parameter estimates computed by using the singular value decomposition, ${\stackrel{^}{\beta }}_{\text{svd}}$.
On exit: the parameter estimates of the parameters with the constraints imposed, ${\stackrel{^}{\beta }}_{\mathrm{c}}$.
7:     $\mathbf{rss}$ – Real (Kind=nag_wp)Input
On entry: the residual sum of squares as returned by g02daf or g02ddf.
Constraint: ${\mathbf{rss}}>0.0$.
8:     $\mathbf{idf}$ – IntegerInput
On entry: the degrees of freedom associated with the residual sum of squares as returned by g02daf or g02ddf.
Constraint: ${\mathbf{idf}}>0$.
9:     $\mathbf{se}\left({\mathbf{ip}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the standard error of the parameter estimates in b.
10:   $\mathbf{cov}\left({\mathbf{ip}}×\left({\mathbf{ip}}+1\right)/2\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the upper triangular part of the variance-covariance matrix of the ip parameter estimates given in b. They are stored packed by column, i.e., the covariance between the parameter estimate given in ${\mathbf{b}}\left(i\right)$ and the parameter estimate given in ${\mathbf{b}}\left(j\right)$, $j\ge i$, is stored in ${\mathbf{cov}}\left(\left(j×\left(j-1\right)/2+i\right)\right)$.
11:   $\mathbf{wk}\left(2×{\mathbf{ip}}×{\mathbf{ip}}+{\mathbf{ip}}×{\mathbf{iconst}}+2×{\mathbf{iconst}}×{\mathbf{iconst}}+4×{\mathbf{iconst}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
Note that a simple upper bound for the size of the workspace is $5×{\mathbf{ip}}×{\mathbf{ip}}$.
12:   $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{iconst}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{iconst}}>0$.
On entry, ${\mathbf{iconst}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{iconst}}<{\mathbf{ip}}$.
On entry, ${\mathbf{idf}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{idf}}>0$.
On entry, ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ip}}\ge 1$.
On entry, ${\mathbf{ldc}}=〈\mathit{\text{value}}〉$ and ${\mathbf{ip}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{ldc}}\ge {\mathbf{ip}}$.
On entry, ${\mathbf{rss}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{rss}}>0.0$.
${\mathbf{ifail}}=2$
c does not give a model of full rank.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

It should be noted that due to rounding errors a parameter that should be zero when the constraints have been imposed may be returned as a value of order machine precision.

## 8Parallelism and Performance

g02dkf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02dkf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

g02dkf is intended for use in situations in which dummy ($0–1$) variables have been used such as in the analysis of designed experiments when you do not wish to change the parameters of the model to give a full rank model. The routine is not intended for situations in which the relationships between the independent variables are only approximate.

## 10Example

Data from an experiment with four treatments and three observations per treatment are read in. A model, including the mean term, is fitted by g02daf and the results printed. The constraint that the sum of treatment effect is zero is then read in and the parameter estimates with this constraint imposed are computed by g02dkf and printed.

### 10.1Program Text

Program Text (g02dkfe.f90)

### 10.2Program Data

Program Data (g02dkfe.d)

### 10.3Program Results

Program Results (g02dkfe.r)