# NAG FL Interfacef08aaf (dgels)

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## 1Purpose

f08aaf solves linear least squares problems of the form
 $minx ‖b-Ax‖2 or minx ‖b-ATx‖2 ,$
where $A$ is an $m×n$ real matrix of full rank, using a $QR$ or $LQ$ factorization of $A$.

## 2Specification

Fortran Interface
 Subroutine f08aaf ( m, n, nrhs, a, lda, b, ldb, work, info)
 Integer, Intent (In) :: m, n, nrhs, lda, ldb, lwork Integer, Intent (Out) :: info Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork)) Character (1), Intent (In) :: trans
#include <nag.h>
 void f08aaf_ (const char *trans, const Integer *m, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, double b[], const Integer *ldb, double work[], const Integer *lwork, Integer *info, const Charlen length_trans)
The routine may be called by the names f08aaf, nagf_lapackeig_dgels or its LAPACK name dgels.

## 3Description

The following options are provided:
1. 1.If ${\mathbf{trans}}=\text{'N'}$ and $m\ge n$: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
 $minx ‖b-Ax‖2 .$
2. 2.If ${\mathbf{trans}}=\text{'N'}$ and $m: find the minimum norm solution of an underdetermined system $Ax=b$.
3. 3.If ${\mathbf{trans}}=\text{'T'}$ and $m\ge n$: find the minimum norm solution of an undetermined system ${A}^{\mathrm{T}}x=b$.
4. 4.If ${\mathbf{trans}}=\text{'T'}$ and $m: find the least squares solution of an overdetermined system, i.e., solve the least squares problem
 $minx ‖b-ATx‖2 .$
Several right-hand side vectors $b$ and solution vectors $x$ can be handled in a single call; they are stored as the columns of the $m×r$ right-hand side matrix $B$ and the $n×r$ solution matrix $X$.

## 4References

Anderson E, Bai Z, Bischof C, Blackford S, Demmel J, Dongarra J J, Du Croz J J, Greenbaum A, Hammarling S, McKenney A and Sorensen D (1999) LAPACK Users' Guide (3rd Edition) SIAM, Philadelphia https://www.netlib.org/lapack/lug
Golub G H and Van Loan C F (1996) Matrix Computations (3rd Edition) Johns Hopkins University Press, Baltimore

## 5Arguments

1: $\mathbf{trans}$Character(1) Input
On entry: if ${\mathbf{trans}}=\text{'N'}$, the linear system involves $A$.
If ${\mathbf{trans}}=\text{'T'}$, the linear system involves ${A}^{\mathrm{T}}$.
Constraint: ${\mathbf{trans}}=\text{'N'}$ or $\text{'T'}$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
3: $\mathbf{n}$Integer Input
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
4: $\mathbf{nrhs}$Integer Input
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrices $B$ and $X$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
5: $\mathbf{a}\left({\mathbf{lda}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array a must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: the $m×n$ matrix $A$.
On exit: if ${\mathbf{m}}\ge {\mathbf{n}}$, a is overwritten by details of its $QR$ factorization, as returned by f08aef.
If ${\mathbf{m}}<{\mathbf{n}}$, a is overwritten by details of its $LQ$ factorization, as returned by f08ahf.
6: $\mathbf{lda}$Integer Input
On entry: the first dimension of the array a as declared in the (sub)program from which f08aaf is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
7: $\mathbf{b}\left({\mathbf{ldb}},*\right)$Real (Kind=nag_wp) array Input/Output
Note: the second dimension of the array b must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nrhs}}\right)$.
On entry: the matrix $B$ of right-hand side vectors, stored in columns; b is $m×r$ if ${\mathbf{trans}}=\text{'N'}$, or $n×r$ if ${\mathbf{trans}}=\text{'T'}$.
On exit: b is overwritten by the solution vectors, $x$, stored in columns:
• if ${\mathbf{trans}}=\text{'N'}$ and $m\ge n$, or ${\mathbf{trans}}=\text{'T'}$ and $m, elements $1$ to $\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ in each column of b contain the least squares solution vectors; the residual sum of squares for the solution is given by the sum of squares of the modulus of elements $\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)+1\right)$ to $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ in that column;
• otherwise, elements $1$ to $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(m,n\right)$ in each column of b contain the minimum norm solution vectors.
8: $\mathbf{ldb}$Integer Input
On entry: the first dimension of the array b as declared in the (sub)program from which f08aaf is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
9: $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$Real (Kind=nag_wp) array Workspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
10: $\mathbf{lwork}$Integer Input
On entry: the dimension of the array work as declared in the (sub)program from which f08aaf is called.
If ${\mathbf{lwork}}=-1$, a workspace query is assumed; the routine only calculates the optimal size of the work array, returns this value as the first entry of the work array, and no error message related to lwork is issued.
Suggested value: for optimal performance, ${\mathbf{lwork}}\ge \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}},{\mathbf{nrhs}}\right)×\mathit{nb}$, where $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge \mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}},{\mathbf{nrhs}}\right)$ or ${\mathbf{lwork}}=-1$.
11: $\mathbf{info}$Integer Output
On exit: ${\mathbf{info}}=0$ unless the routine detects an error (see Section 6).

## 6Error Indicators and Warnings

${\mathbf{info}}<0$
If ${\mathbf{info}}=-i$, argument $i$ had an illegal value. An explanatory message is output, and execution of the program is terminated.
${\mathbf{info}}>0$
Diagonal element $⟨\mathit{\text{value}}⟩$ of the triangular factor of $A$ is zero, so that $A$ does not have full rank; the least squares solution could not be computed.

## 7Accuracy

See Section 4.5 of Anderson et al. (1999) for details of error bounds.

## 8Parallelism and Performance

f08aaf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08aaf 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.

The total number of floating-point operations required to factorize $A$ is approximately $\frac{2}{3}{n}^{2}\left(3m-n\right)$ if $m\ge n$ and $\frac{2}{3}{m}^{2}\left(3n-m\right)$ otherwise. Following the factorization the solution for a single vector $x$ requires $\mathit{O}\left(\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({m}^{2},{n}^{2}\right)\right)$ operations.
The complex analogue of this routine is f08anf.

## 10Example

This example solves the linear least squares problem
 $minx ‖b-Ax‖2 ,$
where
 $A = ( -0.57 -1.28 -0.39 0.25 -1.93 1.08 -0.31 -2.14 2.30 0.24 0.40 -0.35 -1.93 0.64 -0.66 0.08 0.15 0.30 0.15 -2.13 -0.02 1.03 -1.43 0.50 ) and b= ( -2.67 -0.55 3.34 -0.77 0.48 4.10 ) .$
The square root of the residual sum of squares is also output.
Note that the block size (NB) of $64$ assumed in this example is not realistic for such a small problem, but should be suitable for large problems.

### 10.1Program Text

Program Text (f08aafe.f90)

### 10.2Program Data

Program Data (f08aafe.d)

### 10.3Program Results

Program Results (f08aafe.r)