# NAG Library Routine Document

## 1Purpose

f08baf (dgelsy) computes the minimum norm solution to a real linear least squares problem
 $minx b-Ax2$
using a complete orthogonal factorization of $A$. $A$ is an $m$ by $n$ matrix which may be rank-deficient. Several right-hand side vectors $b$ and solution vectors $x$ can be handled in a single call.

## 2Specification

Fortran Interface
 Subroutine f08baf ( m, n, nrhs, a, lda, b, ldb, jpvt, rank, work, info)
 Integer, Intent (In) :: m, n, nrhs, lda, ldb, lwork Integer, Intent (Inout) :: jpvt(*) Integer, Intent (Out) :: rank, info Real (Kind=nag_wp), Intent (In) :: rcond Real (Kind=nag_wp), Intent (Inout) :: a(lda,*), b(ldb,*) Real (Kind=nag_wp), Intent (Out) :: work(max(1,lwork))
#include nagmk26.h
 void f08baf_ (const Integer *m, const Integer *n, const Integer *nrhs, double a[], const Integer *lda, double b[], const Integer *ldb, Integer jpvt[], const double *rcond, Integer *rank, double work[], const Integer *lwork, Integer *info)
The routine may be called by its LAPACK name dgelsy.

## 3Description

The right-hand side vectors are stored as the columns of the $m$ by $r$ matrix $B$ and the solution vectors in the $n$ by $r$ matrix $X$.
f08baf (dgelsy) first computes a $QR$ factorization with column pivoting
 $AP= Q R11 R12 0 R22 ,$
with ${R}_{11}$ defined as the largest leading sub-matrix whose estimated condition number is less than $1/{\mathbf{rcond}}$. The order of ${R}_{11}$, rank, is the effective rank of $A$.
Then, ${R}_{22}$ is considered to be negligible, and ${R}_{12}$ is annihilated by orthogonal transformations from the right, arriving at the complete orthogonal factorization
 $AP= Q T11 0 0 0 Z .$
The minimum norm solution is then
 $X = PZT T11-1 Q1T b 0$
where ${Q}_{1}$ consists of the first rank columns of $Q$.

## 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 http://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:     $\mathbf{m}$ – IntegerInput
On entry: $m$, the number of rows of the matrix $A$.
Constraint: ${\mathbf{m}}\ge 0$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the number of columns of the matrix $A$.
Constraint: ${\mathbf{n}}\ge 0$.
3:     $\mathbf{nrhs}$ – IntegerInput
On entry: $r$, the number of right-hand sides, i.e., the number of columns of the matrices $B$ and $X$.
Constraint: ${\mathbf{nrhs}}\ge 0$.
4:     $\mathbf{a}\left({\mathbf{lda}},*\right)$ – Real (Kind=nag_wp) arrayInput/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 $m$ by $n$ matrix $A$.
On exit: a has been overwritten by details of its complete orthogonal factorization.
5:     $\mathbf{lda}$ – IntegerInput
On entry: the first dimension of the array a as declared in the (sub)program from which f08baf (dgelsy) is called.
Constraint: ${\mathbf{lda}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}}\right)$.
6:     $\mathbf{b}\left({\mathbf{ldb}},*\right)$ – Real (Kind=nag_wp) arrayInput/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 $m$ by $r$ right-hand side matrix $B$.
On exit: the $n$ by $r$ solution matrix $X$.
7:     $\mathbf{ldb}$ – IntegerInput
On entry: the first dimension of the array b as declared in the (sub)program from which f08baf (dgelsy) is called.
Constraint: ${\mathbf{ldb}}\ge \mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{m}},{\mathbf{n}}\right)$.
8:     $\mathbf{jpvt}\left(*\right)$ – Integer arrayInput/Output
Note: the dimension of the array jpvt must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}\right)$.
On entry: if ${\mathbf{jpvt}}\left(i\right)\ne 0$, the $i$th column of $A$ is permuted to the front of $AP$, otherwise column $i$ is a free column.
On exit: if ${\mathbf{jpvt}}\left(i\right)=k$, the $i$th column of $AP$ was the $k$th column of $A$.
9:     $\mathbf{rcond}$ – Real (Kind=nag_wp)Input
On entry: used to determine the effective rank of $A$, which is defined as the order of the largest leading triangular sub-matrix ${R}_{11}$ in the $QR$ factorization of $A$, whose estimated condition number is $\text{}<1/{\mathbf{rcond}}$.
Suggested value: if the condition number of a is not known then ${\mathbf{rcond}}=\sqrt{\left(\epsilon \right)/2}$ (where $\epsilon$ is machine precision, see x02ajf) is a good choice. Negative values or values less than machine precision should be avoided since this will cause a to have an effective $\text{rank}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$ that could be larger than its actual rank, leading to meaningless results.
10:   $\mathbf{rank}$ – IntegerOutput
On exit: the effective rank of $A$, i.e., the order of the sub-matrix ${R}_{11}$. This is the same as the order of the sub-matrix ${T}_{11}$ in the complete orthogonal factorization of $A$.
11:   $\mathbf{work}\left(\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{lwork}}\right)\right)$ – Real (Kind=nag_wp) arrayWorkspace
On exit: if ${\mathbf{info}}={\mathbf{0}}$, ${\mathbf{work}}\left(1\right)$ contains the minimum value of lwork required for optimal performance.
12:   $\mathbf{lwork}$ – IntegerInput
On entry: the dimension of the array work as declared in the (sub)program from which f08baf (dgelsy) 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,
 $lwork ≥ max k + 2 × n + nb × n+1 , 2 × k + nb × nrhs ,$
where $k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$ and $\mathit{nb}$ is the optimal block size.
Constraint: ${\mathbf{lwork}}\ge k+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(2×k,{\mathbf{n}}+1,k+{\mathbf{nrhs}}\right)\text{, where ​}k=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left({\mathbf{m}},{\mathbf{n}}\right)$ or
${\mathbf{lwork}}=-1$.
13:   $\mathbf{info}$ – IntegerOutput
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.

## 7Accuracy

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

## 8Parallelism and Performance

f08baf (dgelsy) is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
f08baf (dgelsy) 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 complex analogue of this routine is f08bnf (zgelsy).

## 10Example

This example solves the linear least squares problem
 $minx b-Ax2$
for the solution, $x$, of minimum norm, where
 $A = -0.09 0.14 -0.46 0.68 1.29 -1.56 0.20 0.29 1.09 0.51 -1.48 -0.43 0.89 -0.71 -0.96 -1.09 0.84 0.77 2.11 -1.27 0.08 0.55 -1.13 0.14 1.74 -1.59 -0.72 1.06 1.24 0.34 and b= 7.4 4.2 -8.3 1.8 8.6 2.1 .$
A tolerance of $0.01$ is used to determine the effective rank of $A$.
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 (f08bafe.f90)

### 10.2Program Data

Program Data (f08bafe.d)

### 10.3Program Results

Program Results (f08bafe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017