# NAG Library Function Document

## 1Purpose

nag_lone_fit (e02gac) calculates an ${l}_{1}$ solution to an over-determined system of linear equations.

## 2Specification

 #include #include
 void nag_lone_fit (Nag_OrderType order, Integer m, double a[], double b[], Integer nplus2, double toler, double x[], double *resid, Integer *rank, Integer *iter, NagError *fail)

## 3Description

Given a matrix $A$ with $m$ rows and $n$ columns $\left(m\ge n\right)$ and a vector $b$ with $m$ elements, the function calculates an ${l}_{1}$ solution to the over-determined system of equations
 $Ax=b.$
That is to say, it calculates a vector $x$, with $n$ elements, which minimizes the ${l}_{1}$ norm (the sum of the absolute values) of the residuals
 $rx=∑i=1mri,$
where the residuals ${r}_{i}$ are given by
 $ri=bi-∑j=1naijxj, i=1,2,…,m.$
Here ${a}_{ij}$ is the element in row $i$ and column $j$ of $A$, ${b}_{i}$ is the $i$th element of $b$ and ${x}_{j}$ the $j$th element of $x$. The matrix $A$ need not be of full rank.
Typically in applications to data fitting, data consisting of $m$ points with coordinates $\left({t}_{i},{y}_{i}\right)$ are to be approximated in the ${l}_{1}$ norm by a linear combination of known functions ${\varphi }_{j}\left(t\right)$,
 $α1ϕ1t+α2ϕ2t+⋯+αnϕnt.$
This is equivalent to fitting an ${l}_{1}$ solution to the over-determined system of equations
 $∑j=1nϕjtiαj=yi, i=1,2,…,m.$
Thus if, for each value of $i$ and $j$, the element ${a}_{ij}$ of the matrix $A$ in the previous paragraph is set equal to the value of ${\varphi }_{j}\left({t}_{i}\right)$ and ${b}_{i}$ is set equal to ${y}_{i}$, the solution vector $x$ will contain the required values of the ${\alpha }_{j}$. Note that the independent variable $t$ above can, instead, be a vector of several independent variables (this includes the case where each ${\varphi }_{i}$ is a function of a different variable, or set of variables).
The algorithm is a modification of the simplex method of linear programming applied to the primal formulation of the ${l}_{1}$ problem (see Barrodale and Roberts (1973) and Barrodale and Roberts (1974)). The modification allows several neighbouring simplex vertices to be passed through in a single iteration, providing a substantial improvement in efficiency.

## 4References

Barrodale I and Roberts F D K (1973) An improved algorithm for discrete ${l}_{1}$ linear approximation SIAM J. Numer. Anal. 10 839–848
Barrodale I and Roberts F D K (1974) Solution of an overdetermined system of equations in the ${l}_{1}$-norm Comm. ACM 17(6) 319–320

## 5Arguments

1:    $\mathbf{order}$Nag_OrderTypeInput
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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{m}$IntegerInput
On entry: the number of equations, $m$ (the number of rows of the matrix $A$).
Constraint: ${\mathbf{m}}\ge n\ge 1$.
3:    $\mathbf{a}\left[\left({\mathbf{m}}+2\right)×{\mathbf{nplus2}}\right]$doubleInput/Output
Note: where ${\mathbf{A}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{a}}\left[\left(j-1\right)×\left(\left({\mathbf{m}}+2\right)\right)+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{a}}\left[\left(i-1\right)×{\mathbf{nplus2}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: ${\mathbf{A}}\left(\mathit{i},\mathit{j}\right)$ must contain ${a}_{\mathit{i}\mathit{j}}$, the element in the $\mathit{i}$th row and $\mathit{j}$th column of the matrix $A$, for $\mathit{i}=1,2,\dots ,m$ and $\mathit{j}=1,2,\dots ,n$. The remaining elements need not be set.
On exit: contains the last simplex tableau generated by the simplex method.
4:    $\mathbf{b}\left[{\mathbf{m}}\right]$doubleInput/Output
On entry: ${\mathbf{b}}\left[\mathit{i}-1\right]$ must contain ${b}_{\mathit{i}}$, the $\mathit{i}$th element of the vector $b$, for $\mathit{i}=1,2,\dots ,m$.
On exit: the $\mathit{i}$th residual ${r}_{\mathit{i}}$ corresponding to the solution vector $x$, for $\mathit{i}=1,2,\dots ,m$.
5:    $\mathbf{nplus2}$IntegerInput
On entry: $n+2$, where $n$ is the number of unknowns (the number of columns of the matrix $A$).
Constraint: $3\le {\mathbf{nplus2}}\le {\mathbf{m}}+2$.
6:    $\mathbf{toler}$doubleInput
On entry: a non-negative value. In general toler specifies a threshold below which numbers are regarded as zero. The recommended threshold value is ${\epsilon }^{2/3}$ where $\epsilon$ is the machine precision. The recommended value can be computed within the function by setting toler to zero. If premature termination occurs a larger value for toler may result in a valid solution.
Suggested value: $0.0$.
7:    $\mathbf{x}\left[{\mathbf{nplus2}}\right]$doubleOutput
On exit: ${\mathbf{x}}\left[\mathit{j}-1\right]$ contains the $\mathit{j}$th element of the solution vector $x$, for $\mathit{j}=1,2,\dots ,n$. The elements ${\mathbf{x}}\left[n\right]$ and ${\mathbf{x}}\left[n+1\right]$ are unused.
8:    $\mathbf{resid}$double *Output
On exit: the sum of the absolute values of the residuals for the solution vector $x$.
9:    $\mathbf{rank}$Integer *Output
On exit: the computed rank of the matrix $A$.
10:  $\mathbf{iter}$Integer *Output
On exit: the number of iterations taken by the simplex method.
11:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{nplus2}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nplus2}}\ge 3$.
NE_INT_2
On entry, ${\mathbf{nplus2}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: $3\le {\mathbf{nplus2}}\le {\mathbf{m}}+2$.
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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NON_UNIQUE
An optimal solution has been obtained, but may not be unique.
NE_TERMINATION_FAILURE
Premature termination due to rounding errors. Try using larger value of toler: ${\mathbf{toler}}=〈\mathit{\text{value}}〉$.
NE_TOO_MANY_ITER
More than $1000*n$ iterations were performed. nag_lone_fit (e02gac) has terminated without calculating a solution. The output data from the function is as computed on the last good iteration. Consider increasing the value of toler. Alternatively, $A$ may be ill conditioned—try scaling its columns.

## 7Accuracy

Experience suggests that the computational accuracy of the solution $x$ is comparable with the accuracy that could be obtained by applying Gaussian elimination with partial pivoting to the $n$ equations satisfied by this algorithm (i.e., those equations with zero residuals). The accuracy therefore varies with the conditioning of the problem, but has been found generally very satisfactory in practice.

## 8Parallelism and Performance

nag_lone_fit (e02gac) is not threaded in any implementation.

The effects of $m$ and $n$ on the time and on the number of iterations in the Simplex Method vary from problem to problem, but typically the number of iterations is a small multiple of $n$ and the total time taken is approximately proportional to $m{n}^{2}$.
It is recommended that, before the function is entered, the columns of the matrix $A$ are scaled so that the largest element in each column is of the order of unity. This should improve the conditioning of the matrix, and also enable the argument toler to perform its correct function. The solution $x$ obtained will then, of course, relate to the scaled form of the matrix. Thus if the scaling is such that, for each $j=1,2,\dots ,n$, the elements of the $j$th column are multiplied by the constant ${k}_{j}$, the element ${x}_{j}$ of the solution vector $x$ must be multiplied by ${k}_{j}$ if it is desired to recover the solution corresponding to the original matrix $A$.

## 10Example

Suppose we wish to approximate a set of data by a curve of the form
 $y=Ket+Le-t+M$
where $K$, $L$ and $M$ are unknown. Given values ${y}_{i}$ at $5$ points ${t}_{i}$ we may form the over-determined set of equations for $K$, $L$ and $M$
 $exiK+e-xiL+M=yi, i=1,2,…,5.$
nag_lone_fit (e02gac) is used to solve these in the ${l}_{1}$ sense.

### 10.1Program Text

Program Text (e02gace.c)

### 10.2Program Data

Program Data (e02gace.d)

### 10.3Program Results

Program Results (e02gace.r)

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