NAG CL Interface
e02gac (glin_l1sol)
1
Purpose
e02gac calculates an ${l}_{1}$ solution to an overdetermined system of linear equations.
2
Specification
void 
e02gac (Nag_OrderType order,
Integer m,
double a[],
double b[],
Integer nplus2,
double toler,
double x[],
double *resid,
Integer *rank,
Integer *iter,
NagError *fail) 

The function may be called by the names: e02gac, nag_fit_glin_l1sol or nag_lone_fit.
3
Description
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 overdetermined system of equations
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
where the residuals
${r}_{i}$ are given by
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
${\varphi}_{j}\left(t\right)$,
This is equivalent to fitting an
${l}_{1}$ solution to the overdetermined system of equations
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.
4
References
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
5
Arguments

1:
$\mathbf{order}$ – Nag_OrderType
Input

On entry: the
order argument specifies the twodimensional storage scheme being used, i.e., rowmajor ordering or columnmajor 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:
$\mathbf{m}$ – Integer
Input

On entry: the number of equations, $m$ (the number of rows of the matrix $A$).
Constraint:
${\mathbf{m}}\ge n\ge 1$.

3:
$\mathbf{a}\left[\left({\mathbf{m}}+2\right)\times {\mathbf{nplus2}}\right]$ – double
Input/Output

Note: where
${\mathbf{A}}\left(i,j\right)$ appears in this document, it refers to the array element
 ${\mathbf{a}}\left[\left(j1\right)\times \left(\left({\mathbf{m}}+2\right)\right)+i1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
 ${\mathbf{a}}\left[\left(i1\right)\times {\mathbf{nplus2}}+j1\right]$ 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:
$\mathbf{b}\left[{\mathbf{m}}\right]$ – double
Input/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:
$\mathbf{nplus2}$ – Integer
Input

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:
$\mathbf{toler}$ – double
Input

On entry: a nonnegative 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:
$\mathbf{x}\left[{\mathbf{nplus2}}\right]$ – double
Output

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 7 in the Introduction to the NAG Library CL Interface).
6
Error 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.
 NE_BAD_PARAM

On entry, argument $\u2329\mathit{\text{value}}\u232a$ had an illegal value.
 NE_INT

On entry, ${\mathbf{nplus2}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{nplus2}}\ge 3$.
 NE_INT_2

On entry, ${\mathbf{nplus2}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
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 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_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:
${\mathbf{toler}}=\u2329\mathit{\text{value}}\u232a$.
 NE_TOO_MANY_ITER

More than
$1000*n$ iterations were performed.
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.
7
Accuracy
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.
8
Parallelism and Performance
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
$A$.
10
Example
Suppose we wish to approximate a set of data by a curve of the form
where
$K$,
$L$ and
$M$ are unknown. Given values
${y}_{i}$ at
$5$ points
${t}_{i}$ we may form the overdetermined set of equations for
$K$,
$L$ and
$M$
e02gac is used to solve these in the
${l}_{1}$ sense.
10.1
Program Text
10.2
Program Data
10.3
Program Results