The function may be called by the names: e02gac, nag_fit_glin_l1sol or nag_lone_fit.
3Description
Given a matrix $A$ with $m$ rows and $n$ columns $(m\ge n)$ and a vector $b$ with $m$ elements, the function calculates an ${l}_{1}$ solution to the overdetermined system of equations
$$Ax=b\text{.}$$
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
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 $({t}_{i},{y}_{i})$ are to be approximated in the ${l}_{1}$ norm by a linear combination of known functions ${\varphi}_{j}\left(t\right)$,
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. ACM17(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 3.1.3 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: $\mathbf{m}$ – IntegerInput
On entry: the number of equations, $m$ (the number of rows of the matrix $A$).
Note: where ${\mathbf{A}}(i,j)$ appears in this document, it refers to the array element
${\mathbf{a}}\left[(j-1)\times \left(\left({\mathbf{m}}+2\right)\right)+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_ColMajor}$;
${\mathbf{a}}\left[(i-1)\times {\mathbf{nplus2}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag\_RowMajor}$.
On entry: ${\mathbf{A}}(\mathit{i},\mathit{j})$ 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.
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 $A$).
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.
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).
6Error 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 $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{nplus2}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nplus2}}\ge 3$.
NE_INT_2
On entry, ${\mathbf{nplus2}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{m}}=\u27e8\mathit{\text{value}}\u27e9$.
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: ${\mathbf{toler}}=\u27e8\mathit{\text{value}}\u27e9$.
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.
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
e02gac is not threaded in any implementation.
9Further Comments
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
$$y=K{e}^{t}+L{e}^{-t}+M$$
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 $M$