NAG Library Routine Document

E02GAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

E02GAF calculates an

l_{1}

solution to an over-determined system of linear equations.

2 Specification

SUBROUTINE E02GAF (

M, A, LDA, B, NPLUS2, TOLER, X, RESID, IRANK, ITER, IWORK, IFAIL)

INTEGER	M, LDA, NPLUS2, IRANK, ITER, IWORK(M), IFAIL
REAL (KIND=nag_wp)	A(LDA,NPLUS2), B(M), TOLER, X(NPLUS2), RESID

3 Description

Given a matrix

A

with

m

rows and

n

columns

(m \geq n)

and a vector

b

with

m

elements, the routine calculates an

l_{1}

solution to the over-determined system of equations

A x = 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

r (x) = \sum_{i = 1}^{m} |r_{i}|,

where the residuals

r_{i}

are given by

r_{i} = b_{i} - \sum_{j = 1}^{n} a_{i j} x_{j}, i = 1, 2, \dots, m .

Here

a_{i j}

is the element in row

i

and column

j

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

ϕ_{j} (t)

α_{1} ϕ_{1} (t) + α_{2} ϕ_{2} (t) + \dots + α_{n} ϕ_{n} (t) .

This is equivalent to fitting an

l_{1}

solution to the over-determined system of equations

\sum_{j = 1}^{n} ϕ_{j} (t_{i}) α_{j} = y_{i}, i = 1, 2, \dots, m .

Thus if, for each value of

i

and

j

, the element

a_{i j}

of the matrix

A

in the previous paragraph is set equal to the value of

ϕ_{j} (t_{i})

and

b_{i}

is set equal to

y_{i}

, the solution vector

x

will contain the required values of the

α_{j}

. Note that the independent variable

t

above can, instead, be a vector of several independent variables (this includes the case where each

ϕ_{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 Parameters

1: M – INTEGERInput: On entry: the number of equations, $m$ (the number of rows of the matrix $A$ ).
Constraint: $M \geq n \geq 1$ .
2: A(LDA,NPLUS2) – REAL (KIND=nag_wp) arrayInput/Output: On entry: $A (i, j)$ must contain $a_{i j}$ , the element in the $i$ th row and $j$ th column of the matrix $A$ , for $i = 1, 2, \dots, m$ and $j = 1, 2, \dots, n$ . The remaining elements need not be set.

On exit: contains the last simplex tableau generated by the simplex method.
3: LDA – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which E02GAF is called.
Constraint: $LDA \geq M + 2$ .
4: B(M) – REAL (KIND=nag_wp) arrayInput/Output: On entry: $B (i)$ must contain $b_{i}$ , the $i$ th element of the vector $b$ , for $i = 1, 2, \dots, m$ .

On exit: the $i$ th residual $r_{i}$ corresponding to the solution vector $x$ , for $i = 1, 2, \dots, m$ .
5: NPLUS2 – INTEGERInput: On entry: $n + 2$ , where $n$ is the number of unknowns (the number of columns of the matrix $A$ ).
Constraint: $3 \leq NPLUS2 \leq M + 2$ .
6: TOLER – REAL (KIND=nag_wp)Input: On entry: a non-negative value. In general TOLER specifies a threshold below which numbers are regarded as zero. The recommended threshold value is $ε^{2 / 3}$ where $ε$ is the machine precision. The recommended value can be computed within the routine 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: X(NPLUS2) – REAL (KIND=nag_wp) arrayOutput: On exit: $X (j)$ contains the $j$ th element of the solution vector $x$ , for $j = 1, 2, \dots, n$ . The elements $X (n + 1)$ and $X (n + 2)$ are unused.
8: RESID – REAL (KIND=nag_wp)Output: On exit: the sum of the absolute values of the residuals for the solution vector $x$ .
9: IRANK – INTEGEROutput: On exit: the computed rank of the matrix $A$ .
10: ITER – INTEGEROutput: On exit: the number of iterations taken by the simplex method.
11: IWORK(M) – INTEGER arrayWorkspace
12: IFAIL – INTEGERInput/Output: On entry: IFAIL must be set to $0$ , $- 1 or 1$ . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $- 1 or 1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is $0$ . When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit: $IFAIL = 0$ unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$: An optimal solution has been obtained but this may not be unique.

$IFAIL = 2$: The calculations have terminated prematurely due to rounding errors. Experiment with larger values of TOLER or try scaling the columns of the matrix (see Section 8).

$IFAIL = 3$

On entry,	$NPLUS2 < 3$ ,
or	$NPLUS2 > M + 2$ ,
or	$LDA < M + 2$ .

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 Further 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 routine 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 parameter 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

9 Example

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}

5

points

t_{i}

we may form the over-determined set of equations for

K

L

and

M

e^{x_{i}} K + e^{- x_{i}} L + M = y_{i}, i = 1, 2, \dots, 5 .

E02GAF is used to solve these in the

l_{1}

sense.

NAG Library Routine DocumentE02GAF