NAG Library Routine Document

E02GCF

+− 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

E02GCF calculates an

l_{\infty}

solution to an over-determined system of linear equations.

2 Specification

SUBROUTINE E02GCF (

M, N, SDA, LDA, A, B, TOL, RELERR, X, RESMAX, IRANK, ITER, IFAIL)

INTEGER	M, N, SDA, LDA, IRANK, ITER, IFAIL
REAL (KIND=nag_wp)	A(LDA,SDA), B(M), TOL, RELERR, X(N), RESMAX

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_{\infty}

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_{\infty}

norm of the residuals (the absolutely largest residual)

r (x) = \max_{1 \leq i \leq 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. The solution is not unique in this case, and may not be unique even if

A

is of full rank.

Alternatively, in applications where a complete minimization of the

l_{\infty}

norm is not necessary, you may obtain an approximate solution, usually in shorter time, by giving an appropriate value to the parameter RELERR.

Typically in applications to data fitting, data consisting of

m

points with coordinates

(t_{i}, y_{i})

is to be approximated in the

l_{\infty}

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 finding an

l_{\infty}

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

above 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 dual formation of the

l_{\infty}

problem (see Barrodale and Phillips (1974) and Barrodale and Phillips (1975)). The modifications are designed to improve the efficiency and stability of the simplex method for this particular application.

4 References

Barrodale I and Phillips C (1974) An improved algorithm for discrete Chebyshev linear approximation Proc. 4th Manitoba Conf. Numerical Mathematics 177–190 University of Manitoba, Canada

Barrodale I and Phillips C (1975) Solution of an overdetermined system of linear equations in the Chebyshev norm [F4] (Algorithm 495) ACM Trans. Math. Software 1(3) 264–270

5 Parameters

1: M – INTEGERInput: On entry: the number of equations, $m$ (the number of rows of the matrix $A$ ).
Constraint: $M \geq N$ .
2: N – INTEGERInput: On entry: the number of unknowns, $n$ (the number of columns of the matrix $A$ ).
Constraint: $N \geq 1$ .
3: SDA – INTEGERInput: On entry: the second dimension of the array A as declared in the (sub)program from which E02GCF is called.
Constraint: $SDA \geq M + 1$ .
4: LDA – INTEGERInput: On entry: the first dimension of the array A as declared in the (sub)program from which E02GCF is called.
Constraint: $LDA \geq N + 3$ .
5: A(LDA,SDA) – REAL (KIND=nag_wp) arrayInput/Output: On entry: $A (j, i)$ 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 $j = 1, 2, \dots, n$ , (that is, the transpose of the matrix). The remaining elements need not be set. Preferably, the columns of the matrix $A$ (rows of the parameter A) should be scaled before entry: see Section 7.

On exit: contains the last simplex tableau.
6: 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 $i = 1, 2, \dots, m$ . Note however that these residuals may contain few significant figures, especially when RESMAX is within one or two orders of magnitude of TOL. Indeed if $RESMAX \leq TOL$ , the elements $B (i)$ may all be set to zero. It is therefore often advisable to compute the residuals directly.
7: TOL – REAL (KIND=nag_wp)Input: On entry: a threshold below which numbers are regarded as zero. The recommended threshold value is $10.0 \times ε$ , where $ε$ is the machine precision. If $TOL \leq 0.0$ on entry, the recommended value is used within the routine. If premature termination occurs, a larger value for TOL may result in a valid solution.

Suggested value: $0.0$ .
8: RELERR – REAL (KIND=nag_wp)Input/Output: On entry: must be set to a bound on the relative error acceptable in the maximum residual at the solution.
If $RELERR \leq 0.0$ , then the $l_{\infty}$ solution is computed, and RELERR is set to $0.0$ on exit.

If $RELERR > 0.0$ , then the routine obtains instead an approximate solution for which the largest residual is less than $1.0 + RELERR$ times that of the $l_{\infty}$ solution; on exit, RELERR contains a smaller value such that the above bound still applies. (The usual result of this option, say with $RELERR = 0.1$ , is a saving in the number of simplex iterations).

On exit: is altered as described above.
9: X(N) – REAL (KIND=nag_wp) arrayOutput: On exit: if $IFAIL = 0$ or $1$ , $X (j)$ contains the $j$ th element of the solution vector $x$ , for $j = 1, 2, \dots, n$ . Whether this is an $l_{\infty}$ solution or an approximation to one, depends on the value of RELERR on entry.
10: RESMAX – REAL (KIND=nag_wp)Output: On exit: if $IFAIL = 0$ or $1$ , RESMAX contains the absolute value of the largest residual(s) for the solution vector $x$ . (See B.)
11: IRANK – INTEGEROutput: On exit: if $IFAIL = 0$ or $1$ , IRANK contains the computed rank of the matrix $A$ .
12: ITER – INTEGEROutput: On exit: if $IFAIL = 0$ or $1$ , ITER contains the number of iterations taken by the simplex method.
13: 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, because for this routine the values of the output parameters may be useful even if $IFAIL \neq 0$ on exit, the recommended value is $- 1$ . 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).

Note: E02GCF may return useful information for one or more of the following detected errors or warnings.

Errors or warnings detected by the routine:

$IFAIL = 1$: An optimal solution has been obtained but this may not be unique (perhaps simply because the matrix $A$ is not of full rank, i.e., $IRANK < N$ ).

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

$IFAIL = 3$

On entry,	$LDA < N + 3$ ,
or	$SDA < M + 1$ ,
or	$M < N$ ,
or	$N < 1$ .

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 + 1

equations which have residuals of largest absolute value. 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 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 TOL 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

This example approximates 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^{t_{i}} K + e^{- t_{i}} L + M = y_{i}, i = 1, 2, \dots, 5 .

E02GCF is used to solve these in the

l_{\infty}

sense.

NAG Library Routine DocumentE02GCF