NAG Library Function Document

nag_linf_fit (e02gcc)

void	nag_linf_fit (Nag_OrderType order, Integer m, Integer n, double a[], double b[], double tol, double relerr, double x[], double resmax, Integer rank, Integer iter, NagError *fail)

3 Description

Given a matrix

A

with

m

rows and

n

columns

(m \geq n)

and a vector

b

with

m

elements, the function 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 argument 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 Arguments

1: 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

order = Nag_RowMajor

. See Section 3.2.1.3 in the Essential Introduction for a more detailed explanation of the use of this argument.

Constraint:

order = Nag_RowMajor

Nag_ColMajor

2: m – IntegerInput

On entry: the number of equations,

m

(the number of rows of the matrix

A

Constraint:

m \geq n

3: n – IntegerInput

On entry: the number of unknowns,

n

(the number of columns of the matrix

A

Constraint:

n \geq 1

4: a[ $\dim$ ] – doubleInput/Output

Note: the dimension, dim, of the array a must be at least

(n + 3) \times (m + 1)

Where

A (j, i)

appears in this document, it refers to the array element

$a [(i - 1) \times (n + 3) + j - 1]$ when $order = Nag_ColMajor$ ;
$a [(j - 1) \times (m + 1) + i - 1]$ when $order = Nag_RowMajor$ .

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 argument a) should be scaled before entry: see Section 7.

On exit: contains the last simplex tableau.

5: b[m] – doubleInput/Output

On entry:

b [i - 1]

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 - 1]

may all be set to zero. It is therefore often advisable to compute the residuals directly.

6: tol – doubleInput

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 function. If premature termination occurs, a larger value for tol may result in a valid solution.

Suggested value:

0.0

7: relerr – double *Input/Output

On entry: must be set to a bound on the relative error acceptable in the maximum residual at the solution.

relerr \leq 0.0

, then the

l_{\infty}

solution is computed, and relerr is set to

0.0

on exit.

relerr > 0.0

, then the function 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.

8: x[n] – doubleOutput

On exit: if an optimal but not necessarily unique solution is found,

x [j - 1]

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.

9: resmax – double *Output

On exit: if an optimal but not necessarily unique solution is found, resmax contains the absolute value of the largest residual(s) for the solution vector

x

. (See b.)

10: rank – Integer *Output

On exit: if an optimal but not necessarily unique solution is found, rank contains the computed rank of the matrix

A

11: iter – Integer *Output

On exit: if an optimal but not necessarily unique solution is found, iter contains the number of iterations taken by the simplex method.

12: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument $⟨value⟩$ had an illegal value.
NE_INT: On entry, $n = ⟨value⟩$ .
Constraint: $n \geq 1$ .
NE_INT_2: On entry, $m = ⟨value⟩$ and $n = ⟨value⟩$ .
Constraint: $m \geq n$ .
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.
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 tol: $tol = ⟨value⟩$ .

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 Parallelism and Performance

Not applicable.

9 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 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 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

10 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 .

nag_linf_fit (e02gcc) is used to solve these in the

l_{\infty}

sense.

NAG Library Function Documentnag_linf_fit (e02gcc)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_linf_fit (e02gcc)