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NAG Toolbox

NAG Toolbox: nag_fit_glin_linf (e02gc)

Purpose

nag_fit_glin_linf (e02gc) calculates an ll solution to an over-determined system of linear equations.

Syntax

[a, b, relerr, x, resmax, irank, iter, ifail] = e02gc(n, a, b, relerr, 'm', m, 'tol', tol)
[a, b, relerr, x, resmax, irank, iter, ifail] = nag_fit_glin_linf(n, a, b, relerr, 'm', m, 'tol', tol)

Description

Given a matrix AA with mm rows and nn columns (mn)(mn) and a vector bb with mm elements, the function calculates an ll solution to the over-determined system of equations
Ax = b.
Ax=b.
That is to say, it calculates a vector xx, with nn elements, which minimizes the ll norm of the residuals (the absolutely largest residual)
r(x) = max |ri|
1im
r(x) = max 1im |ri|
where the residuals riri are given by
n
ri = biaijxj,  i = 1,2,,m.
j = 1
ri = bi - j=1n aij xj ,   i=1,2,,m .
Here aijaij is the element in row ii and column jj of AA, bibi is the iith element of bb and xjxj the jjth element of xx. The matrix AA need not be of full rank. The solution is not unique in this case, and may not be unique even if AA is of full rank.
Alternatively, in applications where a complete minimization of the ll 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 mm points with coordinates (ti,yi)(ti,yi) is to be approximated in the ll norm by a linear combination of known functions φj(t)ϕj(t),
α1φ1(t) + α2φ2(t) + + αnφn(t).
α1ϕ1(t)+α2ϕ2(t)++αnϕn(t).
This is equivalent to finding an ll solution to the over-determined system of equations
n
φj(ti)αj = yi,  i = 1,2,,m.
j = 1
j=1n ϕj (ti) αj = yi ,   i=1,2,,m .
Thus if, for each value of ii and jj the element aijaij of the matrix AA above is set equal to the value of φj(ti)ϕj(ti) and bibi is set equal to yiyi, the solution vector xx will contain the required values of the αjαj. Note that the independent variable tt above can, instead, be a vector of several independent variables (this includes the case where each φiϕi 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 ll 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.

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

Parameters

Compulsory Input Parameters

1:     n – int64int32nag_int scalar
The number of unknowns, nn (the number of columns of the matrix AA).
Constraint: n1n1.
2:     a(lda,sda) – double array
lda, the first dimension of the array, must satisfy the constraint ldan + 3ldan+3.
a(j,i)aji must contain aijaij, the element in the iith row and jjth column of the matrix AA, for i = 1,2,,mi=1,2,,m and j = 1,2,,nj=1,2,,n, (that is, the transpose of the matrix). The remaining elements need not be set. Preferably, the columns of the matrix AA (rows of the parameter a) should be scaled before entry: see Section [Accuracy].
3:     b(m) – double array
m, the dimension of the array, must satisfy the constraint mnmn.
b(i)bi must contain bibi, the iith element of the vector bb, for i = 1,2,,mi=1,2,,m.
4:     relerr – double scalar
Must be set to a bound on the relative error acceptable in the maximum residual at the solution.
If relerr0.0relerr0.0, then the ll solution is computed, and relerr is set to 0.00.0 on exit.
If relerr > 0.0relerr>0.0, then the function obtains instead an approximate solution for which the largest residual is less than 1.0 + relerr1.0+relerr times that of the ll solution; on exit, relerr contains a smaller value such that the above bound still applies. (The usual result of this option, say with relerr = 0.1relerr=0.1, is a saving in the number of simplex iterations).

Optional Input Parameters

1:     m – int64int32nag_int scalar
Default: The dimension of the array b.
The number of equations, mm (the number of rows of the matrix AA).
Constraint: mnmn.
2:     tol – double scalar
A threshold below which numbers are regarded as zero. The recommended threshold value is 10.0 × ε10.0×ε, where εε is the machine precision. If tol0.0tol0.0 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.
Default: 0.00.0.

Input Parameters Omitted from the MATLAB Interface

sda lda

Output Parameters

1:     a(lda,sda) – double array
sdam + 1sdam+1.
ldan + 3ldan+3.
Contains the last simplex tableau.
2:     b(m) – double array
The iith residual riri corresponding to the solution vector xx, for i = 1,2,,mi=1,2,,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 resmaxtolresmaxtol, the elements b(i)bi may all be set to zero. It is therefore often advisable to compute the residuals directly.
3:     relerr – double scalar
Is altered as described above.
4:     x(n) – double array
If ifail = 0ifail=0 or 11, x(j)xj contains the jjth element of the solution vector xx, for j = 1,2,,nj=1,2,,n. Whether this is an ll solution or an approximation to one, depends on the value of relerr on entry.
5:     resmax – double scalar
If ifail = 0ifail=0 or 11, resmax contains the absolute value of the largest residual(s) for the solution vector xx. (See b.)
6:     irank – int64int32nag_int scalar
If ifail = 0ifail=0 or 11, irank contains the computed rank of the matrix AA.
7:     iter – int64int32nag_int scalar
If ifail = 0ifail=0 or 11, iter contains the number of iterations taken by the simplex method.
8:     ifail – int64int32nag_int scalar
ifail = 0ifail=0 unless the function detects an error (see [Error Indicators and Warnings]).

Error Indicators and Warnings

Note: nag_fit_glin_linf (e02gc) may return useful information for one or more of the following detected errors or warnings.
Errors or warnings detected by the function:

Cases prefixed with W are classified as warnings and do not generate an error of type NAG:error_n. See nag_issue_warnings.

W ifail = 1ifail=1
An optimal solution has been obtained but this may not be unique (perhaps simply because the matrix AA is not of full rank, i.e., irank < nirank<n).
  ifail = 2ifail=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 [Further Comments]).
  ifail = 3ifail=3
On entry,lda < n + 3lda<n+3,
orsda < m + 1sda<m+1,
orm < nm<n,
orn < 1n<1.

Accuracy

Experience suggests that the computational accuracy of the solution xx is comparable with the accuracy that could be obtained by applying Gaussian elimination with partial pivoting to the n + 1n+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.

Further Comments

The effects of mm and nn 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 nn and the total time is approximately proportional to mn2mn2.
It is recommended that, before the function is entered, the columns of the matrix AA 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 xx 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,,nj=1,2,,n, the elements of the jjth column are multiplied by the constant kjkj, the element xjxj of the solution vector xx must be multiplied by kjkj if it is desired to recover the solution corresponding to the original matrix AA.

Example

function nag_fit_glin_linf_example
n = int64(3);
a = zeros(6, 6);
for i = 1:5
  a(1, i) = exp((i-1)/5);
  a(2, i) = exp(-(i-1)/5);
  a(3, i) = 1;
end
b = [4.501;
     4.36;
     4.333;
     4.418;
     4.625];
relerr = 0;
[aOut, bOut, relerrOut, x, resmax, irank, iter, ifail] = nag_fit_glin_linf(n, a, b, relerr)
 

aOut =

   -3.0207   -5.5042    8.8604    1.0000    0.3355    6.0000
   -1.4796   -4.9123    5.4459   -0.3289    0.0541   -4.0000
    3.0207    5.5042   -8.3604   -0.0000    0.1645    8.0000
    1.4796    4.9123   -5.9459    0.3289    0.4459   -7.0000
    1.0049    2.0149    1.4822    0.0003    0.0010         0
    1.0000    2.0000    3.0000    5.0000         0         0


bOut =

   -0.0010
    0.0007
    0.0010
   -0.0010
    0.0010


relerrOut =

     0


x =

    1.0049
    2.0149
    1.4822


resmax =

    0.0010


irank =

                    3


iter =

                    4


ifail =

                    0


function e02gc_example
n = int64(3);
a = zeros(6, 6);
for i = 1:5
  a(1, i) = exp((i-1)/5);
  a(2, i) = exp(-(i-1)/5);
  a(3, i) = 1;
end
b = [4.501;
     4.36;
     4.333;
     4.418;
     4.625];
relerr = 0;
[aOut, bOut, relerrOut, x, resmax, irank, iter, ifail] = e02gc(n, a, b, relerr)
 

aOut =

   -3.0207   -5.5042    8.8604    1.0000    0.3355    6.0000
   -1.4796   -4.9123    5.4459   -0.3289    0.0541   -4.0000
    3.0207    5.5042   -8.3604   -0.0000    0.1645    8.0000
    1.4796    4.9123   -5.9459    0.3289    0.4459   -7.0000
    1.0049    2.0149    1.4822    0.0003    0.0010         0
    1.0000    2.0000    3.0000    5.0000         0         0


bOut =

   -0.0010
    0.0007
    0.0010
   -0.0010
    0.0010


relerrOut =

     0


x =

    1.0049
    2.0149
    1.4822


resmax =

    0.0010


irank =

                    3


iter =

                    4


ifail =

                    0



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Chapter Introduction
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