Vector(1..m, datatype=float[8]);

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NAG[e02gcc] NAG[nag_linf_fit] - -approximation by general linear function

Calling Sequence

e02gcc(a, b, relerr, x, resmax, rank, iter, 'm'=m, 'n'=n, 'tol'=tol, 'fail'=fail)

nag_linf_fit(. . .)

Parameters

a - Matrix(1.., 1.., datatype=float[8], order=order);

Note: this array may be supplied in Fortran_order or C_order , as specified by order. All array parameters must use a consistent order.

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

On entry: must contain , the element in the th row and th column of the matrix for, ; (that is, the transpose of the matrix). The remaining elements need not be set. Preferably, the columns of the matrix (rows of the argument a) should be scaled before entry: see Section [Accuracy].

On exit: contains the last simplex tableau.

b - Vector(1..m, datatype=float[8]);

On entry: must contain , the th element of the vector , for .

On exit: the th residual corresponding to the solution vector , for . 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 , the elements may all be set to zero. It is therefore often advisable to compute the residuals directly.

relerr - assignable;

Note: On exit the variable relerr will have a value of type float.

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

If , then the solution is computed, and relerr is set to on exit.

If , then the function obtains instead an approximate solution for which the largest residual is less than times that of the solution; on exit, relerr contains a smaller value such that the above bound still applies. (The usual result of this option, say with , is a saving in the number of simplex iterations).

On exit: is altered as described above.

x - Vector(1..n, datatype=float[8]);

On exit: if an optimal but not necessarily unique solution is found, contains the th element of the solution vector , for . Whether this is an solution or an approximation to one, depends on the value of relerr on entry.

resmax - assignable;

Note: On exit the variable resmax will have a value of type float.

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 . (See b.)

rank - assignable;

Note: On exit the variable rank will have a value of type integer.

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

iter - assignable;

Note: On exit the variable iter will have a value of type integer.

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

'm'=m - integer; (optional)

On entry: the number of equations, (the number of rows of the matrix ).

Constraint: . .

'n'=n - integer; (optional)

On entry: the number of unknowns, (the number of columns of the matrix ).

Constraint: . .

'tol'=tol - float; (optional)

On entry: a threshold below which numbers are regarded as zero. The recommended threshold value is , where is the machine precision. If 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: . (default: )

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_linf_fit (e02gcc) calculates an solution to an over-determined system of linear equations.

Description

Given a matrix with rows and columns and a vector with elements, the function calculates an solution to the over-determined system of equations

That is to say, it calculates a vector , with elements, which minimizes the norm of the residuals (the absolutely largest residual)

where the residuals are given by

Here is the element in row and column of , is the th element of and the th element of . The matrix need not be of full rank. The solution is not unique in this case, and may not be unique even if is of full rank.

Alternatively, in applications where a complete minimization of the 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 points with co-ordinates is to be approximated in the norm by a linear combination of known functions ,

This is equivalent to finding an solution to the over-determined system of equations

Thus if, for each value of and the element of the matrix above is set equal to the value of and is set equal to , the solution vector will contain the required values of the . Note that the independent variable above can, instead, be a vector of several independent variables (this includes the case where each 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 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.

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_INT"

On entry, . Constraint: .

"NE_INT_2"

On entry, , . Constraint: .

"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 consult 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: .

	Accuracy
	Experience suggests that the computational accuracy of the solution is comparable with the accuracy that could be obtained by applying Gaussian elimination with partial pivoting to the 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 and 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 and the total time is approximately proportional to .

It is recommended that, before the function is entered, the columns of the matrix 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 obtained will then, of course, relate to the scaled form of the matrix. Thus if the scaling is such that, for each , the elements of the th column are multiplied by the constant , the element of the solution vector must be multiplied by if it is desired to recover the solution corresponding to the original matrix .

Examples

m := 5:
n := 3:
tol := 0.0:
relerr := 0.0:
a := Matrix([[1, 1.22140275816017, 1.49182469764127, 1.822118800390509, 2.225540928492468, 0], [1, 0.8187307530779818, 0.6703200460356393, 0.5488116360940264, 0.4493289641172216, 0], [1, 1, 1, 1, 1, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0]], datatype=float[8]):
b := Vector([4.501, 4.36, 4.333, 4.418, 4.625], datatype=float[8]):
x := Vector(3, datatype=float[8]):
NAG:-e02gcc(a, b, relerr, x, resmax, rank, iter, 'm' = m, 'n' = n, 'tol' = tol):