Least-squares polynomial fit, values and derivatives may be constrained, arbitrary data points

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NAG[e02agc] NAG[nag_1d_cheb_fit_constr] - Least-squares polynomial fit, values and derivatives may be constrained, arbitrary data points

Calling Sequence

e02agc(k, xmin, xmax, x, y, w, xf, yf, p, a, s, n, resid, 'm'=m, 'mf'=mf, 'fail'=fail)

nag_1d_cheb_fit_constr(. . .)

Parameters

k - integer;

On entry: , the maximum degree required.

Constraint: where is the total number of constraints and is the number of data points with non-zero weights and distinct abscissae which do not coincide with any of the . .

xmin - float;
xmax - float;

On entry: the lower and upper end points, respectively, of the interval . Unless there are specific reasons to the contrary, it is recommended that xmin and xmax be set respectively to the lowest and highest value among the and . This avoids the danger of extrapolation provided there is a constraint point or data point with non-zero weight at each end point.

Constraint: . .

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

On entry: must contain the value of the independent variable at the th data point, for .

Constraint: the must be in non-decreasing order and satisfy . .

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

On entry: must contain , the value of the dependent variable at the th data point, for .

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

On entry: must contain the weight to be applied to the data point , for . For advice on the choice of weights see the e02 Chapter Introduction. Negative weights are treated as positive. A zero weight causes the corresponding data point to be ignored. Zero weight should be given to any data point whose and values both coincide with those of a constraint (otherwise the denominators involved in the root-mean-square residuals will be slightly in error).

xf - Vector(1..mf, datatype=float[8]);

On entry: must contain the th value of the independent variable at which a constraint is specified, for .

Constraint: these values need not be ordered but must be distinct and satisfy . .

yf - Vector(1.. mf+sum(p[i], i = 1 .. mf) , datatype=float[8]);

Note: the dimension, dim, of the array yf must be at least mf+sum(p[i], i = 1 .. mf) .

On entry: the values which the approximating polynomials and their derivatives are required to take at the points specified in xf. For each value of , yf contains in successive elements the required value of the approximation, its first derivative, second derivative, th derivative, for . Thus the value which the th derivative of each approximation ( referring to the approximation itself) is required to take at the point must be contained in , where

for ; . The derivatives are with respect to your variable .

p - Vector(1..mf, datatype=integer[kernelopts('wordsize')/8]);

On entry: must contain , the order of the highest-order derivative specified at , for . implies that the value of the approximation at is specified, but not that of any derivative.

Constraint: , for . .

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 exit: contains the coefficient in the approximating polynomial of degree , for ; .

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

On exit: contains , for , the root-mean-square residual corresponding to the approximating polynomial of degree . In the case where the number of data points with non-zero weight is equal to , is indeterminate: the function sets it to zero. For the interpretation of the values of and their use in selecting an appropriate degree, see Section [General] in the e02 Chapter Introduction.

n - assignable;

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

On exit: contains the total number of constraint conditions imposed: .

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

On exit: contains weighted residuals of the highest degree of fit determined . The residual at is in element , for .

'm'=m - integer; (optional)

On entry: the number of data points to be fitted.

Constraint: . .

'mf'=mf - integer; (optional)

On entry: the number, of values of the independent variable at which a constraint is specified.

Constraint: . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_1d_cheb_fit_constr (e02agc) computes constrained weighted least-squares polynomial approximations in Chebyshev-series form to an arbitrary set of data points. The values of the approximations and any number of their derivatives can be specified at selected points.

Description

nag_1d_cheb_fit_constr (e02agc) determines least-squares polynomial approximations of degrees up to to the set of data points with weights , for . The value of , the maximum degree required, is prescribed by you. At each of the values , for , of the independent variable , the approximations and their derivatives up to order are constrained to have one of the user-specified values , for , where n = mf+sum(p[r], r = 0 .. mf) .

The approximation of degree has the property that, subject to the imposed constraints, it minimizes , the sum of the squares of the weighted residuals , for , where

and is the value of the polynomial approximation of degree at the th data point.

Each polynomial is represented in Chebyshev-series form with normalized argument . This argument lies in the range to and is related to the original variable by the linear transformation

x=(2x-(x[max],+,x[min]))/(x[max],-,x[min])

where and , specified by you, are respectively the lower and upper end points of the interval of over which the polynomials are to be defined.

The polynomial approximation of degree can be written as

where is the Chebyshev polynomial of the first kind of degree with argument . For , the function produces the values of the coefficients , for , together with the value of the root-mean-square residual, , defined as

sqrt(((sigma)[i])/(m',+,n,-,i,-,1))

where is the number of data points with non-zero weight.

Values of the approximations may subsequently be computed using e02aec (nag_1d_cheb_eval) or e02akc (nag_1d_cheb_eval2).

First nag_1d_cheb_fit_constr (e02agc) determines a polynomial , of degree , which satisfies the given constraints, and a polynomial , of degree , which has value (or derivative) zero wherever a constrained value (or derivative) is specified. It then fits , for , with polynomials of the required degree in each with factor . Finally the coefficients of are added to the coefficients of these fits to give the coefficients of the constrained polynomial approximations to the data points , for . The method employed is given in Hayes (1970): it is an extension of Forsythe's orthogonal polynomials method (see Forsythe (1957)) as modified by Clenshaw (see Clenshaw (1960)).

Error Indicators and Warnings

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument had an illegal value.

"NE_ILL_CONDITIONED"

The polynomials and/or cannot be found. The problem is too ill-conditioned.

"NE_INT"

On entry, . Constraint: .

"NE_INT_3"

On entry, , where is the number of data points with non-zero weight and distinct abscissae different from all the xf, and n is the total number of constraints: , , .

"NE_INT_ARRAY"

On entry, , . Constraint: , for .

"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_NOT_MONOTONIC"

On entry, : , , .

"NE_REAL_2"

On entry, : , .

"NE_REAL_ARRAY"

On entry, : , , , .

On entry, lies outside interval : , , , .

On entry, lies outside interval for some .

	Accuracy
	No complete error analysis exists for either the interpolating algorithm or the approximating algorithm. However, considerable experience with the approximating algorithm shows that it is generally extremely satisfactory. Also the moderate number of constraints, of low-order, which are typical of data fitting applications, are unlikely to cause difficulty with the interpolating function.

	Further Comments
	The time taken to form the interpolating polynomial is approximately proportional to , and that to form the approximating polynomials is very approximately proportional to . To carry out a least-squares polynomial fit without constraints, use e02adc (nag_1d_cheb_fit). To carry out polynomial interpolation only, use e01aec (nag_1d_cheb_interp).

Examples

m := 5:
k := 4:
xmin := 0:
xmax := 4:
mf := 2:
x := Vector([0.5, 1, 2, 2.5, 3], datatype=float[8]):
y := Vector([0.03, -0.75, -1, -0.1, 1.75], datatype=float[8]):
w := Vector([1, 1, 1, 1, 1], datatype=float[8]):
xf := Vector([0, 4], datatype=float[8]):
yf := Vector([1, -2, 9], datatype=float[8]):
p := Vector([1, 0], datatype=integer[kernelopts('wordsize')/8]):
a := Matrix(5, 5, datatype=float[8]):
s := Vector(5, datatype=float[8]):
resid := Vector(5, datatype=float[8]):
NAG:-e02agc(k, xmin, xmax, x, y, w, xf, yf, p, a, s, n, resid, 'm' = m, 'mf' = mf):

resid;

	See Also
	Clenshaw C W (1960) Curve fitting with a digital computer Comput. J. 2 170–173 Forsythe G E (1957) Generation and use of orthogonal polynomials for data fitting with a digital computer J. Soc. Indust. Appl. Math. 5 74–88 Hayes J G (ed.) (1970) Numerical Approximation to Functions and Data Athlone Press, London e02 Chapter Introduction. NAG Toolbox Overview. NAG Web Site.