Interpolating function, bicubic spline interpolant, two variables

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NAG[e01dac] NAG[nag_2d_spline_interpolant] - Interpolating function, bicubic spline interpolant, two variables

Calling Sequence

e01dac(x, y, f, spline_data, 'mx'=mx, 'my'=my, 'fail'=fail)

nag_2d_spline_interpolant(. . .)

Parameters

x - Vector(1..mx, datatype=float[8]);
y - Vector(1..my, datatype=float[8]);

On entry: and must contain , for , and , for , respectively.

Constraints:

, for ;

, for .

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

On entry: must contain , for and .

spline_data - table;

A Maple table, which should be generated using NAG[Nag_2dSpline], corresponding to the Nag_2dSpline structure.

The following fields may be accessed via NAG[GetOptions]:

nx - integer

ny - integer

On exit: nx and ny contain and , the total number of knots of the computed spline with respect to the and variables, respectively.

lamda - float

On exit: the pointer to which memory of size nx is internally allocated. lamda contains the complete set of knots associated with the variable, i.e., the interior knots , , , , as well as the additional knots and needed for the B-spline representation.

mu - float

On exit: the pointer to which memory of size ny is internally allocated. mu contains the corresponding complete set of knots associated with the variable.

c - float

On exit: the pointer to which memory of size is internally allocated. c holds the coefficients of the spline interpolant. contains the coefficient described in Section [Description].

'mx'=mx - integer; (optional)
'my'=my - integer; (optional)

Default value: the first dimension of the arrays x, ythe arrays x, y.

On entry: mx and my must specify and respectively, the number of points along the and axis that define the rectangular grid.

Constraint: and . .

'fail'=fail - table; (optional)

The NAG error argument, see the documentation for NagError.

Description

	Purpose
	nag_2d_spline_interpolant (e01dac) computes a bicubic spline interpolating surface through a set of data values, given on a rectangular grid in the - plane.

Description

nag_2d_spline_interpolant (e01dac) determines a bicubic spline interpolant to the set of data points , for and . The spline is given in the B-spline representation

s(x,,,y)=(∑)(∑)c[ij]M[i](x)N[j](y)

such that

where and denote normalized cubic B-splines, the former defined on the knots to and the latter on the knots to , and the are the spline coefficients. These knots, as well as the coefficients, are determined by the function, which is derived from the routine B2IRE in Anthony et al. (1982). The method used is described in Section [Outline of Method Used].

For further information on splines, see Hayes and Halliday (1974) for bicubic splines and De Boor (1972) for normalized B-splines.

Values of the computed spline can subsequently be obtained by calling e02dec (nag_2d_spline_eval) or e02dfc (nag_2d_spline_eval_rect) as described in Section [Evaluation of Computed Spline].

	Error Indicators and Warnings
	"NE_ALLOC_FAIL" Dynamic memory allocation failed. "NE_DATA_ILL_CONDITIONED" An intermediate set of linear equations is singular, the data is too ill-conditioned to compute B-spline coefficients. "NE_INT_ARG_LT" On entry, mx must not be less than 4: . "NE_NOT_STRICTLY_INCREASING" The sequence x is not strictly increasing: , . The sequence y is not strictly increasing: , .

Accuracy

The main sources of rounding errors are in steps 1, 3, 6 and 7 of the algorithm described in Section [Outline of Method Used]. It can be shown (Cox (1975a)) that the matrix formed in step 2 has elements differing relatively from their true values by at most a small multiple of , where is the machine precision. is "totally positive", and a linear system with such a coefficient matrix can be solved quite safely by elimination without pivoting. Similar comments apply to steps 6 and 7. Thus the complete process is numerically stable.

Further Comments

The time taken by nag_2d_spline_interpolant (e01dac) is approximately proportional to .

Outline of Method Used

The process of computing the spline consists of the following steps:

choice of the interior -knots , as , for ,

formation of the system

where is a band matrix of order and bandwidth 4, containing in its th row the values at of the B-splines in , is the by rectangular matrix of values , and denotes an by rectangular matrix of intermediate coefficients,

use of Gaussian elimination to reduce this system to band triangular form,

solution of this triangular system for ,

choice of the interior knots , as , for ,

formation of the system

where is the counterpart of for the variable, and denotes the by rectangular matrix of values of ,

use of Gaussian elimination to reduce this system to band triangular form,

solution of this triangular system for and hence .

For computational convenience, steps 2 and 3, and likewise steps 6 and 7, are combined so that the formation of and and the reductions to triangular form are carried out one row at a time.

Evaluation of Computed Spline

The values of the computed spline at the points , for , may be obtained in the array ff, of length at least n, by the following call:

e02dec (tx, ty, ff, spline, 'm'=n)

where spline_data is a structure of type which is the output argument of nag_2d_spline_interpolant (e01dac).

To evaluate the computed spline on a kx by ky rectangular grid of points in the - plane, which is defined by the co-ordinates stored in , for , and the co-ordinates stored in , for , returning the results in the array fg which is of length at least , the following call may be used:

e02dfc (ky, tx, ty, fg, spline, 'mx'=kx)

where spline_data is a structure of type which is the output argument of nag_2d_spline_interpolant (e01dac). The result of the spline evaluated at grid point is returned in element of the array fg.

Examples

mx := 7:
my := 6:
spline_data := NAG:-Nag_2dSpline():
x := Vector([1, 1.1, 1.3, 1.5, 1.6, 1.8, 2], datatype=float[8]):
y := Vector([0, 0.1, 0.4, 0.7, 0.9, 1], datatype=float[8]):
f := Vector([1, 1.1, 1.4, 1.7, 1.9, 2, 1.21, 1.31, 1.61, 1.91, 2.11, 2.21, 1.69, 1.79, 2.09, 2.39, 2.59, 2.69, 2.25, 2.35, 2.65, 2.95, 3.15, 3.25, 2.56, 2.66, 2.96, 3.26, 3.46, 3.56, 3.24, 3.34, 3.64, 3.94, 4.14, 4.24, 4, 4.1, 4.4, 4.7, 4.9, 5], datatype=float[8]):
NAG:-e01dac(x, y, f, spline_data, 'mx' = mx, 'my' = my):