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 9.1.

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

Values and derivatives of the computed spline can subsequently be computed by calling e02dec, e02dfc and e02dhc as described in Section 9.2.

4 References

Anthony G T, Cox M G and Hayes J G (1982) DASL – Data Approximation Subroutine Library National Physical Laboratory

Cox M G (1975) An algorithm for spline interpolation J. Inst. Math. Appl. 15 95–108

de Boor C (1972) On calculating with B-splines J. Approx. Theory 6 50–62

Hayes J G and Halliday J (1974) The least squares fitting of cubic spline surfaces to general data sets J. Inst. Math. Appl. 14 89–103

5 Arguments

1: $mx$ – Integer Input

2: $my$ – Integer Input

On entry: mx and my must specify

m_{x}

and

m_{y}

respectively, the number of points along the

x

and

y

axis that define the rectangular grid.

Constraint:

mx \geq 4

and

my \geq 4

3: $x [mx]$ – const double Input

4: $y [my]$ – const double Input

On entry:

x [q - 1]

and

y [r - 1]

must contain

x_{q}

, for

q = 1, 2, \dots, m_{x}

, and

y_{r}

, for

r = 1, 2, \dots, m_{y}

, respectively.

Constraints:

$x [q - 1] < x [q]$ , for $q = 1, 2, \dots, m_{x} - 1$ ;
$y [r - 1] < y [r]$ , for $r = 1, 2, \dots, m_{y} - 1$ .

5: $f [mx \times my]$ – const double Input

On entry:

f [m_{y} \times (q - 1) + r - 1]

must contain

f_{q, r}

, for

q = 1, 2, \dots, m_{x}

and

r = 1, 2, \dots, m_{y}

6: $spline$ – Nag_2dSpline *

Pointer to structure of type Nag_2dSpline with the following members:

nx – IntegerOutput
ny – IntegerOutput: On exit: $nx$ and $ny$ contain $m_{x} + 4$ and $m_{y} + 4$ , the total number of knots of the computed spline with respect to the $x$ and $y$ variables, respectively.

lamda – double *Output: On exit: the pointer to which memory of size $nx$ is internally allocated. $lamda$ contains the complete set of knots $λ_{i}$ associated with the $x$ variable, i.e., the interior knots $lamda [4]$ , $lamda [5]$ , $\dots$ , $lamda [nx - 5]$ , as well as the additional knots $lamda [0] = lamda [1] = lamda [2] = lamda [3] = x [0]$ and $lamda [nx - 4] = lamda [nx - 3] = lamda [nx - 2] = lamda [nx - 1] = x [mx - 1]$ needed for the B-spline representation.

mu – double *Output: On exit: the pointer to which memory of size $ny$ is internally allocated. $mu$ contains the corresponding complete set of knots $μ_{i}$ associated with the $y$ variable.

c – double *Output: On exit: the pointer to which memory of size $mx \times my$ is internally allocated. $c$ holds the coefficients of the spline interpolant. $c [m_{y} \times (i - 1) + j - 1]$ contains the coefficient $c_{i j}$ described in Section 3.

Note that when the information contained in the pointers

lamda

mu

and

c

is no longer of use, or before a new call to e01dac with the same spline, you should free these pointers using the NAG macro NAG_FREE. This storage will not have been allocated if this function returns with

fail . code \neq

NE_NOERROR.

7: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 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, $mx = 〈value〉$ .
Constraint: $mx \geq 4$ .

On entry, $my = 〈value〉$ .
Constraint: $my \geq 4$ .
NE_NOT_STRICTLY_INCREASING: The sequence x is not strictly increasing: $x [〈value〉] = 〈value〉$ , $x [〈value〉] = 〈value〉$ .
The sequence y is not strictly increasing: $y [〈value〉] = 〈value〉$ , $y [〈value〉] = 〈value〉$ .

7 Accuracy

The main sources of rounding errors are in steps 1, 3, 6 and 7 of the algorithm described in Section 9.1. It can be shown (Cox (1975)) that the matrix

A_{x}

formed in step 2 has elements differing relatively from their true values by at most a small multiple of

3 ε

, where

ε

is the machine precision.

A_{x}

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.

8 Parallelism and Performance

e01dac is not threaded in any implementation.

9 Further Comments

The time taken by e01dac is approximately proportional to

m_{x} m_{y}

9.1 Outline of Method Used

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

1.choice of the interior $x$ -knots $λ_{5}$ , $λ_{6}, \dots, λ_{m_{x}}$ as $λ_{i} = x_{i - 2}$ , for $i = 5, 6, \dots, m_{x}$ ,
2.formation of the system
$A_{x} E = F,$
where $A_{x}$ is a band matrix of order $m_{x}$ and bandwidth $4$ , containing in its $q$ th row the values at $x_{q}$ of the B-splines in $x$ , $F$ is the $m_{x}$ by $m_{y}$ rectangular matrix of values $f_{q, r}$ , and $E$ denotes an $m_{x}$ by $m_{y}$ rectangular matrix of intermediate coefficients,
3.use of Gaussian elimination to reduce this system to band triangular form,
4.solution of this triangular system for $E$ ,
5.choice of the interior $y$ knots $μ_{5}$ , $μ_{6}, \dots, μ_{m_{y}}$ as $μ_{i} = y_{i - 2}$ , for $i = 5, 6, \dots, m_{y}$ ,
6.formation of the system
$A_{y} C^{T} = E^{T},$
where $A_{y}$ is the counterpart of $A_{x}$ for the $y$ variable, and $C$ denotes the $m_{x}$ by $m_{y}$ rectangular matrix of values of $c_{i j}$ ,
7.use of Gaussian elimination to reduce this system to band triangular form,
8.solution of this triangular system for $C^{T}$ and hence $C$ .

For computational convenience, steps 2 and 3, and likewise steps 6 and 7, are combined so that the formation of

A_{x}

and

A_{y}

and the reductions to triangular form are carried out one row at a time.

9.2 Evaluation of Computed Spline

The values of the computed spline at the points

(tx [r - 1], ty [r - 1])

, for

r = 1, 2, \dots, n

, may be obtained in the array ff, of length at least n, by the following call:

e02dec (n, tx, ty, ff, &spline, &fail)

where spline is a structure of type Nag_2dSpline which is the output argument of e01dac.

To evaluate the computed spline on a kx by ky rectangular grid of points in the

x

y

plane, which is defined by the

x

coordinates stored in

tx [q - 1]

, for

q = 1, 2, \dots, kx

, and the

y

coordinates stored in

ty [r - 1]

, for

r = 1, 2, \dots, ky

, returning the results in the array fg which is of length at least

kx \times ky

, the following call may be used:

e02dfc (kx, ky, tx, ty, fg, &spline, &fail)

where spline is a structure of type Nag_2dSpline which is the output argument of e01dac. The result of the spline evaluated at grid point

(q, r)

is returned in element

[ky \times (q - 1) + r - 1]

of the array fg.

10 Example

This program reads in values of

m_{x}

x_{q}

, for

q = 1, 2, \dots, m_{x}

m_{y}

and

y_{r}

, for

r = 1, 2, \dots, m_{y}

, followed by values of the ordinates

f_{q, r}

defined at the grid points

(x_{q}, y_{r})

. It then calls e01dac to compute a bicubic spline interpolant of the data values, and prints the values of the knots and B-spline coefficients. Finally it evaluates the spline at a small sample of points on a rectangular grid.