NAG Library Routine Document

E01DAF

are the spline coefficients. These knots, as well as the coefficients, are determined by the routine, which is derived from the routine B2IRE in Anthony et al. (1982). The method used is described in Section 8.2.

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 E02DEF, E02DFF or E02DHF as described in Section 8.3.

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 Parameters

1: MX – INTEGERInput

2: MY – INTEGERInput

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) – REAL (KIND=nag_wp) arrayInput

4: Y(MY) – REAL (KIND=nag_wp) arrayInput

On entry:

X (q)

and

Y (r)

must contain

x_{q}

, for

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

, and

y_{r}

, for

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

, respectively.

Constraints:

$X (q) < X (q + 1)$ , for $q = 1, 2, \dots, m_{x} - 1$ ;
$Y (r) < Y (r + 1)$ , for $r = 1, 2, \dots, m_{y} - 1$ .

5: F( $MX \times MY$ ) – REAL (KIND=nag_wp) arrayInput

On entry:

F (m_{y} \times (q - 1) + r)

must contain

f_{q, r}

, for

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

and

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

6: PX – INTEGEROutput

7: PY – INTEGEROutput

On exit: PX and PY 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.

8: LAMDA( $MX + 4$ ) – REAL (KIND=nag_wp) arrayOutput

9: MU( $MY + 4$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: LAMDA contains the complete set of knots

λ_{i}

associated with the

x

variable, i.e., the interior knots

LAMDA (5), LAMDA (6), \dots, LAMDA (PX - 4)

, as well as the additional knots

LAMDA (1) = LAMDA (2) = LAMDA (3) = LAMDA (4) = X (1)

and

LAMDA (PX - 3) = LAMDA (PX - 2) = LAMDA (PX - 1) = LAMDA (PX) = X (MX)

needed for the B-spline representation.

10: C( $MX \times MY$ ) – REAL (KIND=nag_wp) arrayOutput

On exit: the coefficients of the spline interpolant.

C (m_{y} \times (i - 1) + j)

contains the coefficient

c_{i j}

described in Section 3.

11: WRK( $(MX + 6) \times (MY + 6)$ ) – REAL (KIND=nag_wp) arrayWorkspace

12: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$MX < 4$ ,
or	$MY < 4$ .

$IFAIL = 2$: On entry, either the values in the X array or the values in the Y array are not in increasing order if not already there.

$IFAIL = 3$: A system of linear equations defining the B-spline coefficients was singular; the problem is too ill-conditioned to permit solution.

7 Accuracy

The main sources of rounding errors are in steps

2

3

6

and

7

of the algorithm described in Section 8.2. It can be shown (see 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 Further Comments

8.1 Timing

The time taken by E01DAF is approximately proportional to

m_{x} m_{y}

8.2 Outline of Method Used

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

choice of the interior $x$ -knots $λ_{5}$ , $λ_{6}, \dots, λ_{m_{x}}$ as $λ_{i} = x_{i - 2}$ , for $i = 5, 6, \dots, m_{x}$ ,
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,
use of Gaussian elimination to reduce this system to band triangular form,
solution of this triangular system for $E$ ,
choice of the interior $y$ knots $μ_{5}$ , $μ_{6}, \dots, μ_{m_{y}}$ as $μ_{i} = y_{i - 2}$ , for $i = 5, 6, \dots, m_{y}$ ,
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 $c_{i j}$ ,
use of Gaussian elimination to reduce this system to band triangular form,
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.

8.3 Evaluation of Computed Spline

The values of the computed spline at the points

(x_{k}, y_{k})

, for

k = 1, 2, \dots, m

, may be obtained in the real array FF (see E02DEF), of length at least

m

, by the following call:

IFAIL = 0
CALL E02DEF(M,PX,PY,X,Y,LAMDA,MU,C,FF,WRK,IWRK,IFAIL)

where

M = m

and the coordinates

x_{k}

y_{k}

are stored in

X (k)

Y (k)

. PX and PY, LAMDA, MU and C have the same values as PX and PY LAMDA, MU and C output from E01DAF. WRK is a real workspace array of length at least PY, and IWRK is an integer workspace array of length at least

PY - 4

. (See E02DEF.)

To evaluate the computed spline on an

m_{x}

m_{y}

rectangular grid of points in the

x

y

plane, which is defined by the

x

coordinates stored in

X (j)

, for

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

, and the

y

coordinates stored in

Y (k)

, for

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

, returning the results in the real array FF (see E02DFF) which is of length at least

MX \times MY

, the following call may be used:

 IFAIL = 0
 CALL E02DFF(MX,MY,PX,PY,X,Y,LAMDA,MU,C,FG,WRK,LWRK,
*            IWRK,LIWRK,IFAIL)

where

MX = m_{x}

MY = m_{y}

. PX and PY, LAMDA, MU and C have the same values as PX, PY, LAMDA, MU and C output from E01DAF. WRK is a real workspace array of length at least

LWRK = \min (nwrk1, nwrk2)

, for

nwrk1 = MX \times 4 + PX

nwrk2 = MY \times 4 + PY

, and IWRK is an integer workspace array of length at least

LIWRK = MY + PY - 4

nwrk1 > nwrk2

, or

MX + PX - 4

otherwise.

The result of the spline evaluated at grid point

(j, k)

is returned in element (

MY \times (j - 1) + k

) of the array FG.

9 Example

This example 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 E01DAF 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.

NAG Library Routine DocumentE01DAF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

8.1 Timing

8.2 Outline of Method Used

8.3 Evaluation of Computed Spline

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

E01DAF