NAG Library Routine Document

E02DFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

E02DFF calculates values of a bicubic spline from its B-spline representation. The spline is evaluated at all points on a rectangular grid.

2 Specification

SUBROUTINE E02DFF (

MX, MY, PX, PY, X, Y, LAMDA, MU, C, FF, WRK, LWRK, IWRK, LIWRK, IFAIL)

INTEGER	MX, MY, PX, PY, LWRK, IWRK(LIWRK), LIWRK, IFAIL
REAL (KIND=nag_wp)	X(MX), Y(MY), LAMDA(PX), MU(PY), C((PX-4)(PY-4)), FF(MXMY), WRK(LWRK)

3 Description

E02DFF calculates values of the bicubic spline

s (x, y)

on a rectangular grid of points in the

x

y

plane, from its augmented knot sets

\{λ\}

and

\{μ\}

and from the coefficients

c_{i j}

, for

i = 1, 2, \dots, PX - 4

and

j = 1, 2, \dots, PY - 4

, in its B-spline representation

s (x, y) = \sum_{i j} c_{i j} M_{i} (x) N_{j} (y) .

Here

M_{i} (x)

and

N_{j} (y)

denote normalized cubic B-splines, the former defined on the knots

λ_{i}

λ_{i + 4}

and the latter on the knots

μ_{j}

μ_{j + 4}

The points in the grid are defined by coordinates

x_{q}

, for

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

, along the

x

axis, and coordinates

y_{r}

, for

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

, along the

y

axis.

This routine may be used to calculate values of a bicubic spline given in the form produced by E01DAF, E02DAF, E02DCF and E02DDF. It is derived from the routine B2VRE in Anthony et al. (1982).

4 References

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

Cox M G (1978) The numerical evaluation of a spline from its B-spline representation J. Inst. Math. Appl. 21 135–143

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 1

and

MY \geq 1

3: PX – INTEGERInput

4: PY – INTEGERInput

On entry: PX and PY must specify the total number of knots associated with the variables

x

and

y

respectively. They are such that

PX - 8

and

PY - 8

are the corresponding numbers of interior knots.

Constraint:

PX \geq 8

and

PY \geq 8

5: X(MX) – REAL (KIND=nag_wp) arrayInput

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

On entry: X and Y must contain

x_{q}

, for

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

, and

y_{r}

, for

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

, respectively. These are the

x

and

y

coordinates that define the rectangular grid of points at which values of the spline are required.

Constraint:

X

and Y must satisfy

LAMDA (4) \leq X (q) < X (q + 1) \leq LAMDA (PX - 3), q = 1, 2, \dots, m_{x} - 1

and

MU (4) \leq Y (r) < Y (r + 1) \leq MU (PY - 3), r = 1, 2, \dots, m_{y} - 1 .

The spline representation is not valid outside these intervals.

7: LAMDA(PX) – REAL (KIND=nag_wp) arrayInput

8: MU(PY) – REAL (KIND=nag_wp) arrayInput

On entry: LAMDA and MU must contain the complete sets of knots

\{λ\}

and

\{μ\}

associated with the

x

and

y

variables respectively.

Constraint: the knots in each set must be in nondecreasing order, with

LAMDA (PX - 3) > LAMDA (4)

and

MU (PY - 3) > MU (4)

9: C( $(PX - 4) \times (PY - 4)$ ) – REAL (KIND=nag_wp) arrayInput

On entry:

C ((PY - 4) \times (i - 1) + j)

must contain the coefficient

c_{i j}

described in Section 3, for

i = 1, 2, \dots, PX - 4

and

j = 1, 2, \dots, PY - 4

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

On exit:

FF (MY \times (q - 1) + r)

contains the value of the spline at the point

(x_{q}, y_{r})

, for

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

and

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

11: WRK(LWRK) – REAL (KIND=nag_wp) arrayWorkspace

12: LWRK – INTEGERInput

On entry: the dimension of the array WRK as declared in the (sub)program from which E02DFF is called.

Constraint:

LWRK \geq \min (4 \times MX + PX, 4 \times MY + PY)

13: IWRK(LIWRK) – INTEGER arrayWorkspace

14: LIWRK – INTEGERInput

On entry: the dimension of the array IWRK as declared in the (sub)program from which E02DFF is called.

Constraints:

if $4 \times MX + PX > 4 \times MY + PY$ , $LIWRK \geq MY + PY - 4$ ;
otherwise $LIWRK \geq MX + PX - 4$ .

15: 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,	$MX < 1$ ,
or	$MY < 1$ ,
or	$PY < 8$ ,
or	$PX < 8$ .

$IFAIL = 2$

On entry,	LWRK is too small,
or	LIWRK is too small.

$IFAIL = 3$: On entry, the knots in array LAMDA, or those in array MU, are not in nondecreasing order, or $LAMDA (PX - 3) \leq LAMDA (4)$ , or $MU (PY - 3) \leq MU (4)$ .

$IFAIL = 4$: On entry, the restriction $LAMDA (4) \leq X (1) < \dots < X (MX) \leq LAMDA (PX - 3)$ , or the restriction $MU (4) \leq Y (1) < \dots < Y (MY) \leq MU (PY - 3)$ , is violated.

7 Accuracy

The method used to evaluate the B-splines is numerically stable, in the sense that each computed value of

s (x_{r}, y_{r})

can be regarded as the value that would have been obtained in exact arithmetic from slightly perturbed B-spline coefficients. See Cox (1978) for details.

8 Further Comments

Computation time is approximately proportional to

m_{x} m_{y} + 4 (m_{x} + m_{y})

9 Example

This example reads in knot sets

LAMDA (1), \dots, LAMDA (PX)

and

MU (1), \dots, MU (PY)

, and a set of bicubic spline coefficients

c_{i j}

. Following these are values for

m_{x}

and the

x

coordinates

x_{q}

, for

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

, and values for

m_{y}

and the

y

coordinates

y_{r}

, for

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

, defining the grid of points on which the spline is to be evaluated.

NAG Library Routine DocumentE02DFF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

E02DFF