e02df calculates values of a bicubic spline from its B-spline representation. The spline is evaluated at all points on a rectangular grid.

Syntax

C#
public static void e02df( int mx, int my, int px, int py, double[] x, double[] y, double[] lamda, double[] mu, double[] c, double[] ff, out int ifail )

Visual Basic
Public Shared Sub e02df ( _ mx As Integer, _ my As Integer, _ px As Integer, _ py As Integer, _ x As Double(), _ y As Double(), _ lamda As Double(), _ mu As Double(), _ c As Double(), _ ff As Double(), _ <OutAttribute> ByRef ifail As Integer _ )

Visual Basic

Public Shared Sub e02df ( _
	mx As Integer, _
	my As Integer, _
	px As Integer, _
	py As Integer, _
	x As Double(), _
	y As Double(), _
	lamda As Double(), _
	mu As Double(), _
	c As Double(), _
	ff As Double(), _
	<OutAttribute> ByRef ifail As Integer _
)

Visual C++
public: static void e02df( int mx, int my, int px, int py, array<double>^ x, array<double>^ y, array<double>^ lamda, array<double>^ mu, array<double>^ c, array<double>^ ff, [OutAttribute] int% ifail )

F#
static member e02df : mx : int * my : int * px : int * py : int * x : float[] * y : float[] * lamda : float[] * mu : float[] * c : float[] * ff : float[] * ifail : int byref -> unit

Parameters

mx: Type: System..::..Int32
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 $my \geq 1$ .

my: Type: System..::..Int32
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 $my \geq 1$ .

px: Type: System..::..Int32
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 $py \geq 8$ .

py: Type: System..::..Int32
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 $py \geq 8$ .

x

Type: array<System..::..Double>[]()[][]

An array of size [mx]

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 [3] \leq x [q - 1] < x [q] \leq lamda [px - 4], q = 1, 2, \dots, m_{x} - 1

and

mu [3] \leq y [r - 1] < y [r] \leq mu [py - 4], r = 1, 2, \dots, m_{y} - 1 .

The spline representation is not valid outside these intervals.

y

Type: array<System..::..Double>[]()[][]

An array of size [mx]

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 [3] \leq x [q - 1] < x [q] \leq lamda [px - 4], q = 1, 2, \dots, m_{x} - 1

and

mu [3] \leq y [r - 1] < y [r] \leq mu [py - 4], r = 1, 2, \dots, m_{y} - 1 .

The spline representation is not valid outside these intervals.

lamda: Type: array<System..::..Double>[]()[][]
An array of size [px]
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 $lamda [px - 4] > lamda [3]$ and $mu [py - 4] > mu [3]$ .

mu: Type: array<System..::..Double>[]()[][]
An array of size [px]
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 $lamda [px - 4] > lamda [3]$ and $mu [py - 4] > mu [3]$ .

c: Type: array<System..::..Double>[]()[][]
An array of size [ $(px - 4) \times (py - 4)$ ]
On entry: $c [(py - 4) \times (i - 1) + j - 1]$ must contain the coefficient $c_{i j}$ described in [Description], for $i = 1, 2, \dots, px - 4$ and $j = 1, 2, \dots, py - 4$ .

ff: Type: array<System..::..Double>[]()[][]
An array of size [ $mx \times my$ ]
On exit: $ff [my \times (q - 1) + r - 1]$ contains the value of the spline at the point $(x_{q}, y_{r})$ , for $q = 1, 2, \dots, m_{x}$ and $r = 1, 2, \dots, m_{y}$ .

ifail: Type: System..::..Int32%
On exit: $ifail = 0$ unless the method detects an error or a warning has been flagged (see [Error Indicators and Warnings]).

Description

e02df 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 method may be used to calculate values of a bicubic spline given in the form produced by e01da (E02DAF not in this release) (E02DCF not in this release) (E02DDF not in this release). It is derived from the method B2VRE in Anthony et al. (1982).

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

Error Indicators and Warnings

Errors or warnings detected by the method:

$ifail = 1$

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 $lamda [px - 4] \leq lamda [3]$ , or $mu [py - 4] \leq mu [3]$ .

$ifail = 4$: On entry, the restriction $lamda [3] \leq x [0] < \dots < x [mx - 1] \leq lamda [px - 4]$ , or the restriction $mu [3] \leq y [0] < \dots < y [my - 1] \leq mu [py - 4]$ , is violated.

$ifail = -9000$: An error occured, see message report.
$ifail = -8000$: Negative dimension for array $〈value〉$
$ifail = -6000$: Invalid Parameters $〈value〉$

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.

Parallelism and Performance

None.

Further Comments

Computation time is approximately proportional to

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

Example

This example reads in knot sets

lamda [0], \dots, lamda [px - 1]

and

mu [0], \dots, mu [py - 1]

, 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.

Example program (C#): e02dfe.cs

Example program data: e02dfe.d

Example program results: e02dfe.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also