e02de calculates values of a bicubic spline from its B-spline representation.

Syntax

C#
public static void e02de( int m, int px, int py, double[] x, double[] y, double[] lamda, double[] mu, double[] c, double[] ff, out int ifail )

Visual Basic
Public Shared Sub e02de ( _ m 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 e02de ( _
	m 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 e02de( int m, 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 e02de : m : int * px : int * py : int * x : float[] * y : float[] * lamda : float[] * mu : float[] * c : float[] * ff : float[] * ifail : int byref -> unit

Parameters

m: Type: System..::..Int32
On entry: $m$ , the number of points at which values of the spline are required.

Constraint: $m \geq 1$ .

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 [m]

On entry: x and y must contain

x_{r}

and

y_{r}

, for

r = 1, 2, \dots, m

, respectively. These are the coordinates of the points at which values of the spline are required. The order of the points is immaterial.

Constraint:

x

and

y

must satisfy

lamda [3] \leq x [r - 1] \leq lamda [px - 4]

and

mu [3] \leq y [r - 1] \leq mu [py - 4], r = 1, 2, \dots, m .

The spline representation is not valid outside these intervals.

y

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

An array of size [m]

On entry: x and y must contain

x_{r}

and

y_{r}

, for

r = 1, 2, \dots, m

, respectively. These are the coordinates of the points at which values of the spline are required. The order of the points is immaterial.

Constraint:

x

and

y

must satisfy

lamda [3] \leq x [r - 1] \leq lamda [px - 4]

and

mu [3] \leq y [r - 1] \leq mu [py - 4], r = 1, 2, \dots, m .

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 [m]
On exit: $ff [r - 1]$ contains the value of the spline at the point $(x_{r}, y_{r})$ , for $r = 1, 2, \dots, m$ .

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

e02de calculates values of the bicubic spline

s (x, y)

at prescribed points

(x_{r}, y_{r})

, for

r = 1, 2, \dots, m

, 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}

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,	$m < 1$ ,
or	$py < 8$ ,
or	$px < 8$ .

$ifail = 2$: 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 $mu [py - 4] \leq mu [3]$ .

$ifail = 3$: On entry, at least one of the prescribed points $(x_{r}, y_{r})$ lies outside the rectangle defined by $lamda [3]$ , $lamda [px - 4]$ and $mu [3]$ , $mu [py - 4]$ .

$ifail = -9000$: 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 the number of points,

m

, at which the evaluation is required.

Example

This program 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 a value for

m

and the coordinates

(x_{r}, y_{r})

, for

r = 1, 2, \dots, m

, at which the spline is to be evaluated.

Example program (C#): e02dee.cs

Example program data: e02dee.d

Example program results: e02dee.r

Syntax

Parameters

Description

References

Error Indicators and Warnings

Accuracy

Parallelism and Performance

Further Comments

Example

See Also