NAG FL Interface
e01daf (dim2_spline_grid)
1
Purpose
e01daf computes a bicubic spline interpolating surface through a set of data values, given on a rectangular grid in the $x$$y$ plane.
2
Specification
Fortran Interface
Subroutine e01daf ( 
mx, my, x, y, f, px, py, lamda, mu, c, wrk, ifail) 
Integer, Intent (In) 
:: 
mx, my 
Integer, Intent (Inout) 
:: 
ifail 
Integer, Intent (Out) 
:: 
px, py 
Real (Kind=nag_wp), Intent (In) 
:: 
x(mx), y(my), f(mx*my) 
Real (Kind=nag_wp), Intent (Out) 
:: 
lamda(mx+4), mu(my+4), c(mx*my), wrk((mx+6)*(my+6)) 

C Header Interface
#include <nag.h>
void 
e01daf_ (const Integer *mx, const Integer *my, const double x[], const double y[], const double f[], Integer *px, Integer *py, double lamda[], double mu[], double c[], double wrk[], Integer *ifail) 

C++ Header Interface
#include <nag.h> extern "C" {
void 
e01daf_ (const Integer &mx, const Integer &my, const double x[], const double y[], const double f[], Integer &px, Integer &py, double lamda[], double mu[], double c[], double wrk[], Integer &ifail) 
}

The routine may be called by the names e01daf or nagf_interp_dim2_spline_grid.
3
Description
e01daf determines a bicubic spline interpolant to the set of data points
$\left({x}_{\mathit{q}},{y}_{\mathit{r}},{f}_{\mathit{q},\mathit{r}}\right)$, for
$\mathit{q}=1,2,\dots ,{m}_{x}$ and
$\mathit{r}=1,2,\dots ,{m}_{y}$. The spline is given in the Bspline representation
such that
where
${M}_{i}\left(x\right)$ and
${N}_{j}\left(y\right)$ denote normalized cubic Bsplines, the former defined on the knots
${\lambda}_{i}$ to
${\lambda}_{i+4}$ and the latter on the knots
${\mu}_{j}$ to
${\mu}_{j+4}$, and the
${c}_{ij}$ 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 9.2.
For further information on splines, see
Hayes and Halliday (1974) for bicubic splines and
de Boor (1972) for normalized Bsplines.
Values and derivatives of the computed spline can subsequently be computed by calling
e02def,
e02dff or
e02dhf as described in
Section 9.3.
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 Bsplines 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:
$\mathbf{mx}$ – Integer
Input

2:
$\mathbf{my}$ – 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:
${\mathbf{mx}}\ge 4$ and ${\mathbf{my}}\ge 4$.

3:
$\mathbf{x}\left({\mathbf{mx}}\right)$ – Real (Kind=nag_wp) array
Input

4:
$\mathbf{y}\left({\mathbf{my}}\right)$ – Real (Kind=nag_wp) array
Input

On entry: ${\mathbf{x}}\left(\mathit{q}\right)$ and ${\mathbf{y}}\left(\mathit{r}\right)$ must contain ${x}_{\mathit{q}}$, for $\mathit{q}=1,2,\dots ,{m}_{x}$, and
${y}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,{m}_{y}$, respectively.
Constraints:
 ${\mathbf{x}}\left(\mathit{q}\right)<{\mathbf{x}}\left(\mathit{q}+1\right)$, for $\mathit{q}=1,2,\dots ,{m}_{x}1$;
 ${\mathbf{y}}\left(\mathit{r}\right)<{\mathbf{y}}\left(\mathit{r}+1\right)$, for $\mathit{r}=1,2,\dots ,{m}_{y}1$.

5:
$\mathbf{f}\left({\mathbf{mx}}\times {\mathbf{my}}\right)$ – Real (Kind=nag_wp) array
Input

On entry: ${\mathbf{f}}\left({m}_{y}\times \left(\mathit{q}1\right)+\mathit{r}\right)$ must contain ${f}_{\mathit{q},\mathit{r}}$, for $\mathit{q}=1,2,\dots ,{m}_{x}$ and $\mathit{r}=1,2,\dots ,{m}_{y}$.

6:
$\mathbf{px}$ – Integer
Output

7:
$\mathbf{py}$ – Integer
Output

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:
$\mathbf{lamda}\left({\mathbf{mx}}+4\right)$ – Real (Kind=nag_wp) array
Output

9:
$\mathbf{mu}\left({\mathbf{my}}+4\right)$ – Real (Kind=nag_wp) array
Output

On exit:
lamda contains the complete set of knots
${\lambda}_{i}$ associated with the
$x$ variable, i.e., the interior knots
${\mathbf{lamda}}\left(5\right),{\mathbf{lamda}}\left(6\right),\dots ,{\mathbf{lamda}}\left({\mathbf{px}}4\right)$, as well as the additional knots
and
needed for the Bspline representation.
In a similar way,
mu contains the set of knots associated with the
$y$ variable.

10:
$\mathbf{c}\left({\mathbf{mx}}\times {\mathbf{my}}\right)$ – Real (Kind=nag_wp) array
Output

On exit: the coefficients of the spline interpolant.
${\mathbf{c}}\left({m}_{y}\times \left(i1\right)+j\right)$ contains the coefficient
${c}_{ij}$ described in
Section 3.

11:
$\mathbf{wrk}\left(\left({\mathbf{mx}}+6\right)\times \left({\mathbf{my}}+6\right)\right)$ – Real (Kind=nag_wp) array
Workspace


12:
$\mathbf{ifail}$ – Integer
Input/Output

On entry:
ifail must be set to
$0$,
$1\text{or}1$. If you are unfamiliar with this argument you should refer to
Section 4 in the Introduction to the NAG Library FL Interface for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is
$0$.
When the value $\mathbf{1}\text{or}\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit:
${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6
Error Indicators and Warnings
If on entry
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry, ${\mathbf{mx}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{mx}}\ge 4$.
On entry, ${\mathbf{my}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{my}}\ge 4$.
 ${\mathbf{ifail}}=2$

On entry, the
x or the
y mesh points are not in strictly ascending order.
 ${\mathbf{ifail}}=3$

An intermediate set of linear equations is singular – the data is too illconditioned to compute $B$spline coefficients.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 7 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 8 in the Introduction to the NAG Library FL Interface for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 9 in the Introduction to the NAG Library FL Interface for further information.
7
Accuracy
The main sources of rounding errors are in steps
$2$,
$3$,
$6$ and
$7$ of the algorithm described in
Section 9.2. 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\epsilon $, where
$\epsilon $ 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
e01daf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
The time taken by e01daf is approximately proportional to ${m}_{x}{m}_{y}$.
The process of computing the spline consists of the following steps:

1.choice of the interior $x$knots ${\lambda}_{5}$, ${\lambda}_{6},\dots ,{\lambda}_{{m}_{x}}$ as ${\lambda}_{\mathit{i}}={x}_{\mathit{i}2}$, for $\mathit{i}=5,6,\dots ,{m}_{x}$,

2.formation of the system
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 Bsplines in $x$, ${\mathbf{f}}$ 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 ${\mu}_{5}$, ${\mu}_{6},\dots ,{\mu}_{{m}_{y}}$ as ${\mu}_{\mathit{i}}={y}_{\mathit{i}2}$, for $\mathit{i}=5,6,\dots ,{m}_{y}$,

6.formation of the system
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 ${c}_{ij}$,

7.use of Gaussian elimination to reduce this system to band triangular form,

8.solution of this triangular system for ${C}^{\mathrm{T}}$ 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.
The values of the computed spline at the points
$\left({x}_{\mathit{k}},{y}_{\mathit{k}}\right)$, for
$\mathit{k}=1,2,\dots ,m$, 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
$\mathtt{M}=m$ and the coordinates
${x}_{k}$,
${y}_{k}$ are stored in
$\mathtt{X}\left(k\right)$,
$\mathtt{Y}\left(k\right)$.
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
$\mathtt{PY}4$.
(See
e02def.)
To evaluate the computed spline on an
${m}_{x}$ by
${m}_{y}$ rectangular grid of points in the
$x$
$y$ plane, which is defined by the
$x$ coordinates stored in
$\mathtt{X}\left(\mathit{j}\right)$, for
$\mathit{j}=1,2,\dots ,{m}_{x}$, and the
$y$ coordinates stored in
$\mathtt{Y}\left(\mathit{k}\right)$, for
$\mathit{k}=1,2,\dots ,{m}_{y}$, returning the results in the real array
ff (see
e02dff) which is of length at least
${\mathbf{mx}}\times {\mathbf{my}}$, 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
$\mathtt{MX}={m}_{x}$,
$\mathtt{MY}={m}_{y}$.
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
$\mathtt{LWRK}=\mathrm{min}\phantom{\rule{0.125em}{0ex}}\left(\mathit{nwrk1},\mathit{nwrk2}\right)$, for
$\mathit{nwrk1}=\mathtt{MX}\times 4+\mathtt{PX}$,
$\mathit{nwrk2}=\mathtt{MY}\times 4+\mathtt{PY}$, and
IWRK is an integer workspace array of length at least
$\mathtt{LIWRK}=\mathtt{MY}+\mathtt{PY}4$ if
$\mathit{nwrk1}>\mathit{nwrk2}$, or
$\mathtt{MX}+\mathtt{PX}4$ otherwise.
The result of the spline evaluated at grid point $\left(j,k\right)$ is returned in element ($\mathtt{MY}\times \left(j1\right)+k$) of the array FG.
10
Example
This example reads in values of ${m}_{x}$, ${x}_{\mathit{q}}$, for $\mathit{q}=1,2,\dots ,{m}_{x}$, ${m}_{y}$ and ${y}_{\mathit{r}}$, for $\mathit{r}=1,2,\dots ,{m}_{y}$, followed by values of the ordinates ${f}_{q,r}$ defined at the grid points $\left({x}_{q},{y}_{r}\right)$.
It then calls e01daf to compute a bicubic spline interpolant of the data values, and prints the values of the knots and Bspline coefficients. Finally it evaluates the spline at a small sample of points on a rectangular grid.
10.1
Program Text
10.2
Program Data
10.3
Program Results