E01 Chapter Contents
E01 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentE01DAF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 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

 SUBROUTINE E01DAF ( MX, MY, X, Y, F, PX, PY, LAMDA, MU, C, WRK, IFAIL)
 INTEGER MX, MY, PX, PY, IFAIL REAL (KIND=nag_wp) X(MX), Y(MY), F(MX*MY), LAMDA(MX+4), MU(MY+4), C(MX*MY), WRK((MX+6)*(MY+6))

## 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 B-spline representation
 $sx,y=∑i=1mx∑j=1mycijMixNjy,$
such that
 $sxq,yr=fq,r,$
where ${M}_{i}\left(x\right)$ and ${N}_{j}\left(y\right)$ denote normalized cubic B-splines, 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 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: ${\mathbf{MX}}\ge 4$ and ${\mathbf{MY}}\ge 4$.
3:     X(MX) – REAL (KIND=nag_wp) arrayInput
4:     Y(MY) – REAL (KIND=nag_wp) arrayInput
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:     F(${\mathbf{MX}}×{\mathbf{MY}}$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{F}}\left({m}_{y}×\left(\mathit{q}-1\right)+\mathit{r}\right)$ must contain ${f}_{\mathit{q},\mathit{r}}$, for $\mathit{q}=1,2,\dots ,{m}_{x}$ and $\mathit{r}=1,2,\dots ,{m}_{y}$.
6:     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(${\mathbf{MX}}+4$) – REAL (KIND=nag_wp) arrayOutput
9:     MU(${\mathbf{MY}}+4$) – REAL (KIND=nag_wp) arrayOutput
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
 $LAMDA1=LAMDA2=LAMDA3=LAMDA4=X1$
and
 $LAMDAPX-3=LAMDAPX-2=LAMDAPX-1=LAMDAPX=XMX$
needed for the B-spline representation.
10:   C(${\mathbf{MX}}×{\mathbf{MY}}$) – REAL (KIND=nag_wp) arrayOutput
On exit: the coefficients of the spline interpolant. ${\mathbf{C}}\left({m}_{y}×\left(i-1\right)+j\right)$ contains the coefficient ${c}_{ij}$ described in Section 3.
11:   WRK($\left({\mathbf{MX}}+6\right)×\left({\mathbf{MY}}+6\right)$) – REAL (KIND=nag_wp) arrayWorkspace
12:   IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this parameter, 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 ${\mathbf{IFAIL}}={\mathbf{0}}$ or $-{\mathbf{1}}$, 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, ${\mathbf{MX}}<4$, or ${\mathbf{MY}}<4$.
${\mathbf{IFAIL}}=2$
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.
${\mathbf{IFAIL}}=3$
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 $3\epsilon$, 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.

### 8.1  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:
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
 $AxE=F,$
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 $x$, ${\mathbf{F}}$ 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
 $AyCT=ET,$
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.

### 8.3  Evaluation of Computed Spline

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 ${\mathbf{MX}}×{\mathbf{MY}}$, 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 $\mathit{nwrk1}=\mathtt{MX}×4+\mathtt{PX}$, $\mathit{nwrk2}=\mathtt{MY}×4+\mathtt{PY}$, 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 ($\mathtt{MY}×\left(j-1\right)+k$) of the array FG.

## 9  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 B-spline coefficients. Finally it evaluates the spline at a small sample of points on a rectangular grid.

### 9.1  Program Text

Program Text (e01dafe.f90)

### 9.2  Program Data

Program Data (e01dafe.d)

### 9.3  Program Results

Program Results (e01dafe.r)