NAG Library Routine Document
E01DAF
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 |
| double precision | 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
x
q
,
y
r
,
f
q
,
r
, for
q=1,2,…,mx and
r=1,2,…,my. The spline is given in the B-spline representation
such that
where
Mix and
Njy denote normalized cubic B-splines, the former defined on the knots
λi to
λi+4 and the latter on the knots
μj to
μj+4, and the
cij 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 of the computed spline can subsequently be obtained by calling
E02DEF or
E02DFF 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
mx and
my respectively, the number of points along the
x and
y axis that define the rectangular grid.
Constraint:
MX≥4 and MY≥4.
- 3: X(MX) – double precision arrayInput
- 4: Y(MY) – double precision arrayInput
On entry: Xq and Yr must contain xq, for q=1,2,…,mx, and yr, for r=1,2,…,my, respectively.
Constraints:
- Xq<Xq+1, for q=1,2,…,mx-1;
- Yr<Yr+1, for r=1,2,…,my-1.
- 5: F(MX×MY) – double precision arrayInput
On entry: Fmy×q-1+r must contain fq,r, for q=1,2,…,mx and r=1,2,…,my.
- 6: PX – INTEGEROutput
- 7: PY – INTEGEROutput
On exit:
PX and
PY contain
mx+4 and
my+4, the total number of knots of the computed spline with respect to the
x and
y variables, respectively.
- 8: LAMDA(MX+4) – double precision arrayOutput
- 9: MU(MY+4) – double precision arrayOutput
On exit:
LAMDA contains the complete set of knots
λi associated with the
x variable, i.e., the interior knots
LAMDA5,LAMDA6,…,LAMDAPX-4, as well as the additional knots
and
needed for the B-spline representation.
MU contains the corresponding complete set of knots
μi associated with the
y variable.
- 10: C(MX×MY) – double precision arrayOutput
On exit: the coefficients of the spline interpolant.
Cmy×i-1+j contains the coefficient
cij described in
Section 3.
- 11: WRK(MX+6×MY+6) – double precision arrayWorkspace
- 12: 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.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
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.
6 Error Indicators and Warnings
If on entry
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:
- IFAIL=1
-
- 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.
- 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
Ax formed in step
2 has elements differing relatively from their true values by at most a small multiple of
3ε, where
ε is the
machine precision.
Ax 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 Further Comments
8.1 Timing
The time taken by E01DAF is approximately proportional to mxmy.
8.2 Outline of Method Used
The process of computing the spline consists of the following steps:
- choice of the interior x-knots λ5, λ6,…,λmx as λi=xi-2, for i=5,6,…,mx,
- formation of the system
where Ax is a band matrix of order mx and bandwidth 4, containing in its qth row the values at xq of the B-splines in x, F is the mx by my rectangular matrix of values fq,r, and E denotes an mx by my rectangular matrix of intermediate coefficients,
- use of Gaussian elimination to reduce this system to band triangular form,
- solution of this triangular system for E,
- choice of the interior y knots μ5, μ6,…,μmy as μi=yi-2, for i=5,6,…,my,
- formation of the system
where Ay is the counterpart of Ax for the y variable, and C denotes the mx by my rectangular matrix of values of cij,
- use of Gaussian elimination to reduce this system to band triangular form,
- solution of this triangular system for CT and hence C.
For computational convenience, steps 2 and 3, and likewise steps 6 and 7, are combined so that the formation of Ax and Ay 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
xk,yk
, for
k=1,2,…,m, may be obtained in the
double precision 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
M=m and the co-ordinates
xk,
yk are stored in
Xk,
Yk.
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
double precision workspace array of length at least
PY, and
IWRK is an integer workspace array of length at least
PY-4.
(See
E02DEF.)
To evaluate the computed spline on an
mx by
my rectangular grid of points in the
x-
y plane, which is defined by the
x co-ordinates stored in
Xj, for
j=1,2,…,mx, and the
y co-ordinates stored in
Yk, for
k=1,2,…,my, returning the results in the
double precision array
FF (see
E02DFF) which is of length at least
MX×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
MX=mx,
MY=my.
PX and
PY,
LAMDA,
MU and
C have the same values as
PX,
PY,
LAMDA,
MU and
C output from E01DAF.
WRK is a
double precision workspace array of length at least
LWRK
=
minnwrk1,nwrk2
, for
nwrk1
=
MX
×
4
+
PX
,
nwrk2
=
MY
×
4
+
PY
, and
IWRK is an integer workspace array of length at least
LIWRK
=
MY
+
PY
-
4
if
nwrk1
>
nwrk2
, or
MX
+
PX
-
4
otherwise.
The result of the spline evaluated at grid point
j,k
is returned in element (
MY×j-1+k
) of the array FG.
9 Example
This example reads in values of mx, xq, for q=1,2,…,mx, my and yr, for r=1,2,…,my, followed by values of the ordinates fq,r defined at the grid points xq,yr.
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.f)
9.2 Program Data
Program Data (e01dafe.d)
9.3 Program Results
Program Results (e01dafe.r)