| Author(s): W. Mönch, B. Schorr | Library: MATHLIB |
| Submitter: W. Mönch | Submitted: 15.03.1993 |
| Language: Fortran | Revised: |
NORBAS (NORmalized BAsis Splines) is a portable collection of subprograms for various calculations with polynomial splines in one dimension (1D) and in two dimensions (2D). The polynomial splines are represented as linear combinations of normalized basis splines (B-splines).
On computers other than CDC or Cray, only the double-precision versions DSPKN1, etc. are available. On CDC and Cray computers, only the single-precision versions RSPKN1, etc. are available.
The following outline provides the background material and the
notations needed for describing the subprograms
and their parameters. For further information about splines and their
applications see References, in particular Ref. 7.
Case (1D):
.
.
.
,
in non-decreasing order, with multiplicity 
(i.e. no more than k+1 knots coincide).
, 
.
with
index i. The value of 
is identically zero outside the
interval 
, and the normalization of 
is such that
in B-spline
representation
Spline interpolation to a data set:
Given a data set 
; determine
coefficients 
of a polynomial interpolation
spline y = s(x) in B-spline representation with degree k over a set
of m=n+k+1 knots, such that the following relations
(interpolation conditions) hold:
The existence of a solution of this interpolation problem depends on an appropriate choice of the spline-knots (Ref. 7, Theorem XIII.1 (Schoenberg-Whitney)).
Least squares spline approximation to a data set:
Given a data set 
; determine
coefficients 
of a polynomial approximation
spline y = s(x) in B-spline representation with degree k over a set
of 
knots, such that following least squares
problem is solved:
Variation diminishing spline approximation to a function (Schoenberg):
For a given function y = f(x) on [a,b] this spline approximation is
defined by y = s(x), with
(Ref. 7, p. 158-162)
Case (2D):
.
.
.
.
in
x-direction.
in
y-direction.
in x-direction,
in non-decreasing order, with multiplicity 
(i.e. no more than kx+1 knots coincide).
in y-direction,
in non-decreasing order, with multiplicity 
(i.e. no more than ky+1 knots coincide).
, 
.
, 
.
with index i.
with index j.
of
two one-dimensional B-splines.
in B-spline representation
Spline interpolation to a data set:
Given a data set 
; determine coefficients
of a
two-dimensional polynomial interpolation spline z = s(x,y) in B-spline
representation with degrees kx , ky over the sets 
of mx=nx+kx+1 knots in x-direction and 
of my=ny+ky+1
knots in y-direction, such that following relations
(interpolation conditions) hold:
The existence of a solution of this interpolation problem depends
on an appropriate choice of the spline-knots 
, 
in the
two-dimensional interpolation area 
.
Least squares spline approximation to a data set:
Given a data set 
; determine coefficients
of a
two-dimensional polynomial approximation spline z = s(x,y) in B-spline
representation with degrees kx, ky over the sets 
of 
knots in x-direction and 
of
knots in y-direction, such that following
least squares problem is solved:
Variation diminishing spline approximation to a function:
For a given function z = f(x,y) on 
this
two-dimensional spline approximation is defined by z = s(x,y)
on 
, with
The package NORBAS contains FUNCTION and SUBROUTINE subprograms for solving the following problems. To calculate:
of m spline-knots
in the interval [a,b] for normalized B-splines 
of
degree k, use subprogram tSPKN1 (1D).
The knots are in non-decreasing order and determined in such a way that
the first k+1 knots coincide with a, the last k+1 knots coincide
with b, and the remaining 
knots are equidistant
in (a,b).
Two sets 
, 
of spline-knots
in 
and 
for B-splines 
of degrees kx and ky in x- and y-direction, use subprogram
tSPKN2 (2D).
and 
, are calculated by the same formulae
in x-, and y-direction, as in case (1D).
,
the n-th derivative
,
or the integral 
,
with fixed degree k and index i over a set 
of
spline-knots, use subprogram: tSPNB1 (1D).
The function 
,
the partial derivative
,
or the integral 
, with fixed
degrees kx, ky and indices i, j over the sets 
,
of spline-knots, use subprogram tSPNB2 (2D).
the n-th derivative
,
or the integral 
, use subprogram tSPPS1 (1D).
The function s(x,y),
the partial derivative
,
or the integral 
, use
subprogram tSPPS2 (2D).
of a one-dimensional
polynomial interpolation spline y = s(x) in B-spline
representation to a user-supplied data set 
, use
subprogram tSPIN1 (1D).
The coefficients 
of a two-dimensional
polynomial interpolation spline z = s(x,y) in B-spline
representation to a user-supplied data set
, use subprogram
tSPIN2 (2D).
of a one-dimensional
polynomial least squares approximation spline y=s(x) in
B-spline representation to a user-supplied data set 
,
use subprogram tSPAP1 (1D).
The coefficients 
of a two-dimensional
polynomial least squares approximation spline z=s(x,y) in
B-spline representation to a user-supplied data set
,
use subprogram tSPAP2 (2D).
of a one-dimensional
polynomial variation diminishing spline approximation y = s(x)
in B-spline representation to a user-supplied function y = f(x),
use subprogram tSPVD1 (1D).
The coefficients 
of a two-dimensional
polynomial variation diminishing spline approximation z = s(x,y)
in B-spline representation to a user-supplied function
z = f(x,y), use subprogram tSPVD2 (2D).
of a
one-dimensional polynomial spline y = s(x) in B-spline
representation, the corresponding coefficients 
of its n-th
derivative 
in B-spline representation,
use subprogram tSPCD1 (1D).
From given coefficients 
of a
two-dimensional polynomial spline z = s(x,y) in B-spline
representation, the corresponding coefficients 
of its
partial derivative
in B-spline representation,
use subprogram tSPCD2 (2D).
Structure:
SUBROUTINE and FUNCTION subprograms
User Entry Names:
| RSPKN1, | RSPKN2, | RSPNB1, | RSPNB2, | RSPPS1, | RSPPS2, | RSPIN1, | RSPIN2, |
| RSPAP1, | RSPAP2, | RSPVD1, | RSPVD2, | RSPCD1, | RSPCD2, | ||
| DSPKN1, | DSPKN2, | DSPNB1, | DSPNB2, | DSPPS1, | DSPPS2, | DSPIN1, | DSPIN2, |
| DSPAP1, | DSPAP2, | DSPVD1, | DSPVD2, | DSPCD1, | DSPCD2, |
| RGBTRF, | RGBTRS, | RGESVD, | ||||||
| DGBTRF, | DGBTRS, | DGESVD, | ||||||
| RVSET, | RVSUM, | RVCPY, | RVMPY, | |||||
| DVSET, | DVSUM, | DVCPY, | DVMPY, | |||||
| RMCPY, | RMMPY, | DMCPY, | DMMPY, | |||||
| MTLMTR, | ABEND. | |||||||
| @l | User-supplied FUNCTION subprogram | |||||||
Usage:
For 
(type REAL),
(type DOUBLE PRECISION):
(K) Knots
CALL tSPKN1(K,M,A,B,T,NERR)
CALL tSPKN2(KX,KY,MX,MY,AX,BX,AY,BY,TX,TY,NERR)
(B) Normalized B-splines
tSPNB1(K,M,I,NDER,X,T,NERR)
tSPNB2(KX,KY,MX,MY,I,J,NDERX,NDERY,X,Y,TX,TY,NERR)
(P) Polynomial spline
tSPPS1(K,M,NDER,X,T,C,NERR)
tSPPS2(KX,KY,MX,MY,NDERX,NDERY,X,Y,TX,TY,C,NDIMC,W,NERR)
(I) Spline interpolation
CALL tSPIN1(K,N,XI,YI,KNOT,T,C,W,IW,NERR)
CALL tSPIN2(KX,KY,NX,NY,XI,YI,ZI,NDIMZ,KNOT,TX,TY,C,NDIMC,W,IW,NERR)
(A) Least squares spline approximation
CALL tSPAP1(K,M,N,XI,YI,KNOT,T,C,W,NW,NERR)
CALL tSPAP2(KX,KY,MX,MY,NX,NY,XI,YI,ZI,NDIMZ,KNOT,TX,TY,C,NDIMC,W,NW,NERR)
(V) Variation diminishing spline approximation
CALL tSPVD1(F,K,M,T,C,NERR)
CALL tSPVD2(F,KX,KY,MX,MY,TX,TY,C,NDIMC,NERR)
(D) Coefficients of derivatives
CALL tSPCD1(K,M,NDER,T,C,D,NERR)
CALL tSPCD2(KX,KY,MX,MY,NDERX,NDERY,TX,TY,C,NDIMC,D,NERR)
Case (1D):
for tSPVD1,
otherwise).
).
).
( 
for spline interpolation (tSPIN1);
for spline approximation (tSPAP1)).
for tSPNB1 and tSPPS1;
for tSPCD1).
Calculation of the integral of 
(tSPNB1), or the integral of s(x) (tSPPS1).
Calculation of the function value 
(tSPNB1), or the function value s(x) (tSPPS1).
.
. On entry, 
must contain the l-th
data point 
for spline interpolation (tSPIN1) or spline
approximation (tSPAP1). The data points must be in
ascending order.
. On entry, 
must contain the l-th
data point 
for spline interpolation (tSPIN1) or spline
approximation (tSPAP1).
The knots must be supplied by the user.
. 
On exit, T(J) contains the j-th knot.
In the other cases, on entry, T(J) must contain the j-th knot.
The knots must be in non-decreasing order with multiplicity
.
. 
of the polynomial spline s(x) in B-spline
representation. 
of the polynomial interpolation
or approximation spline s(x) in B-spline representation.
. 
of the NDER-th
derivative of a polynomial spline s(x) in B-spline representation.
(tSPIN1), and of length
(tSPAP1); used as working space.
).
, used as working space.
No error or warning detected.
At least one of the parameters I, K, M, N,
NDER is not in range or 
is not true.
for tSPVD2;
otherwise).
.
).
in x-(y-)direction ( 
for spline interpolation tSPIN2;
for spline approximation tSPAP2).
for tSPNB2 and tSPPS2; 
,
for tSPCD2).
Calculation of the integral of 
(tSPNB2), or the integral of s(x,y) (tSPPS2).
Calculation of the function value 
(tSPNB2), or the function value s(x,y) (tSPPS2).
, respectively.
.
and 
, respectively.
On entry, 
and 
must contain the
lx-th data point 
and the ly-th data point 
for spline interpolation (tSPIN2) or spline approximation
(tSPAP2). The data points must be in ascending order.
. On entry, 
must contain
the (lx,ly)-th data point 
for spline interpolation
(tSPIN2) or spline approximation (tSPAP2).
).
The knots must be supplied by the user.
for the calculation of a set
of spline knots in x-(y-)direction, respectively, by tSPKN2.
and 
, repectively. 
On exit, TX(J) and TY(J) contain the
j-th knot in x-(y-)direction. In the other cases, on entry,
TX(J) and TY(J) must contain the j-th knot in
x-(y-)direction. The knots must be in non-decreasing order with
multiplicity 
and 
, respectively.
. 
of the polynomial spline s(x,y)
in B-spline representation. 
of the polynomial interpolation or
approximation spline s(x,y) in B-spline representation.
).
. 
of the
partial derivative of order 
with respect to x and y
of a polynomial spline s(x,y) in B-spline representation.
(tSPPS2), 
(tSPIN2),
and of length 
(tSPAP2), used as
working space.
).
, used as working space.
No error or warning detected.
Examples:
Calculate
of
a polynomial interpolation spline y=s(x) of degree k=2 in B-spline
representation to a given data set 
;
of the first derivative
;
and 
for x=0(0.1)1.
IMPLICIT DOUBLE PRECISION (A-H,O-Z)
DIMENSION XI(6),YI(6),T(9),C(6),D(6),W(42),IW(6)
DATA K,N,NDER,KNOT / 2,6,1,1 /
DATA XI / 0D0,0.20D0,0.40D0,0.60D0,0.80D0,1.00D0 /
DATA YI / 1D0,0.66D0,0.47D0,0.38D0,0.35D0,0.34D0 /
M=N+K+1
CALL DSPIN1(K,N,XI,YI,KNOT,T,C,W,IW,NERR)
CALL DSPCD1(K,M,NDER,T,C,D,NERR)
WRITE(6,1000) K,N,(T(I),I=1,M)
DO 20 J=0,10
X=J/1D1
SPL0=DSPPS1(K,M, 0,X,T,C,NERR)
SPL1=DSPPS1(K,M, 1,X,T,D,NERR)
SPLI=DSPPS1(K,M,-1,X,T,C,NERR)
20 WRITE(6,1010) J,X,SPL0,SPL1,SPLI
1000 FORMAT(...)
1010 FORMAT(...)
END
DEGREE OF POLYNOMIAL SPLINE: 2 NUMBER OF DATA POINTS: 6
KNOTS T : 0.00 0.00 0.00 0.25 0.50 0.75 1.00 1.00 1.00
J X S(X) DS(X) IN(X)
0 0.00 1.000000 -2.119921 0.000000
1 0.10 0.809004 -1.700000 0.090100
2 0.20 0.660000 -1.280079 0.163201
3 0.30 0.550992 -0.940017 0.223467
4 0.40 0.470000 -0.679816 0.274299
5 0.50 0.415028 -0.419615 0.318334
6 0.60 0.380000 -0.280953 0.357970
7 0.70 0.358838 -0.142290 0.394796
8 0.80 0.350000 -0.065306 0.430174
9 0.90 0.344235 -0.050000 0.464873
10 1.00 0.340000 -0.034694 0.499072
Error handling:
| Error E210.1: K|KX,KY not in range, | Error E210.5: NDER|NDERX,NDERY not in range, | |
| Error E210.2: M|MX,MY not in range, | Error E210.6: A,B|AX,BX;AY,BY inconsistent, | |
| Error E210.3: I|I,J not in range, | Error E210.7: NDERX|NDERY inconsistent. | |
| Error E210.4: N|NX,NY not in range, |
(see above). A message is
written on Unit 6, unless subroutine MTLSET (N002) has
been called.
References:
