E211: Cubic Splines and their Integrals

Author(s): K.S. Kölbig, H. Lipps	Library: MATHLIB
Submitter: K.S. Kölbig	Submitted: 01.05.1990
Language: Fortran	Revised:

Subroutines RCSPLN and DCSPLN compute a (vector-valued) cubic spline function F(x) which interpolates between a given set of points. Entries RCSPNT and DCSPNT compute the first and second integral over F(x).

On computers other than CDC or Cray, only the double-precision versions DCSPLN and DCSPNT are available. On CDC and Cray computers, only the single-precision versions RCSPLN and RCSPNT are available.

Given an interval [a,b], a subdivision of this interval into subintervals

and n+1 function values on the n+1 abscissae (called `knots') ( ), RCSPLN and DCSPLN compute a function F(x) of class , defined on [a,b], which assumes the given value at the knot (i.e. ), and which, when restricted to the ith sub-interval is identical with a set of m polynomials , each of degree at most 3. Any function F(x) which satisfies the above two conditions is called a `cubic spline' through the n + 1 points . To define the spline function F(x) uniquely the subroutines impose an additional boundary condition, specified by their MODE parameter:

MODE = 0: (the so-called natural spline).

MODE = 1: and .

Structure:

SUBROUTINE subprograms
User Entry Names: RCSPLN, RCSPNT, DCSPLN, DCSPNT
Files referenced: Unit 6

Usage:

For (type REAL), (type DOUBLE PRECISION),

Spline: CALL tCSPLN(N,X,M,Y,NDIM,MODE,A,B,C,D)

Integrals: CALL tCSPNT(N,X,M,Y,NDIM,MODE,A,B,C,D)

N

(INTEGER) Number n of subintervals . Must contain a value of at least 2 on entry. Unchanged on return.

X

(type according to t) One-dimensional array of dimension (0:d) of at least n + 1 elements. On entry, X(k) must contain the abscissa , . Unchanged on return.

M

(INTEGER) Number m of components of the vector-valued spline function F(x). Must contain a value of at least 1 on entry. Unchanged on return.

Y

(type according to t) Two-dimensional array of dimension (0:NDIM,d) where d is a value not less than m. On entry, Y(k,j) must contain the value of the jth component of the vector , ; . Unchanged on return.

NDIM

(INTEGER) Upper bound of the first dimension of arrays A, B, C, D and Y. Must contain a value of at least n on entry. Unchanged on return.

MODE

(INTEGER) Type of boundary condition (see description above). Must contain the value 0 or 1 on entry. Unchanged on return.

A,B,C,D

(type according to t) Two-dimensional arrays of dimension (NDIM,d), where .
On return from RCSPLN, A(i,j), B(i,j), C(i,j) and D(i,j) will contain the four coefficients and of the polynomial

that determines the jth component of the spline in the ith subinterval , , .
On return from RCSPNT,
A(i,j) = and B(i,j) = ,
with .
Arrays C and D have been used as working space.

Restrictions:

, , , or 1.

Error handling:

Error E211.1: .
Error E211.2: .
Error E211.3: .
Error E211.4: and .
A message is written on Unit 6, unless subroutine MTLSET (N002) has been called.