Chapter 3: Interpolation and

Approximation

3.2.Cubic Spline Evaluation and Integration

3.4.Spline Evaluation, Integration, and Conversion to Piecewise

3.6.Quadratic Polynomial Interpolation Routines for Gridded Data

3.7.Scattered Data Interpolation

3.10.Rational L∞ Approximation

Usage Notes

The majority of the routines in this chapter produce piecewise polynomial or spline functions that either interpolate or approximate given data, or are support routines for the evaluation, integration, and conversion from one representation to another. Two major subdivisions of routines are provided. The cubic spline routines begin with the letters “ CS” and utilize the piecewise polynomial representation described below. The B-spline routines begin with the letters

“ BS” and utilize the B-spline representation described below. Most of the spline routines are based on routines in the book by de Boor (1978).

Piecewise Polynomials

A univariate piecewise polynomial (function) p is specified by giving its breakpoint sequence ξ RQ, the order k (degree k − 1) of its polynomial pieces, and the k × (n − 1) matrix c of its local polynomial coefficients. In terms of this information, the piecewise polynomial (pp) function is given by

The breakpoint sequence ξ is assumed to be strictly increasing, and we extend the pp function to the entire real axis by extrapolation from the first and last intervals. The subroutines in this chapter will consistently make the following identifications for FORTRAN variables:

c = PPCOEF

ξ = BREAK k = KORDER

N = NBREAK

This representation is redundant when the pp function is known to be smooth. For example, if p is known to be continuous, then we can compute c L from the cML as follows

where ΔξL := ξL − ξL. For smooth pp, we prefer to use the irredundant representation in terms of the B-(for ‘basis’)-splines, at least when such a function is first to be determined. The above pp representation is employed for evaluation of the pp function at many points since it is more efficient.

Splines and B-splines

B-splines provide a particularly convenient and suitable basis for a given class of smooth pp functions. Such a class is specified by giving its breakpoint sequence, its order, and the required smoothness across each of the interior breakpoints.

The corresponding B-spline basis is specified by giving its knot sequence t

R0. The specification rule is the following: If the class is to have all derivatives up to and including the j-th derivative continuous across the interior breakpoint ξL, then the number ξL should occur k − j − 1 times in the knot sequence.

Assuming that ξ , and ξQ are the endpoints of the interval of interest, one

chooses the first k knots equal to x and the last k knots equal to xQ. This can be done since the B-splines are defined to be right continuous near x and left continuous near xQ.

When the above construction is completed, we will have generated a knot sequence t of length M; and there will be m := M - k B-splines of order k, say B ,

¼, BP that span the pp functions on the interval with the indicated smoothness. That is, each pp function in this class has a unique representation

p = a B + a B + ¼ + aPBP

as a linear combination of B-splines. The B-spline routines will consistently make use of the following identifiers for FORTRAN variables:

a = BSCOEF t = XKNOT m = NCOEF

M = NKNOT

A B-spline is a particularly compact pp function. BL is a nonnegative function that is nonzero only on the interval [tL, tL N]. More precisely, the support of the i-th B- spline is [tL, tL N]. No pp function in the same class (other than the zero function) has smaller support (i.e., vanishes on more intervals) than a B-spline. This makes B-splines particularly attractive basis functions since the influence of any particular B-spline coefficient extends only over a few intervals. When it is necessary to emphasize the dependence of the B-spline on its parameters, we will use the notation

BL N W

to denote the i-th B-spline of order k for the knot sequence t.

CSDEC (natural spline)

CSINT

CSCON

CSAKM

Figure 3-1 Spline Interpolants of the Same Data

Cubic Splines

Cubic splines are smooth (i.e., C or C ) fourth-order pp functions. For historical and other reasons, cubic splines are the most heavily used pp functions. Therefore, we provide special routines for their construction and evaluation. The routines for their determination use yet another representation (in terms of value and slope at all the breakpoints) but output the pp representation as described above for general pp functions.

We provide seven cubic spline interpolation routines: CSIEZ (page 420), CSINT (page 423), CSDEC (page 425), CSHER (page 429), CSAKM (page 432), CSCON (page 434), and CSPER (page 438). The first routine, CSIEZ, is an easy-to-use version of CSINT coupled with CSVAL. The routine CSIEZ will compute the

value of the cubic spline interpolant (to given data using the ‘not-a-knot’ criterion) on a grid. The routine CSDEC allows the user to specify various endpoint conditions (such as the value of the first or second derivative at the right and left points). This means that the natural cubic spline can be obtained using this routine by setting the second derivative to zero at both endpoints. If function values and derivatives are available, then the Hermite cubic interpolant can be computed using CSHER. The two routines CSAKM and CSCON are designed so that the shape of the curve matches the shape of the data. In particular, CSCON preserves the convexity of the data while CSAKM attempts to minimize oscillations. If the data is periodic, then CSPER will produce a periodic interpolant. The routine CONFT (page 551) allows the user wide latitude in enforcing shapes. This routine returns the B-spline representation.

It is possible that the cubic spline interpolation routines will produce unsatisfactory results. The adventurous user should consider using the B-spline interpolation routine BSINT that allows one to choose the knots and order of the spline interpolant.

In Figure 3-1, we display six spline interpolants to the same data. This data can be found in Example 1 of the IMSL routine CSCON (page 434) Notice the different characteristics of the interpolants. The interpolation routines CSAKM (page 432) and CSCON are the only two that attempt to preserve the shape of the data. The other routines tend to have extraneous inflection points, with the piecewise quartic (k = 5) exhibiting the most oscillation.

Tensor Product Splines

The simplest method of obtaining multivariate interpolation and approximation routines is to take univariate methods and form a multivariate method via tensor products. In the case of two-dimensional spline interpolation, the development proceeds as follows: Let t[ be a knot sequence for splines of order k[, and t\ be a knot sequence for splines of order k\. Let N[ + k[ be the length of t[, and N\ + k\ be the length of t\. Then, the tensor product spline has the form

Ny Nx

å åcnm Bn,kx ,tx (x)Bm,ky ,ty (y) m=1 n=1

Given two sets of points

;xi @iN=x1 and ;yi @iN=y1

for which the corresponding univariate interpolation problem could be solved, the tensor product interpolation problem becomes: Find the coefficients cQP so that

Ny Nx

å åcnm Bn,kx ,tx (xi )Bm,ky ,ty (yi ) = fij m=1 n=1

This problem can be solved efficiently by repeatedly solving univariate interpolation problems as described in de Boor (1978, page 347). Threedimensional interpolation has analogous behavior. In this chapter, we provide routines that compute the two-dimensional tensorproduct spline coefficients given two-dimensional interpolation data (BS2IN, page 459), compute the threedimensional tensor-product spline coefficients given three-dimensional interpolation data (BS3IN, page 464) compute the two-dimensional tensorproduct spline coefficients for a tensor-product least squares problem (BSLS2, page 561), and compute the three-dimensional tensor-product spline coefficients for a tensor-product least squares problem (BSLS3, page 566). In addition, we provide evaluation, differentiation, and integration routines for the twoand threedimensional tensor-product spline functions. The relevant routines are BS2VL (page 479), BS3VL (page 490), BS2DR (page 480), BS3DR (page 491), BS2GD (page 483), BS3GD (page 495), BS2IG (page 487), and BS3IG (page 500).

Quadratic Interpolation

The routines that begin with the letters “ QD” in this chapter are designed to interpolate a one-, two-, or three-dimensional (tensor product) table of values and return an approximation to the value of the underlying function or one of its derivatives at a given point. These routines are all based on quadratic polynomial interpolation.

Scattered Data Interpolation

We have one routine, SURF, that will return values of an interpolant to scattered data in the plane. This routine is based on work by Akima (1978), which utilizes

C piecewise quintics on a triangular mesh.

Least Squares

Routines are provided to smooth noisy data: regression using linear polynomials (RLINE), regression using arbitrary polynomials (RCURV, page 535), and regression using user-supplied functions (FNLSQ, page 539). Additional routines compute the least-squares fit using splines with fixed knots (BSLSQ, page 543) or free knots (BSVLS, page 547). These routines can produce cubic-spline leastsquares fit simply by setting the order to 4. The routine CONFT (page 551) computes a fixed-knot spline weighted least-squares fit subject to linear constraints. This routine is very general and is recommended if issues of shape are important. The twoand three-dimensional tensor-product spline regression routines are (BSLS2, page 561) and (BSLS3, page 566).

Smoothing by Cubic Splines

Two “smoothing spline” routines are provided. The routine CSSMH (page 575) returns the cubic spline that smooths the data, given a smoothing parameter chosen by the user. Whereas, CSSCV (page 578) estimates the smoothing

parameter by cross-validation and then returns the cubic spline that smooths the data. In this sense, CSSCV is the easier of the two routines to use. The routine CSSED (page 572) returns a smoothed data vector approximating the values of the underlying function when the data are contaminated by a few random spikes.

Rational Chebyshev Approximation

The routine RATCH (page 581) computes a rational Chebyshev approximation to a user-supplied function. Since polynomials are rational functions, this routine can be used to compute best polynomial approximations.

Using the Univariate Spline Routines

An easy to use spline interpolation routine CSIEZ (page 420) is provided . This routine computes an interpolant and returns the values of the interpolant on a user-supplied grid. A slightly more advanced routine SPLEZ (page 447) computes (at the users discretion) one of several interpolants or least-squares fits and returns function values or derivatives on a user-supplied grid.

For more advanced uses of the interpolation (or least squares) spline routines, one first forms an interpolant from interpolation (or least-squares) data. Then it must be evaluated, differentiated, or integrated once the interpolant has been formed. One way to perform these tasks, using cubic splines with the ‘not-a-knot’ end condition, is to call CSINT to obtain the local coefficients of the piecewise cubic interpolant and then call CSVAL to evaluate the interpolant. A more complicated situation arises if one wants to compute a quadratic spline interpolant and then evaluate it (efficiently) many times. Typically, the sequence of routines called might be BSNAK (get the knots), BSINT (returns the B-spline coefficients of the interpolant), BSCPP (convert to pp form), and PPVAL (evaluate). The last two calls could be replaced by a call to the B-spline grid evaluator BS1GD, or the last call could be replaced with pp grid evaluator PP1GD. The interconnection of the spline routines is summarized in Figure 3-2.

DATA

OUT

Figure 3-2 Interrelation of the Spline Routines

Choosing an Interpolation Routine

The choice of an interpolation routine depends both on the type of data and on the use of the interpolant. We provide 18 interpolation routines. These routines are depicted in a decision tree in Figure 3-3. This figure provides a guide for selecting an appropriate interpolation routine. For example, if periodic onedimensional (univariate) data is available, then the path through univariate to periodic leads to the IMSL routine CSPER, which is the proper routine for this setting. The general-purpose univariate interpolation routines can be found in

the box beginning with CSINT. Twoand three-dimensional tensor-product interpolation routines are also provided. For two-dimensional scattered data, the appropriate routine is SURF .

Figure 3-3 Choosing an Interplation Routine

CSIEZ/DCSIEZ (Single/Double precision)

Compute the cubic spline interpolant with the ‘not-a-knot’ condition and return values of the interpolant at specified points.

Usage

CALL CSIEZ (NDATA, XDATA, FDATA, N, XVEC, VALUE)

Arguments

NDATA — Number of data points. (Input)

NDATA must be at least 2.

XDATA — Array of length NDATA containing the data point abscissas. (Input) The data point abscissas must be distinct.

FDATA — Array of length NDATA containing the data point ordinates. (Input)

N — Length of vector XVEC. (Input)

XVEC — Array of length N containing the points at which the spline is to be evaluated. (Input)

VALUE — Array of length N containing the values of the spline at the points in XVEC. (Output)

Comments

Automatic workspace usage is

Workspace may be explicitly provided, if desired, by use of C2IEZ/DC2IEZ. The reference is

CALL C2IEZ (NDATA, XDATA, FDATA, N, XVEC, VALUE, IWK, WK1, WK2)

The additional arguments are as follows:

Algorithm

This routine is designed to let the user easily compute the values of a cubic spline interpolant. The routine CSIEZ computes a spline interpolant to a set of data points (xL, fL) for i = 1, ¼, NDATA. The output for this routine consists of a vector of values of the computed cubic spline. Specifically, let n = N, v = XVEC, and y = VALUE, then if s is the computed spline we set

yM = s(vM )j = 1, ¼, n

Additional documentation can be found by referring to the IMSL routines CSINT (page 423) or SPLEZ (page 447).

Example

In this example, a cubic spline interpolant to a function F is computed. The values of this spline are then compared with the exact function values.

10CONTINUE

DO 20 I=1, 2*NDATA - 1

XVEC(I) = FLOAT(I-1)/FLOAT(2*NDATA-2)

Output

CSINT/DCSINT (Single/Double precision)

Compute the cubic spline interpolant with the ‘not-a-knot’ condition.

Usage

CALL CSINT (NDATA, XDATA, FDATA, BREAK, CSCOEF)

Arguments

NDATA — Number of data points. (Input)

NDATA must be at least 2.

XDATA — Array of length NDATA containing the data point abscissas. (Input) The data point abscissas must be distinct.

FDATA — Array of length NDATA containing the data point ordinates. (Input)

BREAK — Array of length NDATA containing the breakpoints for the piecewise cubic representation. (Output)

CSCOEF — Matrix of size 4 by NDATA containing the local coefficients of the cubic pieces. (Output)

Comments

1.Automatic workspace usage is

CSINT NDATA units, or

DCSINT NDATA units.

Workspace may be explicitly provided, if desired, by use of

C2INT/DC2INT. The reference is

CALL C2INT (NDATA, XDATA, FDATA, BREAK, CSCOEF, IWK)

The additional argument is

IWK — Work array of length NDATA.

2.The cubic spline can be evaluated using CSVAL (page 440); its derivative can be evaluated using CSDER (page 441).

3.Note that column NDATA of CSCOEF is used as workspace.

Algorithm

The routine CSINT computes a C cubic spline interpolant to a set of data points (xL, fL) for i = 1, ¼, NDATA = N. The breakpoints of the spline are the abscissas. Endpoint conditions are automatically determined by the program. These conditions correspond to the “not-a-knot” condition (see de Boor 1978), which requires that the third derivative of the spline be continuous at the second and next-to-last breakpoint. If N is 2 or 3, then the linear or quadratic interpolating polynomial is computed, respectively.

If the data points arise from the values of a smooth (say C ) function f, i.e. fL= f(xL), then the error will behave in a predictable fashion. Let ξ be the

breakpoint vector for the above spline interpolant. Then, the maximum absolute error satisfies

For more details, see de Boor (1978, pages 55 −56).

Example

In this example, a cubic spline interpolant to a function F is computed. The values of this spline are then compared with the exact function values.

20CONTINUE END

Output

CSDEC/DCSDEC (Single/Double precision)

Compute the cubic spline interpolant with specified derivative endpoint conditions.

Usage

CALL CSDEC (NDATA, XDATA, FDATA, ILEFT, DLEFT, IRIGHT,

DRIGHT, BREAK, CSCOEF)

Arguments

NDATA — Number of data points. (Input)

XDATA — Array of length NDATA containing the data point abscissas. (Input) The data point abscissas must be distinct.

FDATA — Array of length NDATA containing the data point ordinates. (Input)

ILEFT — Type of end condition at the left endpoint. (Input)

ILEFT Condition

0“Not-a-knot” condition

1First derivative specified by DLEFT

2Second derivative specified by DLEFT

DLEFT — Derivative at left endpoint if ILEFT is equal to 1 or 2. (Input)

If ILEFT = 0, then DLEFT is ignored.

IRIGHT — Type of end condition at the right endpoint. (Input)

IRIGHT Condition

0“Not-a-knot” condition

1First derivative specified by DRIGHT

2Second derivative specified by DRIGHT

DRIGHT — Derivative at right endpoint if IRIGHT is equal to 1 or 2. (Input) If IRIGHT = 0 then DRIGHT is ignored.

BREAK — Array of length NDATA containing the breakpoints for the piecewise cubic representation. (Output)

CSCOEF — Matrix of size 4 by NDATA containing the local coefficients of the cubic pieces. (Output)

Comments

1.Automatic workspace usage is

CSDEC NDATA units, or DCSDEC NDATA units.

Workspace may be explicitly provided, if desired, by use of C2DEC/DC2DEC. The reference is

CALL C2DEC (NDATA, XDATA, FDATA, ILEFT, DLEFT, IRIGHT, DRIGHT, BREAK, CSCOEF, IWK)

The additional argument is

IWK — Work array of length NDATA.

2.The cubic spline can be evaluated using CSVAL (page 440); its derivative can be evaluated using CSDER (page 441).

3.Note that column NDATA of CSCOEF is used as workspace.

Algorithm

The routine CSDEC computes a C cubic spline interpolant to a set of data points (xL, fL) for i = 1, ¼, NDATA = N. The breakpoints of the spline are the abscissas. Endpoint conditions are to be selected by the user. The user may specify not-a- knot, first derivative, or second derivative at each endpoint (see de Boor 1978, Chapter 4).

If the data (including the endpoint conditions) arise from the values of a smooth (say C ) function f, i.e. fL = f(xL), then the error will behave in a predictable fashion. Let x be the breakpoint vector for the above spline interpolant. Then, the maximum absolute error satisfies

where

ξ : = i=2,K,N ξi − ξi−1

For more details, see de Boor (1978, Chapter 4 and 5).

Example 1

In Example 1, a cubic spline interpolant to a function f is computed. The value of the derivative at the left endpoint and the value of the second derivative at the right endpoint are specified. The values of this spline are then compared with the exact function values.

&BREAK, CSCOEF)

20CONTINUE END

Output

Example 2

In Example 2, we compute the natural cubic spline interpolant to a function f by forcing the second derivative of the interpolant to be zero at both endpoints. As in the previous example, we compare the exact function values with the values of the spline.

&BREAK, CSCOEF)

20CONTINUE END

Output

CSHER/DCSHER (Single/Double precision)

Compute the Hermite cubic spline interpolant.

Usage

CALL CSHER (NDATA, XDATA, FDATA, DFDATA, BREAK, CSCOEF)

Arguments

NDATA — Number of data points. (Input)

XDATA — Array of length NDATA containing the data point abscissas. (Input) The data point abscissas must be distinct.

FDATA — Array of length NDATA containing the data point ordinates. (Input)

DFDATA — Array of length NDATA containing the values of the derivative. (Input)

BREAK — Array of length NDATA containing the breakpoints for the piecewise cubic representation. (Output)

CSCOEF — Matrix of size 4 by NDATA containing the local coefficients of the cubic pieces. (Output)

Comments

1.Automatic workspace usage is

CSHER NDATA units, or

DCSHER NDATA units.

Workspace may be explicitly provided, if desired, by use of

C2HER/DC2HER. The reference is

CALL C2HER (NDATA, XDATA, FDATA, DFDATA, BREAK,

CSCOEF, IWK)

The additional argument is

IWK — Work array of length NDATA.

2.Informational error Type Code

42 The XDATA values must be distinct.

3.The cubic spline can be evaluated using CSVAL (page 440); its derivative can be evaluated using CSDER (page 441).

4.Note that column NDATA of CSCOEF is used as workspace.

Algorithm

The routine CSHER computes a C cubic spline interpolant to the set of data points

1xi , fi 6 and 1xi , fi′ 6

for i = 1, ¼, NDATA = N. The breakpoints of the spline are the abscissas.

If the data points arise from the values of a smooth (say C ) function f, i.e.,

fi = f (xi ) and fi′ = f ′(xi )

then the error will behave in a predictable fashion. Let x be the

breakpoint vector for the above spline interpolant. Then, the maximum absolute error satisfies

For more details, see de Boor (1978, page 51).

Example

In this example, a cubic spline interpolant to a function f is computed. The value of the function f and its derivative f ¢ are computed on the interpolation nodes and passed to CSHER. The values of this spline are then compared with the exact function values.

INTEGER NDATA

PARAMETER (NDATA=11)

C

&DFDATA(NDATA), F, FDATA(NDATA), FLOAT, SIN, X,

Output

CSAKM/DCSAKM (Single/Double precision)

Compute the Akima cubic spline interpolant.

Usage

CALL CSAKM (NDATA, XDATA, FDATA, BREAK, CSCOEF)

Arguments

NDATA — Number of data points. (Input)

XDATA — Array of length NDATA containing the data point abscissas. (Input) The data point abscissas must be distinct.

FDATA — Array of length NDATA containing the data point ordinates. (Input)

BREAK — Array of length NDATA containing the breakpoints for the piecewise cubic representation. (Output)

CSCOEF — Matrix of size 4 by NDATA containing the local coefficients of the cubic pieces. (Output)

Comments

1.Automatic workspace usage is

CSAKM NDATA units, or

DCSAKM NDATA units.

Workspace may be explicitly provided, if desired, by use of

C2AKMD/C2AKM. The reference is

CALL C2AKM (NDATA, XDATA, FDATA, BREAK, CSCOEF, IWK)

The additional argument is

IWK — Work array of length NDATA.

2.The cubic spline can be evaluated using CSVAL (page 440); its derivative can be evaluated using CSDER (page 441).

3.Note that column NDATA of CSCOEF is used as workspace.

Algorithm

The routine CSAKM computes a C cubic spline interpolant to a set of data points (xL, fL) for i = 1, ¼, NDATA = N. The breakpoints of the spline are the abscissas. Endpoint conditions are automatically determined by the program; see Akima (1970) or de Boor (1978).

If the data points arise from the values of a smooth (say C ) function f, i.e. fL = f(xL), then the error will behave in a predictable fashion. Let x be the

breakpoint vector for the above spline interpolant. Then, the maximum absolute error satisfies

The routine CSAKM is based on a method by Akima (1970) to combat wiggles in the interpolant. The method is nonlinear; and although the interpolant is a piecewise cubic, cubic polynomials are not reproduced. (However, linear polynomials are reproduced.)

20CONTINUE END

Output

CSCON/DCSCON (Single/Double precision)

Compute a cubic spline interpolant that is consistent with the concavity of the data.

Usage

CALL CSCON (NDATA, XDATA, FDATA, IBREAK, BREAK, CSCOEF)

Arguments

NDATA — Number of data points. (Input)

NDATA must be at least 3.

XDATA — Array of length NDATA containing the data point abscissas. (Input) The data point abscissas must be distinct.

FDATA — Array of length NDATA containing the data point ordinates. (Input)

IBREAK — The number of breakpoints. (Output)

It will be less than 2 * NDATA.

BREAK — Array of length IBREAK containing the breakpoints for the piecewise cubic representation in its first IBREAK positions. (Output)

The dimension of BREAK must be at least 2 * NDATA.

CSCOEF — Matrix of size 4 by N where N is the dimension of BREAK. (Output) The first IBREAK − 1 columns of CSCOEF contain the local coefficients of the cubic pieces.

Comments

1.Automatic workspace usage is

CSCON 11 * NDATA - 14 units, or

DCSCON 21 * NDATA - 28 units.

Workspace may be explicitly provided, if desired, by use of C2CON/DC2CON. The reference is

CALL C2CON (NDATA, XDATA, FDATA, IBREAK, BREAK, CSCOEF, ITMAX, XSRT, FSRT, A, Y, DIVD, ID, WK)

The additional arguments are as follows:

ITMAX — Maximum number of iterations of Newton’s method. (Input)

XSRT — Work array of length NDATA to hold the sorted XDATA values.

FSRT — Work array of length NDATA to hold the sorted FDATA values.

2Informational errors Type Code

316 Maximum number of iterations exceeded, call C2CON/DC2CON to set a larger number for ITMAX.

43 The XDATA values must be distinct.

3.The cubic spline can be evaluated using CSVAL (page 440); its derivative can be evaluated using CSDER (page 441).

4.The default value for ITMAX is 25. This can be reset by calling

C2CON/DC2CON directly.

Algorithm

The routine CSCON computes a cubic spline interpolant to n = NDATA data points {xL, fL} for i = 1, ¼, n. For ease of explanation, we will assume that xL < xL,

although it is not necessary for the user to sort these data values. If the data are

strictly convex, then the computed spline is convex, C , and minimizes the expression

Ixn 0g′′52

x1

over all convex C functions that interpolate the data. In the general case when the data have both convex and concave regions, the convexity of the spline is consistent with the data and the above integral is minimized under the appropriate constraints. For more information on this interpolation scheme, we refer the reader to Micchelli et al. (1985) and Irvine et al. (1986).

One important feature of the splines produced by this subroutine is that it is not possible, a priori, to predict the number of breakpoints of the resulting interpolant. In most cases, there will be breakpoints at places other than data locations. The method is nonlinear; and although the interpolant is a piecewise cubic, cubic polynomials are not reproduced. (However, linear polynomials are reproduced.) This routine should be used when it is important to preserve the convex and concave regions implied by the data.

Example

We first compute the shape-preserving interpolant using CSCON, and display the coefficients and breakpoints. Second, we interpolate the same data using CSINT (page 423) in a program not shown and overlay the two results. The graph of the result from CSINT is represented by the dashed line. Notice the extra inflection points in the curve produced by CSINT.

&XDATA(NDATA)

C

DATA XDATA/0.0, .1, .2, .3, .4, .5, .6, .8, 1./ DATA FDATA/0.0, .9, .95, .9, .1, .05, .05, .2, 1./ DATA RLABEL/’ 1’, ’ 2’, ’ 3’, ’ 4’/

DATA CLABEL/’ ’, ’ 1’, ’ 2’, ’ 3’, ’ 4’, ’ 5’, ’ 6’,

&’ 7’, ’ 8’, ’ 9’, ’10’, ’11’, ’12’, ’13’/

&CLABEL)

Output

13

11.000

20.000

30.000

40.000

Figure 3-4 CSCON vs. CSINT

CSPER/DCSPER (Single/Double precision)

Compute the cubic spline interpolant with periodic boundary conditions.

Usage

CALL CSPER (NDATA, XDATA, FDATA, BREAK, CSCOEF)

Arguments

NDATA — Number of data points. (Input)

NDATA must be at least 4.

XDATA — Array of length NDATA containing the data point abscissas. (Input) The data point abscissas must be distinct.

FDATA — Array of length NDATA containing the data point ordinates. (Input)

BREAK — Array of length NDATA containing the breakpoints for the piecewise cubic representation. (Output)

CSCOEF — Matrix of size 4 by NDATA containing the local coefficients of the cubic pieces. (Output)

Comments

1.Automatic workspace usage is

CSPER 7 * NDATA units, or DCSPER 13 * NDATA units.

Workspace may be explicitly provided, if desired, by use of C2PER/DC2PER. The reference is

CALL C2PER (NDATA, XDATA, FDATA, BREAK, CSCOEF, WK,

IWK)

The additional arguments are as follows:

2.Informational error Type Code

31 The data set is not periodic, i.e., the function values at the smallest and largest XDATA points are not equal. The value at the smallest XDATA point is used.

3.The cubic spline can be evaluated using CSVAL (page 440)and its derivative can be evaluated using CSDER (page 441).

Algorithm

The routine CSPER computes a C cubic spline interpolant to a set of data points (xL, fL) for i = 1, ¼, NDATA = N. The breakpoints of the spline are the abscissas. The program enforces periodic endpoint conditions. This means that the spline s satisfies s(a) = s(b), s¢(a) = s¢(b), and s²(a) = s²(b), where a is the leftmost abscissa and b is the rightmost abscissa. If the ordinate values corresponding to a and b are not equal, then a warning message is issued. The ordinate value at b is set equal to the ordinate value at a and the interpolant is computed.

If the data points arise from the values of a smooth (say C ) periodic function f, i.e. fL = f(xL), then the error will behave in a predictable fashion. Let x be the breakpoint vector for the above spline interpolant. Then, the maximum absolute error satisfies

For more details, see de Boor (1978, pages 320 -322).

Example

In this example, a cubic spline interpolant to a function f is computed. The values of this spline are then compared with the exact function values.

PI = CONST(’PI’)

H = 2.0*PI/15.0/10.0

FDATA(NDATA) = FDATA(1)

99999 FORMAT (13X, ’X’, 9X, ’Interpolant’, 5X, ’Error’) NINTV = NDATA - 1

20CONTINUE END

Output

CSVAL/DCSVAL (Single/Double precision)

Evaluate a cubic spline.

Usage

CSVAL(X, NINTV, BREAK, CSCOEF)

Arguments

X — Point at which the spline is to be evaluated. (Input)

NINTV — Number of polynomial pieces. (Input)

BREAK — Array of length NINTV + 1 containing the breakpoints for the piecewise cubic representation. (Input)

BREAK must be strictly increasing.

CSCOEF — Matrix of size 4 by NINTV + 1 containing the local coefficients of the cubic pieces. (Input)

CSVAL — Value of the polynomial at X. (Output)

Algorithm

The routine CSVAL evaluates a cubic spline at a given point. It is a special case of the routine PPDER (page 507), which evaluates the derivative of a piecewise polynomial. (The value of a piecewise polynomial is its zero-th derivative and a cubic spline is a piecewise polynomial of order 4.) The routine PPDER is based on the routine PPVALU in de Boor (1978, page 89).

Example

For an example of the use of CSVAL, see IMSL routine CSINT (page 423).

CSDER/DCSDER (Single/Double precision)

Evaluate the derivative of a cubic spline.

Usage

CSDER(IDERIV, X, NINTV, BREAK, CSCOEF)

Arguments

IDERIV — Order of the derivative to be evaluated. (Input)

In particular, IDERIV = 0 returns the value of the polynomial.

X — Point at which the polynomial is to be evaluated. (Input)

NINTV — Number of polynomial pieces. (Input)

BREAK — Array of length NINTV + 1 containing the breakpoints for the piecewise cubic representation. (Input)

BREAK must be strictly increasing.

CSCOEF — Matrix of size 4 by NINTV + 1 containing the local coefficients of the cubic pieces. (Input)

CSDER — Value of the IDERIV-th derivative of the polynomial at X. (Output)

Algorithm

The function CSDER evaluates the derivative of a cubic spline at a given point. It is a special case of the routine PPDER (page 507), which evaluates the derivative

of a piecewise polynomial. (A cubic spline is a piecewise polynomial of order 4.) The routine PPDER is based on the routine PPVALU in de Boor (1978, page 89).

Example

In this example, we compute a cubic spline interpolant to a function f using IMSL routine CSINT (page 423). The values of the spline and its first and second derivatives are computed using CSDER. These values can then be compared with the corresponding values of the interpolated function.

INTEGER NDATA

PARAMETER (NDATA=10)

C

&CSDER, DDF, DF, F, FDATA(NDATA), FLOAT, SIN, X,

&XDATA(NDATA)

Output