chbder_c

 Procedure Abstract Required_Reading Keywords Brief_I/O Detailed_Input Detailed_Output Parameters Exceptions Files Particulars Examples Restrictions Literature_References Author_and_Institution Version Index_Entries

#### Procedure

```   void chbder_c ( ConstSpiceDouble * cp,
SpiceInt           degp,
SpiceDouble        x2s,
SpiceDouble        x,
SpiceInt           nderiv,
SpiceDouble      * partdp,
SpiceDouble      * dpdxs )

```

#### Abstract

```   Given the coefficients for the Chebyshev expansion of a
polynomial, this returns the value of the polynomial and its
first nderiv derivatives evaluated at the input X.
```

```   None.
```

#### Keywords

```    INTERPOLATION
MATH
POLYNOMIAL

```

#### Brief_I/O

```    VARIABLE  I/O  DESCRIPTION
--------  ---  --------------------------------------------------
cp         I   degp+1 Chebyshev polynomial coefficients.
degp       I   Degree of polynomial.
x2s        I   Transformation parameters of polynomial.
x          I   Value for which the polynomial is to be evaluated
nderiv     I   The number of derivatives to compute
partdp    I-O  Workspace provided for computing derivatives
dpdxs      O   Array of the derivatives of the polynomial
```

#### Detailed_Input

```    cp         is an array of coefficients a polynomial with respect
to the Chebyshev basis. The polynomial to be
evaluated is assumed to be of the form:

cp(degp+1)*T(degp,S) + cp(degp)*T(degp-1,S) + ...

... + cp(2)*T(1,S) + cp(1)*T(0,S)

where T(I,S) is the I'th Chebyshev polynomial
evaluated  at a number S whose double precision
value lies between -1 and 1. The value of S is
computed from the input variables x2s,
x2s and X.

degp       is the degree of the Chebyshev polynomial to be
evaluated.

x2s        is an array of two parameters. These parameters are
used to transform the domain of the input variable X
into the standard domain of the Chebyshev polynomial.
x2s should be a reference point in the domain of
x; x2s should be the radius by which points are
allowed to deviate from the reference point and while
remaining within the domain of x. The value of
x is transformed into the value S given by

S = ( x - x2s ) / x2s

Typically x2s is the midpoint of the interval over
which x is allowed to vary and x2s is the radius
of the interval.

The main reason for doing this is that a Chebyshev
expansion is usually fit to data over a span
from A to B where A and B are not -1 and 1
respectively. Thus to get the "best fit" the
data was transformed to the interval [-1,1] and
coefficients generated. These coefficients are
not re-scaled to the interval of the data so that
the numerical "robustness" of the Chebyshev fit will
not be lost. Consequently, when the "best fitting"
polynomial needs to be evaluated at an intermediate
point, the point of evaluation must be transformed
in the same way that the generating points were
transformed.

x          Value for which the polynomial is to be evaluated.

nderiv     is the number of derivatives to be computed by the
routine. nderiv should be non-negative.

partdp     Is a work space used by the program to compute
all of the desired derivatives. It should be declared
in the calling program as

SpiceDouble partdp[3 * (nderiv+1)]
```

#### Detailed_Output

```    dpdxs(0)   The value of the polynomial to be evaluated. It
is given by

cp(degp+1)*T(degp,S) + cp(degp)*T(degp-1,S) + ...

... + cp(2)*T(1,S) + cp(1)*T(0,S)

where T(I,S) is the I'th Chebyshev polynomial
evaluated  at a number S = ( x -  x2s )/ x2s.

dpdxs(i)   The value of the i'th derivative of the polynomial at
x. (I ranges from 1 to nderiv) It is given by

[i]
(1/x2s**i) ( cp(degp+1)*T   (degp,S)

[i]
+ cp(degp)*T   (degp-1,S) + ...

.
.
.
[i]
... + cp(2)*T   (1,S)

[i]
+ cp(1)*T   (0,S) )

[i]
where T(k,S) and T   (i,S)  are the k'th Chebyshev
polynomial and its i'th derivative respectively,
evaluated at the number S = ( x - x2s )/x2s.
```

#### Parameters

```    None.
```

#### Exceptions

```   Error free

No tests are performed for exceptional values ( nderiv negative,
degp negative, etc.) This routine is expected to be used at a low
level in ephemeris evaluations. For that reason it has been
elected as a routine that will not participate in error handling.
```

#### Files

```    None.
```

#### Particulars

```    This routine computes the value of a Chebyshev polynomial
expansion and the derivatives of the expansion with respect to X.
The polynomial is given by

cp(degp+1)*T(degp,S) + cp(degp)*T(degp-1,S) + ...

... + cp(2)*T(1,S) + cp(1)*T(0,S)

where

S  =  ( x - x2s ) / x2s

and

T(i,S) is the i'th Chebyshev polynomial of the first kind
evaluated at S.
```

#### Examples

```   Depending upon the user's needs, there are 3 routines available
for evaluating Chebyshev polynomials.

chbval_c for evaluating a Chebyshev polynomial when no
derivatives are desired.

chbint_c for evaluating a Chebyshev polynomial and its
first derivative.

chbder_c for evaluating a Chebyshev polynomial and a user
or application dependent number of derivatives.

Of these 3 the one most commonly employed by NAIF software
is chbint_c as it is used to interpolate ephemeris state
vectors which requires the evaluation of a polynomial
and its derivative. When no derivatives are desired one
should use chbval_c, or when more than one or an unknown
number of derivatives are desired one should use chbder_c.

Example:

#include <stdio.h>
#include "SpiceUsr.h"

int main()
{

/.
Local variables.
./
SpiceDouble      cp [] = {  1., 3., 0.5, 1., 0.5, -1., 1. };
SpiceInt         degp    = 6;
SpiceInt         nderiv = 3;
SpiceDouble      x2s[]  = { .5, 3.};
SpiceDouble      x      = 1.;

/. Dimension partdp as 3 * (nderiv + 1) ./
SpiceDouble      partdp[3 * 4];

/. Dimension dpdxs as nderiv + 1. ./
SpiceDouble       dpdxs [3+1];

int              i;

chbder_c ( cp, degp, x2s, x, nderiv, partdp, dpdxs );

for ( i=0; i<=nderiv; i++ )
{
printf( "dpdxs = %lf\n", dpdxs[i] );
}

return(0);
}

The program outputs:

dpdxs = -0.340878
dpdxs = 0.382716
dpdxs = 4.288066
dpdxs = -1.514403
```

#### Restrictions

```    The user must be sure that the provided workspace is declared
properly in the calling routine. The proper declaration is:

SpiceInt         nderiv = the desired number of derivatives;
SpiceDouble      partdp[3 * (nderiv + 1)];

If for some reason a parameter is not passed to this routine in
nderiv, the user should make sure that the value of nderiv is not
so large that the work space provided is inadequate.

One needs to be careful that the value (X-x2s) / x2s lies
between -1 and 1. Otherwise, the routine may fail spectacularly
(for example with a floating point overflow).

While this routine will compute derivatives of the input
polynomial, the user should consider how accurately the
derivatives of the Chebyshev fit, match the derivatives of the
function it approximates.
```

#### Literature_References

```    "Numerical Recipes -- The Art of Scientific Computing" by
William H. Press, Brian P. Flannery, Saul A. Teukolsky,
Willam T. Vetterling. (See Clenshaw's Recurrence Formula)

"The Chebyshev Polynomials" by Theodore J. Rivlin.

"CRC Handbook of Tables for Mathematics"
```

#### Author_and_Institution

```    N.J. Bachman    (JPL)
W.L. Taber      (JPL)
E.D. Wright     (JPL)
```

#### Version

```   -CSPICE Version 1.0.0, 24-AUG-2015 (EDW)
```

#### Index_Entries

```   derivatives of a chebyshev expansion
```
`Wed Apr  5 17:54:29 2017`