hrmint_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 hrmint_c ( SpiceInt            n,
ConstSpiceDouble  * xvals,
ConstSpiceDouble  * yvals,
SpiceDouble         x,
SpiceDouble       * work,
SpiceDouble       * f,
SpiceDouble       * df )

```

#### Abstract

```   Evaluate a Hermite interpolating polynomial at a specified
abscissa value.
```

```   None.
```

#### Keywords

```   INTERPOLATION
MATH
POLYNOMIAL

```

#### Brief_I/O

```   Variable  I/O  Description
--------  ---  --------------------------------------------------
n          I   Number of points defining the polynomial.
xvals      I   Abscissa values.
yvals      I   Ordinate and derivative values.
x          I   Point at which to interpolate the polynomial.
work      I-O  Work space array.
f          O   Interpolated function value at x.
df         O   Interpolated function's derivative at x.
```

#### Detailed_Input

```   n              is the number of points defining the polynomial.
The arrays xvals and yvals contain n and 2*n
elements respectively.

xvals          is an array of length n containing abscissa values.

yvals          is an array of length 2*n containing ordinate and
derivative values for each point in the domain
defined by xvals and n. The elements

yvals[ 2*i    ]
yvals[ 2*i +1 ]

give the value and first derivative of the output
polynomial at the abscissa value

xvals[i]

where i ranges from 0 to n - 1.

work           is a work space array.  It is used by this routine
as a scratch area to hold intermediate results.
Generally sized at number of elements in yvals
time two.

x              is the abscissa value at which the interpolating
polynomial and its derivative are to be evaluated.
```

#### Detailed_Output

```   f,
df             are the value and derivative at x of the unique
polynomial of degree 2n-1 that fits the points and
derivatives defined by xvals and yvals.
```

#### Parameters

```   None.
```

#### Exceptions

```   1)  The error SPICE(DIVIDEBYZERO) signals from a routine
in the call tree if two input abscissas are equal,

2)  The error SPICE(INVALIDSIZE) signals from a routine
in the call tree if n is less than 1.

3)  This routine does not attempt to ward off or diagnose
arithmetic overflows.
```

#### Files

```   None.
```

#### Particulars

```   Users of this routine must choose the number of points to use
in their interpolation method.  The authors of Reference  have
this to say on the topic:

Unless there is solid evidence that the interpolating function
is close in form to the true function f, it is a good idea to
be cautious about high-order interpolation.  We
enthusiastically endorse interpolations with 3 or 4 points, we
are perhaps tolerant of 5 or 6; but we rarely go higher than
that unless there is quite rigorous monitoring of estimated
errors.

The same authors offer this warning on the use of the
interpolating function for extrapolation:

...the dangers of extrapolation cannot be overemphasized:
An interpolating function, which is perforce an extrapolating
function, will typically go berserk when the argument x is
outside the range of tabulated values by more than the typical
spacing of tabulated points.
```

#### Examples

```   Example:

Fit a 7th degree polynomial through the points ( x, y, y' )

( -1,      6,       3 )
(  0,      5,       0 )
(  3,   2210,    5115 )
(  5,  78180,  109395 )

and evaluate this polynomial at x = 2.

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

int main()
{

/.
Local variables.
./
SpiceDouble      deriv;
SpiceDouble      xvals [] = {-1., 0., 3., 5.};
SpiceDouble      yvals [] = {6., 3., 5., 0.,
2210., 5115., 78180., 109395.};
SpiceDouble      work  [2*8];
SpiceDouble      x = 2;
SpiceInt         n = 4;

hrmint_c ( n, xvals, yvals, x, work, &answer, &deriv );

printf( "deriv  = %lf\n", deriv );

return(0);
}

The returned value of 'answer' should be 141., and the returned
value of 'deriv' should be 456., since the unique 7th degree
polynomial that fits these constraints is

7       2
f(x)  =  x   +  2x  + 5
```

#### Restrictions

```   None.
```

#### Literature_References

```     "Numerical Recipes---The Art of Scientific Computing" by
William H. Press, Brian P. Flannery, Saul A. Teukolsky,
William T. Vetterling (see sections 3.0 and 3.1).

  "Elementary Numerical Analysis---An Algorithmic Approach"
by S. D. Conte and Carl de Boor.  See p. 64.
```

#### Author_and_Institution

```    N.J. Bachman    (JPL)
E.D. Wright     (JPL)
```

#### Version

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

#### Index_Entries

```   interpolate function using Hermite polynomial
Hermite interpolation
```
`Wed Apr  5 17:54:36 2017`