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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

```   hrmint_c ( Hermite polynomial interpolation  )

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
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'. 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
times 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)  If two input abscissas are equal, the error SPICE(DIVIDEBYZERO) is
signaled by a routine in the call tree of this routine.

2)  If `n' is less than 1, the error SPICE(INVALIDSIZE) is
signaled by a routine in the call tree of this routine.

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 [1] 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

```   The numerical results shown for this example may differ across
platforms. The results depend on the SPICE kernels used as
input, the compiler and supporting libraries, and the machine
specific arithmetic implementation.

1) 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.

The returned value should be 141.0, and the returned
derivative value should be 456.0, since the unique 7th degree
polynomial that fits these constraints is

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

Example code begins here.

/.
Program hrmint_ex1
./

#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  = %f\n", deriv );

return ( 0 );
}

When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:

DERIV  = 456.000000
```

Restrictions

```   None.
```

Literature_References

```   [1]  W. Press, B. Flannery, S. Teukolsky and W. Vetterling,
"Numerical Recipes -- The Art of Scientific Computing,"
chapters 3.0 and 3.1, Cambridge University Press, 1986.

[2]  S. Conte and C. de Boor, "Elementary Numerical Analysis -- An
Algorithmic Approach," 3rd Edition, p 64, McGraw-Hill, 1980.
```

Author_and_Institution

```   J. Diaz del Rio     (ODC Space)
E.D. Wright         (JPL)
```

Version

```   -CSPICE Version 1.0.1, 22-FEB-2021 (JDR)

Updated the header to comply with NAIF standard.

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

Index_Entries

```   interpolate function using Hermite polynomial
Hermite interpolation
```
`Fri Dec 31 18:41:08 2021`