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hrmesp_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

```   hrmesp_c ( Hermite polynomial interpolation, equal spacing )

void hrmesp_c ( SpiceInt            n,
SpiceDouble         first,
SpiceDouble         step,
ConstSpiceDouble    yvals  [],
SpiceDouble         x,
SpiceDouble       * f,
SpiceDouble       * df        )

```

#### Abstract

```   Evaluate, at a specified point, a Hermite interpolating polynomial
for a specified set of equally spaced abscissa values and
corresponding pairs of function and function derivative values.
```

```   None.
```

#### Keywords

```   INTERPOLATION
POLYNOMIAL

```

#### Brief_I/O

```   VARIABLE  I/O  DESCRIPTION
--------  ---  --------------------------------------------------
n          I   Number of points defining the polynomial.
first      I   First abscissa value.
step       I   Step size.
yvals      I   Ordinate and derivative values.
x          I   Point at which to interpolate the polynomial.
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 array `yvals' contains 2*n elements.

first,
step        are, respectively, a starting abscissa value and a
step size that define the set of abscissa values

first   +   i * step,     i = 0, ..., n-1

`step' must be non-zero.

yvals       is an array of length 2*n containing ordinate and
derivative values for each point in the domain
defined by `first', `step', 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

first   +   i * step

where `i' ranges from 0 to n-1.

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 2*n-1 that fits the points and
derivatives defined by `first', `step', and `yvals'.
```

#### Parameters

```   None.
```

#### Exceptions

```   1)  If `step' is zero, the error SPICE(INVALIDSTEPSIZE) 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.

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

4)  If memory cannot be allocated to create the temporary variable
required for the execution of the underlying Fortran routine,
the error SPICE(MALLOCFAILED) is signaled.
```

#### 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 )
(  1,      8,      11 )
(  3,   2210,    5115 )
(  5,  78180,  109395 )

and evaluate this polynomial at x = 2.

The returned value of ANSWER 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 hrmesp_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{

SpiceDouble          deriv;
SpiceDouble          first;
SpiceDouble          step;
SpiceDouble          yvals  [8];
SpiceInt             n;

n         =       4;

yvals[0]  =       6.0;
yvals[1]  =       3.0;
yvals[2]  =       8.0;
yvals[3]  =      11.0;
yvals[4]  =    2210.0;
yvals[5]  =    5115.0;
yvals[6]  =   78180.0;
yvals[7]  =  109395.0;

first     =  -1.0;
step      =   2.0;

hrmesp_c ( n, first, step, yvals, 2.0, &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)
```

#### Version

```   -CSPICE Version 1.0.0, 03-AUG-2021 (JDR)
```

#### Index_Entries

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