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lgrind_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

lgrind_c (Lagrange polynomial interpolation with derivative)

void lgrind_c ( SpiceInt            n,
ConstSpiceDouble  * xvals,
ConstSpiceDouble  * yvals,
SpiceDouble       * work,
SpiceDouble         x,
SpiceDouble       * p,
SpiceDouble       * dp )

Abstract

Evaluate a Lagrange interpolating polynomial, for a specified
set of coordinate pairs, at a specified abscissa value. Return
both the value of the polynomial and its derivative.

None.

INTERPOLATION
POLYNOMIAL

Brief_I/O

VARIABLE  I/O  DESCRIPTION
--------  ---  --------------------------------------------------
n          I   Number of points defining the polynomial.
xvals      I   Abscissa values.
yvals      I   Ordinate values.
work      I-O  Work space array.
x          I   Point at which to interpolate the polynomial.
p          O   Polynomial value at `x'.
dp         O   Polynomial derivative at `x'.

Detailed_Input

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

xvals,
yvals       are arrays of abscissa and ordinate values that
together define `n' ordered pairs. The set of points

( xvals[i], yvals[i] )

define the Lagrange polynomial used for
interpolation. The elements of `xvals' must be
distinct and in increasing order.

work        is an n * 2 work space array, where `n' is the same
dimension as that of `xvals' and `yvals'. It is used
by this routine as a scratch area to hold
intermediate results.

x           is the abscissa value at which the interpolating
polynomial is to be evaluated.

Detailed_Output

p           is the value at `x' of the unique polynomial of
degree n-1 that fits the points in the plane
defined by `xvals' and `yvals'.

dp          is the derivative at `x' of the interpolating
polynomial described above.

None.

Exceptions

1)  If any two elements of the array `xvals' 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.

None.

Particulars

Given a set of `n' distinct abscissa values and corresponding
ordinate values, there is a unique polynomial of degree n-1, often
called the "Lagrange polynomial", that fits the graph defined by
these values. The Lagrange polynomial can be used to interpolate
the value of a function at a specified point, given a discrete
set of values of the function.

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

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 cubic polynomial through the points

( -1, -2 )
(  0, -7 )
(  1, -8 )
(  3, 26 )

and evaluate this polynomial at x = 2.

The returned value of `p' should be 1.0, since the
unique cubic polynomial that fits these points is

3      2
f(x)   =   x  + 2*x  - 4*x - 7

The returned value of `dp' should be 16.0, since the
derivative of f(x) is

'           2
f (x)  =  3*x  + 4*x - 4

Example code begins here.

/.
Program lgrind_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"

int main( )
{

/.
Local variables.
./
SpiceDouble      p;
SpiceDouble      dp;
SpiceDouble      xvals [] = { -1., 0., 1., 3. };
SpiceDouble      yvals [] = { -2., -7., -8., 26. };
SpiceDouble      work  [4*2];
SpiceInt         n = 4;

lgrind_c ( n, xvals, yvals, work, 2., &p, &dp );

printf( "P, DP = %f %f\n", p, dp );

return ( 0 );
}

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

P, DP = 1.000000 16.000000

Note that we could also have lgrind_c with the reference

lgrind_c ( n, xvals, yvals, yvals, 2., &p, &dp );

if we wished to; in this case `yvals' would have been
modified on output.

None.

Literature_References

  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.

Author_and_Institution

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

Version

-CSPICE Version 1.0.1, 01-NOV-2021 (JDR)

Edited the header to comply with NAIF standard.

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

Index_Entries

interpolate function using Lagrange polynomial
Lagrange interpolation
Fri Dec 31 18:41:09 2021