| lgresp_c |
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Table of contents
Procedure
lgresp_c ( Lagrange interpolation on equally spaced points )
SpiceDouble lgresp_c ( SpiceInt n,
SpiceDouble first,
SpiceDouble step,
ConstSpiceDouble yvals [],
SpiceDouble x )
AbstractEvaluate a Lagrange interpolating polynomial for a specified set of coordinate pairs whose first components are equally spaced, at a specified abscissa value. Required_ReadingNone. KeywordsINTERPOLATION POLYNOMIAL Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- n I Number of points defining the polynomial. first I First abscissa value. step I Step size. yvals I Ordinate values. x I Point at which to interpolate the polynomial. The function returns the value at `x' of the unique polynomial of degree n-1 that fits the points in the plane defined by `first', `step', and `yvals'. Detailed_Input
n is the number of points defining the polynomial. The
array `yvals' contains `n' elements.
first,
step are, respectively, a starting abscissa value and a
step size that define the set of abscissa values
at which a Lagrange interpolating polynomial is to
be defined. The set of abscissa values is
first + i * step, i = 0, ..., n-1
`step' must be non-zero.
yvals is an array of ordinate values that, together with
the abscissa values defined by `first' and `step',
define `n' ordered pairs belonging to the graph of
a function. The set of points
( first + i*step, yvals(i) )
where `i' ranges from 0 to n-1, define the Lagrange
polynomial used for interpolation.
x is the abscissa value at which the interpolating
polynomial is to be evaluated.
Detailed_OutputThe function returns the value at `x' of the unique polynomial of degree n-1 that fits the points in the plane defined by `first', `step', and `yvals'. ParametersNone. Exceptions
1) If `step' is zero, the error SPICE(INVALIDSTEPSIZE) is signaled
by a routine in the call tree of this routine. The function
will return the value 0.0.
2) If `n' is less than 1, the error SPICE(INVALIDSIZE) is signaled. The
function will return the value 0.0.
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. The function
returns the value result.
FilesNone. 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 [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.
For Lagrange interpolation on unequally spaced abscissa values,
see the CSPICE routine lgrint_c.
Examples
The numerical results shown for these examples 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 )
( 1, -8 )
( 3, 26 )
( 5, 148 )
and evaluate this polynomial at x = 2.
The returned value of lgresp_c should be 1.0, since the
unique cubic polynomial that fits these points is
3 2
f(x) = x + 2*x - 4*x - 7
Example code begins here.
/.
Program lgresp_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"
int main( )
{
SpiceDouble answer;
SpiceDouble first;
SpiceDouble step;
SpiceDouble yvals [4];
SpiceInt n;
n = 4;
first = -1.0;
step = 2.0;
yvals[0] = -2.0;
yvals[1] = -8.0;
yvals[2] = 26.0;
yvals[3] = 148.0;
answer = lgresp_c ( n, first, step, yvals, 2.0 );
printf( "ANSWER = %f\n", answer );
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
ANSWER = 1.000000
2) Solve the same problem using a negative step. In order to
find the solution, set the elements of `yvals' in reverse order.
The returned value of lgresp_c would still be 1.0.
Example code begins here.
/.
Program lgresp_ex2
./
#include <stdio.h>
#include "SpiceUsr.h"
int main( )
{
SpiceDouble answer;
SpiceDouble first;
SpiceDouble step;
SpiceDouble yvals [4];
SpiceInt n;
n = 4;
first = 5.0;
step = -2.0;
yvals[0] = 148.0;
yvals[1] = 26.0;
yvals[2] = -8.0;
yvals[3] = -2.0;
answer = lgresp_c ( n, first, step, yvals, 2.0 );
printf( "ANSWER = %f\n", answer );
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
ANSWER = 1.000000
RestrictionsNone. 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.
Author_and_InstitutionJ. Diaz del Rio (ODC Space) Version-CSPICE Version 1.0.0, 04-AUG-2021 (JDR) Index_Entriesinterpolate function using Lagrange polynomial Lagrange interpolation |
Fri Dec 31 18:41:09 2021