lgrind_c |

## Procedurevoid lgrind_c ( SpiceInt n, ConstSpiceDouble * xvals, ConstSpiceDouble * yvals, SpiceDouble * work, SpiceDouble x, SpiceDouble * p, SpiceDouble * dp ) ## AbstractEvaluate a Lagrange interpolating polynomial for a specified set of coordinate pairs, at a specified abscissa value. Return the value of both polynomial and derivative. ## Required_ReadingNone. ## KeywordsINTERPOLATION MATH POLYNOMIAL ## Brief_I/OVariable 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_Inputn 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 x 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_Outputp 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. ## ParametersNone. ## Exceptions1) The error SPICE(DIVIDEBYZERO) signals from a routine in the call tree if any two elements of the array xvals 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. ## FilesNone. ## ParticularsGiven 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. ## ExamplesExample: Fit a cubic polynomial through the points ( -1, -2 ) ( 0, -7 ) ( 1, -8 ) ( 3, 26 ) and evaluate this polynomial at x = 2. #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; ## RestrictionsNone. ## Literature_References[1] "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). ## Author_and_InstitutionN.J. Bachman (JPL) E.D. Wright (JPL) ## Version-CSPICE Version 1.0.0, 24-AUG-2015 (EDW) ## Index_Entriesinterpolate function using Lagrange polynomial Lagrange interpolation |

Wed Apr 5 17:54:38 2017