lgrind_c |

Table of contents## Procedurelgrind_c (Lagrange polynomial interpolation with derivative) void 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 both the value of the polynomial and its derivative. ## Required_ReadingNone. ## KeywordsINTERPOLATION 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 * 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) 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. ## 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. ## ExamplesThe 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; ## 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_InstitutionN.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_Entriesinterpolate function using Lagrange polynomial Lagrange interpolation |

Fri Dec 31 18:41:09 2021