hrmesp_c |

Table of contents## Procedurehrmesp_c ( Hermite polynomial interpolation, equal spacing ) void hrmesp_c ( SpiceInt n, SpiceDouble first, SpiceDouble step, ConstSpiceDouble yvals [], SpiceDouble x, SpiceDouble * f, SpiceDouble * df ) ## AbstractEvaluate, 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. ## 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 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_Inputn 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_Outputf, 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'. ## ParametersNone. ## Exceptions1) 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. ## FilesNone. ## ParticularsUsers 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 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 answer; 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; ## 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. [2] S. Conte and C. de Boor, "Elementary Numerical Analysis -- An Algorithmic Approach," 3rd Edition, p 64, McGraw-Hill, 1980. ## Author_and_InstitutionJ. Diaz del Rio (ODC Space) ## Version-CSPICE Version 1.0.0, 03-AUG-2021 (JDR) ## Index_Entriesinterpolate function using Hermite polynomial Hermite interpolation |

Fri Dec 31 18:41:08 2021