lgresp_c |

Table of contents## Procedurelgresp_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_Inputn 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. ## Exceptions1) 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. ## 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. For Lagrange interpolation on unequally spaced abscissa values, see the CSPICE routine lgrint_c. ## ExamplesThe 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 ## 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