Table of contents
CSPICE_LGRINT evaluates a Lagrange interpolating polynomial for a
specified set of coordinate pairs, at a specified abscissa value.
Given:
n the number of points defining the polynomial.
[1,1] = size(n); int32 = class(n)
The arrays `xvals' and `yvals' contain `n' elements.
xvals,
yvals arrays of abscissa and ordinate values that together
define `n' ordered pairs.
[n,1] = size(xvals); double = class(xvals)
[n,1] = size(yvals); double = class(yvals)
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.
x the abscissa value at which the interpolating polynomial is
to be evaluated.
[1,1] = size(x); double = class(x)
the call:
[lgrint] = cspice_lgrint( n, xvals, yvals, x )
returns:
lgrint the value at `x' of the unique polynomial of degree n-1 that
fits the points in the plane defined by `xvals' and `yvals'.
[1,1] = size(lgrint); double = class(lgrint)
None.
Any numerical results shown for this example may differ between
platforms as the results depend on the SPICE kernels used as input
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 cspice_lgrint 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.
function lgrint_ex1()
n = 4;
xvals = [ -1.0, 0.0, 1.0, 3.0 ]';
yvals = [ -2.0, -7.0, -8.0, 26.0 ]';
answer = cspice_lgrint( n, xvals, yvals, 2.0 );
fprintf( 'ANSWER = %f\n', answer )
When this program was executed on a Mac/Intel/Octave6.x/64-bit
platform, the output was:
ANSWER = 1.000000
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.
1) 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, an error is signaled the Mice Interface.
3) This routine does not attempt to ward off or diagnose
arithmetic overflows.
4) If any of the input arguments, `n', `xvals', `yvals' or `x',
is undefined, an error is signaled by the Matlab error
handling system.
5) If any of the input arguments, `n', `xvals', `yvals' or `x',
is not of the expected type, or it does not have the expected
dimensions and size, an error is signaled by the Mice
interface.
6) If the number of elements in `xvals' or `yvals' is less than `n',
an error is signaled by the Mice interface.
None.
None.
MICE.REQ
[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.
J. Diaz del Rio (ODC Space)
-Mice Version 1.0.0, 24-JUN-2021 (JDR)
interpolate function using Lagrange polynomial
Lagrange interpolation
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