Table of contents
CSPICE_LGRESP evaluates a Lagrange interpolating polynomial for a
specified set of coordinate pairs whose first components are equally
spaced, at a specified abscissa value.
Given:
n the number of points defining the polynomial.
help, n
LONG = Scalar
The array `yvals' contains `n' elements.
first,
step 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.
help, first
DOUBLE = Scalar
help, step
DOUBLE = Scalar
The set of abscissa values is
first + i * step, i = 0, ..., n-1
`step' must be non-zero.
yvals 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.
help, yvals
DOUBLE = Array[n]
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 the abscissa value at which the interpolating polynomial is
to be evaluated.
help, x
DOUBLE = Scalar
the call:
lgresp = cspice_lgresp( n, first, step, yvals, x )
returns:
lgresp 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'.
help, lgresp
DOUBLE = Scalar
None.
Any numerical results shown for these examples 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 )
( 1, -8 )
( 3, 26 )
( 5, 148 )
and evaluate this polynomial at x = 2.
The returned value of cspice_lgresp 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.
PRO lgresp_ex1
n = 4L
first = -1.D0
step = 2.D0
yvals = [ -2.D0, -8.D0, 26.D0, 148.D0 ]
answer = cspice_lgresp( n, first, step, yvals, 2.D0 )
print, 'ANSWER = ', answer
END
When this program was executed on a Mac/Intel/IDL8.x/64-bit
platform, the output was:
ANSWER = 1.0000000
2) Solve the same problem using a negative step. In order to
find the solution, set the elements of `yvals' in reverse order.
The returned value of cspice_lgresp would still be 1.0.
Example code begins here.
PRO lgresp_ex2
n = 4L
first = 5.D0
step = -2.D0
yvals = [ 148.D0, 26.D0, -8.D0, -2.D0 ]
answer = cspice_lgresp( n, first, step, yvals, 2.D0 )
print, 'ANSWER = ', answer
END
When this program was executed on a Mac/Intel/IDL8.x/64-bit
platform, the output was:
ANSWER = 1.0000000
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.
For Lagrange interpolation on unequally spaced abscissa values,
see the Icy routine cspice_lgrint.
1) If `step' is zero, the error SPICE(INVALIDSTEPSIZE) is signaled
by a routine in the call tree of this routine. The function
will return an undefined value.
2) If `n' is less than 1, the error SPICE(INVALIDSIZE) is signaled
by a routine in the call tree of this routine. The function
will return an undefined value.
3) This routine does not attempt to ward off or diagnose
arithmetic overflows.
4) If any of the input arguments, `n', `first', `step', `yvals'
or `x', is undefined, an error is signaled by the IDL error
handling system.
5) If any of the input arguments, `n', `first', `step', `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 Icy
interface.
6) If the number of elements in `yvals' is less than `n', an error
is signaled by the Icy interface.
None.
None.
ICY.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)
-Icy Version 1.0.0, 09-JUN-2021 (JDR)
interpolate function using Lagrange polynomial
Lagrange interpolation
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