Table of contents
CSPICE_HRMINT evaluates a Hermite interpolating polynomial at a specified
abscissa value.
Given:
n the number of points defining the polynomial.
help, n
LONG = Scalar
The arrays `xvals' and `yvals' contain `n' and 2*n elements
respectively.
xvals an array of length `n' containing abscissa values.
help, xvals
DOUBLE = Array[n]
yvals an array of length 2*n containing ordinate and derivative
values for each point in the domain defined by `xvals'.
help, yvals
DOUBLE = Array[2*n]
The elements
yvals[ 2*i ]
yvals[ 2*i +1 ]
give the value and first derivative of the output
polynomial at the abscissa value
xvals[i]
where `i' ranges from 0 to n - 1.
x the abscissa value at which the interpolating polynomial and
its derivative are to be evaluated.
help, x
DOUBLE = Scalar
the call:
cspice_hrmint, n, xvals, yvals, x, f, df
returns:
f,
df the value and derivative at `x' of the unique polynomial of
degree 2n-1 that fits the points and derivatives defined by
`xvals' and `yvals'.
help, f
DOUBLE = Scalar
help, df
DOUBLE = Scalar
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 7th degree polynomial through the points ( x, y, y' )
( -1, 6, 3 )
( 0, 5, 0 )
( 3, 2210, 5115 )
( 5, 78180, 109395 )
and evaluate this polynomial at x = 2.
The returned value 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.
PRO hrmint_ex1
n = 4L
xvals = [ -1.D0, 0.D0, 3.D0, 5.D0 ]
yvals = [ 6.D0, 3.D0, 5.D0, 0.D0, $
2210.D0, 5115.D0, 78180.D0, 109395.D0 ]
cspice_hrmint, n, xvals, yvals, 2.D0, answer, deriv
print, 'ANSWER = ', answer
print, 'DERIV = ', deriv
END
When this program was executed on a Mac/Intel/IDL8.x/64-bit
platform, the output was:
ANSWER = 141.00000
DERIV = 456.00000
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 two input abscissas 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.
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 IDL 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 Icy
interface.
6) If any of the output arguments, `f' or `df', is not a named
variable, an error is signaled by the Icy interface.
7) If the number of elements in `xvals' is less than `n', an error
is signaled by the Icy interface.
8) If the number of elements in `yvals' is less than 2*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.
[2] S. Conte and C. de Boor, "Elementary Numerical Analysis -- An
Algorithmic Approach," 3rd Edition, p 64, McGraw-Hill, 1980.
J. Diaz del Rio (ODC Space)
-Icy Version 1.0.0, 18-JUN-2021 (JDR)
interpolate function using Hermite polynomial
Hermite interpolation
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