Table of contents
CSPICE_HRMESP evaluates, 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.
Given:
n the number of points defining the polynomial.
[1,1] = size(n); int32 = class(n)
The array `yvals' contains 2*n elements.
first,
step respectively, a starting abscissa value and a
step size that define the set of abscissa values
first + (I-1) * `step', i = 1, ..., n
[1,1] = size(first); double = class(first)
[1,1] = size(step); double = class(step)
`step' must be non-zero.
yvals an array of length 2*n containing ordinate and derivative
values for each point in the domain defined by `first',
`step', and `n'.
[2*n,1] = size(yvals); double = class(yvals)
The elements
yvals( 2*i - 1 )
yvals( 2*i )
give the value and first derivative of the output
polynomial at the abscissa value
first + (i-1) * `step'
where `i' ranges from 1 to `n'.
x the abscissa value at which the interpolating polynomial and
its derivative are to be evaluated.
[1,1] = size(x); double = class(x)
the call:
[f, df] = cspice_hrmesp( n, first, step, yvals, x )
returns:
f,
df 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'.
[1,1] = size(f); double = class(f)
[1,1] = size(df); double = class(df)
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 )
( 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.
function hrmesp_ex1()
n = 4;
yvals = [ 6.0, 3.0, 8.0, 11.0, ...
2210.0, 5115.0, 78180.0, 109395.0 ]';
first = -1.0;
step = 2.0;
[answer, deriv] = cspice_hrmesp( n, first, step, yvals, 2.0 );
fprintf( 'ANSWER = %f\n', answer )
fprintf( 'DERIV = %f\n', deriv )
When this program was executed on a Mac/Intel/Octave6.x/64-bit
platform, the output was:
ANSWER = 141.000000
DERIV = 456.000000
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 `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 by a routine in the call tree of this routine.
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 Matlab 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 Mice
interface.
6) If the number of elements in `yvals' is less than 2*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.
[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)
-Mice Version 1.0.0, 19-JUL-2021 (JDR)
interpolate function using Hermite polynomial
Hermite interpolation
|