| chbigr_c |
|
Table of contents
Procedure
chbigr_c ( Chebyshev expansion integral )
void chbigr_c ( SpiceInt degp,
ConstSpiceDouble cp [],
ConstSpiceDouble x2s [2],
SpiceDouble x,
SpiceDouble * p,
SpiceDouble * itgrlp )
AbstractEvaluate an indefinite integral of a Chebyshev expansion at a specified point `x' and return the value of the input expansion at `x' as well. The constant of integration is selected to make the integral zero when `x' equals the abscissa value x2s[0]. Required_ReadingNone. KeywordsCHEBYSHEV EPHEMERIS INTEGRAL MATH Brief_I/OVARIABLE I/O DESCRIPTION -------- --- -------------------------------------------------- degp I Degree of input Chebyshev expansion. cp I Chebyshev coefficients of input expansion. x2s I Transformation parameters. x I Abscissa value of evaluation. p O Input expansion evaluated at `x'. itgrlp O Integral evaluated at `x'. Detailed_Input
degp is the degree of the input Chebyshev expansion.
cp is an array containing the coefficients of the input
Chebyshev expansion. The coefficient of the i'th
Chebyshev polynomial is contained in element cp[i],
for i = 0 : degp.
x2s is an array containing the "transformation parameters"
of the domain of the expansion. Element x2s[0]
contains the midpoint of the interval on which the
input expansion is defined; x2s[1] is one-half of the
length of this interval; this value is called the
interval's "radius."
The input expansion defines a function f(x) on the
interval
[ x2s[0]-x2s[1], x2s[0]+x2s[1] ]
as follows:
Define s = ( x - x2s[0] ) / x2s[1]
degp
__
\
f(x) = g(s) = / cp[k] T (s)
-- k
k=0
x is the abscissa value at which the function defined by
the input expansion and its integral are to be
evaluated. Normally `x' should lie in the closed
interval
[ x2s[0]-x2s[1], x2s[0]+x2s[1] ]
See the -Restrictions section below.
Detailed_Output
p,
itgrlp Define `s' and f(x) as above in the description of the
input argument `x2s'. Then `p' is f(x), and `itgrlp' is
an indefinite integral of f(x), evaluated at `x'.
The indefinite integral satisfies
d(itgrlp)/dx = f(x)
and
itgrlp( x2s[0] ) = 0
ParametersNone. Exceptions
1) If the input degree is negative, the error SPICE(INVALDDEGREE) is
signaled by a routine in the call tree of this routine.
2) If the input interval radius is non-positive, the error
SPICE(INVALIDRADIUS) is signaled by a routine in the call tree of
this routine.
FilesNone. Particulars
Let
T , n = 0, ...
n
represent the nth Chebyshev polynomial of the first kind:
T (x) = cos( n*arccos(x) )
n
The input coefficients represent the Chebyshev expansion
degp
__
\
f(x) = g(s) = / cp[k] T (s)
-- k
k=0
where
s = ( x - x2s[0] ) / x2s[1]
This routine evaluates and returns the value at `x' of an
indefinite integral F(x), where
dF(x)/dx = f(x) for all `x' in
[x2s[0]-x2s[0], x2s[0]+x2s[1]]
F( x2s[0] ) = 0
The value at `x' of the input expansion
f(x)
is returned as well.
Note that numerical problems may result from applying this
routine to abscissa values outside of the interval defined
by the input parameters x2s[*]. See the -Restrictions section.
To evaluate Chebyshev expansions and their derivatives, use the
CSPICE routines chbint_c or chbder_c.
This routine supports the SPICELIB SPK type 20 and PCK type 20
evaluators SPKE20 and PCKE20.
Examples
The numerical results shown for this example 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) Let the domain of a polynomial to be evaluated be the
closed interval
[20, 30]
Let the input expansion represent the polynomial
6
f(x) = g(s) = 5*s
where
s = (x - 20)/10
Let F(x) be an indefinite integral of f(x) such that
F(20) = 0
Evaluate
f(30) and F(30)
Example code begins here.
/.
Program chbigr_ex1
./
#include <stdio.h>
#include "SpiceUsr.h"
int main( )
{
/.
Local variables
./
SpiceDouble cp [6];
SpiceDouble x;
SpiceDouble x2s [2];
SpiceDouble p;
SpiceDouble itgrlp;
SpiceInt degp;
/.
Let our domain be the interval [10, 30].
./
x2s[0] = 20.0;
x2s[1] = 10.0;
/.
Assign the expansion coefficients.
./
degp = 5;
cp[0] = 0.0;
cp[1] = 3.75;
cp[2] = 0.0;
cp[3] = 1.875;
cp[4] = 0.0;
cp[5] = 0.375;
/.
Evaluate the function and its integral at x = 30.
./
x = 30.0;
chbigr_c ( degp, cp, x2s, x, &p, &itgrlp );
/.
We make the change of variables
s(x) = (1/10) * ( x - 20 )
The expansion represents the polynomial
5
f(x) = g(s) = 6*s
An indefinite integral of the expansion is
6
F(x) = G(s) * dx/ds = 10 * s
where `G' is defined on the interval [-1, 1]. The result
should be (due to the change of variables)
(G(1) - G(0) ) * dx/ds
= (F(30) - F(20)) * 10
= 10
The value of the expansion at `x' should be
f(30) = g(1) = 6
./
printf( "ITGRLP = %f\n", itgrlp );
printf( "P = %f\n", p );
return ( 0 );
}
When this program was executed on a Mac/Intel/cc/64-bit
platform, the output was:
ITGRLP = 10.000000
P = 6.000000
Restrictions
1) The value (x-x2s[0]) / x2s[1] normally should lie within the
interval -1:1 inclusive, that is, the closed interval
[-1, 1]. Chebyshev polynomials increase rapidly in magnitude
as a function of distance of abscissa values from this
interval.
In typical SPICE applications, where the input expansion
represents position, velocity, or orientation, abscissa
values that map to points outside of [-1, 1] due to round-off
error will not cause numeric exceptions.
2) No checks for floating point overflow are performed.
3) Significant accumulated round-off error can occur for input
expansions of excessively high degree. This routine imposes
no limits on the degree of the input expansion; users must
verify that the requested computation provides appropriate
accuracy.
Literature_References
[1] W. Press, B. Flannery, S. Teukolsky and W. Vetterling,
"Numerical Recipes -- The Art of Scientific Computing,"
chapter 5.4, "Recurrence Relations and Clenshaw's Recurrence
Formula," p 161, Cambridge University Press, 1986.
[2] "Chebyshev polynomials," Wikipedia, The Free Encyclopedia.
Retrieved 01:23, November 23, 2013, from
http://en.wikipedia.org/w/index.php?title=
Chebyshev_polynomials&oldid=574881046
Author_and_InstitutionJ. Diaz del Rio (ODC Space) Version-CSPICE Version 1.0.0, 19-JUL-2021 (JDR) Index_Entriesintegral of chebyshev_polynomial_expansion integrate chebyshev_polynomial_expansion |
Fri Dec 31 18:41:02 2021