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Complex polynomial evaluation
- From: Frank Reininghaus <frank78ac at googlemail dot com>
- To: gsl-discuss at sourceware dot org
- Date: Wed, 12 Dec 2007 00:04:05 +0100
- Subject: Complex polynomial evaluation
Hi,
I've written a bit of code that evaluates polynomials with real or
complex coefficients for complex variables. There was some discussion
about this a couple of years ago, see
http://sourceware.org/ml/gsl-discuss/2004-q2/msg00040.html
and follow-ups.
I've followed the example of gsl_poly_eval () in gsl_poly.h and
poly/eval.c and wrote two functions:
gsl_poly_complex_eval () evaluates a polynomial with real coefficients
for a complex variable,
gsl_complex_poly_complex_eval () does the same with complex coefficients.
I'm not quite sure if the choice of names is appropriate, but I think
that it makes sense to have both functions because one often needs to
evaluate real polynomials for complex values (like Taylor series
expansions for some functions).
If you are interested in including this in GSL, you can use the patches
for gsl_poly.h and eval.c which I have attached (an alternative might be
to create a new header file gsl_complex_poly.h). If this is the case, I
could also write a bit of documentation and code for some test cases to
be included in 'make check'. So far, I've checked the results for many
random polynomials with Maxima. Moreover, I've checked with random data
that both functions yield the same results for real polynomials and that
their results coincide with gsl_poly_eval () if the variable is also real.
Regards,
Frank
35a36,68
>
> gsl_complex
> gsl_poly_complex_eval(const double c[], const int len, const gsl_complex x)
> {
> int i;
> gsl_complex ans;
> GSL_SET_COMPLEX (&ans, c[len-1], 0.0);
> for(i=len-1; i>0; i--) {
> /* The following three lines are equivalent to
> ans = gsl_complex_add_real (gsl_complex_mul (x, ans), c[i-1]);
> but faster */
> double tmp = c[i-1] + GSL_REAL (x) * GSL_REAL (ans) - GSL_IMAG (x) * GSL_IMAG (ans);
> GSL_SET_IMAG (&ans, GSL_IMAG (x) * GSL_REAL (ans) + GSL_REAL (x) * GSL_IMAG (ans));
> GSL_SET_REAL (&ans, tmp);
> }
> return ans;
> }
>
> gsl_complex
> gsl_complex_poly_complex_eval(const gsl_complex c[], const int len, const gsl_complex x)
> {
> int i;
> gsl_complex ans = c[len-1];
> for(i=len-1; i>0; i--) {
> /* The following three lines are equivalent to
> ans = gsl_complex_add (c[i-1], gsl_complex_mul (x, ans));
> but faster */
> double tmp = GSL_REAL (c[i-1]) + GSL_REAL (x) * GSL_REAL (ans) - GSL_IMAG (x) * GSL_IMAG (ans);
> GSL_SET_IMAG (&ans, GSL_IMAG (c[i-1]) + GSL_IMAG (x) * GSL_REAL (ans) + GSL_REAL (x) * GSL_IMAG (ans));
> GSL_SET_REAL (&ans, tmp);
> }
> return ans;
> }
44a45,46
>
> /* real polynomial, real x */
46a49,54
> /* real polynomial, complex x */
> gsl_complex gsl_poly_complex_eval (const double c [], const int len, const gsl_complex x);
>
> /* complex polynomial, complex x */
> gsl_complex gsl_complex_poly_complex_eval (const gsl_complex c [], const int len, const gsl_complex x);
>
57a66,100
>
> extern inline
> gsl_complex
> gsl_poly_complex_eval(const double c[], const int len, const gsl_complex x)
> {
> int i;
> gsl_complex ans;
> GSL_SET_COMPLEX (&ans, c[len-1], 0.0);
> for(i=len-1; i>0; i--) {
> /* The following three lines are equivalent to
> ans = gsl_complex_add_real (gsl_complex_mul (x, ans), c[i-1]);
> but faster */
> double tmp = c[i-1] + GSL_REAL (x) * GSL_REAL (ans) - GSL_IMAG (x) * GSL_IMAG (ans);
> GSL_SET_IMAG (&ans, GSL_IMAG (x) * GSL_REAL (ans) + GSL_REAL (x) * GSL_IMAG (ans));
> GSL_SET_REAL (&ans, tmp);
> }
> return ans;
> }
>
> extern inline
> gsl_complex
> gsl_complex_poly_complex_eval(const gsl_complex c[], const int len, const gsl_complex x)
> {
> int i;
> gsl_complex ans = c[len-1];
> for(i=len-1; i>0; i--) {
> /* The following three lines are equivalent to
> ans = gsl_complex_add (c[i-1], gsl_complex_mul (x, ans));
> but faster */
> double tmp = GSL_REAL (c[i-1]) + GSL_REAL (x) * GSL_REAL (ans) - GSL_IMAG (x) * GSL_IMAG (ans);
> GSL_SET_IMAG (&ans, GSL_IMAG (c[i-1]) + GSL_IMAG (x) * GSL_REAL (ans) + GSL_REAL (x) * GSL_IMAG (ans));
> GSL_SET_REAL (&ans, tmp);
> }
> return ans;
> }