Complex polynomial evaluation

Complex polynomial evaluation

[Frank Reininghaus]

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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


[-- Attachment #2: patch_eval.c --]
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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;
> }

[-- Attachment #3: patch_gsl_poly.h --]
[-- Type: text/x-chdr, Size: 1562 bytes --]

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;
> }

Re: Complex polynomial evaluation

[Brian Gough]

At Wed, 12 Dec 2007 00:04:05 +0100,
Frank Reininghaus wrote:
> 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'. 

Yes - that sounds good. Thanks.

-- 
Brian Gough

Re: Complex polynomial evaluation

[Frank Reininghaus]

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Hi,

I've written some documentation for my complex polynomial evaluation 
functions and some test cases to be checked by 'make check'. I've 
created patches containing all my modifications to the current CVS 
version with

diff gsl/poly/ [my directory with new version of 'poly'] > patch-poly
diff gsl/doc/ [my directory with new version of 'doc'] > patch-doc

Let me know if there is a way I could create patch files that are more 
convenient to you.

patch-doc also contains a correction for a typo in the macro @inlinefns 
in doc/gsl-ref.texi. I think it should be "Inline versions of these 
*functions* are used...", not "Inline versions of these *function* are 
used..."

Regards,
Frank


[-- Attachment #2: patch-doc --]
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Common subdirectories: gsl/doc/CVS and /home/frank/tmp/doc-with-complex-polynomials/CVS
Common subdirectories: gsl/doc/examples and /home/frank/tmp/doc-with-complex-polynomials/examples
diff gsl/doc/gsl-ref.texi /home/frank/tmp/doc-with-complex-polynomials/gsl-ref.texi
127c127
< Inline versions of these function are used when @code{HAVE_INLINE} is defined.
---
> Inline versions of these functions are used when @code{HAVE_INLINE} is defined.
diff gsl/doc/poly.texi /home/frank/tmp/doc-with-complex-polynomials/poly.texi
25,26c25
< @deftypefun double gsl_poly_eval (const double @var{c}[], const int @var{len}, const double @var{x})
< This function evaluates the polynomial 
---
> The functions described here evaluate the polynomial 
29c28,39
< Horner's method for stability.  @inlinefn{}
---
> Horner's method for stability. @inlinefns{}
> 
> @deftypefun double gsl_poly_eval (const double @var{c}[], const int @var{len}, const double @var{x})
> This function evaluates a polynomial with real coefficients for the real variable @var{x}.
> @end deftypefun
> 
> @deftypefun gsl_complex gsl_poly_complex_eval (const double @var{c}[], const int @var{len}, const gsl_complex @var{x})
> This function evaluates a polynomial with real coefficients for the complex variable @var{x}.
> @end deftypefun
> 
> @deftypefun gsl_complex gsl_complex_poly_complex_eval (const gsl_complex @var{c}[], const int @var{len}, const gsl_complex @var{x})
> This function evaluates a polynomial with complex coefficients for the complex variable @var{x}.

[-- Attachment #3: patch-poly --]
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Common subdirectories: gsl/poly/CVS and /home/frank/tmp/poly-with-complex-evaluation/CVS
diff gsl/poly/eval.c /home/frank/tmp/poly-with-complex-evaluation/eval.c
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;
> }
diff gsl/poly/gsl_poly.h /home/frank/tmp/poly-with-complex-evaluation/gsl_poly.h
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;
> }
diff gsl/poly/test.c /home/frank/tmp/poly-with-complex-evaluation/test.c
33a34,35
>   /* Polynomial evaluation */
> 
52a55,108
>   {
>     gsl_complex x, y;
>     double c[1] = {0.3};
>     GSL_SET_REAL (&x, 0.75);
>     GSL_SET_IMAG (&x, 1.2);
>     y = gsl_poly_complex_eval (c, 1, x);
> 
>     gsl_test_rel (GSL_REAL (y), 0.3, eps, "y.real, gsl_poly_complex_eval ({0.3}, 0.75 + 1.2i)");
>     gsl_test_rel (GSL_IMAG (y), 0.0, eps, "y.imag, gsl_poly_complex_eval ({0.3}, 0.75 + 1.2i)");
>   }
> 
>   {
>     gsl_complex x, y;
>     double c[4] = {2.1, -1.34, 0.76, 0.45};
>     GSL_SET_REAL (&x, 0.49);
>     GSL_SET_IMAG (&x, 0.95);
>     y = gsl_poly_complex_eval (c, 4, x);
> 
>     gsl_test_rel (GSL_REAL (y), 0.3959143, eps, "y.real, gsl_poly_complex_eval ({2.1, -1.34, 0.76, 0.45}, 0.49 + 0.95i)");
>     gsl_test_rel (GSL_IMAG (y), -0.6433305, eps, "y.imag, gsl_poly_complex_eval ({2.1, -1.34, 0.76, 0.45}, 0.49 + 0.95i)");
>   }
> 
>   {
>     gsl_complex x, y;
>     gsl_complex c[1];
>     GSL_SET_REAL (&c[0], 0.674);
>     GSL_SET_IMAG (&c[0], -1.423);
>     GSL_SET_REAL (&x, -1.44);
>     GSL_SET_IMAG (&x, 9.55);
>     y = gsl_complex_poly_complex_eval (c, 1, x);
> 
>     gsl_test_rel (GSL_REAL (y), 0.674, eps, "y.real, gsl_complex_poly_complex_eval ({0.674 - 1.423i}, -1.44 + 9.55i)");
>     gsl_test_rel (GSL_IMAG (y), -1.423, eps, "y.imag, gsl_complex_poly_complex_eval ({0.674 - 1.423i}, -1.44 + 9.55i)");
>   }
> 
>   {
>     gsl_complex x, y;
>     gsl_complex c[4];
>     GSL_SET_REAL (&c[0], -2.31);
>     GSL_SET_IMAG (&c[0], 0.44);
>     GSL_SET_REAL (&c[1], 4.21);
>     GSL_SET_IMAG (&c[1], -3.19);
>     GSL_SET_REAL (&c[2], 0.93);
>     GSL_SET_IMAG (&c[2], 1.04);
>     GSL_SET_REAL (&c[3], -0.42);
>     GSL_SET_IMAG (&c[3], 0.68);
>     GSL_SET_REAL (&x, 0.49);
>     GSL_SET_IMAG (&x, 0.95);
>     y = gsl_complex_poly_complex_eval (c, 4, x);
> 
>     gsl_test_rel (GSL_REAL (y), 1.82462012, eps, "y.real, gsl_complex_poly_complex_eval ({-2.31 + 0.44i, 4.21 - 3.19i, 0.93 + 1.04i, -0.42 + 0.68i}, 0.49 + 0.95i)");
>     gsl_test_rel (GSL_IMAG (y), 2.30389412, eps, "y.imag, gsl_complex_poly_complex_eval ({-2.31 + 0.44i, 4.21 - 3.19i, 0.93 + 1.04i, -0.42 + 0.68i}, 0.49 + 0.95i)");
>   }
> 

Re: Complex polynomial evaluation

[Brian Gough]

At Thu, 13 Dec 2007 21:05:13 +0100,
Frank Reininghaus wrote:
> Let me know if there is a way I could create patch files that are more 
> convenient to you.

Can you send them with diff -u,  thanks.

Re: Complex polynomial evaluation

[Brian Gough]

At Fri, 14 Dec 2007 13:39:57 +0000,
Brian Gough wrote:
> 
> At Thu, 13 Dec 2007 21:05:13 +0100,
> Frank Reininghaus wrote:
> > Let me know if there is a way I could create patch files that are more 
> > convenient to you.
> 
> Can you send them with diff -u,  thanks.
> 

Thanks, I've added these into the CVS repository now.

-- 
Brian Gough