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<title>General Polynomial Equations - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">6.5 General Polynomial Equations</h3>
<p><a name="index-general-polynomial-equations_002c-solving-234"></a>
The roots of polynomial equations cannot be found analytically beyond
the special cases of the quadratic, cubic and quartic equation. The
algorithm described in this section uses an iterative method to find the
approximate locations of roots of higher order polynomials.
<div class="defun">
— Function: gsl_poly_complex_workspace * <b>gsl_poly_complex_workspace_alloc</b> (<var>size_t n</var>)<var><a name="index-gsl_005fpoly_005fcomplex_005fworkspace_005falloc-235"></a></var><br>
<blockquote><p>This function allocates space for a <code>gsl_poly_complex_workspace</code>
struct and a workspace suitable for solving a polynomial with <var>n</var>
coefficients using the routine <code>gsl_poly_complex_solve</code>.
<p>The function returns a pointer to the newly allocated
<code>gsl_poly_complex_workspace</code> if no errors were detected, and a null
pointer in the case of error.
</p></blockquote></div>
<div class="defun">
— Function: void <b>gsl_poly_complex_workspace_free</b> (<var>gsl_poly_complex_workspace * w</var>)<var><a name="index-gsl_005fpoly_005fcomplex_005fworkspace_005ffree-236"></a></var><br>
<blockquote><p>This function frees all the memory associated with the workspace
<var>w</var>.
</p></blockquote></div>
<div class="defun">
— Function: int <b>gsl_poly_complex_solve</b> (<var>const double * a, size_t n, gsl_poly_complex_workspace * w, gsl_complex_packed_ptr z</var>)<var><a name="index-gsl_005fpoly_005fcomplex_005fsolve-237"></a></var><br>
<blockquote><p>This function computes the roots of the general polynomial
<!-- {$P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1}$} -->
P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1} using
balanced-QR reduction of the companion matrix. The parameter <var>n</var>
specifies the length of the coefficient array. The coefficient of the
highest order term must be non-zero. The function requires a workspace
<var>w</var> of the appropriate size. The n-1 roots are returned in
the packed complex array <var>z</var> of length 2(n-1), alternating
real and imaginary parts.
<p>The function returns <code>GSL_SUCCESS</code> if all the roots are found. If
the QR reduction does not converge, the error handler is invoked with
an error code of <code>GSL_EFAILED</code>. Note that due to finite precision,
roots of higher multiplicity are returned as a cluster of simple roots
with reduced accuracy. The solution of polynomials with higher-order
roots requires specialized algorithms that take the multiplicity
structure into account (see e.g. Z. Zeng, Algorithm 835, ACM
Transactions on Mathematical Software, Volume 30, Issue 2 (2004), pp
218–236).
</p></blockquote></div>
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