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<h3 class="section">6.5 General Polynomial Equations</h3>
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<p><a name="index-general-polynomial-equations_002c-solving-229"></a>
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The roots of polynomial equations cannot be found analytically beyond
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the special cases of the quadratic, cubic and quartic equation. The
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algorithm described in this section uses an iterative method to find the
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approximate locations of roots of higher order polynomials.
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— 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-230"></a></var><br>
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<blockquote><p>This function allocates space for a <code>gsl_poly_complex_workspace</code>
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struct and a workspace suitable for solving a polynomial with <var>n</var>
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coefficients using the routine <code>gsl_poly_complex_solve</code>.
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<p>The function returns a pointer to the newly allocated
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<code>gsl_poly_complex_workspace</code> if no errors were detected, and a null
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pointer in the case of error.
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</p></blockquote></div>
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— 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-231"></a></var><br>
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<blockquote><p>This function frees all the memory associated with the workspace
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</p></blockquote></div>
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— 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-232"></a></var><br>
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<blockquote><p>This function computes the roots of the general polynomial
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<!-- {$P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1}$} -->
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P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_{n-1} x^{n-1} using
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balanced-QR reduction of the companion matrix. The parameter <var>n</var>
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specifies the length of the coefficient array. The coefficient of the
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highest order term must be non-zero. The function requires a workspace
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<var>w</var> of the appropriate size. The n-1 roots are returned in
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the packed complex array <var>z</var> of length 2(n-1), alternating
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real and imaginary parts.
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<p>The function returns <code>GSL_SUCCESS</code> if all the roots are found and
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<code>GSL_EFAILED</code> if the QR reduction does not converge. Note that due
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to finite precision, roots of higher multiplicity are returned as a
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cluster of simple roots with reduced accuracy. The solution of
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polynomials with higher-order roots requires specialized algorithms that
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take the multiplicity structure into account (see e.g. Z. Zeng,
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Algorithm 835, ACM Transactions on Mathematical Software, Volume 30,
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Issue 2 (2004), pp 218–236).
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</p></blockquote></div>
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