# normal equations

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##### 2: 30.2 Differential Equations
The Liouville normal form of equation (30.2.1) is …
##### 4: 3.11 Approximation Techniques
From the equations $\ifrac{\partial S}{\partial a_{k}}=0$, $k=0,1,\dots,n$, we derive the normal equations
##### 5: 29.20 Methods of Computation
The normalization of Lamé functions given in §29.3(v) can be carried out by quadrature (§3.5). A second approach is to solve the continued-fraction equations typified by (29.3.10) by Newton’s rule or other iterative methods; see §3.8. …Subsequently, formulas typified by (29.6.4) can be applied to compute the coefficients of the Fourier expansions of the corresponding Lamé functions by backward recursion followed by application of formulas typified by (29.6.5) and (29.6.6) to achieve normalization; compare §3.6. (Equation (29.6.3) serves as a check.) … §29.15(i) includes formulas for normalizing the eigenvectors. …
##### 6: 28.7 Analytic Continuation of Eigenvalues
The normal values are simple roots of the corresponding equations (28.2.21) and (28.2.22). …
##### 7: 28.5 Second Solutions $\mathrm{fe}_{n}$, $\mathrm{ge}_{n}$
28.5.5 $(C_{n}(q))^{2}\int_{0}^{2\pi}(f_{n}(x,q))^{2}\mathrm{d}x=(S_{n}(q))^{2}\int_{0% }^{2\pi}(g_{n}(x,q))^{2}\mathrm{d}x=\pi.$
##### 8: 3.6 Linear Difference Equations
Let us assume the normalizing condition is of the form $w_{0}=\lambda$, where $\lambda$ is a constant, and then solve the following tridiagonal system of algebraic equations for the unknowns $w_{1}^{(N)},w_{2}^{(N)},\dots,w_{N-1}^{(N)}$; see §3.2(ii). …
##### 9: 28.12 Definitions and Basic Properties
28.12.2 $\lambda_{\nu}\left(-q\right)=\lambda_{\nu}\left(q\right)=\lambda_{-\nu}\left(q% \right).$
If $q$ is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of $z$ and $q$ by the normalization
##### 10: 29.21 Tables
• Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^{2}=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.