# conditioning of linear systems

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##### 2: 32.4 Isomonodromy Problems
$\mbox{P}_{\mbox{\scriptsize I}}$$\mbox{P}_{\mbox{\scriptsize VI}}$ can be expressed as the compatibility condition of a linear system, called an isomonodromy problem or Lax pair. …
##### 3: 28.34 Methods of Computation
• (d)

Solution of the systems of linear algebraic equations (28.4.5)–(28.4.8) and (28.14.4), with the conditions (28.4.9)–(28.4.12) and (28.14.5), by boundary-value methods (§3.6) to determine the Fourier coefficients. Subsequently, the Fourier series can be summed with the aid of Clenshaw’s algorithm (§3.11(ii)). See Meixner and Schäfke (1954, §2.87). This procedure can be combined with §28.34(ii)(d).

• ##### 4: 3.11 Approximation Techniques
With $b_{0}=1$, the last $q$ equations give $b_{1},\dots,b_{q}$ as the solution of a system of linear equations. … (3.11.29) is a system of $n+1$ linear equations for the coefficients $a_{0},a_{1},\dots,a_{n}$. The matrix is symmetric and positive definite, but the system is ill-conditioned when $n$ is large because the lower rows of the matrix are approximately proportional to one another. … … Then the system (3.11.33) is diagonal and hence well-conditioned. …
##### 5: 3.6 Linear Difference Equations
###### §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). …
###### §3.6(vii) Linear Difference Equations of Other Orders
or for systems of $k$ first-order inhomogeneous equations, boundary-value methods are the rule rather than the exception. Typically $k-\ell$ conditions are prescribed at the beginning of the range, and $\ell$ conditions at the end. …
##### 6: Bibliography M
• I. G. Macdonald (1982) Some conjectures for root systems. SIAM J. Math. Anal. 13 (6), pp. 988–1007.
• I. G. Macdonald (2000) Orthogonal polynomials associated with root systems. Sém. Lothar. Combin. 45, pp. Art. B45a, 40 pp. (electronic).
• Maxima (free interactive system)
• J. C. P. Miller (1950) On the choice of standard solutions for a homogeneous linear differential equation of the second order. Quart. J. Mech. Appl. Math. 3 (2), pp. 225–235.
• MuPAD (commercial interactive system and Matlab toolbox)
• ##### 7: 3.8 Nonlinear Equations
###### Regula Falsi
Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation: …
###### §3.8(vi) Conditioning of Zeros
are well separated but extremely ill-conditioned. …
##### 8: 1.13 Differential Equations
For extensions of these results to linear homogeneous differential equations of arbitrary order see Spigler (1984). … Assuming that $u(x)$ satisfies un-mixed boundary conditions of the form …or periodic boundary conditionsA regular Sturm-Liouville system will only have solutions for certain (real) values of $\lambda$, these are eigenvalues. … For a regular Sturm-Liouville system, equations (1.13.26) and (1.13.29) have: (i) identical eigenvalues, $\lambda$; (ii) the corresponding (real) eigenfunctions, $u(x)$ and $w(t)$, have the same number of zeros, also called nodes, for $t\in(0,c)$ as for $x\in(a,b)$; (iii) the eigenfunctions also satisfy the same type of boundary conditions, un-mixed or periodic, for both forms at the corresponding boundary points. …
##### 9: 3.7 Ordinary Differential Equations
Consideration will be limited to ordinary linear second-order differential equationsThe remaining two equations are supplied by boundary conditions of the form … If, for example, $\beta_{0}=\beta_{1}=0$, then on moving the contributions of $w(z_{0})$ and $w(z_{P})$ to the right-hand side of (3.7.13) the resulting system of equations is not tridiagonal, but can readily be made tridiagonal by annihilating the elements of $\mathbf{A}_{P}$ that lie below the main diagonal and its two adjacent diagonals. … The Sturm–Liouville eigenvalue problem is the construction of a nontrivial solution of the systemThe Runge–Kutta method applies to linear or nonlinear differential equations. …
##### 10: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions
###### Bounded and Unbounded Linear Operators
This question may be rephrased by asking: do $f(x)$ and $g(x)$ satisfy the same boundary conditions which are needed to fully specify the solutions of a second order linear differential equation? A simple example is the choice $f(a)=f(b)=0$, and $g(a)=g(b)=0$, this being only one of many. … A boundary value for the end point $a$ is a linear form $\mathcal{B}$ on $\mathcal{D}({\mathcal{L}}^{*})$ of the form …Then, if the linear form $\mathcal{B}$ is nonzero, the condition $\mathcal{B}(f)=0$ is called a boundary condition at $a$. …