Concepts inAlgorithm 810: The SLEIGN2 Sturm-Liouville Code
Sturm¿Liouville theory
In mathematics and its applications, a classical Sturm¿Liouville equation, named after Jacques Charles François Sturm (1803¿) and Joseph Liouville (1809¿), is a real second-order linear differential equation of the form where y is a function of the free variable x. Here the functions p(x) > 0, q(x), and w(x) > 0 are specified at the outset.
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Continuous spectrum
The spectrum of a linear operator is commonly divided into three parts: point spectrum, continuous spectrum, and residual spectrum. If is a topological vector space and is a linear map, the spectrum of is the set of complex numbers such that is not invertible. We divide the spectrum depending on why this is not invertible. If is not injective, we say that is in the point spectrum of .
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Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose. By the finite-dimensional spectral theorem, such operators can be associated with an orthonormal basis of the underlying space in which the operator is represented as a diagonal matrix with entries in the real numbers.
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Eigenfunction
In mathematics, an eigenfunction of a linear operator, A, defined on some function space is any non-zero function f in that space that returns from the operator exactly as is, except for a multiplicative scaling factor. More precisely, one has for some scalar, ¿, the corresponding eigenvalue. The solution of the differential eigenvalue problem also depends on any boundary conditions required of .
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Boundary value problem
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional restraints, called the boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satisfies the boundary conditions. Boundary value problems arise in several branches of physics as any physical differential equation will have them.
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Eigenvalues and eigenvectors
The eigenvectors of a square matrix are the non-zero vectors that, after being multiplied by the matrix, remain parallel to the original vector. For each eigenvector, the corresponding eigenvalue is the factor by which the eigenvector is scaled when multiplied by the matrix. The prefix eigen- is adopted from the German word "eigen" for "self" in the sense of a characteristic description. The eigenvectors are sometimes also called characteristic vectors.
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