Concepts inEquilibrium states of Runge-Kutta schemes: part II
Stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle.
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Jacobian matrix and determinant
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. Suppose F : R → R is a function from Euclidean n-space to Euclidean m-space. Such a function is given by m real-valued component functions, F1(x1,... ,xn), ... , Fm(x1,... ,xn).
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Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs. For example, 3 + 4i and 3 − 4i are complex conjugates. The conjugate of the complex number where and are real numbers, is For example, An alternative notation for the complex conjugate is .
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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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Theory
The English word theory was derived from a technical term in philosophy in Ancient Greek. The word theoria, θεωρία, meant "a looking at, viewing, beholding", and referring to contemplation or speculation, as opposed to action. Theory is especially often contrasted to "practice" (from Greek praxis, πρᾶξις) a Greek term for "doing", which is opposed to theory because theory involved no doing apart from itself.
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Runge–Kutta methods
In numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations. These techniques were developed around 1900 by the German mathematicians C. Runge and M.W. Kutta. See the article on numerical ordinary differential equations for more background and other methods. See also List of Runge–Kutta methods.
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Real number
In mathematics, a real number is a value that represents a quantity along a continuous line. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers such as √2 and π. Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced.
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Behavior
Behavior or behaviour refers to the actions and mannerisms made by organisms, systems, or artificial entities in conjunction with their environment, which includes the other systems or organisms around as well as the physical environment. It is the response of the system or organism to various stimuli or inputs, whether internal or external, conscious or subconscious, overt or covert, and voluntary or involuntary.
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