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In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control , [ 1 ] and can be formulated to construct solutions in a memory-efficient, factored form.
If all the partial derivatives of a function are known (for example, with the gradient), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant. Unlike in the single-variable case, however, not every set of functions can be the set of all (first) partial derivatives of a single function.
The name is in analogy with quadrature, meaning numerical integration, where weighted sums are used in methods such as Simpson's method or the Trapezoidal rule. There are various methods for determining the weight coefficients, for example, the Savitzky–Golay filter. Differential quadrature is used to solve partial differential equations ...
When a partial fraction term has a single (i.e. unrepeated) binomial in the denominator, the numerator is a residue of the function defined by the input fraction. We calculate each respective numerator by (1) taking the root of the denominator (i.e. the value of x that makes the denominator zero) and (2) then substituting this root into the ...
The method of separation of variables is also used to solve a wide range of linear partial differential equations with boundary and initial conditions, such as the heat equation, wave equation, Laplace equation, Helmholtz equation and biharmonic equation.
Name Dim Equation Applications Landau–Lifshitz model: 1+n = + Magnetic field in solids Lin–Tsien equation: 1+2 + = Liouville equation: any + = Liouville–Bratu–Gelfand equation
In mathematics, matrix calculus is a specialized notation for doing multivariable calculus, especially over spaces of matrices.It collects the various partial derivatives of a single function with respect to many variables, and/or of a multivariate function with respect to a single variable, into vectors and matrices that can be treated as single entities.
Thus, the second partial derivative test indicates that f(x, y) has saddle points at (0, −1) and (1, −1) and has a local maximum at (,) since = <. At the remaining critical point (0, 0) the second derivative test is insufficient, and one must use higher order tests or other tools to determine the behavior of the function at this point.