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A mathematical markup language is a computer notation for representing mathematical formulae, based on mathematical notation.Specialized markup languages are necessary because computers normally deal with linear text and more limited character sets (although increasing support for Unicode is obsoleting very simple uses).
When all three terms above are multiplied by an optional amplitude function, A(t) > 0, the left-hand side of the equality is known as the amplitude/phase form, and the right-hand side is the quadrature-carrier or IQ form.
The conjugate gradient method can be applied to an arbitrary n-by-m matrix by applying it to normal equations A T A and right-hand side vector A T b, since A T A is a symmetric positive-semidefinite matrix for any A. The result is conjugate gradient on the normal equations (CGN or CGNR). A T Ax = A T b
Fitting of a noisy curve by an asymmetrical peak model, with an iterative process (Gauss–Newton algorithm with variable damping factor α).Curve fitting [1] [2] is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, [3] possibly subject to constraints.
Methods such as Runge–Kutta take some intermediate steps (for example, a half-step) to obtain a higher order method, but then discard all previous information before taking a second step. Multistep methods attempt to gain efficiency by keeping and using the information from previous steps rather than discarding it.
To satisfy the last three equations, either a = 0 or b, c, and d are all 0. The latter is impossible because a is a real number and the first equation would imply that a 2 = −1. Therefore, a = 0 and b 2 + c 2 + d 2 = 1. In other words: A quaternion squares to −1 if and only if it is a vector quaternion with norm 1.
The homotopy analysis method is also able to combine with other techniques employed in nonlinear differential equations such as spectral methods [7] and Padé approximants. It may further be combined with computational methods, such as the boundary element method to allow the linear method to solve nonlinear systems.
If an airplane's altitude at time t is a(t), and the air pressure at altitude x is p(x), then (p ∘ a)(t) is the pressure around the plane at time t. Function defined on finite sets which change the order of their elements such as permutations can be composed on the same set, this being composition of permutations.