Concepts inComputing multivariable Taylor series to arbitrary order
Binomial coefficient
In mathematics, binomial coefficients are a family of positive integers that occur as coefficients in the binomial theorem. They are indexed by two nonnegative integers; the binomial coefficient indexed by n and k is usually written, and it is the coefficient of the x term in the polynomial expansion of the binomial power (1 + x). Arranging binomial coefficients into rows for successive values of n, and in which k ranges from 0 to n, gives a triangular array called Pascal's triangle.
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Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point. The concept of a Taylor series was formally introduced by the English mathematician Brook Taylor in 1715.
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Elementary function
In mathematics, an elementary function is a function of one variable built from a finite number of exponentials, logarithms, constants, and nth roots through composition and combinations using the four elementary operations (+ – × ÷). By allowing these functions (and constants) to be complex numbers, trigonometric functions and their inverses become included in the elementary functions.
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Automatic differentiation
In mathematics and computer algebra, automatic differentiation (AD), sometimes alternatively called algorithmic differentiation, is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc. ) and elementary functions (exp, log, sin, cos, etc.).
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Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's instantaneous velocity.
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Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and its applications. A "constant" in this context should not be confused with a mathematical constant which is a specific number independent of the scope of the given problem.
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Array data structure
In computer science, an array data structure or simply array is a data structure consisting of a collection of elements, each identified by at least one array index or key. An array is stored so that the position of each element can be computed from its index tuple by a mathematical formula.
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Coefficient
In mathematics, a coefficient is a multiplicative factor in some term of an expression (or of a series); it is usually a number, but in any case does not involve any variables of the expression. For instance in the first three terms respectively have the coefficients 7, −3, and 1.5 (in the third term the variables are hidden, so the coefficient is the term itself; it is called the constant term or constant coefficient of this expression).
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