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Gauss–Kronrod formulas are extensions of the Gauss quadrature formulas generated by adding + points to an -point rule in such a way that the resulting rule is exact for polynomials of degree less than or equal to + (Laurie (1997, p. 1133); the corresponding Gauss rule is of order ).
In computer programming, a nested function (or nested procedure or subroutine) is a named function that is defined within another, enclosing, block and is lexically scoped within the enclosing block – meaning it is only callable by name within the body of the enclosing block and can use identifiers declared in outer blocks, including outer ...
This resulted in compact expressions for the longitude and distance integrals. The expressions were put in Horner (or nested) form, since this allows polynomials to be evaluated using only a single temporary register. Finally, simple iterative techniques were used to solve the implicit equations in the direct and inverse methods; even though ...
To compute the integral, we set n to its value and use the reduction formula to express it in terms of the (n – 1) or (n – 2) integral. The lower index integral can be used to calculate the higher index ones; the process is continued repeatedly until we reach a point where the function to be integrated can be computed, usually when its index is 0 or 1.
Stirling's formula remains centered about a particular data point, for use when the evaluated point is nearer to a data point than to a middle of two data points. Bessel's formula remains centered about a particular middle between two data points, for use when the evaluated point is nearer to a middle than to a data point.
In the case of two nested square roots, the following theorem completely solves the problem of denesting. [2]If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that + = if and only if is the square of a rational number d.
Viète's formula, as printed in Viète's Variorum de rebus mathematicis responsorum, liber VIII (1593). In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant π: = + + + It can also be represented as = = +.
Another method of deriving vector and tensor derivative identities is to replace all occurrences of a vector in an algebraic identity by the del operator, provided that no variable occurs both inside and outside the scope of an operator or both inside the scope of one operator in a term and outside the scope of another operator in the same term ...