Publications by Aaron Schlegel
Set Theory Arbitrary Union and Intersection Operations with R
Part 3 of 3 in the series Set TheoryThe union and intersection set operations were introduced in a previous post using two sets, \(a\) and \(b\). These set operations can be generalized to accept any number of sets. Arbitrary Set Unions Operation Consider a set of infinitely many sets: \( A = \large{\{b_0, b_1, b_2, \cdots \} \large} \) It would ...
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Algebra of Sets in R
Part 4 of 4 in the series Set TheoryThe set operations, union and intersection, the relative complement \(–\) and the inclusion relation (subsets) \(\subseteq\) are known as the algebra of sets. The algebra of sets can be used to find many identities related to set relations that will be discussed later. We turn now to introducing the relative ...
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Set Theory Ordered Pairs and Cartesian Product with R
Part 5 of 5 in the series Set TheoryOrdered and Unordered Pairs A pair set is a set with two members, for example, \(\{2, 3\}\), which can also be thought of as an unordered pair, in that \(\{2, 3\} = \{3, 2\}\). However, we seek a more a strict and rich object that tells us more about two sets and how their elements are ordered. Call this object...
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Lagrangian Polynomial Interpolation with R
Part 2 of 4 in the series Numerical AnalysisPolynomial interpolation is the method of determining a polynomial that fits a set of given points. There are several approaches to polynomial interpolation, of which one of the most well known is the Lagrangian method. This post will introduce the Lagrangian method of interpolating polynomials and how ...
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Neville’s Method of Polynomial Interpolation
Part 1 of 5 in the series Numerical AnalysisNeville’s method evaluates a polynomial that passes through a given set of \(x\) and \(y\) points for a particular \(x\) value using the Newton polynomial form. Neville’s method is similar to a now-defunct procedure named Aitken’s algorithm and is based on the divided differences recursion relatio...
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Divided Differences Method of Polynomial Interpolation
Part of 6 in the series Numerical AnalysisThe divided differences method is a numerical procedure for interpolating a polynomial given a set of points. Unlike Neville’s method, which is used to approximate the value of an interpolating polynomial at a given point, the divided differences method constructs the interpolating polynomial in Newton...
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Numerical Differentiation with Finite Differences in R
Part 1 of 7 in the series Numerical AnalysisNumerical differentiation is a method of approximating the derivative of a function \(f\) at particular value \(x\). Often, particularly in physics and engineering, a function may be too complicated to merit the work necessary to find the exact derivative, or the function itself is unknown, and all that...
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The Trapezoidal Rule of Numerical Integration in R
Part of 8 in the series Numerical AnalysisThe Trapezoidal Rule is another of Closed Newton-Cotes formulas for approximating the definite integral of a function. The trapezoidal rule is so named due to the area approximated under the integral \(\int^a_b f(x) \space dx\) representing a trapezoid. Although there exist much more accurate quadrature ...
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Simpson’s Rule for Approximating Definite Integrals in R
Part 9 of 9 in the series Numerical AnalysisSimpson’s rule is another closed Newton-Cotes formula for approximating integrals over an interval with equally spaced nodes. Unlike the trapezoidal rule, which employs straight lines to approximate a definite integral, Simpson’s rule uses the third Lagrange polynomial, \(P_3(x)\) to approximate the...
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Linear Congruential Generator in R
Part of 1 in the series Random Number GenerationA Linear congruential generator (LCG) is a class of pseudorandom number generator (PRNG) algorithms used for generating sequences of random-like numbers. The generation of random numbers plays a large role in many applications ranging from cryptography to Monte Carlo methods. Linear congruential ge...
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