A representation of the Peano kernel for some quadrature rules and applications

Pečarić, Josip; Ujević, Nenad

doi:10.1098/rspa.2006.1714

Cited by 6 publications

(5 citation statements)

References 10 publications

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“…The corrected trapezoidal rule given in Theorem 3.2 is not the usual one that has traditionally appeared in the literature. For example, in Conte and De Boor [6], Davis and Rabinowitz [9], Dragomir, et al [12], Pečaric and Ujevic [28], and Squire [34], the coefficient is 1/12 in place of our 3/32 in (3.1). The error estimate (b − a) 5 f (4) ∞ /720 is obtained by polynomial interpolation by Conte and De Boor in [6] and with a two-point Taylor expansion by Davis and Rabinowitz in [9].…”

Section: Corrected Trapezoidal Rulementioning

confidence: 93%

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Simple derivation of basic quadrature formulas

Talvila¹,

Wiersma²

2012

Preprint

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Simple proofs of the midpoint, trapezoidal and Simpson's rules are proved for numerical integration on a compact interval. The integrand is assumed to be twice continuously differentiable for the midpoint and trapezoidal rules, and to be four times continuously differentiable for Simpson's rule. Errors are estimated in terms of the uniform norm of second or fourth derivatives of the integrand. The proof uses only integration by parts, applied to the second or fourth derivative of the integrand, multiplied by an appropriate polynomial or piecewise polynomial function. A corrected trapezoidal rule that includes the first derivative of the integrand at the endpoints of the integration interval is also proved in this manner, the coefficient in the error estimate being smaller than for the midpoint and trapezoidal rules. The proofs are suitable for presentation in a calculus or elementary numerical analysis class. Several student projects are suggested.

show abstract

Section: Corrected Trapezoidal Rulementioning

confidence: 93%

“…In their Lemma 2, the error is given with f ′′ ∞ replaced by sup [a,b] f ′′ − inf [a,b] f ′′ . Pečaric and Ujevic [28] give the error estimate as √ 3 (b − a) 3 f ′′ ∞ /54 in their equation (3.3). This also appears in Dedic, et al [10].…”

Section: Corrected Trapezoidal Rulementioning

confidence: 99%

Simple derivation of basic quadrature formulas

Talvila¹,

Wiersma²

2012

Preprint

View full text Add to dashboard Cite

show abstract

“…Newton's method is the most well-known method for solving nonlinear equations. Various numerical methods have been developed using different techniques including finite differences [1][2][3][4], quadrature rules ,QiaolingXue, JianZhu [5], Nenad Ujevic 2006 [6], Taylor's series, decomposition methods, homotopy techniques, Newton theorem (Nasr Al Din IDE, 2013), (Shijun Liao., 1997 [7]) etc., in order to carry out the solution of non-linear equations with different convergence rates. Most commonly used numerical methods for root location of non-linear equations includes, Bisection/interval halving method, Regula-falsi/false position method.…”

Section: Introductionmentioning

confidence: 99%

Using the Least Squares Method with Five Points to Solve Algebraic Equations Nonlinear

2019

International Journal of Scientific and Innovative Mathematical

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In this paper, we introduce an extension of previous research to estimate the parameters of nonlinear algebraic models using the least square method to find the roots of nonlinear algebraic equations that is an important problem in science and engineering later many methods have developed for solving nonlinear equations. The Least square method for five points is used to find the present methods (PM). We applied a number some examples and numerical results obtained show that the present method is faster or slowest than the other methods with the higher degrees of least square polynomial.

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“…Liu [16] assumes the condition f (n−1) ∈ C([a, b]) ∩ BV ([a, b]) and has a quadrature formula with degree of exactness equal to n−1. The problem is tackled using the Peano kernel by Dubeau [9] and Pečarić and Ujević [20]. Ding, Ye and Yang [8] estimate the remainder when f ′′ is Henstock-Kurzweil integrable.…”

Section: Introductionmentioning

confidence: 99%

Higher order corrected trapezoidal rules in Lebesgue and Alexiewicz spaces

Talvila¹

2016

Journal of Classical Analysis

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where φ n is a monic polynomial of degree n and the error is given byThis then gives a quadrature formula for b a f (x) dx. The polynomial φ n is chosen to optimize the error estimate under the assumption that f n) is integrable in the distributional or Henstock-Kurzweil sense. Sharp error estimates are obtained. It is shown that this formula is exact for all such φ n if f is a polynomial of degree at most n − 1. If φ n is a Legendre polynomial then the formula is exact for f a polynomial of degree at most 2n − 1.

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A representation of the Peano kernel for some quadrature rules and applications

Cited by 6 publications

References 10 publications

Simple derivation of basic quadrature formulas

Simple derivation of basic quadrature formulas

Using the Least Squares Method with Five Points to Solve Algebraic Equations Nonlinear

Higher order corrected trapezoidal rules in Lebesgue and Alexiewicz spaces

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