Convergence rates of derivatives of Floater–Hormann interpolants for well-spaced nodes

Cirillo, Emiliano; Hormann, Kai; Sidon, Jean

doi:10.1016/j.apnum.2016.07.008

Cited by 16 publications

(8 citation statements)

References 12 publications

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“…We note that the barycentric basis functions have no real pole, and therefore, the resulting FH interpolant is infinitely smooth. The following theorem from Cirillo's study explains the convergence rate of the FH interpolant and their derivatives.…”

Section: Convergence Analysismentioning

confidence: 99%

“…Definition (Cirillo et al) Let

n \in N

h_{i, j} = false| ξ_{i} - ξ_{j} false|, h_{i} = h_{i + 1, i}, h = {normalmax}_{0 \leq i \leq n - 1} h_{i}

, and

X_{n} = {ξ_{0}, ξ_{1}, \dots, ξ_{n} normal}

be a set of interpolation nodes. We then say that

X = {false(X_{n} false)}_{n \in ℕ}

is a family of well‐spaced nodes, if there exist constants

R_{1}

R_{2}

≥

1

, independent of

n

, such that the conditions…”

Section: Convergence Analysismentioning

confidence: 99%

“…Theorem Suppose that

θ (ξ) \in C^{d + 2 + k} [a, b]

θ_{n} (ξ) = \sum_{i = 0}^{n - d} normal‍ λ_{i} false(ξ false) p_{i} false(ξ) / \sum_{i = 0}^{n - d} normal‍ λ_{i} false(ξ false)

and

n

d \leq n

, and

0 \leq k \leq d

are positive integers. Then, for any set of well‐spaced interpolation nodes,…”

Section: Convergence Analysismentioning

confidence: 99%

See 2 more Smart Citations

Novel numerical solutions of nonlinear heat transfer problems using the linear barycentric rational interpolation

Torkaman

Loghmani

Heydari

et al. 2019

Heat Trans. Asian Res.

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In this investigation, a numerical technique to obtain the approximate solutions of four well‐known nonlinear differential equations in the area of heat transfer is presented. This method is based on the operational matrix of derivative of linear barycentric rational interpolation. The main advantages of this approach are that it uses the Floater‐Hormann weights, which are very efficient in practice, and reduces the governing differential equation to a system of algebraic equations. The results are compared with the obtained numerical results of the fourth‐order Runge‐Kutta method along with the shooting method and some of the previously existing methods. The acquired results reveal that the derivative operational matrix method with barycentric rational basis functions is very efficient and can be implemented easily and fast.

show abstract

Section: Convergence Analysismentioning

confidence: 99%

“…Definition (Cirillo et al) Let

n \in N

h_{i, j} = false| ξ_{i} - ξ_{j} false|, h_{i} = h_{i + 1, i}, h = {normalmax}_{0 \leq i \leq n - 1} h_{i}

, and

X_{n} = {ξ_{0}, ξ_{1}, \dots, ξ_{n} normal}

be a set of interpolation nodes. We then say that

X = {false(X_{n} false)}_{n \in ℕ}

is a family of well‐spaced nodes, if there exist constants

R_{1}

R_{2}

≥

1

, independent of

n

, such that the conditions…”

Section: Convergence Analysismentioning

confidence: 99%

See 1 more Smart Citation

Novel numerical solutions of nonlinear heat transfer problems using the linear barycentric rational interpolation

Torkaman

Loghmani

Heydari

et al. 2019

Heat Trans. Asian Res.

View full text Add to dashboard Cite

show abstract

“…Further results on the convergence of the derivatives of the Floater-Hormann family of interpolants have just been published [13].…”

Section: The Rfd Formulasmentioning

confidence: 99%

“…Next we studied the equation Table 4 shows the numerical results for (n, d) equal to (8,5) and (13,6). There is no Runge phenomenon, as the latter does not occur with this kind of rational interpolants and fixed d [14].…”

Section: Numerical Verificationsmentioning

confidence: 99%