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# Scientific Computing

The following section presupposes Set Theory and Nonstandard Analysis. Let $$\varepsilon \in [\tilde{\nu}, 1 – \tilde{\nu}]$$.

Definition: Let according to the trapezoidal rule $${}^{T}{\uparrow}_{z\in A}{f(z){\downarrow}z:={+}_{z\in A}{\tilde{2}(f(z)+f({}^\curvearrowright\,z))({}^\curvearrowright\,z-z)}}.$$

Let according to the midpoint rule $${}^{M}{\uparrow}_{z\in A}{f(z){\downarrow}z:={+}_{z\in A}{f(\tilde{2}(z\,+{}^\curvearrowright z}))({}^\curvearrowright\,z-z)}.\triangle$$

Remark: In the first fundamental theorem, the derivative $${\downarrow}(F(z))/{\downarrow}z$$ can be tightened to the arithmetic mean $$\tilde{2}(f(z) + f({}^\curvearrowright z))$$ resp. $$f(\tilde{2}(z + {}^\curvearrowright z))$$, and similarly, in the second fundamental theorem, $$F(\gamma(b)) – F(\gamma(a))$$ can be tightened to $$\tilde{2}(F(\gamma(b)) + F({}^\curvearrowleft \gamma(b))) – \tilde{2}(F(\gamma(a)) + F({}^\curvearrowright \gamma(a)))$$ resp. $$F(\tilde{2}(\gamma(b) + {}^\curvearrowleft \gamma(b))) – F(\tilde{2}(\gamma(a) + {}^\curvearrowright \gamma(a)))$$. This yields approximately the original results when $$f$$ and $$F$$ are sufficiently $$\alpha$$-continuous at the boundary.

Definition: Let $$f_n^*(z) = f(\eta_nz)$$ sisters of the TS $$f(z) \in \mathcal{O}(D)$$ centred on 0 on the domain $$D \subseteq {}^{\omega}\mathbb{C}$$ where $$m, n \in {}^{\omega}\mathbb{N}^{*}$$ and $$\eta_n^m := i^{2^{\lceil m/n \rceil}}$$. Then let $$\delta_n^*f = \check{f} – \check{f}_n^*$$ the halved sister distances of $$f$$. Let $$\mu_n^m := m!n!/(m + n)!$$ and $$\widetilde{(-n)!} := 0$$. For $$j, k \in {}^{\omega}\mathbb{N}, f \in \mathcal{C}_{\pi}^{j+2}$$ and Fourier coefficients $$c_k := \tilde{\hat{\pi}}{\uparrow}_{-\pi}^{\pi}{f(t)\tilde{e}^{ikt}{{\downarrow}t}}$$1cf. Walter, Wolfgang: Analysis 2; 5., erw. Aufl.; 2002; Springer; Berlin, p. 358 ff., the series $${+}_{k=-\omega}^{\omega}{c_ke^{ikt}}$$ is called Fourier series. On the level of TS, $$\mu$$ and $$\eta$$ form a $$\mu$$-$$\eta$$-calculus.$$\triangle$$

Remark: The latter allows representing integrals and derivatives easily and finitely closed2cf. Remmert, Reinhold: Funktionentheorie 1; 3., verb. Aufl.; 1992; Springer; Berlin, p. 165 f.. A conversion to other period lengths than $$\hat{\pi }$$ is simply possible3see loc. cit., p. 364.

Example: $$\text{Re} \; c \in [\tilde{\nu}, 1 + \tilde{\nu}[, c \in {}^{\omega}\mathbb{C}$$ and $$\varepsilon := \tilde{2}^j, j \in {}^{\omega}\mathbb{N}^{*}$$ imply $$\zeta (n+c)=a_n{+}_{k=1}^{n}{\delta_n^* v_c(\varepsilon u^k)}$$ for $$z \in \mathbb{B}_{1-\tilde{\nu}}(0) \subset D$$4cf. Remmert, Reinhold: Funktionentheorie 2; 1. unveränd. Nachdruck der 1. Aufl.; 1992; Springer; Berlin, p. 42 and $$v_c(z):={+}_{m=1}^{\omega }{\zeta (m+c){{z}^{m}}}=z{+}_{m=1}^{\omega }{{{{\widetilde{m}}}^{c}}\widetilde{z-m}}.\square$$

Example: A Richardson extrapolation determines the digamma function $$\psi$$ and$\zeta(\grave{n}) =\widetilde{\varepsilon^n\hat{n}}{+}_{k=1}^n{(\psi(\varepsilon u^k i^{2\tilde{n}}) – \psi(\varepsilon u^k))} + \mathcal{O}(\varepsilon^{\hat{n}})$ for $$n = 2$$ and $$\varepsilon = \tilde{2}^{13}$$ as $$\zeta(3) = 2^{24}{+}_{k=1}^2{(\psi(\varepsilon u^ki) – \psi(\varepsilon u^k))} + \mathcal{O}(10^{-16})$$.

Representation theorem for integrals: The TS (see below) $$f(z) \in \mathcal{O}(D)$$ centred on 0 for $$D \subseteq {}^{\omega}\mathbb{C}$$ gives for $$\grave{m}, n \in {}^{\omega}\mathbb{N}^*$$${\uparrow}_0^z…{\uparrow}_0^{\zeta_2}{f(\zeta_1){\downarrow}\zeta_1\;…\;{\downarrow}\zeta_n} = \widetilde{n!} f(z\mu_n) z^n.\square$Example: For the TS $$f(x), g(x) \in {}^{\omega}\mathbb{R}$$, it holds that $${\uparrow}_0^x{f(v){\downarrow}v}{\uparrow}_0^x{\uparrow}_0^{y}{g(v){\downarrow}v{\downarrow}y} = f(x\mu_1)\check{g}(x\mu_2)x^3.$$

Representation theorem for derivatives: For $$\mathbb{B}_{\tilde{\nu}}(0) \subset D \subseteq {}^{\omega}\mathbb{C}, n$$-th unit roots, TS$f(z):=f(0) + {+}_{m=1}^{\omega }{\widetilde{m!}\,{{f}^{(m)}}(0){z^m}},$$$\varepsilon := \tilde{2}^j\tilde{r}, j \in {}^{\omega}\mathbb{Z}, n \in {}^{\omega}\mathbb{N}^{*}, u :=e^{\tilde{n} \hat{\pi}i}$$ and $$f$$’s radius of convergence $$r \in {}^{\nu}{\mathbb{R}}_{>0}$$ imply${{f}^{(n)}}(0)=2^{jn}\acute{n}!{+}_{k=1}^{n}{\delta_n^* f(\tilde{2}^j u^k)}.\square$Universal multistep theorem: For $$n \in {}^{\nu}\mathbb{N}_{\le p}, k, m, p \in {}^{\nu}\mathbb{N}^{*}, {\downarrow}_{{}^\curvearrowright B} x \in\, ]0, 1[, x \in [a, b] \subseteq {}^{\omega}\mathbb{R}, y : [a, b] \rightarrow {}^{\omega}\mathbb{R}^q, f : [a, b]\times{}^{\omega}\mathbb{R}^{q \times n} \rightarrow {}^{\omega}\mathbb{R}^q, g_k({}^\curvearrowright x) := g_{\acute{k}}(x)$$, and $$g_0(a) = f(({}^\curvearrowleft)a, y_0, … , y_{\acute{n}})$$, the TS of the initial value problem $$y^\prime(x) = f(x, y(({}^\curvearrowright)^0 x), … , y(({}^\curvearrowright)^{\acute{n}} x))$$ of order $$n$$ implies$y({}^\curvearrowright x) = y(x) + {\downarrow}_{{}^\curvearrowright}x{\pm}_{k=1}^{p}{\left (g_{p-k}(({}^\curvearrowright) x){+}_{m=k}^{p}{\widetilde{m!}\tbinom{\acute{m}}{\acute{k}}}\right )} + \mathcal{O}(({\downarrow}_{{}^\curvearrowright} x)^{\grave{p}}).\square$Theorem for (anti-) derivatives of TS: For $$j \in {}^{\omega}\mathbb{Z}$$, $$q = \tilde{\varepsilon}(z-a)$$, $$a \in D$$ and $$k, m \in \mathbb{N}_{<n}$$, modular arithmetic5cf. Knuth, Donald Ervin: The Art of Computer Programming Volume 2; 3rd Ed.; 1997; Addison Wesley; Reading, p. 302 – 311 and $$n$$-th unit roots result in the corresponding DFT form:${\updownarrow}^jf_n(z) := \tilde{n}(q^k)^T(\delta_{km}\widetilde{\varepsilon q}^j\widetilde{(k-j)!}k!)({\tilde{u}}^{km})(f({\varepsilon u}^m+a))+\mathcal{O}(\varepsilon^n).\square$Theorem for derivatives of Fourier series: For $$f \in \mathcal{C}_{\pi}^{j+2}$$6cf. Walter, loc. cit., $$j \in {}^{\omega}\mathbb{N}, k \in [-\acute{n}, n] \cap \mathbb{Z}, t \in [-\pi, \pi]$$ and $$m \in \mathbb{N}_{<\hat{n}}$$, the following DFT form is implied allowing integration of individual terms:${\downarrow}^jf_n(t):=(u^{\tilde{\pi}knt})^T(\delta_{(k+\acute{n})m} (ik)^j)(\tilde{u}^{km})(f(\tilde{n}\pi m – \pi))/{\hat{n}}+\mathcal{O}(\tilde{n}).\square$Conclusion: Supporting points $$mr := \tilde{n}\pi m$$ of the smooth $$f(mr)$$ yield for $$k$$ like $$m$$ interpolating in $$\mathcal{O}(_en n)$$:${\downarrow}^jf_n(t):=(u^{\tilde{r}kt})^T(\delta_{km}(ik)^j)(\tilde{u}^{km})(f(mr))/{\hat{n}}.\square$Theorem (DFT fixed-point method): Every $$z \in {}^{\omega}\mathbb{C}$$ of an arbitrary $$m$$-polynomial $$p(z) = 0$$ with $$m \in [2, n] \cap \mathbb{N}$$ for $$n := 2^r, r \in {}^{\nu}\mathbb{N}^*$$ and coefficients from $${}^{\nu}\mathbb{C}$$ can be (initiating) determined also in $$\mathcal{O}(_en n).$$

Proof and algorithm: Let $$U = (\tilde{u}^{jk})$$ for $$j, k \in \mathbb{N}_{<n}, u :=e^{\tilde{n} \hat{\pi} i}, q := \hat{z}$$ and $$s_k := p(\tilde{2}u^k)$$. A simple transform achieves $$|q| < \tilde{2}$$ for all zeros $$\zeta$$ of $$p(z)$$ and $$p(0) = 1$$. It follows from $$p(z) = \tilde{n}(q^j)^TUs = \tilde{n}\mu^Ts = 0$$ the simplified iteration $$\mu^* = U_{1}^{-T}\mu U((\delta_{jk}\tilde{u}^j)U^{-1}\mu-(U_{\acute{n}}^{-T}\mu+\beta s^T\mu, \beta s^T\mu, …, \beta s^T\mu)^T)$$. Here let $$\delta_{jk}$$ the Kronecker delta, starting point $$q := \tilde{2}$$ and $$\beta \in {}^{\nu}\mathbb{C}^*$$ such that each time $$||\mu^*-\mu||$$ is roughly halved and $$\mu^Ts = 0$$ holds. Finish by polynomial division where $$m > 2.\square$$

Conclusion: DFT-zero methods iterate zeros $$a \in {}^{\omega}\mathbb{C}$$ for every function $$f(z) \in {}^{\omega}\mathbb{C}$$ that can be developed into a TS at defaults $$z_0 \in {}^{\omega}\mathbb{C}$$ with each time the same convergence as in methods similar to Simpson’s, if $${\updownarrow}^0f_n(z)$$ is differentiated (several times) for sufficiently precise $$|z_{\grave{m}} – z_m|$$ and $$m \in {}^{\omega}\mathbb{N}.\square$$

Remark: The identity instead of $$\delta_n^*$$ provides arbitrarily precise approximations for the $$f^{(n)}$$. The last theorems are equally valid for multidimensional TS and Laurent series. The error is $$\mathcal{O}(\varepsilon^n)$$ instead of $$\mathcal{O}(\varepsilon)$$ for analogously defined $$m$$-dimensional DFT forms of the TS with $$\binom{m+n}{n}$$ derivatives where the effort is comparable. A good choice for the crown points $$\varepsilon u^m + a$$ around the crown centre $$a$$ is $$\varepsilon = \tilde{2}$$ and $$n = 64$$.

Conclusion for differential equations: Let $$d(z) \in {}^{\omega}\mathbb{C}^{n+2}$$ vector of DFT forms for $$n \in {}^{\omega}\mathbb{N}$$. If the system $$y^{(\grave{n})} = \left(y^{(n)},y^{(\acute{n})}, …, y, 1\right)d(z) \in {}^{\omega}\mathbb{C}$$ where (not) $$\left(y^{(n)}(a),\ y^{(\acute{n})}(a), …, y(a)\right)^T\in {}^{\omega}\mathbb{C}^{\grave{n}}$$ and $$a, z \in {}^{\omega}\mathbb{C}$$ has the solution $$y_0$$, this can be determined in $$\mathcal{O}(\grave{n}^3)$$ giving the error mentioned above. Non-linear or partial (systems of) differential equations may be solved by Newton’s method.$$\square$$

Series theorem for integrals: For $$c, x \in [0, a], f \in {}^{\nu}\mathcal{C}^{n}$$ and $$n, \hat{p}, r = 2^p \in {}^{\nu}2\mathbb{N}^*$$, DFT forms of TS show${\uparrow}_0^a f(x){\downarrow}x = {\uparrow}_0^1 g(y){\downarrow}y = {+}_{\check{q}=1}^{\check{r}}{{+}_{\check{m}=0}^{\check{n}-1}{\widetilde{\grave{m}!}\hat{g}^{(m)}\left (\tilde{r}\acute{q} \right){\tilde{r}}^{\grave{m}}}}+\mathcal{O}\left(\widetilde{\grave{n}!}g^{(n)}(\tilde{a}c){\tilde{r}}^n\right).\square$Remarks: DFT equivalents may replace any $$g^{(m)}$$ for $$g(y) := af(ay)$$ and $$a > 0$$. Transforming $$z := e^{\pm x}$$ makes here infinite bounds of integration finite. A finite integral decreases the remainder’s modulus sufficiently. The midpoint rule is here more advantageous than the trapezoidal rule.

Example: For $$g_{\mp}(\vartheta) := \check{\pi}(1 – {\varepsilon}^2 {\sin}^2(\check{\pi}\vartheta))^{\mp\tilde{2}}$$, complete elliptic integrals of the first and second kind are ${\uparrow}_0^{1} g_{\mp}(\vartheta) {\downarrow}\vartheta = g_{\mp}(\check{1}) + \widetilde{24} g_{\mp}^{(2)}(\check{1}) + \widetilde{1920} g_{\mp}^{(4)}(\check{1}) + \mathcal{O}\left(\widetilde{9!}g_{\mp}^{(6)}(\check{1})\right).$Second Euler-Maclaurin formula: For $$f(\check{q}) = g(\tilde{r}\acute{q})$$ and $$k = \grave{a} = \grave{r}$$, the preceding theorem yields in $$\mathcal{O}(_en n)$$${+}_{\check{q}=1}^{\check{r}} f(\check{q}) = {\uparrow}_{\check{1}}^{\check{k}} f(x){\downarrow}x + {+}_{\check{m}=1}^{\check{n}-1} H_m \left (f^{(\acute{m})}(\check{k}) – f^{(\acute{m})}(\check{1}) \right ) + \mathcal{O}\left (H_n \left (f^{(\acute{n})}(\check{k}) – f^{(\acute{n})}(\check{1}) \right )\right ).$Proof: For $$h(x) = x/$$sin $$x$$ and $$H_m := {i}^m\widetilde{m!}h^{(m)}(0)=\widetilde{m!}B_m(2-2^m) \rightarrow 2(i\tilde{\pi})^m$$,${+}_{\check{q}=1}^{\check{r}} g(\tilde{r}\acute{q}) = \check{r}{\uparrow}_0^1 g(y){\downarrow}y – {+}_{\check{q}=1}^{\check{r}}{+}_{\check{m}=1}^{\check{n}-1} {\widetilde{\grave{m}!}g^{(m)}\left (\tilde{r}\acute{q} \right)} + \mathcal{O}\left(\widetilde{\grave{n}!}g^{(n)}(\tilde{a}c){\tilde{r}}^n\right)$results in the claim by inserting and combining the factorials that are counted depending on partitions where $$B_m$$ are Bernoulli numbers and $$B_{\grave{m}} = 0, B_0=1$$ as well as $$B_1=-\tilde{2}.\square$$

Conclusion: It holds $$n! \sim \left (\hat{\pi} – \widetilde{3k}\pi \right )^{\tilde{2}}\left (\tilde{e}\check{k}\right )^{\check{k}}$$ for $$f(n) = {}_en$$ and $$r = \hat{n}$$.

Proof: Regard $$(1+\tilde{r})^{\check{k}} \sim e^{\tilde{2}}$$ and $$(1-\tilde{n})^{\tilde{2}} \sim 1-\tilde{r}$$ (cf. Nonstandard Analysis) as well as\begin{aligned}{}_en! &={\uparrow}_{\check{3}}^{\check{k}} f(x){\downarrow}x + {+}_{\check{m}=1}^{\check{n}} H_m \left (f^{(\acute{m})} (\check{k}) – f^{(\acute{m})} (\check{3}) \right ) + \mathcal{O}\left( H_{n+2} \left (f^{(\grave{n})} (\check{k}) – f^{(\grave{n})} (\check{3})\right)\right) \\ &= \check{k}\left (_e{\check{k}} – 1 \right ) – \check{3}\left (_e{\check{3}} – 1 \right ) – \left (\tilde{k} – \tilde{3} \right )/12 + 7\left (\tilde{k}^3 – \widetilde{27} \right )/360 \mp \cdots .\square\end{aligned}First Euler-Maclaurin formula: Mathematical induction for $$n$$ shows that7cf. Mollin, Richard A.: Advanced Number Theory with Applications; 1st Ed.; 2017; Routledge; Milton Park, p. 193 f.${+}_{\check{q}=0}^{\check{r}}f(\check{q})=\uparrow_0^{\check{r}}{f\left(x\right){\downarrow}x}+\check{f}(\check{r})+\check{f}(0)+{+}_{\check{m}=1}^{\check{n}-1}{\widetilde{m!}}B_m\left(f^{(\acute{m})}\left(\check{r}\right)-f^{(\acute{m})}\left(0\right)\right)+\mathcal{O}\left({\widetilde{n!}}B_n\left(f^{(\acute{n})}\left(\check{r}\right)-f^{(\acute{n})}\left(0\right)\right)\right).\square$

code of the FFT form