How Are Fields Characterized by Continuous Function

Continuous Function

Stability

Derong Liu , in The Electrical Engineering Handbook, 2005

2.6.1 Definiteness of a Function

A continuous function v: Rⁿ → R [resp., v: B(h) → R] is said to be positive definite if:

1)

v (0) = 0, and

2)

v(x) > 0 for all x ≠ 0 [resp., 0 < ||x|| ≤ r for some r > 0].

A continuous function v is said to be negative definite if − v is a positive definite function.

Positive semidefinite: A continuous function v: Rⁿ → R [resp., v: B(h) → R] is said to be positive semidefinite if:

1)

v (0) = 0, and

2)

v(x) ≥ 0 for all x ∈ B(r) for some r > 0.

A continuous function v is said to be negative semidefinite if −v is a positive semidefinite function.

A continuous function v: Rⁿ → R is said to be radially unbounded if:

1)

v(0) = 0,

2)

v(x) > 0 for all x ≠ 0, and

3)

v(x) → ∞ as ||x|| → ∞.

Example 9

The function v: R ³ → R is given by:

$v\left(x\right)=x_1^2+x_2^2+x_3^2,$

which is positive definite and radially unbounded.

Example 10

The function v: R ³ → R is given by:

$v\left(x\right)=x_1^2+{\left(x_2-x_3\right)}^2,$

which is positive semidefinite. It is not positive definite because it is zero for all x ∈ R ³ such that x ₁ = 0 and x ₂ = x ₃.

Example 11

The function v: R ³ → R is given by:

$v\left(x\right)=x_1^2+x_2^2+x_3^2-{\left(x_1^2+x_2^2+x_3^2\right)}^3,$

which is positive definite in the interior of the ball given by $x_1^2+x_2^2+x_3^2<1$ . It is not radially unbounded since v(x) < 0 when $x_1^2+x_2^2+x_3^2>1$ .

Example 12

The function v: R ³ → R is given by:

$v\left(x\right)=\frac{x_1^4}{1+x_1^4}+x_2^4+x_3^4,$

which is positive definite but not radially unbounded.

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'BEST' APPROXIMATION

G.M. PHILLIPS , P.J. TAYLOR , in Theory and Applications of Numerical Analysis (Second Edition), 1996

Definition 5.8

A continuous function E is said to equioscillate on n points of [a, b] if there exist n points x _i with a ⩽ x ₁ < x ₂ < … < x _n ⩽ b, such that

$\left|E\left(x_i\right)\right|=\underset{a⩽x⩽b}{\max }\left|E\left(x\right)\right|,i=1,\cdots ,n,$

and

$E\left(x_i\right)=-E\left(x_{i+1}\right),i=1,\cdots ,n-1.$

Thus for the minimax approximation from P _n in Example 5.13 the error function equioscillates on n + 2 points and the same is true with n = 1 in Example 5.14. The equioscillation of the error on n + 2 points or more turns out to be the property which characterizes minimax approximation from P _n for any f ∈ C[a, b].

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Special Volume: Mathematical Modeling and Numerical Methods in Finance

Olivier Pironneau , Yves Achdou , in Handbook of Numerical Analysis, 2009

Theorem 4.1

Consider a continuous function P ∈ C ^2,1( $R_{+}^d$ x [0, T)) such that |S _i $\frac{\partial P}{\partial S_i}$ | ≤ C(1 + S _i ), with C independent of t. Assume that P satisfies

(4.5) $\left(\frac{\partial P}{\partial t}+LP-rP\right)(S,t)=0,t<T,S\in R_{+}^d$

and

(4.6) $P(S,T)=P_0(S),S\in R_{+}^d,$

then the price of the European option given by (4.2) satisfies

(4.7) $P_t=P(S_{1,t},\mathrm{\dots },S_{d,t},t).$

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Sequences and series

Mary Attenborough , in Mathematics for Electrical Engineering and Computing, 2003

Example 12.5

A triangular wave of period 2 is given by the function

$\begin{array}{ll}f(t)=t& 0\le t<1\\ f(t)=2-t& 1\le t<2.\end{array}$

Draw a graph of the function and give a sequence of values for t ≥ 0 at a sampling interval of 0.1.

Solution To draw the continuous function, use the definition y = t between t = 0 and 1 and draw the function y = 2 − t in the region where t lies between 1 and 2. The function is of period 2 so that section of the graph is repeated between t = 2 and 4, t = 4 and 6, etc.

The sequence of values found by using a sampling interval of 0.1 is given by substituting t = Tn = 0.1n into the function definition, giving

$\begin{array}{ll}a(n)=0.1n& 0\le 0.1n<1(\text{for}n\text{between}0\text{\hspace{0.17em}and}10)\\ a(n)=2-0.1n& 1\le 0.1n<2(\text{for}n\text{between}10\text{\hspace{0.17em}and}20).\end{array}$

The sequence then repeats periodically.

This gives the sequence:

$0,0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1,0.9,0.8,0.7,0.6,0.5,0.4,0.3,0.2,0.1,0,0.1,0.2,...$

The continuous function is plotted in Figure 12.4(a) and the digital function in Figure 12.4(b).

Figure 12.4. (a) A triangular wave of period 2 given by f(t) = t, 0 &lt; t ≤ 1, f(t) = 2 − t, 1 &lt; t &lt; 2. (b) The function sampled at a sampling interval of 0.1.

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Calculus

Seifedine Kadry , in Mathematical Formulas for Industrial and Mechanical Engineering, 2014

5.16 Continuity of a Function

The concept of a continuous function is that it is a function, whose graph has no break. For this reason, continuous functions are chosen, as far as possible, to model the real world problems. If a function is such that its limiting value at a point equals the functional value at that point, then we say that the function is continuous there.

Definition

A function $f(x)$ is said to be continuous at a point $x=c$ , if the following conditions hold true:

1.

$f(x)$ is defined at $x=c$

2.

$\underset{x\to c}{\lim }f(x)$ exists

3.

$\underset{x\to c}{\lim }f(x)=f(c).$

If at least one of these conditions is not satisfied, then the function will be discontinuous at $x=c$ . We say that a function is continuous on an interval, if it is continuous at each point of that interval.

Examples

1.

Let a function be such that $f(x)=x^2+1$ for x<1 and $f(x)=x$ for x≥1. Draw the graph of this function and discuss its continuity at the point x=1.

2.

Given the function $f(x)=(x^2-4)/(x-2)$ for $x\ne 2$ and $f(2)=0$ . Decide whether this function is continuous on the interval [0,4]. Justify your answer.

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Neural Network Approximation of Piecewise Continuous Functions: Application to Friction Compensation

RASTKO R. ŠELMIĆ , FRANK L. LEWIS , in Soft Computing and Intelligent Systems, 2000

Theorem 1

(Approximation of Piecewise Continuous Functions ). Let there be given any bounded function $f:S\to \text{ℜ}$ that is continuous and analytic on a compact set $S\subset \text{R,}$ except at x = c where the function f has a finite jump and is continuous from the right. Then, given any ε > 0, there exists a sum F of the form

(23) $F\left(x\right)=g\left(x\right)+{\displaystyle \sum _{k=0}^N{a}_k{f}_k\left(x-c\right)}$

such that

(24) $\left|F\left(x\right)-f\left(x\right)\right|<\epsilon$

for every x in S, where g is a function in C ^∞(S), and the polynomial jump approximation basis functions f _k are defined as

(25) $f_k\left(x\right)=\{\begin{array}{ll}0,\hfill & \text{for}x<0\hfill \\ \frac{x^k}{k!},\hfill & \text{for}x\ge 0\hfill \end{array}$

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Introductory Article: Functional Analysis

S. Paycha , in Encyclopedia of Mathematical Physics, 2006

Fredholm operators

A complex-valued continuous function K on [0, 1] × [0, 1] gives rise to an integral operator

$A:f\to {\int }_0^{1}K(x,y)f(y)\text{d}y$

on complex-valued continuous functions on [0, 1] (equipped with the supremum norm ∥·∥_∞) with the following upper bound property:

${||Af||}_{\infty }\le {\text{Sup}}_{[0,1]\times [0,1]}\left|K(x,y)\right|{||f||}_{\infty }$

In other words, A is a bounded linear operator with norm bounded from above by sup_[0,1]×[0,1]|K(x,y)|; a linear operator A:E → F from a normed linear space (E,∥·∥ _E ) to a normed linear space (F,∥·∥ _F ) is bounded (or continuous) if and only if its (operator) norm $\left|||A||\right|:={sup}_{{||u||}_E\le 1}{||Au||}_F$ is bounded.

An integral operator

$A:f\to {\int }_0^{1}K(x,y)f(y)\text{d}y$

defined by a continuous kernel K is, moreover, compact; a compact operator is a bounded operator of normed spaces that maps bounded sets to a precompact sets, that is, to sets whose closure is compact. Other examples of compact operators on normed spaces are finite-rank operators, operators with finite-dimensional range. In fact, any compact operator on a separable Hilbert space can be approximated in the topology induced by the operator norm |∥·∥| by a sequence of finite-rank operators.

Inspired by the work of Volterra, who, in the case of the integral operator defined above, produced continuous solutions ϕ = (I − A)⁻¹ f of the equation f = (I − A)ϕ for f ∈ C([0, 1]), Fredholm in 1900 (Sur une classe d'équations fonctionnelles) studied the equation f = (I − λA)ϕ, introducing a complex parameter λ. He proved what is since then called the Fredholm alternative, which states that either the equation f = (I − λA)ϕ has a unique solution for every f ∈ C([0, 1]) or the corresponding homogeneous equation (I − λA)ϕ = 0 has nontrivial solutions. In modern language, it means that the resolvent R(A,μ) = (A − μI)⁻¹ of a compact linear operator A is surjective if and only if it is injective. The Fredholm alternative is a powerful tool to solve partial differential equations among which the Dirichlet problem, the solutions of which are harmonic functions u (i.e., Δu = 0, where $\Delta =-{\sum }_{i=1}^n{\partial }^{2}u/\partial x_i^2$ ) on some domain $\Omega \in {\mathbb{R}}^n$ with Dirichlet boundary conditions $u_{{|}_{\partial \Omega }}=f$ , where f is a continuous function on the boundary ∂Ω. The Dirichlet problem has geometric applications, in particular to the nonlinear Plateau problem, which minimizes the area of a surface in ${\mathbb{R}}^d$ with given boundary curves and which reduces to a (linear) Dirichlet problem.

The operator B = I − A built from the compact operator A is a particular Fredholm operator, namely a bounded linear operator B:E → F which is invertible "up to compact operators," that is, such that there is a bounded linear operator C:F → E with both BC − I _F and CB − I _E compact. A Fredholm operator B has a finite-dimensional kernel Ker B and when (E,〈·,·〉 _E ) and (F,〈·,·〉 _F ) are Hilbert spaces its cokernel Ker B*, where B* is the adjoint of B defined by

${〈Bu,v〉}_F={〈u,B^{*}v〉}_E\forall u\in E,\forall v\in F$

is also finite dimensional, so that it has a well-defined index ind(B) = dim(KerB) − dim(KerB*), a starting point for index theory. Töplitz operators T _ϕ , where ϕ is a continuous function on the unit circle S ¹, provide first examples of Fredholm operators; they act on the Hardy space ${\mathcal{H}}^2(S^1)$ by

$T_{e_{-n}}\left(\sum _{m\ge 0}{a}_m{e}_m\right)=\sum _{m\ge 0}{a}_{m+n}{e}_m$

under the identification ${\mathcal{H}}^2(S^1)\simeq l^2(\mathbb{N})\subset l^2(\mathbb{Z})$ , with $l^2(\mathbb{Z})$ equipped with the canonical complete orthonormal basis $(e_n,n\in \mathbb{Z})$ . The Fredholm index $\text{ind}(T_{e_{-n}})$ is exactly the integer n so that the index of its adjoint is −n, as a consequence of which the index map from Fredholm operators to integers is onto.

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Subband Coding and Wavelet Transform

Anke Meyer-Baese , Volker Schmid , in Pattern Recognition and Signal Analysis in Medical Imaging (Second Edition), 2014

3.3.2 The Continuous Wavelet Transform

The CWT transforms a continuous function into a highly redundant function of two continuous variables, translation and scale. The resulting transformation is important for time-frequency analysis and is easy to interpret.

The CWT is defined as the mapping of the function $f(t)$ on the timescale space by

(3.46) $W_f(a\text{,}b)={\int }_{-\infty }^{\infty }{\psi }_{\mathit{ab}}(t)f(t)\mathit{dt}=〈{\psi }_{\mathit{ab}}(t)\text{,}f(t)〉$

The CWT is invertible if and only if the resolution of identity holds:

(3.47) $f(t)={\displaystyle \underset{\text{summation}}{\underbrace{\frac{1}{C_{\psi }}{\int }_{-\infty }^{\infty }{\int }_0^{\infty }\frac{\mathit{dadb}}{a^2}}}}{\displaystyle \underset{\text{Wavelet coefficients}}{\underbrace{W_f(a\text{,}b)}}}{\displaystyle \underset{\mathrm{Wavelet}}{\underbrace{{\psi }_{\mathit{ab}}(t)}}}$

where

(3.48) $C_{\psi }={\int }_o^{\infty }\frac{\mid \mathrm{\Psi }(\omega ){\mid }^2}{\omega }d\omega$

assuming that a real-valued $\psi (t)$ fulfills the admissibility condition. If $C_{\psi }<\infty$ , then the wavelet is called admissible. Then we get for the DC gain

(3.49) $\mathrm{\Psi }(0)={\int }_{-\infty }^{\infty }\psi (t)\mathit{dt}=0$

We immediately see that $\psi (t)$ corresponds to the impulse response of a bandpass filter and has a decay rate of $\mid t{\mid }^{1-∊}$ . It is important to note that based on the admissibility condition, it can be shown that the CWT is complete if $W_f(a\text{,}b)$ is known for all $a\text{,}b$ .

The Mexican-hat wavelet

(3.50) $\psi (t)=\left(\frac{2}{\sqrt{3}}{\pi }^{-\frac{1}{4}}\right)(1-t^2)e^{-\frac{t^2}{2}}$

is visualized in Fig. 3.16. It has a distinctive symmetric shape, and it has an average value of zero and dies out rapidly as $\mid t\mid \to \infty$ . There is no scaling function associated with the Mexican-hat wavelet.

Figure 3.16. Mexican-hat wavelet.

Figure 3.17 illustrates the multiscale coefficients describing a spiculated mass. Figure 3.17a shows the scanline through a mammographic image with a mass (8   mm) while Fig. 3.17b visualizes the multiscale coefficients at various levels.

Figure 3.17. Continuous wavelet transform: (a) scan line, (b) multiscale coefficients.

Images courtesy of Dr. A. Laine, Columbia University.

The short-time Fourier transform finds a decomposition of a signal into a set of equal-bandwidth functions across the frequency spectrum. The WT provides a decomposition of a signal based on a set of bandpass functions that are placed over the entire spectrum. The WT can be seen as a signal decomposition based on a set of constant-Q bandpasses. In other words, we have an octave decomposition, logarithmic decomposition, or constant-Q decomposition on the frequency scale. The bandwidth of each of the filters in the bank is the same in a logarithmic scale, or equivalently, the ratio of the filters' bandwidth to the respective central frequency is constant.

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Digital Images and Image Manipulation

TOM McREYNOLDS , DAVID BLYTHE , in Advanced Graphics Programming Using OpenGL, 2005

4.2 Digital Filtering

Consider again the original continuous function representing a primitive. The function drops to zero abruptly at the edge of the polyon, representing a step function at the polygon boundaries. Representing a step function in the frequency domain results in frequency components with non-zero values at infinite frequencies. Avoiding creating undersampling artifacts when reconstructing a sampled step function requires changing the input function, or the way it is sampled. In essence, the boundaries of the polygon must be "smoothed" so that the transition can be represented by a bounded frequency representation. The frequency bound is chosen so that it can be captured by the samples. This process is an application of filtering.

As alluded to in the discussion above, filtering goes hand in hand with the concept of sampling and reconstruction. Conceptually, filtering applies a function to an input signal to produce a new one. The filter modifies some of the properties of the original signal, such as removing frequency components above or below some threshold (low- and high-pass filters). With digital images, filtering is often combined with reconstruction followed by resampling. Reconstruction produces a continuous signal for the filter to operate on and resampling produces a set of sample values from the filtered signal, possibly at a different sample rate. The term filter is frequently used to mean all three parts: reconstruction, filtering, and resampling. The objective of applying the filter is most often to transform the spectral composition of the signal.

As an example, consider the steps to produce a new version of an image that is half the size in the x and y dimensions. One way to generate the new image is to copy every second pixel into the new image. This process can be viewed as a reconstruction and resampling process. By skipping every other pixel (which represents a sample of the original image), we are sampling at half the rate used to capture the original image. Reducing the rate is a form of undersampling, and will introduces new signal aliases.

These aliased signals can be avoided by eliminating the frequency components that cannot be represented at the new, lower sampling rate. This is done by applying a low-pass filter during signal reconstruction, before the new samples are computed. There are many useful low-pass filter functions; one of the simplest is the box filter. The 2×2-box filter computes a new sample by taking an equally weighted average of four adjacent samples. The effect of the box filter on the spectrum of the signal can be evaluated by converting it to the frequency domain. Although simple, the box filter isn't a terrific low-pass filter, it corresponds to multiplying the spectrum by a sinc function in the frequency domain (Figure 4.4). This function doesn't cut off the high frequencies very cleanly, and leads to its own set of artifacts.

Figure 4.4. Box filter in spatial and frequency domain.

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Basic concepts

Miodrag S. Petković , ... Jovana Džunić , in Multipoint Methods for Solving Nonlinear Equations, 2013

1.4 Initial approximations

Every iterative method for solving a nonlinear equation $f(x)=0$ requires the knowledge of an initial approximation $x_0$ to the sought zero $\alpha$ . Many one-point zero-finding methods as well as multipoint iterative methods are based on Newton's method, which is famous for its simplicity and good local convergence properties. However, a good convergence of Newton's method cannot be expected when the initial guess is not properly chosen, especially when the slope of the function $f$ is extremely flat or very steep near the root, or $f$ is of oscillatory type.

The significance of the choice of initial approximations becomes even more important if higher-order iterative methods are applied due to their sensitivity to perturbations. If the initial approximation is not close enough to the sought zero, then these methods may converge slowly at the beginning of the iterative process, which consequently decreases their computational efficiency.

The determination of a reasonably good approximation $x_0$ that guarantees the convergence of the sequence of approximations $\{x_k\}{}_{k\in \mathit{N}}$ to the zero of $f$ is a nontrivial task. There are many methods (mainly of non-iterative nature) and strategies for finding sufficiently good initial approximations. The well-known bisection method and its modifications belong to the simplest but not sufficiently efficient techniques. There is a vast literature on this subject so that we omit the details here.

In this section we present an efficient non-iterative method originally proposed by Yun (2008) and later discussed in more details in the papers (Yun and Petković, 2009; Petković and Yun, 2008; Yun, 2010). This method is based on numerical integration briefly referred to as NIM, where tanh, arctan, and signum functions are involved. The NIM requires neither any knowledge of the derivative $f^{\prime }(x)$ nor any iterative process. In non-pathological cases it is not necessary to have a close approximation to the zero; instead, a real interval (not necessarily tight) that contains the root is sufficient.

We consider three kinds of the so-called sigmoid transformations of $f:\mathcal{S}$ -transformation, $\mathcal{T}$ -transformation, and $\mathcal{A}$ -transformation. The term "sigmoid" comes from the similarity of the graphs of these sigmoid-like functions to the letter $S$ , or sigma in Greek. The functions

$\mathrm{sgn}x\text{,}\tanh \mathit{mx}\text{,}\mathrm{arc\; tan}\mathit{mx}$

are sigmoid-like functions, a kind of the so-called logistic function used for mathematical modeling in a range of fields (http://en.wikipedia.org/ wiki/Logistic_function; von Seggern, 2007), including biology, sociology, economics, neural network, probability, statistics, biomathematics, mathematical psychology, ecology, and medicine (for instance, for modeling non-sinusoidal circadian rhythms, Marler et al., 2006).

Let $f$ be a continuous function having a unique zero $\alpha$ in an interval $[a\text{,}b]$ with $f(a)f(b)<0$ .

Definition 1.4

$\mathcal{S}$ -transformation of $f$ is

$\mathcal{S}(f(x)):=\mathrm{sgn}f(x)\text{,}$

where $\mathrm{sgn}t$ is the signum function defined by

$\mathrm{sgn}t:=\left\{\begin{array}{cc}1\text{;}\hfill & t>0\text{,}\hfill \\ 0\text{;}\hfill & t=0\text{,}\hfill \\ -1\text{;}\hfill & t<0\text{.}\hfill \end{array}\right.$

Definition 1.5

Let $m>0$ be sufficiently large and let $f$ be a real continuous function. $\mathcal{T}$ -transformation of $f$ with the multiplicand $m$ , denoted as ${\mathcal{T}}_m(f(x))$ , is defined by

${\mathcal{T}}_m(f(x)):=\tanh \mathit{mf}(x)\text{.}$

The prefix " $\mathcal{T}$ " comes from the tanh function.

Definition 1.6

Let $m>0$ and let $f$ be a real continuous function. $\mathcal{A}$ -transformation of $f$ with the multiplicand $m$ , denoted as ${\mathcal{A}}_m(f(x))$ , is defined by

${\mathcal{A}}_m(f(x))=\frac{2}{\pi }\mathrm{arc\; tan}\mathit{mf}(x)\text{.}$

In practice, the multiplicand $m$ in Definitions 5 and 6 takes values from the interval [3,   30].

Below, we give a review of some basic properties of $\mathcal{T}$ -transformation:

1° ${\mathcal{T}}_m(-\infty )=-1\text{,}{\mathcal{T}}_m(+\infty )=1$ .

2° $-1<{\mathcal{T}}_m(f(x))<1$ .

3° If $f$ is monotonically increasing (decreasing), then ${\mathcal{T}}_m(f(x))$ is also monotonically increasing (decreasing).

4° $f(\alpha )=0\Rightarrow {\mathcal{T}}_m(f(\alpha ))=0$ , that is, the zeros of ${\mathcal{T}}_m(f(x))$ coincide with the zeros of $f$ .

5° ${\mathcal{T}}_m(f(x))$ converges to $\mathrm{sgn}f(x)$ as $m\to \infty$ .

6° If $\alpha$ is a zero of $f$ then ${\left.\frac{d}{\mathit{dx}}{\mathcal{T}}_m(f(x))\right|}_{x=\alpha }={\mathit{mf}}^{\prime }(\alpha )$ .

7° If the original function $f$ is defined on $[a\text{,}b]$ then ${\mathcal{T}}_m(f(x))$ is defined on the same interval.

The function $x\mapsto \frac{2}{\pi }\mathrm{arc\; tan}x$ behaves very similar to the hyperbolic function $\tanh x$ . The Properties 1°–7° are also valid for the $\mathcal{A}$ -transformation, where

$6°{\left.\frac{d}{\mathit{dx}}{\mathcal{A}}_m(f(x))\right|}_{x=\alpha }=\frac{2}{\pi }{\mathit{mf}}^{\prime }(\alpha )\text{.}$

From 1° and 2° we see that $\mathcal{T}$ - and $\mathcal{A}$ -transformation "amortize" and adjust possible peaks, oscillations, or other improper behavior of the considered function $f$ within the range interval (−1,   1) (see Figure 1.1 for $f(x)=e^x\sin 5x-2$ and ${\mathcal{T}}_8(f(x))$ ).

Figure 1.1. The graphs of the functions $f(x)=e^x\sin 5x-2$ and ${\mathcal{T}}_8(f(x))=\tan h8f(x)$

The Properties 4°–6° give the idea for efficient approximation of a zero of $f$ . Namely, according to 6° there follows that the slope of ${\mathcal{T}}_m(f(x))$ and ${\mathcal{A}}_m(f(x))$ in the vicinity of $\alpha$ is very steep for $m$ large enough (see Figure 1.2). Furthermore, following 5°, we observe that the graphs of ${\mathcal{T}}_m(f(x))$ and ${\mathcal{A}}_m(f(x))$ fit very well in the graph of the signum function for large $m$ (see Figure 1.2 for ${\mathcal{T}}_3(f(x))$ and $\mathrm{sgn}f(x)$ ). Hence, numerical integration over very convenient "almost rectangular-shaped" regions becomes a fruitful tool in a zero approximating scheme.

Figure 1.2. The graphs of the functions $f_1(x)=\mathrm{sgn}f(x)$ and ${\mathcal{T}}_3(f(x))=\tan h3f(x)$

The main features of the mentioned sigmoid-like functions are: (1) they take values within a limited interval [−1,   1] clustering almost all points toward the limits $\pm 1$ and (2) their slope near the zero is very steep. This can be seen from the examples presented by Figures 1.1 and 1.2. Since the graph of the sigmoid-like function is almost part by part rectangular, numerical integration of such functions over any interval would give very accurate result.

Consider the situation as in Figure 1.3, where $\alpha$ is a simple zero of a real function $f$ , isolated in an interval $[a\text{,}b]$ . For $m$ sufficiently large we then have

Figure 1.3. Numerical quadrature approach to the approximation of zeros

$\begin{array}{cc}\hfill & I_m(f):={\int }_a^b{\mathcal{T}}_m(f(x))\mathit{dx}={\int }_a^b\tanh \mathit{mf}(x)\mathit{dx}\hfill \\ \hfill & =\mathrm{sgn}f(a)[(\alpha -a)-(b-\alpha )]+{\epsilon }_m\text{,}\hfill \end{array}$

where ${\epsilon }_m$ is the error due to the approximation of $\tanh \mathit{mf}(x)$ by $\mathrm{sgn}f(x)$ . Let ${\tilde{I}}_m(f)$ be an approximate value of the integral $I_m(f)$ , calculated by numerical integration. Then from the last relation we find

$\begin{array}{cc}\hfill & \alpha =\frac{1}{2}\left[a+b+\mathrm{sgn}f(a)I_m(f)\right]+{\epsilon }_m\hfill \\ \hfill & \approx \frac{1}{2}\left[a+b+\mathrm{sgn}f(a){\tilde{I}}_m(f)\right]+{\epsilon }_m\hfill \\ \hfill & \approx \frac{1}{2}\left[a+b+\mathrm{sgn}f(a){\tilde{I}}_m(f)\right]=:x_0\text{.}\hfill \end{array}$

As noticed previously, the shape of the range of integration is quite close to rectangle, which is very convenient for numerical integration. Various standard quadrature formulae can be applied, including those in computational software packages such as Maple and Mathematica.

Example 2

For demonstration, simple statements in the computational software package Mathematica, applied to the function

$f(x)=(x-2)(x^{10}+x+1)e^{-x-1}$

and the interval [0,   5]

f[x_] =(x-2)(xˆ(10)+x+1)*Exp[-x-1]; a=0; b=5; m=10;

x0=0.5*(a+b+Sign[f[a]]*NIntegrate[Tanh[m*f[x]],{x,a,b}])

give a pretty good initial approximation $x_0=1.99962$ to the zero $\alpha =2$ of $f$ .

The main advantages of NIM are:

(1)

the possibility of obtaining relatively good approximation to the sought zero in one step using numerical integration, without iterating, and

(2)

unlike many iterative methods, the applied procedure does not require sufficiently close initial approximation but only an interval (often not tight) containing the zero.

The $\mathcal{S}$ -transformation is most frequently used because it is simple and gives the best initial approximation. However, since $\mathrm{sgn}f(x)$ is a discontinuous function, some problems may occur when numerical quadrature rule is employed. In practice, we seldom apply $\mathcal{T}$ -transformation or $\mathcal{A}$ -transformation.

Remark 4

From Property 6° there follows that $\mathcal{T}$ -transformation is slightly steeper than $\mathcal{A}$ -transformation. Besides, $|\frac{2}{\pi }\arctan \mathit{mx}|<|\tanh \mathit{mx}|$ for any $m$ and $x\in (-\infty \text{,}+\infty )$ . Therefore, $\tanh \mathit{mx}$ is a better approximation to $\mathrm{sgn}x$ than $\frac{2}{\pi }\mathrm{arc\; tan}\mathit{mx}$ is. Somewhat more precise initial approximations are obtained when $\mathcal{T}$ -transformation is applied, compared to $\mathcal{A}$ -transformation. $\mathcal{A}$ -transformation is more convenient for the detection of clusters of zeros, see Petković and Yun (2008).

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Source: https://www.sciencedirect.com/topics/computer-science/continuous-function

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