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# Removal of spurious solutions encountered in Helmholtz scattering resonance computations in R^d

In this paper we consider a sorting scheme for the removal of spurious scattering resonant pairs in two-dimensional electromagnetic problems and in three-dimensional acoustic problems. The novel sorting scheme is based on a Lippmann-Schwinger type of volume integral equation and can therefore be applied to graded material properties as well as piece-wise constant material properties. For TM/TE polarized electromagnetic waves and for acoustic waves, we compute first approximations of scattering resonances with finite elements. Then, we apply the novel sorting scheme to the computed eigenpairs and use it to remove spurious solutions in electromagnetic and acoustic scattering resonances computations at low computational cost. Several test cases with Drude-Lorentz dielectric resonators as well as with graded material properties are considered.

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10/15/2021

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## 1 Introduction

The most common approach to approximate scattering resonances is to truncate the domain with a perfectly matched layer (PML) and discretize the differential equations with a finite element method. This result in approximations of the true resonances but in practice also a large number of solutions that are unrelated to the true resonances. Those spurious eigenvalues are a major problem in resonance computations. The origin of unphysical eigenvalues in scattering resonance computations is spectral instability, which is common for non-normal operators trefethen07 ; davies97 . Spectral instability is known to be less problematic with volume integral equations compared with formulations based on differential operators. Therefore, we proposed in araujo+engstrom+2017 to use a Lippmann-Schwinger type of integral equation for removing spurious solutions in a one-dimensional setting.

In this paper, we show that it is possible to extended the approach in araujo+engstrom+2017 to higher dimensions. In particular, we show that the used sorting scheme can be computed cheaply and give valuable information on the location of spurious eigenvalues. This test is motivated by spectral stability properties of the Lippmann-Schwinger formulation of the problem. The idea is then to test each computed pair and obtain an pseudospectral indicator . Then we sort all computed pairs from the smallest to the largest value of . Finally, we can choose a user defined tolerance , and drop all pairs with . In araujo+engstrom+2017 we presented computations on several test cases for space dimension . The strategy was successful, since the proposed filtering scheme was able to identify physical pairs. However, the application of the ideas presented in araujo+engstrom+2017 for turn out to be more challenging. The effective application of these ideas in higher dimensions requires further considerations that are addressed in the current work. The new results confirm the efficiency of the proposed sorting scheme in , .

## 2 Electromagnetic and acoustic scattering resonances

Assume that is independent of and consider electromagnetic waves propagating in the -plane. The -independent electromagnetic field is then decomposed into transverse electric (TE) polarized waves and transverse magnetic (TM) polarized waves Cessenat1996 . This decomposition reduces Maxwell’s equations to one scalar equation for and one scalar equation for . The TM-polarized waves and the TE-polarized waves satisfy formally

 −ΔE3−ω2ϵE3=0and−∇⋅(1ϵ∇H3)−ω2H3=0, (1)

respectively. For the scattering resonance problems, and are assumed to be locally integrable functions that satisfy an outgoing condition MR1350074 ; MR1037774 .

Let the physical domain be an open ball of radius with boundary , and let be the bounded domain defining the resonators. Hence, we assume that the relative permittivity in is one. Furthermore, let denote the union of disjoint resonators , satisfying for all , as shown in figure 1. A scattering resonance was in gopal08 defined as a complex number for which the Lippmann-Schwinger equation

 T(ω)u:=u−K(ω)u=0 (2)

has a non-zero solution . The integral operator in (2) is for TM/TE waves given by

 TM:K(ω)u:=ω2∫ΩrΦ(x,y)(ϵ−1)udyTE:K(ω)u:=∇⋅∫ΩrΦ(x,y)(1ϵ−1)∇udy,Φ(x,y):=⎧⎨⎩i2ωeiω|x−y|,d=1i4H(1)0(ω|x−y|),d=2. (3)

Here, is known as the outgoing Green function in free space for the Helmholtz equation MR1350074 ; MR1037774 . Notice that while we are interested in , the integration in (2) is only performed over , since the integration over the air region is zero.

The scattering resonance problem (2) is a highly non-linear eigenvalue problem, where the matrices after discretization are large and full. It is possible to accurately solve the non-linear eigenvalue problem in using a standard laptop; see e.g. bra13 ; araujo+engstrom+2017 . However, accurate computations of eigenpairs of (2) in higher dimensions would require huge computer resources; See MR2406839 and the discussion in Section 7.6.

### Acoustic scattering resonances in R3

Sound-soft materials are characterized by the speed of sound and we assume that acoustic resonators are defined by . Then, the acoustic pressure satisfies formally the Helmholtz equation

 −Δu−ω2c(x)2u=0, (4)

where the outgoing condition can be expressed as satisfying an expansion in spherical harmonics outside the open ball ihl98 . Moreover, is a scattering resonance pair if (2) holds with

 K(ω)u:=ω2∫ΩrΦ(x,y)(1c(y)2−1)u(y)dy,Φ(x,y)=eiω|x−y|4π|x−y|. (5)

Note that the acoustic problem in , is analogous to TM-polarized electromagnetic waves with .

### 2.1 Alternative formulations

Understanding the resonance behavior of structures in unbounded domains is important and many different approaches have been proposed. Graded material properties are increasingly popular in applications Wang2016 and we will therefore not consider boundary integral equations. However, boundary integral equation based methods are a good alternative for cases with piecewise constant coefficients and not too complicated geometry MR3622414 . The most popular method to compute resonances in , is the finite element (FE) method with a perfectly matched layer (PML). In recent years, finite element methods based on Hardy space infinite elements (HIF) MR2485441 and DtN maps araujo+engstrom+jarlebring+2017 have also been proposed as strong alternatives to compute resonances in higher dimensions. For the DtN map, recent developments in computational linear algebra are a key to the high performance of the method jar14 ; jar15J . Discretization with FE of the PML and HIP formulations result in sparse matrices, and a formulation in terms of a DtN result in sparse matrices except a small dense block corresponding to the DtN map. Moreover, the PML and HIF formulations result in a standard generalized eigenvalue problem if is -independent and in the general case the non-linearity in is completely determined by . Hence, the PML and HIF formulations seem to have the most attractive properties of the considered methods. However, it is very important to also take into account the so called spectral instability. Then, the picture changes completely, as discussed in the next section.

### 2.2 Spectral instability and pseudospectra

Let denote an unbounded closed linear operator in a Hilbert space with domain , spectrum , and resolvent set . Then exhibits high spectral instability if for a very small there exist many and such that

 ∥(A−ω2)u∥≤δ∥u∥ (6)

even though is not close to MR1912874 . This is closely related to the pseudospectrum which is defined as the union of and all in the resolvent set for which it exists an such that (6) holds. The generalization of those results to an operator function is straightforward and we will in some of the numerical computations rely on the following alternative characterization of the pseudospectrum:

 σδ(T)=σ(T)∪{ω∈ρ(T):∥T−1(ω)∥>δ−1}.

It is well known that PML and HIF based methods encounter high spectral instability MR2485441 ; araujo+engstrom+2017 . Methods based on a DtN map encounter medium spectral instability araujo+engstrom+jarlebring+2017 ; araujo+engstrom+2017 and integral equation based methods encounter low spectral instability araujo+engstrom+2017 . Hence, the Lippmann-Schwinger equation is in our setting the preferred method in terms of spectral stability. This will be further discussed in the paper.

### 2.3 Domain and material properties

In electromagnetics, the material properties of non-magnetic metals are characterized by the complex relative permittivity function , which changes rapidly at optical frequencies . The most common accurate material model is then the Drude-Lorentz model

 ϵmetal(ω):=ϵ∞+Np∑j=0fjω2pω2j−ω2−iωγj, (7)

where and , , are non-negative Cessenat1996 . Hence, the Maxwell eigenvalue problem in the spectral parameter is nonlinear for metal-dielectric nanostructures. Assume that the domain of the resonators can be written in the form and let

denote the characteristic function of the subset

. For material properties that are piecewise constant in , we assume a permittivity function in the form

 ϵ(x,ω):=Nr∑m=0ϵm(ω)χΩm(x),x∈Ωa,ω∈D, (8)

where the dependencies on in for are of Drude-Lorentz type (7). In addition, we will consider graded material properties, meaning that is a continuous function in . In linear acoustics, the speed of sound is assumed to be independent of the frequency.

## 3 DtN and PML based methods

In the next sections, we will describe two common approaches to compute scattering resonances and the restriction of resonance modes to a compact subset of . In the following, we use the notation

 −∇⋅(ρ∇u)−ω2ηu=0, (9)

where for the TM-case and for the TE-case.

We define for the forms

 a(ω)[u,v]:=∫Ωaρ∇u⋅∇¯¯¯vdx,b(ω)[u,v]:=∫Ωaηu¯¯¯vdx, (10)

where in (10), and are functions of . Let denote the set of values that are zeros or poles of and set .

### 3.1 DtN based methods

Scattering resonances and quasi-normal modes restricted to can be determined from a problem with a Dirichlet-to-Neumann (DtN) map lenoir92 ; schenk2011optimization ; araujo+engstrom+jarlebring+2017 . Below we present variational formulations for , . Formally, is a scattering resonance pair if (9) holds in and

 ∂u∂n=G(ω)uon Γ:=∂Ωa, (11)

where is the normal derivative.

#### 3.1.1 DtN formulation in R

In one space dimension the scattering resonance problem restricted to is formally: Find a non-zero and a complex such that

 −(ρu′)′−ω2ηu=0forx∈Ωa, (12)

where the DtN-map at is

 u′(−a)=−iωu(−a),u′(a)=iωu(a). (13)

Define for and the forms as in (10), and

 g1(ω)[u,v]:=iω(u(a)¯¯¯v(a)+u(−a)¯¯¯v(−a)). (14)

The nonlinear eigenvalue problem is then as follows: Find vectors

and satisfying

 q1(ω)[u,v]:=a(ω)[u,v]−ω2b(ω)[u,v]−g1(ω)[u,v]=0, (15)

for all . Note that (15) is a quadratic eigenvalue problem if is independent of and a rational eigenvalue problem for Drude-Lorentz type of materials (7).

#### 3.1.2 DtN formulation in R2

In this subsection, we present a DtN formulation in polar coordinates . Let denote the Hankel function of first kind, then the DtN operator (11) on the circle has the explicit form

 G(ω)u:=12π∞∑ν=−∞ωH(1)′ν(ωa)H(1)ν(ωa)eiνθ∫2π0u(a,θ′)e−iνθ′dθ′ (16)

and is bounded lenoir92 .

The resonance problem restricted to is formally to find non-trivial solutions such that (9) and (11) with (16) holds. The theory presented in lenoir92 can with minor changes be used in the present case to derive properties of a variational formulation of the problem.

Variational formulation: Let denote the union of the set of zeros of , and denote the operator (11) truncated after . The eigenvalues of the truncated version of (9)-(11) are determined by the following variational problem: Find and such that for all

 q(ω)[u,v]:=a(ω)[u,v]−ω2b(ω)[u,v]−g(ω)[u,v]=0, (17)

where the forms are defined as in (10), and

 g(ω)[u,v]:=(Gνmax(ω)u,v)Γa=νmax∑ν=−νmaxωaH(1)′ν(ωa)H(1)ν(ωa)^uν¯^vν,^φν=1√2π∫2π0φ(a,θ)e−iνθdθ. (18)

### 3.2 PML based methods

In the previous section, a DtN-map was used to reduce the exterior Helmholtz problem to a bounded domain. In this section, we consider an alternative approach based on a complex coordinate stretching (the PML method), which results in a linear eigenvalue problem for non dispersive material coefficients kim09 . The method consists on attaching to a buffer layer of thickness , where outgoing solutions decay rapidly. The buffer domain is referred to as and the full computational domain is enlarged as shown in the Figure 1.

#### 3.2.1 PML formulation in R

Let , and , with . The action of the PML is defined through the stretch function

 ~σ(r):=⎧⎪⎨⎪⎩0,ifrb (19)

with , and where the polynomial is required to be increasing in , and is sufficiently smooth . For this we introduce the fifth order polynomial satisfying: and .

The PML problem is restricted to and the PML strength function has then the profile shown in Fig. 1. In the following sections, we consider the complex change of variable and transformation rule

 ~x=∫xx0~α(y)dy,dd~x=1~α(x)ddx,with~α(x)=1+i~σ(x), (20)

where in and in .

For finite element computations we restrict the domain to , and define and choose similarly as in kim09 homogeneous Dirichlet boundary conditions. Formally, the finite PML problem is then: Find the eigenpairs such that

 −ddx(ρ~αdudx)−ω2η~αu=0,x∈Ωℓ,u(rℓ)=0andu(−rℓ)=0. (21)

In the following, we consider a variational formulation of (21).

Find and such that for all

 t1(ω)[u,v]:=a(ω)[u,v]−ω2b(ω)[u,v]+^t1(ω)[u,v]=0, (22)

where , and the forms are defined as in (10).

#### 3.2.2 PML formulation for Rd

Approximation of resonances using a radial PML was analyzed in kim09 and we will here only consider the PML problem truncated to the ball . The used complex stretching functions are as in one-dimension of the form (19). Let denote the set of values that are zeros or poles of and let denote a partition into the PML-region and the part of the domain containing the resonators.

In the sequel we need the following definitions

 ~α(r):=1+i~σ(r),~r(r):=(1+i~σ)r=~α(r)r,σ(r):=~σ(r)+r∂~σ∂r,α(r):=∂~r∂r=1+iσ(r), (23)

with the properties and .

It is clear that the PML coefficients are designed such that for , we obtain and , which is the same as . Hence, the PML operator restricted to corresponds to the original operator in problem (9) (no PML effect).

Variational formulation: The eigenvalues of (9) are then determined by the following variational problem

Find and such that for all

 t(ω)[u,v]:=a(ω)[u,v]−ω2b(ω)[u,v]+^t(ω)[u,v]=0, (24)

where , and the forms are defined as in (10).

As an example for , direct transformation of (24) from polar to Cartesian coordinates results in

 A=⎛⎜ ⎜ ⎜⎝~ααcos2θ+α~αsin2θ(~αα−α~α)sinθcosθ(~αα−α~α)sinθcosθ~ααsin2θ+α~αcos2θ⎞⎟ ⎟ ⎟⎠,B:=α~α. (25)

Even though , and are defined in the whole , their action takes place only in .

## 4 Discretization of the Lippmann-Schwinger equation

In this section we present a collocation method for discretization of the Lippmann-Schwinger equation (2), which will be used to compute resonances in one-dimension and is the base for the numerical sorting algorithm in Section 6. Further computational details are given in Section 7.6.

### 4.1 A Galerkin-Nyström method

We present a Galerkin-Nyström discretization method for linear Fredholm integral equations of the second kind, with kernels satisfying for in the compact set . In ikebe72 , this method is referred to as case (A) of the Galerkin methods, and convergence for the problem with sources is also discussed. In (gopal08, , Sect. 3.2) the method was used for resonance computations for .

Let be piecewise polynomial functions with the property , . We introduce the representation , and with the use of (2) we obtain the nonlinear eigenvalue problem: Find and such that

 Tγ(ωγ)ξ=(I−K(ωγ))ξ=0,withTM:Kij(ω):=ω2∫ΩrΦ(xi,y)(ϵ−1)φjdy,TE:Kij(ω):=∇⋅∫ΩrΦ(xi,y)(1ϵ−1)∇φjdy, (26)

The Nyström method consists in choosing the collocation points as the nodes of a high order quadrature rule. By doing this, the convergence of the scheme is considerably improved. In Section 7.6 we describe some of the implementation details of the Galerkin-Nyström discretization.

The resulting nonlinear matrix eigenvalue problem (26) is solved by using a contour integration based method Sakurai09 ; Beyn12 ; MR3423605 .

###### Remark 1.

The formulation in (2) uses information of the exact solution of the problem at every discretization node , through its fundamental solution . From where the numerical scheme is flexible in the way that it can be posed in the smallest domain , as well as in larger domains , without taking any special treatment for boundary conditions.

## 5 FE discretization of the DtN and PML based formulations

In this section we discuss briefly the details involved in the assembly of the matrices corresponding to the discretization of the formulations given in (9), and (2).

### 5.1 Discretization with the finite element method

Let the domain be covered with a regular and quasi uniform finite element mesh consisting of elements . The mesh is designed such that the permittivity function is continuous in each . Let be the length of the largest diagonal of the non-curved primitive and denote by the maximum mesh size .

Let denote the space of polynomials on of degree and define the dimensional finite element space

 Sγ(Ω):={u∈H1(Ωa):u|Kj∈Pp(Kj)forKj∈T}. (27)

The computations of discrete resonance pairs are for performed in the approximated domain using curvilinear elements Babuska92 . The meshes used are shape regular in the sense of (Schwab1998, , Sec. 4.3), and consist of quadrilateral/brick elements with curvilinear edges/surfaces that deviate slightly from their non-curved primitives.

### 5.2 Assembly of the FE matrices

In this section we refer to domains . Let be a basis of . Then , and the entries in the finite element matrices are of the form

 uγ=N∑j=1ξjφj,Aij=(ρ∇φj,∇φi)Ωγa,Mij=(ηφj,φi)Ωγa. (28)

The matrix eigenvalue problem is then: Find the eigenpars such that

 F(ω)ξ:=(A−ω2M+Q)(ω)ξ=0, (29)

where the corresponding matrix valued function is

 Qij(ω):={−g1(ω)[φj,φi],DtN and d=1−g(ω)[φj,φi],DtN and d=2,or{^t1(ω)[φj,φi],PML and d=1^t(ω)[φj,φi],PML and d=2,3. (30)

In the case where is given as piecewise smooth function of space, we write (29) as

 F(ω)ξ:=(Nr∑m=0{ρm(ω)~Am−ω2ηm(ω)~Mm}+Q(ω))ξ=0, (31)

with matrices

###### Remark 2.

Truncation of the DtN: Let be the smallest integer greater than or equal to . We use the rule according to the results in araujo+engstrom+jarlebring+2017 , where from the considered spectral window, is the largest real part allowed for computations of eigenvalues.

###### Remark 3.

Truncation of the PML: The PML is set up following the discussions in kim09 ; araujo+engstrom+2017 , which accounts for large enough and such that the search region is feasible. Additionally, we use the space for computations with the PML formulation.

Finally, we mention that all discretization methods presented in this work use the same FE space over .

###### Remark 4.

All formulations (LS, DtN, PML) use the FE triangulation , which is the restriction of to .

By using the FE mesh suggested in Remark 4, we ensure that the approximation properties in the physical domain are the same for all formulations.

## 6 Numerical sorting of resonances

In this section we derive a discrete form of (2) that allow us to identify resonances from spurious solutions once we have computed FE solutions to (29) or to (31). The resulting expression for the sorting scheme is a discrete form of the condition , where is a FE solution restricted to . Let be a basis for and let be the -projection on . Then, the discrete Lippmann-Schwinger equation (26) is written in the form

 T(ω)uγ=uγ−K(ω)uγ,uγ:=N∑j=1ξjφj,with∥uγ∥L2(Ωa)=1.
###### Definition 5.

Pseudospectrum indicator: The computed eigenvalue belongs, for given , to the -psudospectrum if the pair satisfies . Then, for a given domain , we define the pseudospectrum indicator as

 δγ(Ωa):=∥Tγ(ωγ)uγ∥Ωa. (32)

Particularly, we want to be able to measure whether the computed eigenpair is converging to a physical pair. Naturally, non convergent pairs exhibit large values.

Additionally, we have that certain spurious eigenpairs introduced by using the PML method are easily identified and can be removed by using the following definition.

###### Definition 6.

PML added eigenpairs: The use of the coordinate stretching technique in formulations (22), (24) introduces new eigenpairs to problem (9). These new eigenvalues accumulate close to the critical line of the modified PML problem araujo+engstrom+2017

, and eigenfunctions

exhibit oscillations in , but decay in the physical region . Then, by using the normalization , the PML critical eigenvalues exhibit

 0<∥vm∥L2(Ωa)|Ωa|<∥vm∥L2(ΩPML)|ΩPML|,

which can be succesfully used as a filtering criterion for removing PML added eigenpairs.

## 7 Computational details

In this section we outline the computational details for the FE discretization of the DtN and PML based formulations in Section 5. Furthermore, we present results on integration of weakly singular kernels that are used in the new numerical sorting scheme in Subsection 7.7. For convenience of the reader we provide a summary of the used standard techniques.

### 7.1 Master element and transfinite interpolation

Consider a physical element and let denote the master element. Numerical quadrature is used to integrate a function over and when high order polynomial spaces are used, it is convenient to compute information from the shape functions in the master element and then store it. In this way we gain in performance as computations from high polynomials are expensive. Consequently, functions defined over a physical element are mapped to functions over , where we perform integration. Then, the mapping transforms coordinates as . The action of the mapping is enforced by the Jacobian’s determinant (Strang2008, , Sec. 3.3), (pavel-solin04, , Sec. 3.4). Then, we have

 ∫Kf(x)dx=∫Kf∘XK(y)J(y)dy. (33)

For the case where is a line, quadrilateral or a brick element, the explicit expression for is a known bilinear transformation. When has curved edges,

can be described by the so-called theory of Transfinite Interpolation

Gordon1973 , and the implementation and computational details can be found in Gordon73 , (pavel-solin04, , Sec. 3.2). A general rule of thumb is that the bending of the edges must be small compared to the diameter of the element, and that the angles at the element corners should be close to

. For further details and explicit error estimates on curved elements the reader is referred to

(Strang2008, , Sec. 3.3), (Oden76, , Sec. 6.7). For the description on how transform from to , and other related details, the reader is refereed to (pavel-solin04, , Sec. 3.3).

### 7.2 Evaluation of integrals

In this subsection, we revise briefly numerical integration by Gaussian-Legendre quadratures. In the one dimensional case, integration over the master element is approximated by formulas of the form , where are the quadrature weights, the quadrature nodes, and is the quadrature error or remainder. The coefficients are all positive (hildebrand87, , Sec. 8.4). These type of quadrature rules are derived under the assumption that , then, the Weierstrass approximation theorem (hildebrand87, , Sec 1.2) guarantees the existence of a polynomial such that , for an specified . In this way can be set to minimize , and is integrated exactly. The effective way of reducing is by increasing the polynomial degree until the residual is below . In the quadrature formula, increasing is equivalent to increasing the number of evaluation points . The remainder for the -point Gaussian quadrature satisfies

 |E|≤C|f(2m)(ξ)|,forξ∈K, (34)

where we see that if is a polynomial of order , the remainder vanishes and the quadrature gives the exact integral value. Further details on Gaussian quadratures can be revised in e.g. (hildebrand87, , Ch. 8), and implementation details are provided in (solin04, , Ch 4).

In , , we describe quadrature formulas for integration of , when a physical element is allowed to be curved. Then, we resource to the formula

 ∫Kf(x)dx=∫Kf∘XK(y)J(y)dy=m2∑j=1wjf(xj)+E(K), (35)

where the new weights and nodes

are the piled up tensor product version of vector of the corresponding one dimensional values. For example, with

, and , the new index is , and . Then, we obtain the transformed nodes and the corresponding weights as .

Finally, we discuss the composite of a quadrature rule, when integration is performed over a domain defined by the union of several elements . The integrand is now required to be piecewise smooth , for . Then, we obtain

 ∫Ωf(x)dx=∑i∫Kif(x)dx=∑i∑kwk(Ki)f(xk(Ki))+E(Ω), (36)

where for each element , we have the quadrature pairs , similarly as in (35).

The polynomial spaces that we use for are based on the tensor product of one dimensional finite element spaces (solin04, , Sec. 2.2), dealII90 . As we work with piecewise smooth coefficients, we make sure that the jumps of coincide with the possibly curved element edges , such that for we have . This allow us to use quadrature rules in each individual element and guarantee convergence of the error of the numerical integration.

#### 7.2.1 Integrating weakly singular kernels

In the discretization of the Lippmann-Schwinger formulation (26), we encounter the situation where the integrand will contain both evaluation points in an element . For one-dimensional problems (), the kernel is continuous, but has a jump in the derivative at points . In the troublesome element , we can always split the integration interval and perform two separate quadrature integrations. Then, by using Gauss-type of quadratures, it is possible to avoid the evaluation of araujo+engstrom+2017 .

In higher dimensions () the kernel is weakly singular (Colton+Kress1983, , Sec. 2.3), what makes the integration in (26) more demanding. This difficulty can be overcome by specializing the quadratures as done for example in Kaneko1994 , Duan2009 , and Anand2016 . There, extra effort was spent in refining adaptively on elements , where the integrand is unbounded. Then a Nyström type of high order quadratures, combined with interpolation in polar coordinates along with other techniques were used in order to keep small to desired order. As expected, the challenge becomes more pronounced in higher dimension as can be seen in Duan2009 and in Anand2016 . In those papers the aim was to solve a scattering problem through the Lippmann-Schwinger formulation for a given incoming wave. However, our case is very different as we look for scattering resonances, where the corresponding eigensolver is computationally much more demanding than a linear solve.

### 7.3 Solution of the nonlinear eigenvalue problems

The approximation of resonances based on the DtN formulation for leads to the matrix problem in (29). The solution of this NEP is based on the solution strategy presented in araujo+engstrom+jarlebring+2017 , where we use a specialization of the Infinite Arnoldi method Jarlebring:2012:INFARNOLDI ; Jarlebring:2015:WTIARTR called the tensor infinite Arnoldi method (TIAR). In particular we introduce a pole cancellation technique in order to increase the radius of convergence for computation of eigenvalues that lie close to the poles of the matrix-valued function.

For the approximation of resonances when the permittivity function is described by the rational model (7), we prefer to solve the corresponding matrix NEP by the techniques presented in araujo+campos+engstrom+roman , which is a specialization of the solver in guttel17 implemented in the SLEPc library slepc05+roman .

### 7.4 Properties of volume integral equations for resonance computation

We discretize volume integral equations by using the scheme presented in (26), which does not require any boundary conditions as mentioned in Remark 1. In Section 8.2 we will numerically show the desired stability of the spectrum of the resulting discrete operator for perturbations of .

###### Remark 7.

The resulting system matrices has dimensions comparable to FE matrices: , with . However, the matrices are dense ( storage), non-symmetric, and the elements of are transcendental functions of .

The consequences of Remark 7 in a NEP solution strategy is that the matrix from (26) must be re-assembled for each new iteration , which is very expensive especially for problems of dimension .

### 7.5 Memory requirements

Let be the number of cells in a triangulation in space dimension , the polynomial degree of the basis functions in use, and bytes is the memory required to store a complex number in double precision. Given the number of non zeros elements in a matrix, the memory required to store it is .

In the collocation method given in Sec. 4.1, matrices are dense and we get . In turn, the FE matrices from Sec. 5.2 are sparse, and each cell in the triangulation contributes with a block of size support points. Additionally, we have scattered connections of order with neighboring cells that we omit for simplicity. A simple estimation gives . Furthermore, by assuming the division in a one dimensional partition, it is reasonable to have , and for a small, moderate and large problem respectively.


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Backtesting a Moving Average Crossover in Python with pandas

In the previous article on Research Backtesting Environments In Python With Pandas we created an object-oriented research-based backtesting environment and tested it on a random forecasting strategy. In this article we will make use of the machinery we introduced to carry out research on an actual strategy, namely the Moving Average Crossover on AAPL.

## Moving Average Crossover Strategy

The Moving Average Crossover technique is an extremely well-known simplistic momentum strategy. It is often considered the "Hello World" example for quantitative trading.

The strategy as outlined here is long-only. Two separate simple moving average filters are created, with varying lookback periods, of a particular time series. Signals to purchase the asset occur when the shorter lookback moving average exceeds the longer lookback moving average. If the longer average subsequently exceeds the shorter average, the asset is sold back. The strategy works well when a time series enters a period of strong trend and then slowly reverses the trend.

For this example, I have chosen Apple, Inc. (AAPL) as the time series, with a short lookback of 100 days and a long lookback of 400 days. This is the example provided by the zipline algorithmic trading library. Thus if we wish to implement our own backtester we need to ensure that it matches the results in zipline, as a basic means of validation.

## Implementation

Make sure to follow the previous tutorial here, which describes how the initial object hierarchy for the backtester is constructed, otherwise the code below will not work. For this particular implementation I have used the following libraries:

• Python - 2.7.3
• NumPy - 1.8.0
• pandas - 0.12.0
• matplotlib - 1.1.0

The implementation of ma_cross.py requires backtest.py from the previous tutorial. The first step is to import the necessary modules and objects:

# ma_cross.py

import datetime
import matplotlib.pyplot as plt
import numpy as np
import pandas as pd

from backtest import Strategy, Portfolio

As in the previous tutorial we are going to subclass the Strategy abstract base class to produce MovingAverageCrossStrategy, which contains all of the details on how to generate the signals when the moving averages of AAPL cross over each other.

The object requires a short_window and a long_window on which to operate. The values have been set to defaults of 100 days and 400 days respectively, which are the same parameters used in the main example of zipline.

The moving averages are created by using the pandas rolling_mean function on the bars['Close'] closing price of the AAPL stock. Once the individual moving averages have been constructed, the signal Series is generated by setting the colum equal to 1.0 when the short moving average is greater than the long moving average, or 0.0 otherwise. From this the positions orders can be generated to represent trading signals.

# ma_cross.py

class MovingAverageCrossStrategy(Strategy):
"""
Requires:
symbol - A stock symbol on which to form a strategy on.
bars - A DataFrame of bars for the above symbol.
short_window - Lookback period for short moving average.
long_window - Lookback period for long moving average."""

def __init__(self, symbol, bars, short_window=100, long_window=400):
self.symbol = symbol
self.bars = bars

self.short_window = short_window
self.long_window = long_window

def generate_signals(self):
"""Returns the DataFrame of symbols containing the signals
to go long, short or hold (1, -1 or 0)."""
signals = pd.DataFrame(index=self.bars.index)
signals['signal'] = 0.0

# Create the set of short and long simple moving averages over the
# respective periods
signals['short_mavg'] = pd.rolling_mean(bars['Close'], self.short_window, min_periods=1)
signals['long_mavg'] = pd.rolling_mean(bars['Close'], self.long_window, min_periods=1)

# Create a 'signal' (invested or not invested) when the short moving average crosses the long
# moving average, but only for the period greater than the shortest moving average window
signals['signal'][self.short_window:] = np.where(signals['short_mavg'][self.short_window:]
> signals['long_mavg'][self.short_window:], 1.0, 0.0)

# Take the difference of the signals in order to generate actual trading orders
signals['positions'] = signals['signal'].diff()

return signals

The MarketOnClosePortfolio is subclassed from Portfolio, which is found in backtest.py. It is almost identical to the implementation described in the prior tutorial, with the exception that the trades are now carried out on a Close-to-Close basis, rather than an Open-to-Open basis. For details on how the Portfolio object is defined, see the previous tutorial. I've left the code in for completeness and to keep this tutorial self-contained:

# ma_cross.py

class MarketOnClosePortfolio(Portfolio):
"""Encapsulates the notion of a portfolio of positions based
on a set of signals as provided by a Strategy.

Requires:
symbol - A stock symbol which forms the basis of the portfolio.
bars - A DataFrame of bars for a symbol set.
signals - A pandas DataFrame of signals (1, 0, -1) for each symbol.
initial_capital - The amount in cash at the start of the portfolio."""

def __init__(self, symbol, bars, signals, initial_capital=100000.0):
self.symbol = symbol
self.bars = bars
self.signals = signals
self.initial_capital = float(initial_capital)
self.positions = self.generate_positions()

def generate_positions(self):
positions = pd.DataFrame(index=signals.index).fillna(0.0)
positions[self.symbol] = 100*signals['signal']   # This strategy buys 100 shares
return positions

def backtest_portfolio(self):
portfolio = self.positions*self.bars['Close']
pos_diff = self.positions.diff()

portfolio['holdings'] = (self.positions*self.bars['Close']).sum(axis=1)
portfolio['cash'] = self.initial_capital - (pos_diff*self.bars['Close']).sum(axis=1).cumsum()

portfolio['total'] = portfolio['cash'] + portfolio['holdings']
portfolio['returns'] = portfolio['total'].pct_change()
return portfolio

Now that the MovingAverageCrossStrategy and MarketOnClosePortfolio classes have been defined, a __main__ function will be called to tie all of the functionality together. In addition the performance of the strategy will be examined via a plot of the equity curve.

The pandas DataReader object downloads OHLCV prices of AAPL stock for the period 1st Jan 1990 to 1st Jan 2002, at which point the signals DataFrame is created to generate the long-only signals. Subsequently the portfolio is generated with a 100,000 USD initial capital base and the returns are calculated on the equity curve.

The final step is to use matplotlib to plot a two-figure plot of both AAPL prices, overlaid with the moving averages and buy/sell signals, as well as the equity curve with the same buy/sell signals. The plotting code is taken (and modified) from the zipline implementation example.

# ma_cross.py

if __name__ == "__main__":
# Obtain daily bars of AAPL from Yahoo Finance for the period
# 1st Jan 1990 to 1st Jan 2002 - This is an example from ZipLine
symbol = 'AAPL'
bars = DataReader(symbol, "yahoo", datetime.datetime(1990,1,1), datetime.datetime(2002,1,1))

# Create a Moving Average Cross Strategy instance with a short moving
# average window of 100 days and a long window of 400 days
mac = MovingAverageCrossStrategy(symbol, bars, short_window=100, long_window=400)
signals = mac.generate_signals()

# Create a portfolio of AAPL, with $100,000 initial capital portfolio = MarketOnClosePortfolio(symbol, bars, signals, initial_capital=100000.0) returns = portfolio.backtest_portfolio() # Plot two charts to assess trades and equity curve fig = plt.figure() fig.patch.set_facecolor('white') # Set the outer colour to white ax1 = fig.add_subplot(211, ylabel='Price in$')

# Plot the AAPL closing price overlaid with the moving averages
bars['Close'].plot(ax=ax1, color='r', lw=2.)
signals[['short_mavg', 'long_mavg']].plot(ax=ax1, lw=2.)

ax1.plot(signals.ix[signals.positions == 1.0].index,
signals.short_mavg[signals.positions == 1.0],
'^', markersize=10, color='m')

# Plot the "sell" trades against AAPL
ax1.plot(signals.ix[signals.positions == -1.0].index,
signals.short_mavg[signals.positions == -1.0],
'v', markersize=10, color='k')

# Plot the equity curve in dollars
ax2 = fig.add_subplot(212, ylabel='Portfolio value in \$')
returns['total'].plot(ax=ax2, lw=2.)

ax2.plot(returns.ix[signals.positions == 1.0].index,
returns.total[signals.positions == 1.0],
'^', markersize=10, color='m')
ax2.plot(returns.ix[signals.positions == -1.0].index,
returns.total[signals.positions == -1.0],
'v', markersize=10, color='k')

# Plot the figure
fig.show()

The graphical output of the code is as follows. I made use of the IPython %paste command to put this directly into the IPython console while in Ubuntu, so that the graphical output remained in view. The pink upticks represent purchasing the stock, while the black downticks represent selling it back:

AAPL Moving Average Crossover Performance from 1990-01-01 to 2002-01-01

As can be seen the strategy loses money over the period, with five round-trip trades. This is not surprising given the behaviour of AAPL over the period, which was on a slight downward trend, followed by a significant upsurge beginning in 1998. The lookback period of the moving average signals is rather large and this impacted the profit of the final trade, which otherwise may have made the strategy profitable.

In subsequent articles we will create a more sophisticated means of analysing performance, as well as describing how to optimise the lookback periods of the individual moving average signals.

# Just Getting Started with Quantitative Trading?

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# A new approach to finite semifields

Simeon Ball

Universitat Politècnica de Catalunya

A finite semifield is a finite set with two operations, addition and multiplication, such that satisfies all the axioms of a field except (possibly) associativity of multiplication.

A semifield can be used to coordinatise a projective plane of order and we are interested in finding semifields that produce non-isomorphic projective planes. Two semifields are said to be isotopic if they coordinatise isomorphic planes. There are less than roughly 20 known families of (mutually non-isotopic) semifields. Unless it is immediate that two semifields are not isotopic it is generally difficult to establish whether or not they are. Above all, the goal in this area is to construct many more families of non-isotopic semifields. The first semifields were discovered by Dickson, roughly 100 years ago with more examples given later by Albert (1950's), Knuth (1960's), Cohen and Ganley (1980's) and various families due to Kantor, amongst others, have been constructed in the last twenty years.

If is finite then it can be shown that for some prime power and that can viewed as a vector space of rank over , where multiplication is given by by the rule

where is a basis for over . Knuth was first to note that any permutation of the subscripts produces another semifield, so there are six semifields associated with any semifield.

In this talk I shall present a new way to construct finite semifields of order from two subspaces of a vector space of rank over , for some . In fact any finite semifield can be constructed in this way for some , moreover any known semifield of order can be constructed from two subspaces of a vector space of rank or rank over .

The construction also provides us with a new operation (not one of the six due to Knuth) which produces more semifields in the case when .

2005-05-23


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# Numerical integration

Solving differential equations numerically is a very mature field with multiple algorithms carefully explained in multiple textbooks. However, when it comes down to actually solving them in practice by nonspecialists, a lot of people, myself included, will often resort to the Euler method. There are two main reasons for this. The first is that it is trivial to remember and code and the second is that Moore’s law has increased the speed of computing sufficiently that we can get away with it. However, I think it’s time to get out of the eighteenth century and at least move into the nineteenth.

Suppose you have a differential equation $\dot{x}=f(x,t)$, where $x$ can be a scalar or a vector in an arbitrary number of dimensions.  The Euler method simply discretizes the time derivative via

$\frac{dx}{dt} \rightarrow \frac{x(t+h)-x(h)}{h}$

so that $x$ at time step $t+h$ is given by

$x(t+h)=x(t)+h*f(x(t),t)$

Now what could be simpler than that? There are two problems with this algorithm. The first is that it is only accurate to order $h$ so you need to have very small steps to simulate accurately. The second is that the Euler method is unstable if the step size is too large. The way around these limitations is to use very small time steps, which is computationally expensive.

The second most popular algorithm is probably (4th order) Runge-Kutta, which can generally use larger time steps for the same accuracy. However, a higher order algorithm is not necessarily faster because what really slows down a numerical scheme is the number of function evaluations it needs to make and 4th order Runge-Kutta needs to make a lot of function evaluations. Runge-Kutta is also plagued by the same stability problems as Euler and that can also limit the maximum allowed step size.

Now, these problems have been known for decades if not centuries and work-arounds do exist. In fact, the most sophisticated solvers such as CVODE tend to use adaptive multi-step methods, like Adams-Bashforth (AB) or it’s implicit cousin Adams-Moulton and this is what we should all be implementing in our own codes. Below I will give simple pseudo-code to show how a 2nd order Adams-Bashforth scheme is not much more complicated to code than Euler and uses the same number of function evaluations. In short, with not much effort, you can speed your code immensely by simply switching to AB.

The cleverness of AB is that it uses previous steps to refine the estimate for the next step. In this way, you get a higher order scheme without having to re-evaluate the right hand side of your ODE. You simply reuse your computations. It’s the ultimate green scheme. For the same ODE example above 2nd order AB is simply

$x(t+2h)=x(t+h)+h\left(\frac{3}{2}f(x(t+h),t+h)-\frac{1}{2}f(x(t),t)\right)$

You just need to store the previous two evaluations to compute the next step.  To start the algorithm, you can just use an Euler algorithm to get the first step.

Here is the 2nd order AB algorithm written in pseudo Julia code for a one dimensional version of the ODE above.

# make x a two dimensional vector for the two time steps

# Set initial condition

x[1]=0;

# Store evaluation of function f(x,t) for later use

fstore[1]= f(x[1],h)

# Take Euler step

x[2] = x[1] + h*fstore[1]

# store function at time = 2h

fstore[2]  = f(x[2],2*h)

# Store computed values for output

xout[1]=x[1]

xout[2]=x[2]

# Take AB steps

for t = 3:Tfinal

# set update indices for x based on parity of t

index2 = t%2+1

index1 = (t-1)%2+1

# 2nd order AB step

x[index1] = x[index2]+h*(1.5*fstore[2]-0.5*fstore[1]))

# update stored function

fstore[index1] = f(x[index1],t*h)

xout[t]=x[index1]

end

return xout

## 2 thoughts on “Numerical integration”

1. Rick G says:

Very cool! Is there a rule of thumb about what order of Adams-Bashforth you need in order to get good accuracy and stability, based on inspection of the ODE or of the size or path of the deltas from the Euler method?

Like

2. @RickG 2nd order Adams-Bashforth is basically the logical 2nd order version of Euler. Hence, if Euler is accurate at some given time step then AB2 can be run at approximate square root of that step for the same accuracy. It won’t be exact because the constant prefactor will matter. You can estimate that prefactor by linearizing your equations around some point.

Like


text: Since it might not be able to display many digits on its screen, it can be limited in terms of the size of numbers it can add. Which ordered pair below would prevent this table from being a function? So it should point to some other value. an idea ? 1. The following table contains the supported operations and functions: Required only for trigonometric functions. You may recognize this trick from the linear equation y = mx + b. Table 8-1 lists each Calculator function with its keyboard equivalent. This means the function is not returning any value. （ex. 2. If no return type is given, the compiler will normally assume the function returns an int. ; The graph of the linear function is a straight line. What a Four-Function Calculator Does. The function exclud… Otherwise, if displayValue is a non-zero number, the digit is appended to it through string concatenation.. Learn vocabulary, terms, and more with flashcards, games, and other study tools. This is a function. When you enter a function, the calculator will begin by expanding (simplifying) it. Next, the calculator will plot the function over the range that is given. Definition: A function is odd if the equality $$f(x) = -f(-x)$$ is true for all $x$ from the domain of definition. Example. Graphing in the Coordinate Plane. If so, calculator.displayValue is overwritten with whatever digit was clicked. Its simplest form is the linear consumption function used frequently in simple Keynesian models: = + ⋅ where is the autonomous consumption that is independent of disposable income; in other words, consumption when income is zero. You can use function notation as an easy, efficient way of using equations without re-writing them. Start studying Function Notation, Evaluating Function Notation -- GRAPHS, Domain and Range, Function or Not a Function?. Function calculators usually work by allowing users to enter a function to analyze. Thanks to your feedback and relevant comments, dCode has developed the best 'Even or Odd Function' tool, so feel free to write! Graphically, this involves that opposed abscissae have the same ordinates, this means that the ordinate y-axis is an axis of symmetry of the curve representing $f$. The correct way to indicate that a function does not return a value is to use the return type "void". If the calculator did not compute something or you have identified an error, please write it in Another common use of the NOT function in Excel is to reverse the behavior of some other function. Example: Determine whether the function is even or odd: $f(x) = x^2$ (square function) in $\mathbb{R}$, the calculation is $f(-x) = (-x)^2 = x^2 = f(x)$, so the square function $f(x)$ is even. Note that this module is NOT about general plotting of graphs!It discusses functions, their domains, and inverse functions. On the other hand, y 2 + 3x = 6 is not a function, because you can not solve for a unique y: I mean, yes, this is solved for "y =", but it's not unique. Functions []. User defined functions can be further divided into two groups depending on whether a function does or does not return a value. To be a function or not to be a function Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. Regardless, function calculators are designed to shine a spotlight on some function you’ve been considering. Calculator for determining whether a function is an even function and an odd function. (i) Domain of f is A. 0 ⋮ Vote. So this right over here not a function, because it's not clear if you input x what member of the range you're going to get. Developments in convergent power series or polynomials of even (respectively odd) functions have even degrees (respectively odd). 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A function 'CALCULATE' has been used in a True/False expression - Simple Date comparison 03-05-2018 02:00 PM. 3. Optionally, you can also create a custom calculation. How do you figure out if a relation is a function? Limit of a function: limit_calculator. By using this website, you agree to our Cookie Policy. An odd function will provide an opposite image for opposite values. Thank you! This would not be a function. So: If its vertex is pointing downwards, its vertex will mark the lowest point on the y-axis The following *does work* but will get a little more cumbersome when doing MTD or YTD calculations: Adding the IF function, the formula below would display “Text included in dataset” if any cell in the range B2-B13 were not a number. Hi There, I'm looking for a more elegant way sum based on a prior business day calculation. In this book, if a function does not have a need, we will leave its parentheses empty. Ordered pairs are a fundamental part of graphing. Even if your algebra skills are very … What is an Ordered Pair? If you continue browsing the site, you agree to the use of cookies on this website. If it exists, then you have the derivative, or else you know the function is not differentiable. To get tan(x)sec^3(x), use parentheses: tan(x)sec^3(x). This type of function is often referred to as the "void" function. Start studying Function or Not a Function?. For example, the function name corresponding to the square-root key Q is sqrt. If you get an error, double-check your expression, add parentheses and multiplication signs where needed, and consult the table below. =NETWORKDAYS.INTL(start_date, end_date, [weekend], [holidays]) The NETWORKDAYS.INTL function includes the following arguments: 1. Such a function would look like: $$Y = X + 1$$ In this case, X in the input value, and Y is the output (this is a common convention). 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Except explicit open source licence (indicated CC / Creative Commons / free), any algorithm, applet or snippet (converter, solver, encryption / decryption, encoding / decoding, ciphering / deciphering, translator), or any function (convert, solve, decrypt / encrypt, decipher / cipher, decode / encode, translate) written in any informatic language (PHP, Java, C#, Python, Javascript, Matlab, etc.) The result is the output. It transitions row context to filter context. is used to check if the current value displayed on the calculator is zero. Examples. Students will determine if a graph represents a function by using a moving vertical line. The NOT function reverses the value of its argument. Amazon. The OR function returns TRUE if any of its arguments evaluate to TRUE, and returns FALSE if all of its arguments evaluate to FALSE. So this is not a function! It also shows the step-by-step solution, plots of the function and the domain and range. These needs could be as varied as possible. 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For instance, you can combine NOT and ISBLANK functions to create the ISNOTBLANK formula that Microsoft Excel lacks. Watch this video to learn how to tell which relations are functions and which are not. write sin x (or even better sin(x)) instead of sinx. However, you can change the summary function. Weekend (optional argument) –If we omit the weekend parameter, it will assume that the weekends will contain Sunday & Saturday. Example. The second topic of this lab is function parameters. More than just an online tool to explore the continuity of functions. Example 1 : You could set up the relation as a table of ordered pairs. Limit of a function : limit_calculator . The final logical function to discuss in this article is the NOT function, and we have left the simplest for last. The problem is not with CurrMonthYear measure, it's with your second formula - CALCULATE function does not accept measures as criteria, only values. Graphically, this involves that opposed abscissae have opposed ordinates, this means that the origin (central point) (0,0) is a symmetry center of the curve representing $f$. In general, a function is not differentiable for four reasons: Corners, Cusps, Vertical tangents, Jump discontinuities. All suggestions and improvements are welcome. End_date (required argument) – The end date. Or maybe it could point to e or whatever else. Also, be careful when you write fractions: 1/x^2 ln(x) is 1/x^2 ln(x), and 1/(x^2 ln(x)) is 1/(x^2 ln(x)). If x is 1 or TRUE then NOT (x) returns 0 (= FALSE). On the other hand, relation #2 has TWO distinct y values 'a' and 'c' for the same x value of '5' . So this cannot be a function. Learn vocabulary, terms, … Start_date(required argument) – Start_date will either be earlier than end_date or later than end_date or the same as end_date. The first way of calculating the derivative of a function is by simply calculating the limit that is stated above in the definition. (ii) For each x ∈ A, there is only one y ∈ B such that (x, y) ∈ f. Let us look at some examples to understand how to determine whether a relation is a function or not. For example, you can evaluate a function at a certain point: You can use the notation f(x,y), for example, to define a function with more than one variable: Example 1 : Does the following relation represent a function ? And you can see that right here. Function calculator, linear function calculator, input output. By way of example, say you will need to achieve the second story window of your residence, which is 10 feet off the ground. What is an Ordered Pair? Start studying Function or Not a Function?. The funtool app is a visual function calculator that manipulates and displays functions of one variable. If so, you have a function! Can you refactor it, and then call the function to calculate the area with base of 5 and height of 6? The value of the function at the stationary and critical points and the points where the second derivative is zero or does not exist ; 2. If x does not have an exponent, the function is growing at a constant, steady, unchanging rate. Example input. A mapping diagram can be used to represent a relationship between input values and output values. f(x,y) is inputed as "expression". even,odd,function,parity,symmetry,trigonometric,cosine,sine, Source : https://www.dcode.fr/even-odd-function. ( This is a way of explicitly saying that the function returns nothing. ) What is the parity of trigonometric functions (cos, sin, tan)? where: y y y - function value (the function value at single point x, often marked as f(x)),; x x x - function argument (called also independent value),; a a a, b b b - linear function coefficients (slope and free parameter). Same as for when we learned how to compute the domain, there is not one recipe to find the range, it really depends on the structure of the function $$f(x)$$. Qgis-custom functions not showing in field calculator. Online Discontinuity Calculator Find discontinuities of a function with Wolfram|Alpha. SURVEY . A function assigns only output to each input. Every function is a relation, but not every relation is a function! Therefore, relation #2 does not satisfy the definition of a mathematical function. From the table below, you can notice that sech is not supported, but you can still enter it using the identity sech(x)=1/cosh(x). Some function calculators allow users to enter in output information first in order to determine what function(s) would yield this information. One common use for the OR function is to expand the usefulness of other functions that perform logical tests. It can: A dd; Subtract; Multiply ; Divide; D epending on the complexity of the operation, a four-function calculator can be limiting. A mapping diagram represents a function if each input value is paired with only one output value. where: y y y - function value (the function value at single point x, often marked as f(x)),; x x x - function argument (called also independent value),; a a a, b b b - linear function coefficients (slope and free parameter). How to Figure Out When a Function is Not Differentiable. This function is not supported for use in DirectQuery mode when used in calculated columns or row-level security (RLS) rules. Learn vocabulary, terms, and more with flashcards, games, and other study tools. comments below. IF(condition, x, y) returns x if the condition is TRUE (=1), but returns y if the condition is FALSE (=0).This function is also useful for conversion of continuous data into discrete data.. dCode is free and its tools are a valuable help in games, maths, geocaching, puzzles and problems to solve every day!A suggestion ? If you like the website, please share it anonymously with your friend or teacher by entering his/her email: In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. ; Slope of a linear function defines the degree of slope of the line to the OX axis ("horizontal"). Answered: kokeb Dese on 31 Jul 2018 Accepted Answer: Geoff Hayes. Type in any function above then use the table below to input any value to determine the output: How to check if a function neither even nor odd? Sometimes I see expressions like tan^2xsec^3x: this will be parsed as tan^(2*3)(x sec(x)). What a Four-Function Calculator Does. Public Function doubleMe(d As Variant) doubleMe = d * 2 End Function You can also use Application.Volatile , but this has the disadvantage of making your UDF always recalculate - even when it does not need to because the referenced data has not changed. Students will understand the definition of function and use it to identify whether or not an input-output pairing represents a function. It would not be a function if it says, well, it could point to y. Then, test to see if each element in the domain is matched with exactly one element in the range. Calculator for determining whether a function is an even function and an odd function. The value that is put into a function is the input. Since relation #1 has ONLY ONE y value for each x value, this relation is a function. I'll put an exclamation mark. Purpose of use academic assignment Comment/Request huh, i was using this to check an assignment using a taylor series approximation and i think i found a misprint in my textbook's function when i checked it against this and another source's taylor series approximation =IF(ISNUMBER(B2:B13),AVERAGE(B2:B13),"Text included in dataset") Vote. Function . Calculator policy updated july 2018. Students will determine if a table of x- and y-values represents a function. Algebra calculator mathpapa. To compute a square root in a formula, you would use the notation ‘sqrt(x)’. dCode retains ownership of the online 'Even or Odd Function' tool source code. The limit calculator allows the calculation of the limit of a function with the detail and the calculation steps. In trigonometry, the functions are often symmetrical: The cosine function $\cos (x)$ is even. However, a simple function might return the input plus one. In this program, the sqrt() library function is used to calculate the square root of a number. If you need oo, type. The NOT Function. The needs of a function are provided between parentheses. Follow 275 views (last 30 days) Marnie on 11 Apr 2015. At the click of a button, for example, funtool draws a graph representing the sum, product, difference, or ratio of two functions that you specify.funtool includes a function memory that allows you to store functions for later retrieval. So if the logical value is TRUE, then it returns FALSE. The Algebraic Way of Finding the Range of a Function. The Line Test for Mapping Diagrams To check if a relation is a function, given a mapping diagram of the relation, use the following criterion: If each input has only one line connected to it, then the outputs are a function of the inputs. Every function in the Calculator application has a keyboard equivalent associated with it. As you know, the formula =ISBLANK(A2) returns TRUE of if the cell A2 is blank. ; The graph of the linear function is a straight line. For example, 2y + 3x = 6 is a function, because you can solve for y: 2y + 3x = 6 2y = –3x + 6 y = (–3/2)x + 3. Having proved equality for a single value like $f(2) = -f(-2)$ does not allow us to conclude that there is imparity, only to say that 2 and -2 have opposite images by the function $f$. Or it could point to z. 120 seconds . Ordered pairs are a fundamental part of graphing. See the manual page for calctool for a more complete description of each function. Please leave them in comments. By putting any number in for X, we calculate a corresponding output Y by simply adding one. It cannot be a function if for some input into the function you could give me two different values. Even or odd function calculator: is_odd_or_even_function. answer choices . Having proved this equality for a single value like $f(1) = f(-1)$ does not allow to conclude that there is parity, only to say that 1 and -1 have the same image by the function $f$. Most common functions are understood by this graph calculator. The OR function returns TRUE if any of its arguments evaluate to TRUE, and returns FALSE if all of its arguments evaluate to FALSE.. One common use for the OR function is to expand the usefulness of other functions that perform logical tests. An even function will provide an identical image for opposite values. 0. In this lab, we learn how to write user-defined functions that do not return a value. A common solution is to wrap the criteria into FILTER function, something like: x^2*y+x*y^2 ） The reserved functions are located in " Function … ; Slope of a linear function defines the degree of slope of the line to the OX axis ("horizontal"). Creating a function to calculate median. As a function, we take f(x) = x 2. The function NOT (x) reverses the value of its boolean argument x. Watch this video to learn how to tell which relations are functions and which are not. If you skip parentheses or a multiplication sign, type at least a whitespace, i.e. Functions may depend on variables passed to them, called … Wolfram|Alpha is a great tool for finding discontinuities of a function. The calculator will find the domain, range, x-intercepts, y-intercepts, derivative, integral, asymptotes, intervals of increase and decrease, critical points, extrema (minimum and maximum, local, absolute, and global) points, intervals of concavity, inflection points, limit, Taylor polynomial, and graph of the single variable function. x^2*y+x*y^2 ） The reserved functions are located in " Function … Start studying Function VS Not a Function. Details. Free functions calculator - explore function domain, range, intercepts, extreme points and asymptotes step-by-step This website uses cookies to ensure you get the best experience. Instructions: Use this Function Grapher Online to type in the function you want to plot. For example, the IF function performs a logical test and then returns one value if the test evaluates to TRUE and another value if the … Example input. A function is neither odd nor even if neither of the above two equalities are true, that is to say: $$f(x) \neq f(-x)$$ and $$f(x) \neq -f(-x)$$. If a relation is a function, it has to satisfy the following conditions. In some references, instead of leaving the parentheses empty, the programmer would write void. : y is not a function of x (x = 1 has multiple outputs), x is not a function of y (y = 2 has multiple outputs). The graph of a quadratic function is a parabola. The limit calculator allows the calculation of the limit of a function with the detail and the calculation … BYJU’S online function calculator tool makes the calculations faster, and it displays the graph of the function by calculating the x and y-intercept values, slope values in a fraction of seconds. Com: sharp el233sb standard function calculator. Example: Determine whether the function is even or odd: $f(x) = x^2$ (square function) in $\mathbb{R}$, the calculation is $f(-x) = (-x)^2 = x^2 = f(x)$, so the square function $f(x)$ is even. IF function Description. How to Calculate the Derivative of a Function. This function to calculate the area of a rectangle is not very readable. In general, you can skip parentheses, but be very careful: e^3x is e^3x, and e^(3x) is e^(3x). If-then-else function. That's why it works when you put "201704" explicitly but fails when you use the measure. Data in the Values area summarize the underlying source data (not the value that is showing) in the PivotChart report in the following way: numeric values use the SUM function and text values use the COUNT function. Online Discontinuity Calculator Find discontinuities of a function with Wolfram|Alpha. SURVEY . Tags: Question 15 . All intervals between and around the points mentioned in 1 ; 3. NB: if an odd function is defined in 0, then the curve passes at the origin: $f(0) = 0$. Wolfram|Alpha is a great tool for finding discontinuities of a function. It's also possible to return a value from a function. Hi, I am using CALCULATE function to extract sum out of a column if it matches 1 out of 5 criteria. （ex. Free math problem solver answers your algebra, geometry, trigonometry, calculus, and statistics homework questions with step-by-step explanations, just like a math tutor. These are the calculation methods utilized by the calc to locate the derivatives. Check out these examples: (4x)' = 4 (x)' = 1 (-23x)' = -23 Variables. I used an OR() function to see if it matches any of those values. A number multiplied by a variable with no exponent: The derivative of a function of this form is always the number. Every function is a relation, but not every relation is a function! Is this graph a function or not a function? To get tan^2(x)sec^3(x), use parentheses: tan^2(x)sec^3(x). Similarly, tanxsec^3x will be parsed as tan(xsec^3(x)). A function, which can also be referred to as subroutine, procedure, subprogram or even method, carries out tasks defined by a sequence of statements called a statement block that need only be written once and called by a program as many times as needed to carry out the same task.. A History of Inverse Calculator Refuted . IF(C2<0,"NEG","POS") returns the string value "NEG" if the number in cell C2 is less than 0 (C2<0 = TRUE). Functions calculator symbolab. Example: Determine whether the function is even or odd: $f(x) = x^3$ (cube function) in $\mathbb{R}$, the calculation is $-f(-x) = -(-x)^3 = x^3 = f(x)$, so the cube function $f(x)$ is odd. You must use … Keywords: Background Tutorials. Parity will also be determined. Calculator Function Keys. More than just an online tool to explore the continuity of functions. Use the following guidelines to enter functions into the calculator. answer choices (10, 10) (2, 4) (6, 1) (-2, 0) Tags: Question 16 . The above mentioned features aren't supported. Functions which do not have operator symbols like ‘+’ and ‘*’ must be entered in formulas using function-call notation. Q. Yes, the function $f(x) = 0$ (constant zero function) is both even and odd because it respects the 2 equalities $f(x) = f(-x) = 0$ and $f(x) = -f(-x) = 0$. If x is 0 or FALSE then NOT (x) returns 1 (= TRUE). a set of mathematical operations performed on one or more inputs (variables) that results in an output To calculate the image of the second degree functions, you could also calculate it by obtaining the mastery of its inverse function, but we will calculate it with another method. Not a Function. This value is later used to calculate the ratio of Internet sales compared to all sales for the year 2006. Watch this tutorial to see how you can determine if a relation is a function. Whether the function is increasing/decreasing between the stationary and critical points 4. Definition: A function is even if the equality $$f(x) = f(-x)$$ is true for all $x$ from the domain of definition.. An even function will provide an identical image for opposite values.. For example, the ISNUMBER formula will calculate to TRUE if all the cells in the dataset are numbers, otherwise it will calculate to FALSE. Keywords: Background Tutorials. a bug ? 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# Standard Error

The statistics is a wide subject in which we study about terms like data, sample and population, dependent and independent variables, mean, median, mode, standard deviation, variance and many many more. For standard deviation, one more term is used - called standard error

The standard error actually refers to the standard deviation of a statistic. The sample mean of a data is usually deviated from actual population mean. The amount of this deviation is termed as the standard error.
Standard error is used to measure the amount of accuracy by which the given sample represents its population.

For the sample to be more representative over the given population, the standard error must be smaller. Also, the standard error and sample size are inversely proportional to each other. It means that the smaller the sample size, the bigger the standard error and the bigger the sample size, the smaller the standard error.

Let us go ahead in this article and learn details about standard error as well as standard error of the mean, methods of their computations with solved examples.

 Related Calculators calculate standard error calculate percent error Error Function Calculator Margin of Error Calculator

## Standard Error of the Mean

The term "standard erroris also utilized for the estimate of the standard deviation which is derived from a given sampleStandard error of the mean is defined as the standard deviation of the estimate of sample mean of the mean of population.

For Instance: Usually, the estimator of population mean is sample mean is the usual estimator of a population mean. But if we draw another sample from the same population, it may provide different value.

Thus, there would be a
distribution of the sampled means having its different variance and mean. Therefore, standard error of the mean (abbreviated as SEM) may be defined as the standard deviation of such sample means of all the possible samples taken from the same given population. The standard error of mean defines an estimate of standard deviation which has been computed from the sample.

Standard error of mean is calculated by dividing sample standard deviation by the square root of sample size; i.e. the formula for SEM is given by -
$SEM$ = $\frac{s}{\sqrt{n}}$
In which,
SEM stands for standard error of the mean,
s represents sample standard deviation

n denotes the sample size.

This value estimated by above formula is sometimes compared with true standard deviation of sample mean given by the following formula -
$SE$ = $\frac{\sigma}{\sqrt{n}}$
Where, $\sigma$ is population standard deviation.

## Standard Error of the Estimate

There is one more concept related to standard error is "standard error of the estimate". This term refers to deviation of some estimate from intended values.
The formula for the calculation of standard error of the estimate of a sample is written below:
$SEE$ = $\sqrt{\frac{\sum (x_{i} - \bar{x})^{2}}{n-2}}$
In this formula, SEE stands for the standard error of the estimate, $x_{i}$ denotes data values, $\bar{x}$ is the mean where Y refers to individual data sets, Y' is mean of the given sample and N is the sample size.
The students can remember this formula very easily as it similar to the formula of standard deviation formula with a difference of n - 2 for n -1 in the formula for sample standard deviation.

## Difference Between SE and SEM

We usually get confused between the standard error and the standard error of mean (SEM). The differences between them are given below:

i) The standard error quantifies the scatter of the data. It is means how far the data values disperse from one another. While standard error of mean quantifies precisely the true value of mean of given population.

ii) Standard error is standard deviation. On the other hand, SEM takes standard deviation and sample size both in consideration.

iii)
The value of standard error of the mean is smaller than of standard error.

iv) As the sample gets bigger, the value of SEM gets smaller. Thus, SEM is inversely proportional to size of the sample. But sample size does not impact of the standard error. It is not changed
predictably as more data is available. It can be be predicted even if sample size is big or small.

## How to Calculate Standard Error of the Mean

The student are required to follow the steps mentioned below for the calculation of standard error of the mean:

Let us consider the example of an exam in which the scores of 5 students are given as 12, 74, 55, 79, 90.

Step 1:
Find the mean of given data by dividing the sum of samples by total number of samples.

Mean of 12, 74, 55, 79, 90

$\bar{x}$ = $\frac{12 + 74 + 55 + 79 + 90}{5}$ = 62

Step 2: Construct a table for standard deviation of the given data. This table would be 3 columns - first for given data value, second for the deviation of each data from the mean and third one for square of these deviations, as shown in the example below.

 $x_{i}$ $x_{i}-\bar{x}$ $(x_{i}-\bar{x})^{2}$ 12 -50 2500 74 12 144 55 -7 49 79 17 289 90 28 784 $\sum (x_{i}-\bar{x})^{2}$ = 3766

Step 3: By using the formula:

$\sigma$ = $\sqrt{\frac{\sum (x_{i} - \bar{x})^{2}}{n}}$

calculate the standard deviation.

$\sigma$ = $\sqrt{\frac{3766}{5}}$ = 27.44

Step 4: Now, the standard deviation is to be divided by the square root of sample size.
That gives you the

“standard error”.

Step 5: Subtract the standard error from the mean and record that number. Then add the standard

error to the mean and record that number. You have plotted mean± 1 standard error (S. E.),

the distance from 1 standard error below the mean to 1 standard error above the mean

$SEM$ = $\frac{\sigma}{\sqrt{n}}$

$SEM$ = $\frac{27.44}{\sqrt{5}}$

= $\frac{27.44}{2.24}$

= 12.25
 Related Topics Math Help Online Online Math Tutor
*AP and SAT are registered trademarks of the College Board.


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Monitoring of the adaptive capacity of different age spirulina to oxidative stress induced by hypothermia in the presence of chemical stimulators under laboratory conditions
 Articolul precedent Articolul urmator 123 1 Ultima descărcare din IBN: 2019-05-15 17:30 SM ISO690:2012CHIRIAC, Tatiana; DJUR (MAXACOVA), Svetlana; CODREANU, Svetlana; DUMBRĂVEANU, Veronica. Monitoring of the adaptive capacity of different age spirulina to oxidative stress induced by hypothermia in the presence of chemical stimulators under laboratory conditions. In: Microbial Biotechnology. Ediția 4, 11-12 octombrie 2018, Chișinău. Chișinău, Republica Moldova: Institutul de Microbiologie şi Biotehnologie, 2018, pp. 89-90. ISBN 978-9975-3178-8-7. EXPORT metadate: Google Scholar Crossref CERIF BibTeXDataCiteDublin Core
Microbial Biotechnology
Ediția 4, 2018
Conferința "Microbial Biotechnology"
Chișinău, Moldova, 11-12 octombrie 2018

 Monitoring of the adaptive capacity of different age spirulina to oxidative stress induced by hypothermia in the presence of chemical stimulators under laboratory conditions

Pag. 89-90

 Chiriac Tatiana, Djur (Maxacova) Svetlana, Codreanu Svetlana, Dumbrăveanu Veronica Institutul de Microbiologie şi Biotehnologie Disponibil în IBN: 19 februarie 2019

Teza

In order to determine the impact of the inclusion of coordination compounds in biosynthetic activity of cyanobacterium Spirulina platensis, it was studied the change of protein content into biomass in the presence of zinc acetate under short-duration hypothermia. Hypothermic stress was induced by continued cultivation of spirulina at 40C over a period of 1, 2 and 3 hours. Spirulina has been subjected to hypothermic stress at the beginning of lag phase or one day culture, at the beginning of exponential phase or three days culture and at the end of exponential phase or six days culture. Spirulina was cultivated under optimum conditions in the presence of zinc acetate concentration of 15 mg/l. The compound has been added to mineral medium from the first day of cultivation. As a control sample, spirulina culture of the appropriate age was grown under optimal conditions (continuous illumination regime, temperature 28-320C) in the absence of chemical stimulator and in the presence of chemical stimulators. Hypothermic conditions were induced for spirulina cultivated in the absence and presence of zinc acetate. As a result, it was determined that in spirulina grown in the presence of zinc acetate on the first day of cultivation, hypothermia did not alter the protein content in the first two hours of heat stress. The reduction in the protein content has been insignificant after three hours of hypothermia. In control, reducing protein content in biomass with 16% was established after the first hour and three hours of heat stress. Spirulina culture of 3 days, in the absence of chemical stimulator, reacted to hypothermia by reducing the protein content with 28% after two hours of hypothermic stress without subsequent reduction. In the culture of spirulina grown in the presence of zinc acetate hypothermia also has altered protein content after two hours of heat stress, its value being with 24% lower compared to control sample. After three hours of hypothermia, protein content in biomass was reduced by 30% relative to control. Spirulina culture, control sample, at the end of exponential phase (day 6 of cultivation), reacting to hypothermia by reducing the protein content with 15% after the first hour of hypothermic stress and 13% - after a two-hours exposure to low temperatures. Protein content in biomass decreased with 21% after three hours of hypothermia. In spirulina grown in the presence of zinc acetate, at the age of six days, hypothermia significantly modified the protein content. Thus, after the first hour the protein content was reduced by 64%, after two hours of heat stress –with 55%. After three hours of hypothermia, reduction of protein content in biomass was lower by 44% relative to control. Therefore, the most resistant to hypothermia in terms of protein synthesis was culture at the beginning of lag phase and exponential phase. The most vulnerable age for spirulina culture was the end of exponential phase. The most significant reductions in the levels of protein were determined after the first and the third hour of hypothermia. For the experimental variant of spirulina cultivation in the presence of zinc acetate, it was established that hypothermia induced over a period of three hours did not altered the protein content in spirulina biomass at the beginning of lag phase. Spirulina, at the beginning of exponential phase, lost proteins after two hours of hypothermia. The most vulnerable has proved to be spirulina in stationary phase, which lost more than half of the protein content. Zinc acetate from cultivation medium prevented protein loss in spirulina biomass, being in adaptation period and subject to hypothermic stress. In the case of spirulina at the beginning of stationary phase, involvement of zinc acetate in biosynthetic processes has weakened the culture of spirulina which, under hypothermic conditions, lost more than half of protein content.

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