Physical geodesy

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Plot of the error function

In mathematics, the error function (also called the Gauss error function) is a special function (non-elementary) of sigmoid shape which occurs in probability, statistics and partial differential equations. It is defined as:

$\mathrm{erf}(x)=\frac{2}{\sqrt{\pi }}\underset{0}{\overset{x}{\int }}{e}^{-t^2}\mathrm{d}t.$

The complementary error function, denoted erfc, is defined as

$\begin{array}{rl}\mathrm{erfc}(x)& =1-\mathrm{erf}(x)\\ & =\frac{2}{\sqrt{\pi }}\underset{x}{\overset{\mathrm{\infty }}{\int }}{e}^{-t^2}\mathrm{d}t=e^{-x^2}\mathrm{erfcx}(x),\end{array}$

which also defines erfcx, the scaled complementary error function (which can be used instead of erfc to avoid arithmetic underflow).

The imaginary error function, denoted erfi, is defined as

$\mathrm{erfi}(z)=-i\mathrm{erf}(iz)=\frac{2}{\sqrt{\pi }}{e}^{x^2}D(x)$,

where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow).

When the error function is evaluated for arbitrary complex arguments z, the resulting complex error function is usually discussed in scaled form as the Faddeeva function:

$w(z)=e^{-z^2}\mathrm{erfc}(-iz)=\mathrm{erfcx}(-iz).$

The name "error function"

The error function is used in measurement theory (using probability and statistics), and although its use in other branches of mathematics has nothing to do with the characterization of measurement errors, the name has stuck.

The error function is related to the cumulative distribution $\mathrm{\Phi }$, the integral of the standard normal distribution, by

$\mathrm{\Phi }(x)=\frac{1}{2}+\frac{1}{2}\mathrm{erf}(x/\sqrt{2}).$

The error function, evaluated at $\frac{x}{\sigma \sqrt{2}}$ for positive x values, gives the probability that a measurement, under the influence of normally distributed errors with standard deviation $\sigma$, has a distance less than x from the mean value. This function is used in statistics to predict behavior of any sample with respect to the population mean. This usage is similar to the Q-function, which in fact can be written in terms of the error function.

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The property $\mathrm{erf}(-z)=-\mathrm{erf}(z)$ means that the error function is an odd function.

For any complex number z:

$\mathrm{erf}(\bar{z})=\overline{\mathrm{erf}(z)}$

where $\bar{z}$ is the complex conjugate of z.

The integrand ƒ = exp(−z²) and ƒ = erf(z) are shown in the complex z-plane in figures 2 and 3. Level of Im(ƒ) = 0 is shown with a thick green line. Negative integer values of Im(ƒ) are shown with thick red lines. Positive integer values of Im(f) are shown with thick blue lines. Intermediate levels of Im(ƒ) = constant are shown with thin green lines. Intermediate levels of Re(ƒ) = constant are shown with thin red lines for negative values and with thin blue lines for positive values.

At the real axis, erf(z) approaches unity at z → +∞ and −1 at z → −∞. At the imaginary axis, it tends to ±i∞.

Taylor series

The error function is an entire function; it has no singularities (except that at infinity) and its Taylor expansion always converges.

The defining integral cannot be evaluated in closed form in terms of elementary functions, but by expanding the integrand e^−z2 into its Taylor series and integrating term by term, one obtains the error function's Taylor series as:

$\mathrm{erf}(z)=\frac{2}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n{z}^{2n+1}}{n!(2n+1)}=\frac{2}{\sqrt{\pi }}(z-\frac{z^3}{3}+\frac{z^5}{10}-\frac{z^7}{42}+\frac{z^9}{216}-\cdots )$

which holds for every complex number z. The denominator terms are sequence A007680 in the OEIS.

For iterative calculation of the above series, the following alternative formulation may be useful:

$\mathrm{erf}(z)=\frac{2}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}\left(z\prod _{k=1}^n\frac{-(2k-1)z^2}{k(2k+1)}\right)=\frac{2}{\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}\frac{z}{2n+1}\prod _{k=1}^n\frac{-z^2}{k}$

because $\frac{-(2k-1)z^2}{k(2k+1)}$ expresses the multiplier to turn the k^th term into the (k + 1)^th term (considering z as the first term).

The error function at +∞ is exactly 1 (see Gaussian integral).

The derivative of the error function follows immediately from its definition:

$\frac{\mathrm{d}}{\mathrm{d}z}\mathrm{e}\mathrm{r}\mathrm{f}(z)=\frac{2}{\sqrt{\pi }}{e}^{-z^2}.$

An antiderivative of the error function is

$z\mathrm{erf}(z)+\frac{e^{-z^2}}{\sqrt{\pi }}.$

Inverse functions

The inverse error function can be defined in terms of the Maclaurin series

${\mathrm{erf}}^{-1}(z)=\sum _{k=0}^{\mathrm{\infty }}\frac{c_k}{2k+1}{\left(\frac{\sqrt{\pi }}{2}z\right)}^{2k+1},$

where c₀ = 1 and

$c_k=\sum _{m=0}^{k-1}\frac{c_m{c}_{k-1-m}}{(m+1)(2m+1)}=\{1,1,\frac{7}{6},\frac{127}{90},\frac{4369}{2520},\dots \}.$

So we have the series expansion (note that common factors have been canceled from numerators and denominators):

${\mathrm{erf}}^{-1}(z)=\frac{1}{2}\sqrt{\pi }(z+\frac{\pi }{12}{z}^3+\frac{7{\pi }^2}{480}{z}^5+\frac{127{\pi }^3}{40320}{z}^7+\frac{4369{\pi }^4}{5806080}{z}^9+\frac{34807{\pi }^5}{182476800}{z}^{11}+\cdots ).$

(After cancellation the numerator/denominator fractions are entries A092676/A132467 in the OEIS; without cancellation the numerator terms are given in entry A002067.) Note that the error function's value at ±∞ is equal to ±1.

The inverse complementary error function is defined as

${\mathrm{erfc}}^{-1}(1-z)={\mathrm{erf}}^{-1}(z).$

Asymptotic expansion

A useful asymptotic expansion of the complementary error function (and therefore also of the error function) for large real x is

$\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(x)=\frac{e^{-x^2}}{x\sqrt{\pi }}[1+\sum _{n=1}^{\mathrm{\infty }}(-1{)}^n\frac{1\cdot 3\cdot 5\cdots (2n-1)}{(2x^2{)}^n}]=\frac{e^{-x^2}}{x\sqrt{\pi }}\sum _{n=0}^{\mathrm{\infty }}(-1{)}^n\frac{(2n-1)!!}{(2x^2{)}^n},$

where (2n – 1)!! is the double factorial: the product of all odd numbers up to (2n – 1). This series diverges for every finite x, and its meaning as asymptotic expansion is that, for any $N\in \mathbb{\mathbb{N}}$ one has

$\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(x)=\frac{e^{-x^2}}{x\sqrt{\pi }}\sum _{n=0}^{N-1}(-1{)}^n\frac{(2n-1)!!}{(2x^2{)}^n}+R_N(x)$

where the remainder, in Landau notation, is

$R_N(x)=O(x^{-2N+1}{e}^{-x^2})$ as $x\to \mathrm{\infty }$.

Indeed, the exact value of the remainder is

$R_N(x):=\frac{(-1{)}^N}{\sqrt{\pi }}{2}^{-2N+1}\frac{(2N)!}{N!}\underset{x}{\overset{\mathrm{\infty }}{\int }}{t}^{-2N}{e}^{-t^2}\mathrm{d}t,$

which follows easily by induction, writing $e^{-t^2}=-(2t{)}^{-1}(e^{-t^2}{)}^{\prime }$ and integrating by parts.

For large enough values of x, only the first few terms of this asymptotic expansion are needed to obtain a good approximation of erfc(x) (while for not too large values of x note that the above Taylor expansion at 0 provides a very fast convergence).

Continued fraction expansion

A continued fraction expansion of the complementary error function is:

$\mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(z)=\frac{z}{\sqrt{\pi }}{e}^{-z^2}\frac{a_1}{z^2+\frac{a_2}{1+\frac{a_3}{z^2+\frac{a_4}{1+\cdots }}}}{a}_1=1,a_m=\frac{m-1}{2},m\ge 2.$

Approximation with elementary functions

Abramowitz and Stegun give several approximations of varying accuracy (equations 7.1.25–28). This allows one to choose the fastest approximation suitable for a given application. In order of increasing accuracy, they are:

$\mathrm{erf}(x)\approx 1-\frac{1}{(1+a_{1}x+a_2{x}^2+a_3{x}^3+a_4{x}^4{)}^4}$    (maximum error: 5·10⁻⁴)

where a₁ = 0.278393, a₂ = 0.230389, a₃ = 0.000972, a₄ = 0.078108

$\mathrm{erf}(x)\approx 1-(a_{1}t+a_2{t}^2+a_3{t}^3)e^{-x^2},t=\frac{1}{1+px}$    (maximum error: 2.5·10⁻⁵)

where p = 0.47047, a₁ = 0.3480242, a₂ = −0.0958798, a₃ = 0.7478556

$\mathrm{erf}(x)\approx 1-\frac{1}{(1+a_{1}x+a_2{x}^2+\cdots +a_6{x}^6{)}^{16}}$    (maximum error: 3·10⁻⁷)

where a₁ = 0.0705230784, a₂ = 0.0422820123, a₃ = 0.0092705272, a₄ = 0.0001520143, a₅ = 0.0002765672, a₆ = 0.0000430638

$\mathrm{erf}(x)\approx 1-(a_{1}t+a_2{t}^2+\cdots +a_5{t}^5)e^{-x^2},t=\frac{1}{1+px}$    (maximum error: 1.5·10⁻⁷)

where p = 0.3275911, a₁ = 0.254829592, a₂ = −0.284496736, a₃ = 1.421413741, a₄ = −1.453152027, a₅ = 1.061405429

All of these approximations are valid for x ≥ 0. To use these approximations for negative x, use the fact that erf(x) is an odd function, so erf(x) = −erf(−x).

Another approximation is given by

$\mathrm{erf}(x)\approx \mathrm{sgn}(x)\sqrt{1-\exp (-x^2\frac{4/\pi +a{x}^2}{1+a{x}^2})}$

where

$a=\frac{8(\pi -3)}{3\pi (4-\pi )}\approx 0.140012.$

This is designed to be very accurate in a neighborhood of 0 and a neighborhood of infinity, and the error is less than 0.00035 for all x. Using the alternate value a ≈ 0.147 reduces the maximum error to about 0.00012.

This approximation can also be inverted to calculate the inverse error function:

${\mathrm{erf}}^{-1}(x)\approx \mathrm{sgn}(x)\sqrt{\sqrt{{(\frac{2}{\pi a}+\frac{\ln (1-x^2)}{2})}^2-\frac{\ln (1-x^2)}{a}}-(\frac{2}{\pi a}+\frac{\ln (1-x^2)}{2})}.$

A pure exponential approximation for the complementary error function is given by Template:Harvtxt

$\mathrm{erfc}(x)\approx \frac{1}{6}{e}^{-x^2}+\frac{1}{2}{e}^{-\frac{4}{3}{x}^2}x>0.$

Numerical approximation

Over the complete range of values, there is an approximation with a maximal error of $1.2\cdot 10^{-7}$, as follows:

$\mathrm{erf}(x)=\{\begin{array}{ll}1-\tau & \mathrm{f}\mathrm{o}\mathrm{r}x\ge 0\\ \tau -1& \mathrm{f}\mathrm{o}\mathrm{r}x<0\end{array}$

with

$\begin{array}{rcl}\tau & =& t\cdot \exp (-x^2-1.26551223+1.00002368\cdot t+0.37409196\cdot t^2+0.09678418\cdot t^3\\ & & -0.18628806\cdot t^4+0.27886807\cdot t^5-1.13520398\cdot t^6+1.48851587\cdot t^7\\ & & -0.82215223\cdot t^8+0.17087277\cdot t^9)\end{array}$

and

$t=\frac{1}{1+0.5\left|x\right|}$

Applications

When the results of a series of measurements are described by a normal distribution with standard deviation $\sigma$ and expected value 0, then $\mathrm{erf}\left(\frac{a}{\sigma \sqrt{2}}\right)$ is the probability that the error of a single measurement lies between −a and +a, for positive a. This is useful, for example, in determining the bit error rate of a digital communication system.

The error and complementary error functions occur, for example, in solutions of the heat equation when boundary conditions are given by the Heaviside step function.

Related functions

The error function is essentially identical to the standard normal cumulative distribution function, denoted Φ, also named norm(x) by software languages, as they differ only by scaling and translation. Indeed,

$\mathrm{\Phi }(x)=\frac{1}{\sqrt{2\pi }}\underset{-\mathrm{\infty }}{\overset{x}{\int }}{e}^{\frac{-t^2}{2}}\mathrm{d}t=\frac{1}{2}[1+\mathrm{erf}\left(\frac{x}{\sqrt{2}}\right)]=\frac{1}{2}\mathrm{erfc}(-\frac{x}{\sqrt{2}})$

or rearranged for erf and erfc:

$\begin{array}{rl}\mathrm{e}\mathrm{r}\mathrm{f}(x)& =2\mathrm{\Phi }\left(x\sqrt{2}\right)-1\\ \mathrm{e}\mathrm{r}\mathrm{f}\mathrm{c}(x)& =2\mathrm{\Phi }(-x\sqrt{2})=2(1-\mathrm{\Phi }\left(x\sqrt{2}\right)).\end{array}$

Consequently, the error function is also closely related to the Q-function, which is the tail probability of the standard normal distribution. The Q-function can be expressed in terms of the error function as

$Q(x)=\frac{1}{2}-\frac{1}{2}\mathrm{erf}\left(\frac{x}{\sqrt{2}}\right)=\frac{1}{2}\mathrm{erfc}\left(\frac{x}{\sqrt{2}}\right).$

The inverse of $\mathrm{\Phi }$ is known as the normal quantile function, or probit function and may be expressed in terms of the inverse error function as

$\mathrm{probit}(p)={\mathrm{\Phi }}^{-1}(p)=\sqrt{2}{\mathrm{erf}}^{}(2p-1)=-\sqrt{2}{\mathrm{erfc}}^{}(2p).$

The standard normal cdf is used more often in probability and statistics, and the error function is used more often in other branches of mathematics.

The error function is a special case of the Mittag-Leffler function, and can also be expressed as a confluent hypergeometric function (Kummer's function):

$\mathrm{e}\mathrm{r}\mathrm{f}(x)=\frac{2x}{\sqrt{\pi }}{}_1{F}_1(\frac{1}{2},\frac{3}{2},-x^2).$

It has a simple expression in terms of the Fresnel integral.Template:Elucidate

In terms of the Regularized Gamma function P and the incomplete gamma function,

$\mathrm{erf}(x)=\mathrm{sgn}(x)P(\frac{1}{2},x^2)=\frac{\mathrm{sgn}(x)}{\sqrt{\pi }}\gamma (\frac{1}{2},x^2).$

$\mathrm{sgn}(x)$ is the sign function.

Generalized error functions

Graph of generalised error functions E_n(x):
grey curve: E₁(x) = (1 − e ^−x)/$\sqrt{\pi }$
red curve: E₂(x) = erf(x)
green curve: E₃(x)
blue curve: E₄(x)
gold curve: E₅(x).

Some authors discuss the more general functions:Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

$E_n(x)=\frac{n!}{\sqrt{\pi }}\underset{0}{\overset{x}{\int }}{e}^{-t^n}\mathrm{d}t=\frac{n!}{\sqrt{\pi }}\sum _{p=0}^{\mathrm{\infty }}(-1{)}^p\frac{x^{np+1}}{(np+1)p!}.$

Notable cases are:

• E₀(x) is a straight line through the origin: $E_0(x)=\frac{x}{e\sqrt{\pi }}$
• E₂(x) is the error function, erf(x).

After division by n!, all the E_n for odd n look similar (but not identical) to each other. Similarly, the E_n for even n look similar (but not identical) to each other after a simple division by n!. All generalised error functions for n > 0 look similar on the positive x side of the graph.

These generalised functions can equivalently be expressed for x > 0 using the Gamma function and incomplete Gamma function:

$E_n(x)=\frac{\mathrm{\Gamma }(n)(\mathrm{\Gamma }\left(\frac{1}{n}\right)-\mathrm{\Gamma }(\frac{1}{n},x^n))}{\sqrt{\pi }},x>0.$

Therefore, we can define the error function in terms of the incomplete Gamma function:

$\mathrm{erf}(x)=1-\frac{\mathrm{\Gamma }(\frac{1}{2},x^2)}{\sqrt{\pi }}.$

Iterated integrals of the complementary error function

The iterated integrals of the complementary error function are defined by

${\mathrm{i}}^n\mathrm{erfc}(z)=\underset{z}{\overset{\mathrm{\infty }}{\int }}{\mathrm{i}}^{n-1}\mathrm{erfc}(\zeta )\mathrm{d}\zeta .$

They have the power series

${\mathrm{i}}^n\mathrm{erfc}(z)=\sum _{j=0}^{\mathrm{\infty }}\frac{(-z{)}^j}{2^{n-j}j!\mathrm{\Gamma }(1+\frac{n-j}{2})},$

from which follow the symmetry properties

${\mathrm{i}}^{2m}\mathrm{erfc}(-z)=-{\mathrm{i}}^{2m}\mathrm{erfc}(z)+\sum _{q=0}^m\frac{z^{2q}}{2^{2(m-q)-1}(2q)!(m-q)!}$

and

${\mathrm{i}}^{2m+1}\mathrm{erfc}(-z)={\mathrm{i}}^{2m+1}\mathrm{erfc}(z)+\sum _{q=0}^m\frac{z^{2q+1}}{2^{2(m-q)-1}(2q+1)!(m-q)!}.$

Implementations

• C: C99 provides the functions double erf(double x) and double erfc(double x) in the header math.h. The pairs of functions {erff(),erfcf()} and {erfl(),erfcl()} take and return values of type float and long double respectively. For complex double arguments, the function names cerf and cerfc are "reserved for future use"; the missing implementation is provided by the open-source project libcerf, which is based on the Faddeeva package.

• C++: C++11 provides erf() and erfc() in the header cmath. Both functions are overloaded to accept arguments of type float, double, and long double. For complex<double>, the Faddeeva package provides a C++ complex<double> implementation.

• Fortran: The Fortran 2008 standard provides the ERF, ERFC and ERFC_SCALED functions to calculate the error function and its complement for real arguments. Fortran 77 implementations are available in SLATEC.

• Google search: Google's search also acts as a calculator and will evaluate "erf(...)" and "erfc(...)" for real arguments.

• Haskell: An erf package exists that provides a typeclass for the error function and implementations for the native (real) floating point types.

• IDL: provides both erf and erfc for real and complex arguments.

• Java: Apache commons-math provides implementations of erf and erfc for real arguments.

• Maple: Maple implements both erf and erfc for real and complex arguments.

• MathCAD provides both erf(x) and erfc(x) for real arguments.

• Mathematica: erf is implemented as Erf and Erfc in Mathematica for real and complex arguments, which are also available in Wolfram Alpha.

• Matlab provides both erf and erfc for real arguments, also via W. J. Cody's algorithm.

• Maxima provides both erf and erfc for real and complex arguments.

• Perl: erf (for real arguments, using Cody's algorithm) is implemented in the Perl module Math::SpecFun

• Python: Included since version 2.7 as math.erf() for real arguments. For previous versions or for complex arguments, SciPy includes implementations of erf, erfc, erfi, and related functions for complex arguments in scipy.special. A complex-argument erf is also in the arbitrary-precision arithmetic mpmath library as mpmath.erf()

• R: "The so-called 'error function'" is not provided directly, but is detailed as an example of the normal cumulative distribution function (?pnorm), which is based on W. J. Cody's rational Chebyshev approximation algorithm.

• Ruby: Provides Math.erf() and Math.erfc() for real arguments.

See also

Related functions

• Gaussian integral, over the whole real line
• Gaussian function, derivative
• Dawson function, renormalized imaginary error function

In probability

• Normal distribution
• Normal cumulative distribution function, a scaled and shifted form of error function
• Probit, the inverse or quantile function of the normal CDF
• Q-function, the tail probability of the normal distribution

References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010.

External links

• MathWorld – Erf
• Error-function numerical table and calculator