Gaussian Error Function Matlab
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X = erfinv(Y) Inverse of the error function Definition The error function erf(X) is twice the integral of complementary error function the Gaussian distribution with 0 mean and variance of:
The erfc matlab complementary error function erfc(X) is defined as: The scaled complementary error function erfcx(X) is defined erf function calculator as: For large X, erfcx(X) is approximately . Description Y = erf(X) returns the value of the error function for each element of real array error function table X. Y = erfc(X) computes the value of the complementary error function. Y = erfcx(X) computes the value of the scaled complementary error function. X = erfinv(Y) returns the value of the inverse error function for each element of Y. The elements of Y must fall within the domain Examples erfinv(1)Erf(2/sqrt(2))
is Inf
erfinv(-1) is -Inf. For abs(Y) > 1, erfinv(Y) is NaN. Remarks The relationship between the error function and the standard normal probability distribution is: x = -5:0.1:5; standard_normal_cdf = (1 + (erf(x/sqrt(2))))./2; Algorithms For the error functions, the MATLAB code is a translation of a Fortran program by W. J. Cody, Argonne National Laboratory, NETLIB/SPECFUN, March 19, 1990. The main computation evaluates near-minimax rational approximations from [1]. For the inverse of the error function, rational approximations accurate to approximately six significant digits are used to generate an initial approximation, which is then improved to full accuracy by two steps of Newton's method. The M-file is easily modified to eliminate the Newton improvement. The resulting code is about three times faster in execution, but is considerably less accurate. References [1] Cody, W. J., "Rational Chebyshev Approximations for the Error Function," Math. Comp., pgs. 631-638, 1969 [ Previous | Help Desk | Next ]Search All Support Resources Support Documentation MathWorks Search MathWorks.com MathWorks Documentation Support Documentation Toggle navigation Trial Software Product Updates Documentation Home Symbolic Math Toolbox Examples Functions and Other Reference
Inverse Error Function
Release Notes PDF Documentation Mathematics Mathematical Functions Symbolic Math Toolbox Functions erf On erf(1) this page Syntax Description Examples Error Function for Floating-Point and Symbolic Numbers Error Function for Variables and Expressions Error erf function excel Function for Vectors and Matrices Special Values of Error Function Handling Expressions That Contain Error Function Plot Error Function Input Arguments X More About Error Function Tips Algorithms References See Also http://cens.ioc.ee/local/man/matlab/techdoc/ref/erf.html This is machine translation Translated by Mouse over text to see original. Click the button below to return to the English verison of the page. Back to English × Translate This Page Select Language Bulgarian Catalan Chinese Simplified Chinese Traditional Czech Danish Dutch English Estonian Finnish French German Greek Haitian Creole Hindi Hmong Daw Hungarian Indonesian Italian Japanese Korean Latvian Lithuanian Malay Maltese Norwegian https://www.mathworks.com/help/symbolic/erf.html Polish Portuguese Romanian Russian Slovak Slovenian Spanish Swedish Thai Turkish Ukrainian Vietnamese Welsh MathWorks Machine Translation The automated translation of this page is provided by a general purpose third party translator tool. MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation. Translate erfError functioncollapse all in page Syntaxerf(X) exampleDescriptionexampleerf(X
) represents the error function of X. If X is a vector or a matrix, erf(X) computes the error function of each element of X.ExamplesError Function for Floating-Point and Symbolic Numbers Depending on its arguments, erf can return floating-point or exact symbolic results. Compute the error function for these numbers. Because these numbers are not symbolic objects, you get the floating-point results:A = [erf(1/2), erf(1.41), erf(sqrt(2))]A = 0.5205 0.9539 0.9545Compute the error function for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, erf returns unresolved symbolic calls:symA = [erf(sym(1/2)), erf(sym(1.41)), erf(sqrt(sym(2)))]symA = [ erf(1/2), erf(141/100), erf(2^(1/2))]Use vpa to approximate symbolic results with the required number of digits:d = digits(10); vpa(symA) digits(d)ans = [ 0.5204998778, 0.9538524394, 0.9544997361]Error Function for Variables and Expressions For most symbol
Search All Support Resources Support Documentation MathWorks Search MathWorks.com MathWorks Documentation Support Documentation Toggle navigation Trial Software Product Updates Documentation https://www.mathworks.com/help/matlab/ref/erfc.html Home MATLAB Examples Functions Release Notes PDF Documentation Mathematics Elementary Math Special Functions MATLAB Functions erfc On this page Syntax Description Examples Find Complementary Error Function Find https://en.wikipedia.org/wiki/Error_function Bit Error Rate of Binary Phase-Shift Keying Avoid Roundoff Errors Using Complementary Error Function Input Arguments x More About Complementary Error Function Tall Array Support Tips See error function Also This is machine translation Translated by Mouse over text to see original. Click the button below to return to the English verison of the page. Back to English × Translate This Page Select Language Bulgarian Catalan Chinese Simplified Chinese Traditional Czech Danish Dutch English Estonian Finnish French German Greek Haitian Creole Hindi Hmong gaussian error function Daw Hungarian Indonesian Italian Japanese Korean Latvian Lithuanian Malay Maltese Norwegian Polish Portuguese Romanian Russian Slovak Slovenian Spanish Swedish Thai Turkish Ukrainian Vietnamese Welsh MathWorks Machine Translation The automated translation of this page is provided by a general purpose third party translator tool. MathWorks does not warrant, and disclaims all liability for, the accuracy, suitability, or fitness for purpose of the translation. Translate erfcComplementary error functioncollapse all in page Syntaxerfc(x) exampleDescriptionexampleerfc(x
) returns the Complementary Error Function evaluated for each element of x. Use the erfc function to replace 1 - erf(x) for greater accuracy when erf(x) is close to 1.Examplescollapse allFind Complementary Error FunctionOpen ScriptFind the complementary error function of a value.erfc(0.35) ans = 0.6206 Find the complementary error function of the elements of a vector.V = [-0.5 0 1 0.72]; erfc(V) ans = 1.5205 1.0000 0.1573 0.3086 Find the complementary error function of the elements of a matrix.M = [0.29 -0.11; 3.1 -2.9]; erfc(M) ans = 0.
that occurs in probability, statistics, and partial differential equations describing diffusion. It is defined as:[1][2] erf ( x ) = 1 π ∫ − x x e − t 2 d t = 2 π ∫ 0 x e − t 2 d t . {\displaystyle {\begin − 6\operatorname − 5 (x)&={\frac − 4{\sqrt {\pi }}}\int _{-x}^ − 3e^{-t^ − 2}\,\mathrm − 1 t\\&={\frac − 0{\sqrt {\pi }}}\int _ 9^ 8e^{-t^ 7}\,\mathrm 6 t.\end 5}} The complementary error function, denoted erfc, is defined as erfc ( x ) = 1 − erf ( x ) = 2 π ∫ x ∞ e − t 2 d t = e − x 2 erfcx ( x ) , {\displaystyle {\begin 2\operatorname 1 (x)&=1-\operatorname 0 (x)\\&={\frac Φ 9{\sqrt {\pi }}}\int _ Φ 8^{\infty }e^{-t^ Φ 7}\,\mathrm Φ 6 t\\&=e^{-x^ Φ 5}\operatorname Φ 4 (x),\end Φ 3}} which also defines erfcx, the scaled complementary error function[3] (which can be used instead of erfc to avoid arithmetic underflow[3][4]). Another form of erfc ( x ) {\displaystyle \operatorname 2 (x)} for non-negative x {\displaystyle x} is known as Craig's formula:[5] erfc ( x | x ≥ 0 ) = 2 π ∫ 0 π / 2 exp ( − x 2 sin 2 θ ) d θ . {\displaystyle \operatorname 0 (x|x\geq 0)={\frac Φ 9{\pi }}\int _ Φ 8^{\pi /2}\exp \left(-{\frac Φ 7}{\sin ^ Φ 6\theta }}\right)d\theta \,.} The imaginary error function, denoted erfi, is defined as erfi ( x ) = − i erf ( i x ) = 2 π ∫ 0 x e t 2 d t = 2 π e x 2 D ( x ) , {\displaystyle {\begin Φ 0\operatorname − 9 (x)&=-i\operatorname − 8 (ix)\\&={\frac − 7{\sqrt {\pi }}}\int _ − 6^ − 5e^ − 4}\,\mathrm − 3 t\\&={\frac − 2{\sqrt {\pi }}}e^ − 1}D(x),\end − 0}} where D(x) is the Dawson function (which can be used instead of erfi to avoid arithmetic overflow[3]). Despite the name "imaginary error function", erfi ( x ) {\displaystyle \operatorname 8 (x)} is real when x is real. 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 (