erfinv

Inverse error function

Syntax

erfinv(X)

Description

erfinv(X) computes the inverse error function of X. If X is a vector or a matrix, erfinv(X) computes the inverse error function of each element of X.

Examples

Inverse Error Function for Floating-Point and Symbolic Numbers

Depending on its arguments, erfinv can return floating-point or exact symbolic results.

Compute the inverse error function for these numbers. Because these numbers are not symbolic objects, you get floating-point results:

A = [erfinv(1/2), erfinv(0.33), erfinv(-1/3)]

A =
    0.4769    0.3013   -0.3046

Compute the inverse error function for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, erfinv returns unresolved symbolic calls:

symA = [erfinv(sym(1)/2), erfinv(sym(0.33)), erfinv(sym(-1)/3)]

symA =
[ erfinv(1/2), erfinv(33/100), -erfinv(1/3)]

Use vpa to approximate symbolic results with the required number of digits:

d = digits(10);
vpa(symA)
digits(d)

ans =
[ 0.4769362762, 0.3013321461, -0.3045701942]

Inverse Error Function for Variables and Expressions

For most symbolic variables and expressions, erfinv returns unresolved symbolic calls.

Compute the inverse error function for x and sin(x) + x*exp(x). For most symbolic variables and expressions, erfinv returns unresolved symbolic calls:

syms x
f = sin(x) + x*exp(x);
erfinv(x)
erfinv(f)

ans =
erfinv(x)
 
ans =
erfinv(sin(x) + x*exp(x))

Inverse Error Function for Vectors and Matrices

If the input argument is a vector or a matrix, erfinv returns the inverse error function for each element of that vector or matrix.

Compute the inverse error function for elements of matrix M and vector V:

M = sym([0 1 + i; 1/3 1]);
V = sym([-1; inf]);
erfinv(M)
erfinv(V)

ans =
[           0, NaN]
[ erfinv(1/3), Inf]

ans =
 -Inf
  NaN

Special Values of Inverse Complementary Error Function

erfinv returns special values for particular parameters.

Compute the inverse error function for x = –1, x = 0, and x = 1. The inverse error function has special values for these parameters:

[erfinv(-1), erfinv(0), erfinv(1)]

ans =
  -Inf     0   Inf

Handling Expressions That Contain Inverse Complementary Error Function

Many functions, such as diff and int, can handle expressions containing erfinv.

Compute the first and second derivatives of the inverse error function:

syms x
diff(erfinv(x), x)
diff(erfinv(x), x, 2)

ans =
(pi^(1/2)*exp(erfinv(x)^2))/2
 
ans =
(pi*exp(2*erfinv(x)^2)*erfinv(x))/2

Compute the integral of the inverse error function:

int(erfinv(x), x)

ans =
-exp(-erfinv(x)^2)/pi^(1/2)

Plot Inverse Error Function

Plot the inverse error function on the interval from -1 to 1.

syms x
fplot(erfinv(x),[-1,1])
grid on

Input Arguments

X — Input

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.

More About

Inverse Error Function

The inverse error function is defined as erf ^-1(x), such that erf(erf ^-1(x)) = erf ^-1(erf(x)) = x. Here

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

is the error function.

Tips

• Calling erfinv for a number that is not a symbolic object invokes the MATLAB^® erfinv function. This function accepts real arguments only. If you want to compute the inverse error function for a complex number, use sym to convert that number to a symbolic object, and then call erfinv for that symbolic object.

• If x < –1 or x > 1, or if x is complex, then erfinv(x) returns NaN.

Algorithms

The toolbox can simplify expressions that contain error functions and their inverses. For real values x, the toolbox applies these simplification rules:

• erfinv(erf(x)) = erfinv(1 - erfc(x)) = erfcinv(1 - erf(x)) = erfcinv(erfc(x)) = x

• erfinv(-erf(x)) = erfinv(erfc(x) - 1) = erfcinv(1 + erf(x)) = erfcinv(2 - erfc(x)) = -x

For any value x, the toolbox applies these simplification rules:

• erfcinv(x) = erfinv(1 - x)

• erfinv(-x) = -erfinv(x)

• erfcinv(2 - x) = -erfcinv(x)

• erf(erfinv(x)) = erfc(erfcinv(x)) = x

• erf(erfcinv(x)) = erfc(erfinv(x)) = 1 - x

References

[1] Gautschi, W. “Error Function and Fresnel Integrals.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

Version History

Introduced in R2012a