erfcinv
Inverse complementary error function
Syntax
Description
erfcinv(
computes
the inverse
complementary error function of X
)X
. If X
is
a vector or a matrix, erfcinv(X)
computes the inverse
complementary error function of each element of X
.
Examples
Inverse Complementary Error Function for Floating-Point and Symbolic Numbers
Depending on its arguments, erfcinv
can
return floating-point or exact symbolic results.
Compute the inverse complementary error function for these numbers. Because these numbers are not symbolic objects, you get floating-point results:
A = [erfcinv(1/2), erfcinv(1.33), erfcinv(3/2)]
A = 0.4769 -0.3013 -0.4769
Compute the inverse complementary error function for the same
numbers converted to symbolic objects. For most symbolic (exact) numbers, erfcinv
returns
unresolved symbolic calls:
symA = [erfcinv(sym(1/2)), erfcinv(sym(1.33)), erfcinv(sym(3/2))]
symA = [ -erfcinv(3/2), erfcinv(133/100), erfcinv(3/2)]
Use vpa
to approximate symbolic results
with the required number of digits:
d = digits(10); vpa(symA) digits(d)
ans = [ 0.4769362762, -0.3013321461, -0.4769362762]
Inverse Complementary Error Function for Variables and Expressions
For most symbolic variables and expressions, erfcinv
returns
unresolved symbolic calls.
Compute the inverse complementary error function for x
and sin(x)
+ x*exp(x)
. For most symbolic variables and expressions, erfcinv
returns
unresolved symbolic calls:
syms x f = sin(x) + x*exp(x); erfcinv(x) erfcinv(f)
ans = erfcinv(x) ans = erfcinv(sin(x) + x*exp(x))
Inverse Complementary Error Function for Vectors and Matrices
If the input argument is a vector or a matrix, erfcinv
returns
the inverse complementary error function for each element of that
vector or matrix.
Compute the inverse complementary error function for elements
of matrix M
and vector V
:
M = sym([0 1 + i; 1/3 1]); V = sym([2; inf]); erfcinv(M) erfcinv(V)
ans = [ Inf, NaN] [ -erfcinv(5/3), 0] ans = -Inf NaN
Special Values of Inverse Complementary Error Function
erfcinv
returns special
values for particular parameters.
Compute the inverse complementary error function for x = 0, x = 1, and x = 2. The inverse complementary error function has special values for these parameters:
[erfcinv(0), erfcinv(1), erfcinv(2)]
ans = Inf 0 -Inf
Handling Expressions That Contain Inverse Complementary Error Function
Many functions, such as diff
and int
,
can handle expressions containing erfcinv
.
Compute the first and second derivatives of the inverse complementary error function:
syms x diff(erfcinv(x), x) diff(erfcinv(x), x, 2)
ans = -(pi^(1/2)*exp(erfcinv(x)^2))/2 ans = (pi*exp(2*erfcinv(x)^2)*erfcinv(x))/2
Compute the integral of the inverse complementary error function:
int(erfcinv(x), x)
ans = exp(-erfcinv(x)^2)/pi^(1/2)
Plot Inverse Complementary Error Function
Plot the inverse complementary error function on the interval from 0 to 2.
syms x fplot(erfcinv(x),[0 2]) grid on
Input Arguments
More About
Tips
Calling
erfcinv
for a number that is not a symbolic object invokes the MATLAB®erfcinv
function. This function accepts real arguments only. If you want to compute the inverse complementary error function for a complex number, usesym
to convert that number to a symbolic object, and then callerfcinv
for that symbolic object.If x < 0 or x > 2, or if x is complex, then
erfcinv(x)
returnsNaN
.
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