matlab erfcinv,Inverse complementary error function
erfcinv
Inverse complementary error function
Description
erfcinv(X) computes
the inverse
complementary error function of 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
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 Complementary Error Function
The inverse complementary error function is defined as erfc-1(x), such that erfc(erfc-1(x)) = x. Here
erfc(x)=2π∫x∞e−t2dt=1−erf(x)
is the complementary error function.
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, use sym to
convert that number to a symbolic object, and then call erfcinv for
that symbolic object.
If x 2, or if x is
complex, then erfcinv(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.
Introduced in R2012a