erfc
Complementary error function
Description
erfc(
represents
the complementary
error function of X
)X
, that is,erfc(X)
= 1 - erf(X)
.
erfc(
represents
the iterated
integral of the complementary error function of K
,X
)X
,
that is, erfc(K, X) = int(erfc(K - 1, y), y, X, inf)
.
Examples
Complementary Error Function for Floating-Point and Symbolic Numbers
Depending on its arguments, erfc
can
return floating-point or exact symbolic results.
Compute the complementary error function for these numbers. Because these numbers are not symbolic objects, you get the floating-point results:
A = [erfc(1/2), erfc(1.41), erfc(sqrt(2))]
A = 0.4795 0.0461 0.0455
Compute the complementary error function for the same numbers
converted to symbolic objects. For most symbolic (exact) numbers, erfc
returns
unresolved symbolic calls:
symA = [erfc(sym(1/2)), erfc(sym(1.41)), erfc(sqrt(sym(2)))]
symA = [ erfc(1/2), erfc(141/100), erfc(2^(1/2))]
Use vpa
to approximate symbolic results
with the required number of digits:
d = digits(10); vpa(symA) digits(d)
ans = [ 0.4795001222, 0.04614756064, 0.0455002639]
Error Function for Variables and Expressions
For most symbolic variables and expressions, erfc
returns
unresolved symbolic calls.
Compute the complementary error function for x
and sin(x)
+ x*exp(x)
:
syms x f = sin(x) + x*exp(x); erfc(x) erfc(f)
ans = erfc(x) ans = erfc(sin(x) + x*exp(x))
Complementary Error Function for Vectors and Matrices
If the input argument is a vector or a matrix, erfc
returns
the complementary error function for each element of that vector or
matrix.
Compute the complementary error function for elements of matrix M
and
vector V
:
M = sym([0 inf; 1/3 -inf]); V = sym([1; -i*inf]); erfc(M) erfc(V)
ans = [ 1, 0] [ erfc(1/3), 2] ans = erfc(1) 1 + Inf*1i
Compute the iterated integral of the complementary error function
for the elements of V
and M
,
and the integer -1
:
erfc(-1, M) erfc(-1, V)
ans = [ 2/pi^(1/2), 0] [ (2*exp(-1/9))/pi^(1/2), 0] ans = (2*exp(-1))/pi^(1/2) Inf
Special Values of Complementary Error Function
erfc
returns special values
for particular parameters.
Compute the complementary error function for x = 0, x = ∞, and x = –∞. The complementary error function has special values for these parameters:
[erfc(0), erfc(Inf), erfc(-Inf)]
ans = 1 0 2
Compute the complementary error function for complex infinities.
Use sym
to convert complex infinities to symbolic
objects:
[erfc(sym(i*Inf)), erfc(sym(-i*Inf))]
ans = [ 1 - Inf*1i, 1 + Inf*1i]
Handling Expressions That Contain Complementary Error Function
Many functions, such as diff
and int
,
can handle expressions containing erfc
.
Compute the first and second derivatives of the complementary error function:
syms x diff(erfc(x), x) diff(erfc(x), x, 2)
ans = -(2*exp(-x^2))/pi^(1/2) ans = (4*x*exp(-x^2))/pi^(1/2)
Compute the integrals of these expressions:
syms x int(erfc(-1, x), x)
ans = erf(x)
int(erfc(x), x)
ans = x*erfc(x) - exp(-x^2)/pi^(1/2)
int(erfc(2, x), x)
ans = (x^3*erfc(x))/6 - exp(-x^2)/(6*pi^(1/2)) +... (x*erfc(x))/4 - (x^2*exp(-x^2))/(6*pi^(1/2))
Plot Complementary Error Function
Plot the complementary error function on the interval from -5 to 5.
syms x fplot(erfc(x),[-5 5]) grid on
Input Arguments
More About
Tips
Calling
erfc
for a number that is not a symbolic object invokes the MATLAB®erfc
function. This function accepts real arguments only. If you want to compute the complementary error function for a complex number, usesym
to convert that number to a symbolic object, and then callerfc
for that symbolic object.For most symbolic (exact) numbers,
erfc
returns unresolved symbolic calls. You can approximate such results with floating-point numbers usingvpa
.At least one input argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, then
erfc
expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.
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 system 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 R2011b