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%html
<h1>
TP1    
</h1>

TP1 

s=1
f(x)=(1/(s*sqrt(2*pi)))*exp(-x**2/(2*s**2))

show(f)
x ↦ 2e(−x22 s2)2 πs\displaystyle x \ {\mapsto}\ \frac{\sqrt{2} e^{\left(-\frac{x^{2}}{2 \, s^{2}}\right)}}{2 \, \sqrt{\pi} s}
F(x)=integrate(f,x,-infinity,x)
show(F)
x ↦ 2(2πerf(12 2x)+2π)4 π\displaystyle x \ {\mapsto}\ \frac{\sqrt{2} {\left(\sqrt{2} \sqrt{\pi} \text{erf}\left(\frac{1}{2} \, \sqrt{2} x\right) + \sqrt{2} \sqrt{\pi}\right)}}{4 \, \sqrt{\pi}}
plot(f(x),x,-10,10)


plot(F(x),x,-10,10)

m=2
f1(x)=f(x-m)
plot(f1(x),x,-10,10)

solve(F(x)==1/2,x)
[x == 0] 
solve(0.95==2*(F(x)-1/2),x)
[x == sqrt(2)*inverse_erf(19/20)] 


%html
<h1>
TP2 Devoir Maison
</h1>

TP2 Devoir Maison 

erf
Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-7.5/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 982, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "<string>", line 1 solve(erf ^ SyntaxError: unexpected EOF while parsing 
erf?

File: /projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/functions/other.py
Signature : erf(self, coerce=True, hold=False, dont_call_method_on_arg=False, *args)
Docstring :
The error function, defined for real values as

erf(x) = frac{2}{sqrt{pi}} int_0^x e^{-t^2} dt.

This function is also defined for complex values, via analytic
continuation.

EXAMPLES:

We can evaluate numerically:

   sage: erf(2)
   erf(2)
   sage: erf(2).n()
   0.995322265018953
   sage: erf(2).n(100)
   0.99532226501895273416206925637
   sage: erf(ComplexField(100)(2+3j))
   -20.829461427614568389103088452 + 8.6873182714701631444280787545*I

Basic symbolic properties are handled by Sage and Maxima:

   sage: x = var("x")
   sage: diff(erf(x),x)
   2*e^(-x^2)/sqrt(pi)
   sage: integrate(erf(x),x)
   x*erf(x) + e^(-x^2)/sqrt(pi)

ALGORITHM:

Sage implements numerical evaluation of the error function via the
"erf()" function from mpmath. Symbolics are handled by Sage and
Maxima.

REFERENCES:

* http://en.wikipedia.org/wiki/Error_function

*
  http://mpmath.googlecode.com/svn/trunk/doc/build/functions/expin
  tegrals.html#error-functions



erf(100).n()
1.00000000000000 
erf(2).n()
0.995322265018953 
erf(1).n()
erf(1) 
f(x)= (1/(1*sqrt(2*pi)))*exp(-x**2/(2*1**2))
plot(f)

#on cherche a tel que a=F^-1(1.95/2) ou F(x)=integrate(f,x,-infinity,x)
f(x)= (1/(1*sqrt(2*pi)))*exp(-x**2/(2*1**2))
F(x)=integrate(f,x,-infinity,x)
a=(1/F(1.95/2))
show(a)
2 2π2πerf(0.487500000000000 2)+2π\displaystyle \frac{2 \, \sqrt{2} \sqrt{\pi}}{\sqrt{2} \sqrt{\pi} \text{erf}\left(0.487500000000000 \, \sqrt{2}\right) + \sqrt{2} \sqrt{\pi}}
erf(0.4875*sqrt(2)).n()
0.670439740039379 
sqrt(2)
sqrt(2) 
a.n(100)
1.1972895232681974325711429172 
plot(F,-5,5)

plot(f,-5,5)

f(x)= (1/(1*sqrt(2*pi)))*exp(-x**2/(2*1**2))

%r

f <- function(x) {(1/(1*sqrt(2*pi)))*exp(-x**2/(2*1**2))}
F <- function(x) {integrate(f,lower=-Inf,upper=1)$value}

Approx <- function(n){
x0 = 0
x1 = 4
f <- function(x) {(1/(1*sqrt(2*pi)))*exp(-x**2/(2*1**2))}
F <- function(x) {integrate(f,lower=-Inf,upper=1)$value}
a=1
while (x1-x0 > 10**-3 && a<=n ){
if ( F((x0+x1)/2) > 0 ){
    x1=(x0+x1)/2
    x0=x0
    a=a+1
}
if ( F((x0+x1)/2) < 0 ){
    x0=(x0+x1)/2
    x1=x1
    a=a+1
}
else{}
}
vec <- c(x0,x1)
return(vec)
}

Approx(100)
  1. 0
  2. 0.0009765625
%r

s=1
f <- function(x){(1/(s*sqrt(2*pi)))*exp(-x**2/(2*s**2))}
F <- function(x){integrate(f,lower=-Inf,upper=1)$value}
a = 0
b = 4
n=10
plot(F,-5,5)

for (i in 1:n){
if ( F((a+b)/2) > 0 ){
    b=(a+b)/2
    }
if ( F((a+b)/2) < 0 ){
    a=(a+b)/2
    }
else{}
}
return(c(a,b))
Error in curve(expr = x, from = from, to = to, xlim = xlim, ylab = ylab, : 'expr' did not evaluate to an object of length 'n' Traceback: 1. plot(F, -5, 5) 2. plot.function(F, -5, 5) 3. curve(expr = x, from = from, to = to, xlim = xlim, ylab = ylab, . ...) 4. stop("'expr' did not evaluate to an object of length 'n'") 
s=1
f(x)=(1/(s*sqrt(2*pi)))*exp(-x**2/(2*s**2))
F(x)=integrate(f,x,-infinity,x)
a = 0
b = 4
n=10
plot(F,-5,5)

%r
for (i in 1:n){
if ( F((a+b)/2) > 0 ){
    b=(a+b)/2
    print(b)
    }
if ( F((a+b)/2) < 0 ){
    a=(a+b)/2
    print(a)
    }
else{}
}
return(c(a,b))
[1] 2 [1] 1 [1] 0.5 [1] 0.25 [1] 0.125 [1] 0.0625 [1] 0.03125 [1] 0.015625 [1] 0.0078125 [1] 0.00390625
  1. 0
  2. 0.00390625