Cálculo Numérico, Semestre 2017.2, Prova I

from math import *

def f_1(x):
    return x**4-x**3+2.0*x**2-4.0*x-7.0

def df_1(x):
    return 4.0*x**3-3.0*x**2+4.0*x-4.0

def d2f_1(x):
    return 12.0*x**2-6.0*x+4.0

def f_2(x):
    return x*cos(x)+2.0

def df_2(x):
    return cos(x)-x*sin(x)

def d2f_2(x):
    return -2.0*sin(x)-x*cos(x)

def f_3(x):
    return exp(x)-cos(x)-1.0

def df_3(x):
    return exp(x)+sin(x)

def d2f_3(x):
    return exp(x)+cos(x)

def f_4(x):
    return (x-1)*exp(x)-2.0

def df_4(x):
    return x*exp(x)

def d2f_4(x):
    return (x+1.0)*exp(x)

def f_5(x):
    return sin(x)-x*cos(x)-1.0

def df_5(x):
    return x*sin(x)

def d2f_5(x):
    return sin(x)+x*cos(x)

from math import *

from N01 import *


xs=[-3.0,-2.0,-1.0,0.0,1.0,2.0,3.0 ]

fs=[
    f_1,f_2,f_3,f_4,f_5
]

titles=["$x$","$f_1(x)$","$f_2(x)$","$f_3(x)$","$f_4(x)$","$f_5(x)$"]

print '\\begin{tabular}{c|rrrrr}'
print "\t",' & '.join(titles)+'\\\\\n\t\\hline'

for x in xs:
    text=[ "%.6f" % x  ]
    for f in fs:
        text.append( "%.6f" % f(x) )
    print "\t"," & ".join(text)+'\\\\'
print '\\end{tabular}'

\begin{tabular}{c|rrrrr}
    $x$ & $f_1(x)$ & $f_2(x)$ & $f_3(x)$ & $f_4(x)$ & $f_5(x)$\\
    \hline
    -3.000000 & 131.000000 & 4.969977 & 0.039780 & -2.199148 & -4.111097\\
    -2.000000 & 33.000000 & 2.832294 & -0.448518 & -2.406006 & -2.741591\\
    -1.000000 & 1.000000 & 1.459698 & -1.172423 & -2.735759 & -1.301169\\
    0.000000 & -7.000000 & 2.000000 & -1.000000 & -3.000000 & -1.000000\\
    1.000000 & -9.000000 & 2.540302 & 1.177980 & -2.000000 & -0.698831\\
    2.000000 & 1.000000 & 1.167706 & 6.805203 & 5.389056 & 0.741591\\
    3.000000 & 53.000000 & -0.969977 & 20.075529 & 38.171074 & 2.111097\\
\end{tabular}

from math import *

from N01 import *
from Derivatives import *
from Residual import *

xs=[-3.0,-2.0,-1.0,0.0,1.0,2.0,3.0 ]

fs=[
    f_1,f_2,f_3,f_4,f_5
]

dfs=[
    df_1,df_2,df_3,df_4,df_5
]

d2fs=[
    d2f_1,d2f_2,d2f_3,d2f_4,d2f_5
]

names=["f1","f2","f3","f4","f5"]

titles=[
    "$x$",
    "$d^2f_{ana}$","$d^2f_{num}$","$r^2_a$","$r^2_r$",
]
for n in range( len(dfs) ):
    f=fs[n]
    df=dfs[n]
    d2f=d2fs[n]
    print 'Derivative: '+names[n]
    
    print '\\begin{tabular}{c|rrrr}'
    print ' & '.join(titles)+'\\\\\n\\hline'
    for x in xs:
        text=[ "%.6f" % x ]
    
        d2f_ana=d2f(x)
        d2f_num=d2_f(f,x)

        r2_a=R_Abs(d2f_ana,d2f_num)
        r2_r=R_Rel(d2f_ana,d2f_num)
        
        text.append( "%.6f" % d2f_ana )
        text.append( "%.6f" % d2f_num )
        text.append( "%.6e" % r2_a )
        text.append( "%.6e" %  r2_r)
        
        print " & ".join(text)+'\\\\'
        
    print '\\end{tabular}'

Derivative: f1
\begin{tabular}{c|rrrr}
$x$ & $d^2f_{ana}$ & $d^2f_{num}$ & $r^2_a$ & $r^2_r$\\
\hline
-3.000000 & 130.000000 & 130.000899 & 8.989347e-04 & 6.914882e-06\\
-2.000000 & 64.000000 & 64.007466 & 7.465994e-03 & 1.166562e-04\\
-1.000000 & 22.000000 & 21.999513 & 4.866778e-04 & 2.212172e-05\\
0.000000 & 4.000000 & 3.999912 & 8.848688e-05 & 2.212172e-05\\
1.000000 & 10.000000 & 10.000001 & 8.274037e-07 & 8.274037e-08\\
2.000000 & 40.000000 & 40.002224 & 2.223756e-03 & 5.559389e-05\\
3.000000 & 94.000000 & 93.997699 & 2.301486e-03 & 2.448390e-05\\
\end{tabular}
Derivative: f2
\begin{tabular}{c|rrrr}
$x$ & $d^2f_{ana}$ & $d^2f_{num}$ & $r^2_a$ & $r^2_r$\\
\hline
-3.000000 & -2.687737 & -2.687850 & 1.124689e-04 & 4.184521e-05\\
-2.000000 & 0.986301 & 0.986655 & 3.540214e-04 & 3.589385e-04\\
-1.000000 & 2.223244 & 2.223277 & 3.284248e-05 & 1.477232e-05\\
0.000000 & -0.000000 & -0.000056 & 5.551115e-05 & 0.000000e+00\\
1.000000 & -2.223244 & -2.223333 & 8.835363e-05 & 3.974086e-05\\
2.000000 & -0.986301 & -0.986655 & 3.540214e-04 & 3.589385e-04\\
3.000000 & 2.687737 & 2.687850 & 1.124689e-04 & 4.184521e-05\\
\end{tabular}
Derivative: f3
\begin{tabular}{c|rrrr}
$x$ & $d^2f_{ana}$ & $d^2f_{num}$ & $r^2_a$ & $r^2_r$\\
\hline
-3.000000 & -0.940205 & -0.940137 & 6.857098e-05 & 7.293191e-05\\
-2.000000 & -0.280812 & -0.280775 & 3.615038e-05 & 1.287354e-04\\
-1.000000 & 0.908182 & 0.908218 & 3.619826e-05 & 3.985794e-05\\
0.000000 & 2.000000 & 2.000011 & 1.126771e-05 & 5.633855e-06\\
1.000000 & 3.258584 & 3.258505 & 7.955705e-05 & 2.441461e-05\\
2.000000 & 6.972909 & 6.973977 & 1.067689e-03 & 1.531196e-04\\
3.000000 & 19.095544 & 19.094060 & 1.484760e-03 & 7.775426e-05\\
\end{tabular}
Derivative: f4
\begin{tabular}{c|rrrr}
$x$ & $d^2f_{ana}$ & $d^2f_{num}$ & $r^2_a$ & $r^2_r$\\
\hline
-3.000000 & -0.099574 & -0.099587 & 1.286857e-05 & 1.292361e-04\\
-2.000000 & -0.135335 & -0.135336 & 9.034652e-07 & 6.675755e-06\\
-1.000000 & 0.000000 & 0.000000 & 0.000000e+00 & 0.000000e+00\\
0.000000 & 1.000000 & 1.000089 & 8.890058e-05 & 8.890058e-05\\
1.000000 & 5.436564 & 5.436540 & 2.354993e-05 & 4.331768e-06\\
2.000000 & 22.167168 & 22.168933 & 1.765059e-03 & 7.962492e-05\\
3.000000 & 80.342148 & 80.342843 & 6.957965e-04 & 8.660417e-06\\
\end{tabular}
Derivative: f5
\begin{tabular}{c|rrrr}
$x$ & $d^2f_{ana}$ & $d^2f_{num}$ & $r^2_a$ & $r^2_r$\\
\hline
-3.000000 & 2.828857 & 2.828848 & 9.214997e-06 & 3.257498e-06\\
-2.000000 & -0.077004 & -0.077161 & 1.567465e-04 & 2.035569e-03\\
-1.000000 & -1.381773 & -1.381784 & 1.028577e-05 & 7.443893e-06\\
0.000000 & 0.000000 & 0.000000 & 0.000000e+00 & 0.000000e+00\\
1.000000 & 1.381773 & 1.381784 & 1.028577e-05 & 7.443893e-06\\
2.000000 & 0.077004 & 0.077161 & 1.567465e-04 & 2.035569e-03\\
3.000000 & -2.828857 & -2.828959 & 1.018073e-04 & 3.598884e-05\\
\end{tabular}

from math import *

from N01 import *
from Derivatives import *
from Isolate import *
from Bisection import *
from False_Position import *
from Newton_Raphson import *
from Secant import *

xs=[-3.0,-2.0,-1.0,0.0,1.0,2.0,3.0 ]

fs=[
    f_1,f_2,f_3,f_4,f_5
]

dfs=[
    df_1,df_2,df_3,df_4,df_5
]

#Intervals deduced from Q2.
Is=[
    [-1.0, 0.0 ],
    [ 2.0, 3.0 ],
    [ 0.0, 1.0 ],
    [ 1.0, 2.0 ],
    [ 1.0, 2.0 ],
]
names=["f1","f2","f3","f4","f5"]

titles1=[
    "\\multicolumn{3}{c}{}",
    "\\multicolumn{3}{c}{Bisection}",
    "\\multicolumn{3}{c}{False Position}",
    "\\multicolumn{3}{c}{Newton Raphson}",
    "\\multicolumn{3}{c}{Secant}",
]
titles2=[
    "$f$",
    "$a$","$b$",
    "$I$","$x$","$f(x)$",
    "$I$","$x$","$f(x)$",
    "$I$","$x$","$f(x)$",
    "$I$","$x$","$f(x)$",
]

print '\\begin{tabular}{crr|rrr|rrr|rrr|rrr}'
print ' & '.join(titles1)+'\\\\'
print ' & '.join(titles2)+'\\\\\n\\hline'

for n in range( len(dfs) ):
    f=fs[n]
    df=dfs[n]

    a=Is[n][0]
    b=Is[n][1]
    

    text=[
        names[n],
        "%.6f" % a,
        "%.6f" % b,
    ]

    xs=Bisection(f,a,b)
    x=xs[ len(xs)-1 ]
    
    text.append( "%d"   % len(xs) )
    text.append( "%.6f" % x)
    text.append( "%.6e" % abs(f(x)) )
    
    xs=False_Position(f,a,b)
    x=xs[ len(xs)-1 ]
    
    text.append( "%d"   % len(xs) )
    text.append( "%.6f" % x)
    text.append( "%.6e" % abs(f(x)) )

    xs=Newton_Raphson(f,df,a)
    x=xs[ len(xs)-1 ]
    
    text.append( "%d"   % len(xs) )
    text.append( "%.6f" % x)
    text.append( "%.6e" % abs(f(x)) )

    xs=Secant(f,a,b)
    x=xs[ len(xs)-1 ]
    
    text.append( "%d"   % len(xs) )
    text.append( "%.6f" % x)
    text.append( "%.6e" % abs(f(x)) )

    print " & ".join(text)+'\\\\'
        
print '\\end{tabular}'

\begin{tabular}{crr|rrr|rrr|rrr|rrr}
\multicolumn{3}{c}{} & \multicolumn{3}{c}{Bisection} & \multicolumn{3}{c}{False Position} & \multicolumn{3}{c}{Newton Raphson} & \multicolumn{3}{c}{Secant}\\
$f$ & $a$ & $b$ & $I$ & $x$ & $f(x)$ & $I$ & $x$ & $f(x)$ & $I$ & $x$ & $f(x)$ & $I$ & $x$ & $f(x)$\\
\hline
f1 & -1.000000 & 0.000000 & 20 & -0.929837 & 1.564530e-07 & 6 & -0.929837 & 1.100318e-07 & 5 & -0.929837 & 0.000000e+00 & 7 & -0.929837 & 2.791980e-10\\
f2 & 2.000000 & 3.000000 & 19 & 2.498755 & 7.070456e-07 & 2 & 2.498756 & 9.697914e-07 & 5 & 2.498756 & 0.000000e+00 & 6 & 2.498756 & 5.012699e-09\\
f3 & 0.000000 & 1.000000 & 18 & 0.601347 & 4.825563e-07 & 8 & 0.601347 & 4.480312e-07 & 7 & 0.601347 & 2.220446e-16 & 8 & 0.601347 & 1.633400e-10\\
f4 & 1.000000 & 2.000000 & 19 & 1.463056 & 6.147964e-07 & 5 & 1.463055 & 9.453087e-08 & 7 & 1.463056 & 8.881784e-16 & 8 & 1.463055 & 1.278058e-07\\
f5 & 1.000000 & 2.000000 & 19 & 1.570796 & 4.930988e-07 & 5 & 1.570796 & 7.479757e-07 & 6 & 1.570796 & 1.110223e-16 & 7 & 1.570796 & 2.239831e-09\\
\end{tabular}