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第１回回答

1.Newton–@‚Ì 
                         f(x), df(x)
‚ÌŠÖŒW‚ðŽ¦‚·Ž®‚𓱂¯D
restart;
y-f(x1) = (df(x1))(x-x1); 
                  y - f(x1) = df(x1)(x - x1)
y = (df(x1))(x-x1)+f(x1); 
                  y = df(x1)(x - x1) + f(x1)
y=0;
df(x1)(x-x1)+f(x1)=0;
df(x1)(x-x1)=-f(x1);
x-x1=-f(x1)/df(x1);
x=-f(x1)/df(x1)+x1; 
                            y = 0
                  df(x1)(x - x1) + f(x1) = 0
                   df(x1)(x - x1) = -f(x1)
                                 f(x1) 
                      x - x1 = - ------
                                 df(x1)
                            f(x1)      
                      x = - ------ + x1
                            df(x1)
“ñ•ª–@Eƒjƒ…[ƒgƒ“–@
restart; 
Digits := 10; f := unapply(exp(-x)-2*exp(-2*x), x); plot(f(x), x = -1 .. 2);
x -> exp(-x) - 2 exp(-2 x)

“ñ•ª–@
x1:=-1.0: x2:=2.0:
f1:=f(x1): f2:=f(x2):
epsilon:=0.001: delta:=0.001:
for i from 1 to 100 do 
 x3:=(x1+x2)/2;
 f3:=f(x3);
 if (evalf(f1*f3)>=0.0) then
   f1:=f3;
   x1:=x3;
 else
   f2:=f3;
   x2:=x3;
 end if;
 printf("%10.5f, %10.5f, %10.5f, %10.5f\n",
         x1, x2, f1, f2);
 if (abs(x2-x1)<delta) then
   break;
 end if;
 if (abs(f2-f1)<epsilon) then
   break;
 end if;
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x3, f(x3)); 
  0.50000,    2.00000,   -0.12923,    0.09870
  0.50000,    1.25000,   -0.12923,    0.12233
  0.50000,    0.87500,   -0.12923,    0.06931
  0.68750,    0.87500,   -0.00285,    0.06931
  0.68750,    0.78125,   -0.00285,    0.03861
  0.68750,    0.73438,   -0.00285,    0.01938
  0.68750,    0.71094,   -0.00285,    0.00866
  0.68750,    0.69922,   -0.00285,    0.00301
  0.68750,    0.69336,   -0.00285,    0.00011
  0.69043,    0.69336,   -0.00136,    0.00011
  0.69189,    0.69336,   -0.00063,    0.00011
11回目, x = 0.69189, f(x) = -0.00063
Newton–@ 
x4 := -1.0; df := unapply(diff(f(x), x), x); epsilon := 0.1e-2; delta := 0.1e-2; for i to 100 do x5 := x4-f(x4)/df(x4); printf("%10.5f,~%10.5f\n", x5, f(x5)); if abs(x5-x4) < delta then break end if; if abs(f(x5)-f(x4)) < epsilon then break end if; x4 := x5 end do; printf("%d&aring;›ž&ccedil;›&reg;,~x~=~%.5f,~f(x)~=~%.5f\n", i, x4, f(x4));
x -> -exp(-x) + 4 exp(-2 x) 
 -0.55064,   -4.28169
 -0.13485,   -1.47479
  0.22538,   -0.47607
  0.49737,   -0.13151
  0.64833,   -0.02397
  0.69032,   -0.00142
  0.69314,   -0.00001
  0.69315,    0.00000
8回目, x = 0.69314, f(x) = -0.00001
abs‚͐â‘Î’l



3(2004Šú––)

3(a)
restart;
with(plots): with(plottools):
Digits:=10:
f:=unapply(exp(-x)-x^2,x);
plot(f(x),x=-1..1); 
               2
x -> exp(-x) - x

x1:=-1.0: x2:=1.0:
f1:=f(x1): f2:=f(x2):
epsilon:=0.001: delta:=0.001:
for i from 1 to 100 do 
 x3:=(x1+x2)/2;
 f3:=f(x3);
 if (evalf(f1*f3)>=0.00) then
   f1:=f3;
   x1:=x3;
 else
   f2:=f3;
   x2:=x3;
 end if;
 printf("%10.5f, %10.5f, %10.5f, %10.5f\n",
         x1, x2, f1, f2);
 if (abs(x2-x1)<delta) then
   break;
 end if;
 if (abs(f2-f1)<epsilon) then
   break;
 end if;
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x3, f(x3)); 
  0.00000,    1.00000,    1.00000,   -0.63212
  0.50000,    1.00000,    0.35653,   -0.63212
  0.50000,    0.75000,    0.35653,   -0.09013
  0.62500,    0.75000,    0.14464,   -0.09013
  0.68750,    0.75000,    0.03018,   -0.09013
  0.68750,    0.71875,    0.03018,   -0.02924
  0.70312,    0.71875,    0.00065,   -0.02924
  0.70312,    0.71094,    0.00065,   -0.01425
  0.70312,    0.70703,    0.00065,   -0.00679
  0.70312,    0.70508,    0.00065,   -0.00307
  0.70312,    0.70410,    0.00065,   -0.00121
11回目, x = 0.70410, f(x) = -0.00121
x4:=-1.0:
df:=unapply(diff(f(x),x),x);
epsilon:=0.001: delta:=0.001:
for i from 1 to 100 do 
 x5:=x4-f(x4)/df(x4);
 printf("%10.5f, %10.5f\n", x5, f(x5));
 if (abs(x5-x4)<delta) then
   break;
 end if;
 if (abs(f(x5)-f(x4))<epsilon) then
   break;
 end if;
 x4:=x5;
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x4, f(x4));
x -> -exp(-x) - 2 x 
  1.39221,   -1.68973
  0.83509,   -0.26353
  0.70983,   -0.01214
  0.70348,   -0.00003
  0.70347,   -0.00000
5回目, x = 0.70348, f(x) = -0.00003
3(b)
二分法
x1:=-1.0: x2:=1.0:
f1:=f(x1): f2:=f(x2):
x_f:=0.7034674225;
lst1:=[]:
for i from 1 to 100 do 
 x3:=(x1+x2)/2;
 f3:=f(x3);
 if (evalf(f1*f3)>=0.00) then
   f1:=f3;
   x1:=x3;
 else
   f2:=f3;
   x2:=x3;
 end if;
 lst1:=[op(lst1), log(abs(x_f-x3))];
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x3, f(x3)); 
                         0.7034674225
101回目, x = 0.70347, f(x) = 0.00000
Newton法
x4:=-1.0:
df:=unapply(diff(f(x),x),x);
x_f:=0.7034674225;
lst2:=[]:
for i from 1 to 100 do 
 x5:=x4-f(x4)/df(x4);
 lst2:=[op(lst2), log(abs(x_f-x5))];
 x4:=x5;
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x4, f(x4));
x -> -exp(-x) - 2 x 
                         0.7034674225
101回目, x = 0.70347, f(x) = 0.00000
l1:=listplot(lst1,linestyle=solid,legend="二分法""):
l2:=listplot(lst2,linestyle=dashdot,legend="Newton法""):
display([l1,l2]);





4(2008”N“x)
restart;
Digits:=10:
f:=unapply(x^4-x-0.12,x);
plot(f(x),x=0..2); 
     4
x -> x - x - 0.12

x1:=0.0: x2:=2.0:
f1:=f(x1): f2:=f(x2):
epsilon:=1.0*10^(-8): delta:=1.0*10^(-8):
for i from 1 to 1000 do 
 x3:=(x1+x2)/2;
 f3:=f(x3);
 if (evalf(f1*f3)>=0.00) then
   f1:=f3;
   x1:=x3;
 else
   f2:=f3;
   x2:=x3;
 end if;
 if (abs(x2-x1)<delta) then
   break;
 end if;
 if (abs(f2-f1)<epsilon) then
   break;
 end if;
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x3, f(x3));
28回目, x = 1.03717, f(x) = 0.00000
6(Š„ü–@)
restart;
Digits:=10:
func:=x->x^2-4*x+1;
x1:=0: x2:=2:
f1:=func(x1): f2:=func(x2):
plot({(z-x1)*(f1-f2)/(x1-x2)+f1,func(z)},z=0..2);
x_f:=fsolve(func(x)=0,x)[1]; 
     2
x -> x - 4 x + 1 

                         0.2679491924
with(plots): with(plottools):
f:=func:
N:=40:
割線法
x1:=0: x2:=2:
f1:=f(x1): f2:=f(x2):
lst1:=[]:
for i from 1 to N do 
 y1:=((f2-f1)/(x2-x1))*(x-x1)+f1:
 x3:=fsolve(y1=0,x);
 f3:=f(x3);
 if (evalf(f1*f3)>=0.00) then
   f1:=f3;
   x1:=x3;
 else
   f2:=f3;
   x2:=x3;
 end if;
 lst1:=[op(lst1), log(abs(x_f-x3))];
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x3, f(x3));
41回目, x = 0.26795, f(x) = 0.00000
二分法
x1:=0.0: x2:=2.0:
f1:=f(x1): f2:=f(x2):
lst2:=[]:
for i from 1 to N do 
 x3:=(x1+x2)/2;
 f3:=f(x3);
 if (evalf(f1*f3)>=0.00) then
   f1:=f3;
   x1:=x3;
 else
   f2:=f3;
   x2:=x3;
 end if;
 lst2:=[op(lst2), log(abs(x_f-x3))];
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x3, f(x3));
41回目, x = 0.26795, f(x) = -0.00000
Newton法
x4:=0.0:
df:=unapply(diff(f(x),x),x):
lst3:=[]:
for i from 1 to N do 
 x5:=x4-f(x4)/df(x4);
 lst3:=[op(lst3), log(abs(x_f-x5))];
 x4:=x5;
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x4, f(x4));
41回目, x = 0.26795, f(x) = 0.00000
l1:=listplot(lst1,linestyle=solid,legend="Š„ü–@"):
l2:=listplot(lst2,linestyle=dashdot,legend="“ñ•ª–@"):
l3:=listplot(lst3,linestyle=spacedash,legend="Newton"):
display([l1,l2,l3]);

7(2009”N“x)
restart;
Digits:=10:
with(plots): with(plottools):
f:=unapply(cos(x)-x^2,x);
x_f:=fsolve(f(x)=0,x);
N:=50:
plot(f(x),x=0..1); 
                                        2
                    f := x -> cos(x) - x 
                     x_f := 0.8241323123

二分法
x1:=0.0: x2:=1.0:
f1:=f(x1): f2:=f(x2):
lst1:=[]:
for i from 1 to N do 
 x3:=(x1+x2)/2;
 f3:=f(x3);
 if (evalf(f1*f3)>=0.00) then
   f1:=f3;
   x1:=x3;
 else
   f2:=f3;
   x2:=x3;
 end if;
 lst1:=[op(lst1), log(abs(x_f-x3))];
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x3, f(x3));
51回目, x = 0.82413, f(x) = -0.00000
割線法
x1:=0.0: x2:=1.0:
f1:=f(x1): f2:=f(x2):
lst2:=[]:
for i from 1 to N do 
 y1:=((f2-f1)/(x2-x1))*(x-x1)+f1:
 x3:=fsolve(y1=0,x);
 f3:=f(x3);
 if (evalf(f1*f3)>=0.00) then
   f1:=f3;
   x1:=x3;
 else
   f2:=f3;
   x2:=x3;
 end if;
 lst2:=[op(lst2), log(abs(x_f-x3))];
end do:
printf("%d回目, x = %.5f, f(x) = %.5f\n", i, x3, f(x3));
51回目, x = 0.82413, f(x) = -0.00000
l1:=listplot(lst1,linestyle=solid,legend="“ñ•ª–@"):
l2:=listplot(lst2,linestyle=dashdot,legend="Š„ü–@"):
display([l1,l2]);

8(2010”N“x)
restart;
Digits:=10:
f:=unapply(x^3-3*x+3,x);
plot(f(x),x=-5..5); 

     3
x -> x - 3 x + 3

x1:=-3;
df:=unapply(diff(f(x),x),x);
epsilon:=0.001: delta:=0.001:
for i from 1 to 100 do 
 x2:=x1-f(x1)/df(x1);
 printf("%10.5f, %10.5f\n", x2, f(x2));
 if (abs(x2-x1)<delta) then
   break;
 end if;
 if (abs(f(x2)-f(x1))<epsilon) then
   break;
 end if;
 x1:=x2;
end do:
printf("%d‰ñ–Ú, x = %.5f, f(x) = %.5f\n", i, x1, f(x1)); 

                              -3
       2
x -> 3 x - 3 
 -2.37500,   -3.27148
 -2.14001,   -0.38046
 -2.10458,   -0.00801
 -2.10380,   -0.00000
4‰ñ–Ú, x = -2.10458, f(x) = -0.00801
x3:=2;
df:=unapply(diff(f(x),x),x);
epsilon:=0.001: delta:=0.001:
for i from 1 to 100 do 
 x4:=x3-f(x3)/df(x3);
 printf("%10.5f, %10.5f\n", x3, f(x3));
 if (abs(x4-x3)<delta) then
   break;
 end if;
 if (abs(f(x4)-f(x3))<epsilon) then
   break;
 end if;
 x3:=x4;
end do:
printf("%d‰ñ–Ú, x = %.5f, f(x) = %.5f\n", i, x3, f(x3)); 
                              2
       2
x -> 3 x - 3 
  2.00000,    5.00000
  1.44444,    1.68038
  0.92887,    1.01482
  3.39447,   31.92927
  2.38301,    9.38346
  1.71449,    2.89625
  1.21672,    1.15109
  0.41805,    1.81891
  1.15275,    1.07357
  0.06453,    2.80668
  1.00400,    1.00005
-40.56380, -66619.86560
-27.05959, -19729.42266
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最終更新：2013年01月09日 18:48