现代数值数学和计算习题答案(同济版)
时间：2025-07-13
1.
22
π
113
:
,
22
113=3.14159292035398···,π=3.14159265358979···.
,
22
355
:
22π
≤5×10 4
5×10 4
355π
≤9×10 8,
9×10 8
2.
100cm50cm,
0.15cm2.
:
xcm
2
50±
x
2 2
0.15,
x=0.002. x 2
,3.
x
x
√n
n.
:
Taylor
,
√n
x+
1
x
n2
n
ξ2
(x x)2,
1
x
ξ
x
x
√ξ
n
xx
=1
n 1
x
ξ
2
(x x)2,
4.MatLab
1.7977e+308,
:[1e 8,4e+3],[1e+308,1.5e+308],[ 1.4e+308,1.4e+308].(
)
a+b
:
2
a
2
(
b
[1.0e 308,1.01e 308]
)
5.
[a,b]
n
x0=a,
xk=xk 1+h,k=1,···,n
xk=a+kh,
k=0,···,n
h=b a
a,
xk+1=xk+h
8.888888888888888e-0011.777777777777778e+0002.666666666666667e+0003.555555555555555e+0004.444444444444445e+0005.333333333333334e+0006.222222222222223e+0007.111111111111113e+0008.000000000000002e+000
A
sort
A
MatLab
A
(
)
n=size(A,1);fori=1:n-1,
forj=i+1:n,
ifabs(A(i,i))<abs(A(j,j)),
A(:,[i,j])=A(:,[j,i]);A([i,j],:)=A([j,i],:);endendend
A
7.
x=(x1,x2,...,xn)
Euclide1
x =
nx2i
i=1
2
2
=
,
MatLab
x = x(1:n 1) ,x(n) ,
x=(x1,x2)
1
x1,
1,x2
functions=nrmv(x)x=abs(x(x~=0));n=length(x);ifn==0,
s=0;elseifn==1,
s=abs(x);else
scale=0;ssq=1;forj=1:n,
if(scale<x(j)),
ssq=1+ssq*(scale/x(j))^2;scale=x(j);else
ssq=ssq+(x(j)/scale)^2;endend
s=scale*sqrt(ssq);end
BLAS
dnrm2
http://www.77cn.com.cn/blas/dnrm2.f.
8.
ax2√
+bx+x=0
x= b±2a
x=2ca=0cb2= 0
4ac
.a
c10
6e+154
-4e+154
1
1
1
-4
-1e155
1e+155
5
MatLab
:
b2 4ac
1
c=0
b
(Vieta)
)
a=0
b=0
(
function[x,flag]=root2qe(a,b,c)
ifnargin==1,
c=a(3);b=a(2);a=a(1);end
ifa~=0,%Solve2-orderequation
M=max(abs([a,b,c]));%scalinga=a/M;b=b/M;c=c/M;s=sign(b);
ifs==0,s=1;end
x1=(-b-s*sqrt(b^2-4*a*c))/2/a;x2=c/(a*x1);%avoidlargeerrorx=[x1x2]’;
flag=’twosolutions’;elseifb~=0,%Solve1-orderequation
x=-c/b;
flag=’onesolution’;elseifc~=0,%nonzeroconstant
x=[];
flag=’noxisasolution’;else%equality0=0
x=[1];
flag=’allxaresolutions’;end
6
a
10
6e+154
1
1
-4
-1e155
c
-2.00000000000000
-4e+154
-1
1
2.00100000000007
1e+155
x2
0.33333333333333
0.0000100000
0.14005494464026
9.
x1=1
1 k4xk=
12,xk+1=2.25xk 0.5xk 1.
3
c14 2+c222=1
c1=4
3
1
4
3(
2p 1 52 ≤
fl(x)
x
0.64421768720.6518337710
2-9
cosx
sin0.705
cos0.702
:
sin0.705
L1(x)=
x 0.70
=0.76160838x+0.1110918212
8
0.70 0.71×0.6442176872
x=0.705
sin0.705≈L1(0.705)=0.6480257291.
cos0.702
L1(x)=
x 0.70
= 0.64803113x+1.2184639782
0.70 0.71×0.7648421872
x=0.702
2.
cos0.702≈L1(0.702)=0.76354612494.
(
2-10).
0.4
0.6
sinx
0.47943
0.64422
0.6 0.5
×0.56464+x 0.6(0.57891(0.5 00.+
.6)5)××(0(0..557891 0.7)
0.7)
×0.47943(0.7 0.5)×(0.7 0.6)×0.64422
≈
0.54714.
9
x0=0.4,x1=0.5,x3=0.6,
sin0.57891
sin0.57891≈
=
L2(0.57891)
(0.57891 0.5)×(0.57891 0.6)(0.5 0.4)×(0.5 0.6)(0.57891 0.4)×(0.57891 0.5)
×0.47943
+
xf(x)
19
432-11
(1)(2)
f(4.2)
:
(1)
f(xk)1
8
9
14
23
-10
3
128
259
3
-2.75
-8
8.5
34.5
1.875
(2)
N(x)=1+8x+3x(x 1) 2.75x(x 1)(x 2)
+1.875x(x 1)(x 2)(x 4),
x=4.2
f(4.2)≈N(4.2)=4.696.
4.
p(x)=x4 x3+x2 x+1
-2p(x)
(
2-12).2
05
1
61
x
31
-112-13
111
3
:
2-12
2-13
p(x) q(x)
x=0,±1,±2
p(x) q(x)=Ax(x 1)(x+1)(x 2)(x+2),
x=3
A=31
120
x(x 1)(x+1)(x 2)(x+2).
5.
(
2-14)
-2y
4
16
2
-4
:
x= 1,0,1,2
L(x)=4+7(x+1) x(x+1) x(x 1)(x+1).
x= 2,3
y=L(x),
x= 2,3
y=L(x),
y=1, 4,
:
f(xk)-2
3
-1
7
5
1
-3
2
-17
3
-1-1-1
21!
k=0
,f(x)=e x,f(21)(x)= e x,ξ
20
(x xk),
2
|f(21)(ξ)|=|e ξ|<1.
12
x0,x1,···,x20
k=10,
max|(x xk)(x x20 k)|≤1;
x
k=10
,
|x x10|≤1.
|R20(x)|<
1
9,0≤t≤9.
R(x)=R(x0+th)=
f(10)(ξ)
x10
|f(10)(ξ)|<9!.
,
R¯(t) <
1
10·910
=
1
2
Mh.8
f(x)=sin(x),
h
13
1
2!
(x x0)(x x1),
x
x0
x1
,
1
|(x x0)(x x1)|≤
12!
8
.
f(x)=sin(x),
1
|f (x)|=| sin(x)|≤ …… 此处隐藏：1891字，全部文档内容请下载后查看。喜欢就下载吧 ……
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