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8/16/2019 Pract Mat2005
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LU
QR
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X = 1
X
1
X
1 × 1
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1.23 ∗ 10−2
10−368 10368
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(2.2204 ·10−16)
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2.2251 · 10−308
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1.7977 · 10308
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1 × 1
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n×n
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4 × 4
8 × 8
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m × n
n × k
n × m
m × k
5 × 5
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n × n
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m×n
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n × n
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m
×n
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z (x, y) = x2 + y2
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y(x) = 1 − x2
2 .
x1(t) = cos(2πt), x2(t) = sin(2πt).
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sin(2 ∗ x) x
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m × n
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{xi, f i}N i=0
i = 0 , 1 , ..., N
x0 < x1 < ... <
xN
f (x)
f (xi) = f i
i =
0 , 1 , ..., N
{xi , f i }N i=0
xi
f i
p(x) =N
j=0
f jk= j
x − xk
x j − xk
.
p(x) =N
k=0
f 012...k
k−1i=0
(x − xi) .
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f 012...k
1
f ij = f (xi, x j ) = f i−f jxi−xj
;
2
f ijk = f (xi, x j, xk) = f ij−f jk
xi−xk;
. . . . . . . . . . . . . . . . . . . . . . . .
k
f β 0,β 1...β k = f β0β1...βk−1
−f β1β2...βkx0−xk
.
N
N − 1
f (x) = exp(x)
−1 ≤ x ≤ 1
xk
f (x)
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10−15
f (x)
x
1.5 · 10−3
−5 ≤ x ≤ 5
•
f (x) = exp(x − x0)
x0 − 1 ≤ x ≤ x0 + 1 x0 = 100 1000 10000
•
•
x0
•
x0
• x0 = 10000
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f (x) = exp(x − x0)
x0 − 1 ≤ x ≤ x0 + 1
•
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exp(x)
−5 ≤x ≤ 5
N
N
•
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• N ≈ 22
•
•
xk = 5 sin
π
k − 1N − 1
− 12
, k = 1, 2, . . . , N
•
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•
•
sin(x)
T N (x)
xk = cos
π
2
2k − 1
N
, k = 1, 2, . . . , N .
•
|N k(x)| ≤ 1, |x| ≤ 1.
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•
x
y
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x1
y1
p
x1
x2
xn
aij
i = 1, . . . n
j = 0, . . . , N
N
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f (x) =
a10(x
−x1)N + a11(x
−x1)N −1 + . . .
+ a1 N −1(x − x1) + a1N , x < x2
a20(x − x2)N + a21(x − x2)N −1 + . . .
+ a2 N −1(x − x2) + a2N , x2 < x < x3
· · · · · ·ak0(x − xk)N + ak1(x − xk)N −1 + . . .
+ ak N −1(x − xk) + akN , xk < x < xk+1
· · · · · ·an−1 0 (x − xn−1)N + an−1 1(x − xn−1)N −1 + . . .
+ an−1
N −1
(x−
xn−1
) + an−1
N , xn−1
< x
x0
S 13
{x j , y j}
S 13
= {x j}
= {y j}
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S 13
{x j, y j}
sin
[0, π]
sin
sin
[−π/2, 3π/2]
•
•
•
S 13
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•
•
•
x0
xN
P 3
x0
x1
x2
x3
dP 3(x0)
dx =
dS (x0)
dx .
F j
δ ij
F 0
F 0(x0) = 1, F 0(x1) = 0, . . . , F 0(xN ) = 0.
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F k
F k
(x0
) = 0, . . . , F k
(xk−1
) = 0, F k
(xk
) = 1, F k
(xk+1
) = 0,
. . . , F 0(xN ) = 0.
f (x)
S (x) =N
j=0
f (x j)F j(x).
f (x) = cos(5x)exp(−0.3x2)
f (x) = sin(5x)exp(−0.3x2)
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t
0
U
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25 0
f
v
t
0
p
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f (x) = 0 x = F (x)
√ 2
x = 1 12
x2
x
x = 1 14
x2
x
1
1
2 ; 1
1
4 ; 1
3
8; 1
5
16; 1
13
32 ; 1
27
64; . . .
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f (x) = 0
[a, b]
f (x) = 0
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f (x) = 0
[a, b]
1. f = x3 − 2x − 5, [0, 3], 2. f = x3 − 0.001, [−1, 1],
3. f = ln(x + 2/3), [0, 1], 4. f =
(x − 2)
|x − 2|, [1, 4],
5. f = arctan(x) − π3
, [0, 5], 6. f = 1/(x − π), [0, 5].
x = F (x),
F : [a, b] → [a, b]
F (x)
q = supx∈[a,b]
|F (x)| < 1,
x∗
x∗ = limn→∞
xn, xn+1 = F (xn),
x0
[a, b]
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x = cos(x)
x∗ = 0.73908
10−5
k = 29
f (x) = 2x − 3 − ln(x)
f (x) = 0
f (x) = 0
•
f (x) = 0
x = F (x)
F (x) = (ln(x) + 3)/2
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•
|F (x)| < 1
x = 1.8
|F
(x)|
•
f (x) = 0
x = F (x)
10−8
x = 0.05
F (x)
f (x) = 0
F (x) = exp(2x − 3)
10−8
10−8
1. F (x) = cos(sin(x)); 2. F (x) = x2 − sin(x + 0.15);
3. F (x) = x5 − 3x3 − 2x2 + 2; 4. F (x) = tan(x).
xn
q = 1
x = F (x) = 2√
x − 1
x = 2
|xn − xn+1|
xn
x∗
f (x) = 0
f
∈ C 1
F (x) = x − f (x)/f (x)
x j+1 = x j − f (x j )
f (x j ).
f (x)
x j
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f ∈ C 2
|F (x)| < 1
f (x) = 0
x∗
τ k = |xk+1 − x∗|
τ k ≈ δf (xk)
f (x∗) ,
δf (xk)
f
xk
xk
x = x∗
s
xk+1 = xk − s f (xk)
f (xk) .
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x∗ = 2
x j+1 = x j − f jx j − x j−1
f j − f j−1,
f j = f (x j)
(√ 5 + 1)/2 ≈ 1.62
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|xk+1 − x∗| ≈ N (xk − x∗)2,
|xk+2 − x∗| ≈ N 1.62(xk − x∗)2.618.
log(
|x j
−x∗
|)
j
x j+1
x j
x j−1
x j−2
f (x)
(x j , f j)
(x j−1, f j−1)
(x j−1, f j−1)
p2(x) = f j + f j−1,j(x − x j) + f j−2,j−1,j(x − x j )(x − x j−1).
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x j+1
x j
f (a)
f (b)
f (c)
√ 2
f (x) = x2
a = −2
b = 0
c = 2
f (a) = f (c)
a = −2.001
b = 0
c = 1.999
x2 = 500
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0.7391
f (x) = 816x3 − 3835x2 + 6000x − 3125 .
E − e sin E = M .
e
E
M
E
E = M + 2∞
m=1
1
mJ m(me) sin(mM )
J m(x) m
E
M = 24.851090
e = 0.1
C (1)
∗X N + ... + C (N )
∗X + C (N + 1),
C
N + 1
0 + 1.0000i
0 − 1.0000i
z 3 − 1 = 0
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(x1, y1), (x2, y2), . . . , (xm, ym)
xi = x j
i = j
yi
F (x)
yi = F (xi) − ηi
i = 1, . . . , m
ηi
f j
F (x) =m
j=1
c jf j(x)
c j
A
m×n
A =
f 1(x1) f 2(x1) . . . f n(x1)
f 1(x2) f 2(x2) . . . f n(x2)
. . .
f 1(xm) f 2(xm) . . . f n(xm)
c
n
Ac =
f 1(x1) f 2(x1) . . . f n(x1)
f 1(x2) f 2(x2) . . . f n(x2)
. . .
f 1(xm) f 2(xm) . . . f n(xm)
c1
c2
. . .
cn
=
F (x1)
F (x2)
. . .
F (xn)
m
η = Ac − y
η
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min ||η||2
2 = ||Ac − y||2
2 =
n
i=1 (
n
j=1 aijc − yi)
2
ck
δ ||η||22
δck
=n
i=1
2(n
j=1
aij c − yi)aik
n
k = 1, 2, . . . , n
(Ac − y)T A = 0,
AT (Ac − y) = 0.
AT Ac = AT y,
AT A
A
(AT A)−1
c = (AT A)−1AT y = A+y.
A+
A
AA+A = A,
A+AA+ = A+,
(AA+)T = AA+.
A
A+ = A−1
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A+ = A−1
m×n
A
(−1, 2)
(1, 1)
(2, 1)
(3, 0)
(5, 3)
F (x) = c1 + c2x + c3x2
x
y
A
A
f 1 =
1
f 2 = x
f 3 = x2
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f (x) = c1 + c2x log x + c3ex
y = β 1 exp λ1t + β 2 exp(λ2t).
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β 1,2
λ1,2
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z = ax2 + bxy + cy2 + dx + ey + f
x, y
(x, y)
z = 0
b2 − 4ac
(x, y)
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•
•
•
f (x) = sin(x)/x, x = 0
1, x = 0
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f (x) =
1, x ≥ 1
x, −1 ≤ x < 1
−1, x < −1
•
df
dx
∆f
∆x
•
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•
•
•
∆x ∆xopt
∆x
•
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f (x) = sin x
x , f (x) = xα sin x, f (x) =
a1x2 + b1x + c1
a2x2 + b2x + c2
[−10, 10]
f (x) = limh→0
f (x + h) − f (x)
h
h df/dx x
∆f /∆x
x
f (x) = f (x + h) − f (x)
h
y(t) = sin2πF t
t F
•
y(t)
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•
y(t)
1015
•
y(t)
•
f (x) =
x sin(1/x), x = 0
0, x = 0
x
h
h
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•
f (x + h) − f (x)
h = f (x) +
1
2!f (x)h +
1
3!f (x)h2 + . . .
•
∆f ∆x
h
•
•
h
•
h
∆f ∆x = f (x + h
2)−
f (x−
h
2)
h
•
h
•
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h
•
h
101
102
h
•
h
•
•
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•
•
•
h
•
d2f (x)/dx2
df (x)/dx
d2f (x)
dx2 ≈ f (x + h) − f (x)
h =
f (x + 2h) − 2f (x + h) + f (x)
h2
x
x + h
x
d2f (x)
dx2 ≈ f (x + h) − 2f (x) + f (x − h)
h2
•
h
h
h
•
•
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• h
•
•
h
• h
•
•
•
h
h
•
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f i
f (x)
xi
xi
df (x)/dx
∆f i/∆xi
xi
∆f i = f i+1 −f i
∆x = xi+1 − xi
df (x)
dx ≈ ∆f i
∆xi
= f i+1 − f ixi+1 − xi
= f (x + h) − f (x)
h
•
f i
f (xi)
df (x)/dx
•
•
[a, b]
•
•
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•
•
•
•
•
•
•
f (x) = sin(ω(x)x)
ω(x)
•
•
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I =
b a
f (x)dx.
•
[a, b]
xk
k = 1, . . . , N
•
f (x)
xk
f (x)
•
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•
I
I = Q(b)
−Q(a).
π 0
sin(x)dx,
= 2, . . . , 50
b a
f (x)dx ≈N
k=1
λkf (xk),
λk
λk =b
a
LkN −1(x)dx,
LkN −1
k
{x j}N j=1
λk
f (x j) = δ jk
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N k=1
= b − a.
P N (x)
x
P N (x) =N
k=0
ckxk.
P N
xm
m = 0, . . . , N − 1
ck
N
k=0
ck
b
a
xk+mdx
≡
N
k=0
ck
bk+m+1 − ak+m+1
k + m + 1
= 0, m = 0, 1, . . . , N
−1.
cN = 1
Ac = b,
c =
c0
c1
cN −1
, b = −
a1
a2
aN
, A =
a1 a2 · · · aN
a2 a3 · · · aN +1
aN aN +1 · · · a2N −1
,
a = b − a
.
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N − 1
P N (x)
P N (xk) = 0.
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N
[a, b]
b − a
λk = λN −k
N
N
N
xk = a + k − 1
N − 1(b − a), k = 1, . . . , N ,
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E X
X −1 = E X
X
X
A
B
A
A∗B∗A = A
B∗A∗B = B
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Ax = b
x = inv(A) ∗ b
Ax = B
A
m × n
B
m
x
n
A
n
A
n
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A
m
k
n
x = x0 + c1x1 + c2x2 + . . . + ckxk
x0
k
x1
x2
. . .
xk
A
c1
c2
c3
x0 = A b
x1 x2 . . . xk
x0
x0
k
x1
x2
. . .
xk
k − 1
1
k
A
m
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LU
QR
LU
QR
X
LR
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•
U
L
X = L ∗ U
•
U
L
P
L ∗ U
P ∗ X
•
•
QR
QR
Q
R
• R
X Q X = Q ∗ R
•
E
R
Q
X ∗ E = Q ∗ R
E
abs(diag(R))
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•
E
Q ∗ R = X (:, E )
E
abs(diag(R))
•
QR
Q
Q∗
Q = QQ∗
= 1
x = Qx
Q
(x, y) = (Qx, Qy) = (x, y)
x, y
Q R
QR X
Q
R
QR
X
j
X
Q
R
X
X
j
x
Q
R
QR
X
x
X
j
x
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dy(t)
d t = f (t, y(t))
t ∈ [a, b] y(a) = y0
∆t = h
tk = tk−1 + h = a + kh,
yk y(tk) = y(a + kh), k = 0, 1, . . . , N ,
f k = f (tk, yk) y
(xk).
yk+1 − y(xk+1)
y(x0), . . . , y(xk)
y0, . . . , yk
yk+1 − y(xk+1)
y(xk+1) yk+1 − yk
h
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yk+1
= yk
+ h f (tk
, yk
), k = 0, 1, . . . , N .
h
y = 3 + t − y , t ∈ [0, 1], y(0) = 1 .
m
m
h
N = (b − a)/h
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m
y(t)
•
yk+1 = yk + h
2
f (tk, yk) + f (tk + h, yk + h f (tk, yk))
;
•
yk+1 = yk + h f
tk +
h
2, yk +
h
2 f (tk, yk)
;
•
yk+1 = yk + h
4
f (tk, yk) + 3f
tk +
2
3h, yk +
2
3hf (tk, yk)
;
•
yk+1 = yk + h
6 (k1 + 4k2 + k3) ,
k1 = f (tk, yk), k2 = f
tk +
h
2, yk +
k1
2
,
k3 = f (tk + h, yk + 2k2 − k1) .
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[t0, tfinal]
dy
dt = y − t2 + 1, t ∈ [0, 4], y(0) = 0.5
h = 1
h = 0.1
dy
dt = sin(t), t ∈ [1, 5], y(1) = 0
h = 1
h = 0.1
dy
dt =
2y
t + t
2
exp(t); t ∈ [1, 3], y(1) = 0
h = 1
h = 0.1
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•
•
•
•
•
•
•
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•
y = f (t, y)
F (y(a), y(b), p) =
0
•
y = F (t, y).
M (t)y = F (t, y),
M
M (t, y)y = F (t, y).
M (t, y)y
−F (t, y).
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•
•
[t0, tfinal]
t0, tl, . . . , tfinal
tspan = [t0, tl, . . . , tfinal]
• y0
• p1, p2, . . .
F
• T, Y
Y
T
•
y = F (t, y)
tspan
y0
@F
m
Y
T
•
1e−3
1e−6
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•
p1, p2, . . .
F
options = []
m
•
of f
on
|||| <= max(RelTol||y||,AbsTol).
•
10−3
(i) <= max(RelTol ∗ |y(i|,AbsTol(i));
•
10−6
•
1
4
•
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•
of f
•
My = f (t, y)
M
•
y
none
weak
strong
none
M (t)y = F (t, y)
weak
strong
M (t, y)
y
weak
•
•
tspan
x(t) = −x(t) t ∈ [0, 10π]
x(0) = 1
x(0) = 0
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|x(0)−x(10π)|
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dyi
dt = f i(y), i = 1, . . . , n ,
F F ij = df i/dy j F
|λi| 1
h
y = y 2
−y3, y(0) = δ, t
∈[0, 2/δ ] .
δ
δ
δ = .01
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RelTol = 10−4
1/δ
δ
t > 1/δ
10−4
t > 1/δ
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1
y + log
1
y − 1
=
1
δ + log
1
δ − 1
− t .
y
y(t) =
1
W (aea−t) + 1 , a =
1
δ − 1,
W
W
π
F
δ = .01
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δ = r0
y1 = y2, y2 = ε(1 − y1)2y2 − y1, ε = 100,
y1(0) = 0
y2(0) = 1
→
→
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m
y1 = y(1)
y2 = y(2)
ε
y|y| = 0, y(0) = 0, y(4) = −2.
[0, 4]
y = −|y|, y(0) = 0, y(4) + 2 = 0.
m
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d
dtF (t) = G(t, x1, x2, . . . , xn)
F (t)
(x1, x2, . . . , xn)
G
1e − 3
1e− 6
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•
•
•
•
•
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x1 = x3
x2 = x4
x3 = − M 1(x1−C 1x)
((x1−C 1x)2+(x2−C 1y)2)3/2 − M 2(x2−C 2x)
((x1−C 2x)2+(x2−C 2y)2)3/2
x4 = − M 1(x1−C 1y )
((x1−C 1x)2+(x2−C 1y)2)3/2 − M 2(x2−C 2y)
((x1−C 2x)2+(x2−C 2y)2)3/2
M 1 = 50
M 2 = 0
C 1 = (5, 0)
C 2 = (0, 10)
M 2 = 0
M 1
M 2
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ma = G + F
m V = mg − ρ| V |2
2 S
V
| V | .
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ρ
m
S
V x = −ρ√ V 2x +V 2y
2m SV x
V y = −g − ρ√
V 2x +V 2y
2m SV y
x(t) = x0 +
t t0
V x(τ )dτ
y(t) = y0 +t
t0
V y(τ )dτ
t = 0
xi =i
k=2V kx (tk − tk−1)
yi =i
k=2V ky (tk − tk−1)
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t
V x = −ρ√
V 2x +V 2y
2m SV x
V y = −g − ρ√
V 2x +V 2y
2m SV y
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x = V x
y = V y
V x = −ρ√
V 2x +V 2y
2m SV x
V y = −g − ρ√
V 2x +V 2y
2m SV y
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m1, m2
1, 2
x1, y1
x2, y2
θ1, θ2
x1 = 1 sin θ1; y1 = −1 cos θ1;
x2 = 1 sin θ1 + 2 sin θ2; y2 = −1 cos θ1 − 2 cos θ2
(m1 + m2)1θ1 + m22θ2 cos(θ1
−θ2) =
−g(m1 + m2)sin θ1
−m22
θ22 sin(θ1
−θ2)
m21θ1 cos(θ1 − θ2) + m22θ2 = −gm2 sin θ2 + m21 θ2
2 sin(θ1 − θ2)
m1 = m2 = 1
1 = 2 = 1
u(t) = [θ1, θ2, θ1, θ2]T , c = cos(θ1 − θ2, s = sin(θ1 − θ2),
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M u = f,
M =
1 0 0 0
0 1 0 0
0 0 2 c
0 0 c 1
, f =
u3
u4
−g sin u1 − su23
−g sin u2 + su24
.
M
f
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x, y
m1,2
l1,2
t
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∆t
∆t
[0, T ]
t = T
t = 0
Rn
dx(t)
dt
= f (x), x = (x1, x2, . . . , xn) .
M
x0 ∈ M
x(t, x0) ⊂ M
x0
M
M
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x0, x0
x(t, x0)
t → ∞
U (t) = x4
4 +
(q (t) − 1)
2 x2,
q (t)
G
F = −αdx(t)
dt
α
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d2
dt
x(t) + 0.7α d
dt
x(t) + x(t)3 + q (t)−
1x(t) = 0 .
ddt
q (t) = α−0.16q (t) + x(t)2
.
α = 0.1
α = 10
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α = 0.1
α = 10
R3
dx1
dt = −βx1 + x2x3,
dx2
dt = −σx2 + σx3,
dx3
dt = −x1x2 + ρx2 − x1.
x1(t)
σ
ρ
β
x(t) = Ax, A =
−β 0 x2
0 −σ σ
−x2 ρ −1
η = y2(t) η = ±√ β (ρ − 1)
x0 =
ρ − 1
η
η
, x(t, x0) = 0, ∀t .
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β = 8/3
σ = 10
ρ
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ρ
ρ
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x1 = −(x2 + x3), x2 = x1 + ax2, x3 = b + x1x3 − cx3
a = 0.17
b = 0.4
c = 8.5
3x + y + x = 1
x + 4y + 3x = 0x(0) = 0, y(0) = 0
x − x − 2y = t
y − 2x − y = tx(0) = 2, y(0) = 4
2x − x + 9x − y − y − 3y = 0, x(0) = 1, x(0) = 1
2x + x + 7x − y + y − 5y = 0, y(0) = 0, y(0) = 0
y + y − 2y = et, y(0) = −1, y(0) = 0
x − x + y + z = 0, x(0) = 1, x(0) = 0
y − y + x + z = 0, y(0) = 0, y(0) = 0
z − z + x + y = 0, z (0) = 0, z (0) = 0
y(4) + y(3) = cos t, y(0) = 0, y(0) = 0, y(0) = 0, y(3)(0) = 2
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y + y2 = x2
y(4) + 4y = t2, y(0) = 0, y(0) = 1, y(0) = 2, y(3)(0) = 3
4y(3) − 8y(2) − 2y = 8et, y(0) = 1, y(0) = 1, y(0) = 1
y(3)
− 6y(2)
+ 11y − 6y = 0, y(0) = 0, y(0) = 0, y(0) = 10
y(4) + 2y(2) = t sin t, y(0) = 0, y(0) = 10, y(0) = 0.1, y(3)(0) = 0.01
x − x + 2y = 0, x(0) = 0, x(0) = −1
x
−2y = 0, y(0) = 1/2
V (P ) = V 0(1 + 2λ(P ))
λ(P ) = P H −P P H −P atm
H = 500m
P H
H
P atm
f = ρv2
2 S
ρ
v
S
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q (t)
Ld2
q dt2 + R dq dt + q C = V (t)
V (t) = LdI
dt + RI +
q
C
I = dqdt
V (t) = 3
V (t) = 3 − a exp(t) a
t0 = 10
y = −g + α(t)
m , y(0) = 700, y(0) = 0 .
g = 9.81 m/s2
α(t)
α(t) = k1 y(t)2 t < t0
α(t) = k2 y(t)2
k1 = 1/150
k2 = 4/150
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(1 − v)y + a2
3yy + b
1
6y = 0,
y + G4
3πρr3(y)y−2 = 0,
r(y) =
|y|, |y| < R,
R, |y| > R.
x(0) = y (0) = 0
x(6) = 20, y(6) = 0
ρ
ρ0
F = (ρ − ρ0) 43
πr3g.
r g
F c = −6πηrv
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η
v
y = Ay + B.
y(0) = y0, y(0) = y 0
d2x
dt2 + w2x = F 0sin(γ t), x(0) = 0, x(0) = 0
w2π
γ 2π
ν = 112π
ν = 132π
F 0 = 48