Pract Mat2005

8/16/2019 Pract Mat2005

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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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◦

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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

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Pract Mat2005