Esercizi di Calcolo Numerico

Esercizio 3

x₁ + 1.001 x₂ = 2.001x₁ + x₂ = 2

A = ^{( 1 1.001 )}     ( 1 1 )

A^-1 = ¹/_{1 - 1.001} ^{( 1 -1.001 )}                    ( -1 1 )

||A|| = 2.001

||A^-1|| = 2001

K(A) = 2.001 * 2001 = 4.204 x 10³

problema mal condizionato

Esercizio 6

10^-6 x₁ + 10^-6 x₂ = 2 * 10^-6x₁ + 2 x₂ = 3

A = ^{( 10-6 10-6 )}    ( 1 2 )

A-1 = 1/2 * 10-6 - 10-6 ( 2 -10-6 )                    ( -1 10-6 )

||A|| = 3

||A^-1|| = 2000001

K(A) = 6 * 10⁶

||A|| = 2.001

||A⁻¹|| = 2001

k(A) = 2.001⋅2001 ≈ 4.004⋅10³ (problema mal condizionato)

ESERCIZIO 6

¹⁰Cx₁ + ¹⁰Cx₂ = 2⋅10⁶

x₁ + 2x₂ = 3

A = ( ¹⁰CU__¹⁰C)

A⁻¹ = ¹ ∕ _2⋅10⁶ ( -2 ^-10C)

||A|| = 3

|| Δ⁻¹|| = 2000001

k(A) = 6⋅10⁶

problema ben condizionato

D = (_{10⁶⦺ ())}

à = DA = ()

|| Ã || = 3

Ã⁻¹ = ( ⦺1 ^{2 ﹣1})

|| Ã⁻¹ || = 3

k(Ã) = 9 problema non equilibrato

ESERCIZIO 3

A⁽¹⁾ = ( 1 0 0 0)( 0 1 0 2)( 0 1 0 2)b = ( 2 )( 3 )( 3 )

J₁ = ( 1 0 0 )( 0 1 0 )( 0 0 1 )

Δ⁽¹⁾ = J_A = ( 1 0 0 )( 0 1 0 )( 0 0 2 )

b⁽¹⁾ = J_b =( 2 )( 2 )( 3 )

J₂ = ( 0 0 0 )( 0 1 0 )( 0 0 1 )

A⁽²⁾ = J₂A⁽¹⁾ = ( 0 0 0 )( 0 1 0 )( 0 0 2 )

b⁽²⁾ = ( 2 )( 1 )( 1 )

J₃ = ( 1 0 0 )( 0 0 1 )( 0 0 0 )

A⁽⁴⁾ = J₅A₃ = ( 0 0 0 )( 1 0 0 )( 0 0 0 )

b⁽³⁾ = ( 1 )( 2 )( 1 )

J₄ = ( 0 1 0 )( 1 0 0 )( 0 0 0 )

D = Δ⁽⁵⁾J₆(Δ⁽⁴⁾)^-1 =( 0 0 0 )( 1 0 0 )( 0 0 0 )

b⁽⁴⁾ = J_b(b³)^-1 = ( 2 )( 1 )( 1 )

y = b⁽⁵⁾J₆b⁽⁵⁾ =( 1 )( 1 )( 1 )

D x = y

x₁ = ( )x₂ = ( )x₃ = ( )x₄ = ( )

α = (1, 1, 1, 1) ^T

A = JD

J = J^-1J₅J₃J₁J₂^-1 =( 0 0 1 )( 0 1 0 )( 0 0 0 )

= ( 0 0 1 )( 0 1 0 )( 0 0 0 )- ( 0 1 0 )( 0 0 1 )( 1 0 0 )= ( 1 0 0 )( 0 1 0 )( 0 0 2 )

C = AΔ = ( 1 0 1 ) ( 0 0 0 ) ( 0 1 0 )

Δ - ( 1 0 0 )( 0 1 0 )( 0 0 1 ) =( 1 0 0 )( 0 1 0 )( 0 1 0 )

Esercizio 12

Esercizio 1

• x₀ = -1
• x₁ = 1
• x₂ = 2
• x₃ = 2,5
• y₀ = 1,5
• y₁ = 2
• y₂ = 4,5
• y₃ = 1,5

L₀ = (x - x₁)(x - x₂) / (x₀ - x₁)(x₀ - x₂)

L₁ = (x - x₀)(x - x₂) / (x₁ - x₀)(x₁ - x₂)

L₂ = (x - x₀)(x - x₁) / (x₂ - x₀)(x₂ - x₁)

1 / ((x + 1)(x - 2)) = 1 / 6

1 / ((x + 1)(x - 2)) = 1 / 6

1 / ((x + 1)(x - 2)) = 1 / 6

P₂(x) = 1/5 . (x - 1)(x - 2) / 6 + 1 / (x + 1)(x - 2) / 6 + 2 (x² - 1) = 1 / 6

= x² - 3x + 2 / 4 - (x² - x - 2) + 2 x² - x / 3

= 3 - 12 + 8 x² -3 + 4 / 12 + x / 4 + 1 / 2 / (x + 2) = 2 / 3 = - 1 / 12 x² x / 4 - 11 / 6

MATRICI

1a)

A = ²⁄_△ ³⁄₂

B = ¹⁄₂ 1 0

(A+B-C)(A+B-C)^T

(A+B-C)^T(A+B-C)

(A+B)+C

AB

(AB)^T = B^TA^T

BA

||A|| = 2+2+3 = 7

||B|| = 1+2+1 = 4

||C|| = 1+2 = 3

ESERCIZI 10

Esercizio 1

f'(x)₁ = ¹/_n [f(x+n) - f(x)]

f'(x)₁ = -¹/_n [f(x) - f(x-n)] indietro

f'(x)₂ = -¹/_2n [f(x+n) - f(x-n)]

f''(x)₁ = ^{f(x+n) - 2f(x) + f(x-n)}/_n²

• h = 0.2

avanti

f'(x₀) = ¹/_0.2 [f(1.6) - f(1.4)] = ¹/_0.2 [2.3756 - 1.9043] = 2.3565

indietro

f'(x₀) = ¹/_0.2 [1.9043 - 1.5095] = 2.1974

centrale

f'(x₀) = ¹/_{2 · 0.2} [2.3756 - 1.5095] = 2.16525

f''(x₀) = ^{2 · 1.3756 - 2 · 1,9043 + 1,5095}/_0.2² = 1,9125

• h = 0.1

avanti

f'(x₁) = -¹/_0,1 [2,1293 - 1,9043] = 2.25 2.25

indietro

f'(x₁) = ¹/_0,1 [1,9043 - 1,6982] = 2.059

centrale

f'(x₁) = ¹/_2h [2,1235 - 1,6982] = 2,1545

f''(x₁) = ^{2 · 1,2933 - 2 · 1,9043 + 1,6984}/_0,1² = 1,91

CAPITOLO 4

n = 8 F(x) = x(2πt−x) xₖ = 2πt x m = 4

x₀ = 0 x₁ = π/4 x₂ = π/2 x₃ = 3/4π x₄ = π x₅ = 5/4π x₆ = 3/2π x₇ = 7/4π x₈ = 2π

y₀ = 0 y₁ = 3/16π² y₂ = 1/2π² y₃ = 15/16π² y₄ = π² y₅ = 15/16π² y₆ = 1/2π² y₇ = 3/16π² y₈ = 0

β₀x = α₀ + α₁cosx + β₁sinx + α₂cos2x + β₂sin2x + α₃cos3x + β₃sin3x + α₄cos4x

= 2/π

x₁ = 3/16π²(1/2) - 15/16(√2/2) - 15/16 = 2/2π

θ₁ = 15/16(√2/2) - 3/16 = 0

θ₂ = 3/2π(1/2) - 15/16(√2/2) + 15cos(5/4π) - 3/8cos(3π) - 1 = -π²/8

x₃ = 7/16π²(√2/2) - 3/8 = -2(√2/8)π²

β₂ = 1/2

x₄ = 1/2(3/16π²) + 3 = 0

β₃ = 7/16π²(√2/2) = 0

β₀x (x) = -π₂/32π + -π₂/8 cos2x + 2/2π₃/x₃

CAPITOLO 1

ESERCIZIO 2

• (110011)₂ p = 6
• −(11.11)₂ p = 2
• (10.1)₁₂ r = 3
• (0.11)₂ = (0.5)₁₀
• (0,0001)₁₂ = 0 - .1/2¹ + .2/2² - .1/2⁴ - (1/10)₁₀

Esercizio 5

0.003x₁ + 59.14x₂ = 59.17

5.291x₁ + 6.13x₂ = 66.78

R = (0.003   59.140   0.1043 ⋅ 10⁴)

y = (59.17)(0,01014 ⋅ 10⁴)

L₁ = (0.1963 ⋅ 10⁴   0   1)

0.003x₁ + 59.14x₂ = 59.170.1043 ⋅ 10⁴x₂ = 0.1046 ⋅ 10⁶

{ x₂ = 0.1001 ⋅ 10¹x₁ = -9.713

P₁ = (1 0)  (0 1)

P.A = (5.291   -6.130.003   59.14)

P_b = (46.78) (59.17)

L₁ = (1   0   0,5670 ⋅ 10²   1)

R = (5.291   -6.130   -0,2382 ⋅ 10³)

y - (46.78)  (0.2912 ⋅ 10⁴)

5.291x₁ - 6.13x₂ = 46.78

-0,2884 ⋅ 10³x₂ = 0.2712 ⋅ 10⁴

{ x₂ = -0,9403 ⋅ 10x₁ = -0,2054 ⋅ 10

CAPITOLO 2

Esercizio 2

x₀ = -2

x₁ = 0

x₂ = 1

x₃ = 6

P_l(x) = 4x⁴ ; < x = 1, 2, 3

P_c(x) = - ^x2⁄₃ - 4 x + ^{64 x4}⁄_{64 x - 0,3 x²}