El ajuste de datos a fórmulas: Apéndice: Coeficientes de Polinomios Ortogonales

n=2:

Polinomio linear:

_Para X=1: Z_{_I} = -1
_Para X=2: Z_{_I} = 1

_ΣZ² = (-1)² + (1)² = 2
_λ = 2

n=3:

Polinomio linear:

_Para X=1: Z_{_I} = -1
_Para X=2: Z_{_I} = 0
_Para X=3: Z_{_I} = 1

_ΣZ² = (-1)² + (0)² + (1)² = 2
_λ = 1

Polinomio cuadrático:

_Para X=1: Z_{_II} = 1
_Para X=2: Z_{_II} = -2
_Para X=3: Z_{_II} = 1

_ΣZ² = (1)² + (-2)² + (1)² = 6
_λ = 3

n=4:

Polinomio linear:

_Para X=1: Z_{_I} = -3
_Para X=2: Z_{_I} = -1
_Para X=3: Z_{_I} = 1
_Para X=4: Z_{_I} = 3

_ΣZ² = (-3)² + (-1)² + (1)² + (3)² = 20
_λ = 2

Polinomio cuadrático:

_Para X=1: Z_{_II} = 1
_Para X=2: Z_{_II} = -1
_Para X=3: Z_{_II} = -1
_Para X=4: Z_{_II} = 1

_ΣZ² = (1)² + (-1)² + (-1)² + (1)² = 4
_λ = 1

Polinomio cúbico:

_Para X=1: Z_{_III} = -1
_Para X=2: Z_{_III} = 3
_Para X=3: Z_{_III} = -3
_Para X=4: Z_{_III} = 1

_ΣZ² = (-1)² + (3)² + (-3)² + (1)² = 20
_λ = 10/3

n=5:

Polinomio linear:

_Para X=1: Z_{_I} = -2
_Para X=2: Z_{_I} = -1
_Para X=3: Z_{_I} = 0
_Para X=4: Z_{_I} = 1
_Para X=5: Z_{_I} = 2

_ΣZ² = (-2)² + (-1)² + (0)² + (1)² + (2)² = 10
_λ = 1

Polinomio cuadrático:

_Para X=1: Z_{_II} = 2
_Para X=2: Z_{_II} = -1
_Para X=3: Z_{_II} = -2
_Para X=4: Z_{_II} = -1
_Para X=5: Z_{_II} = 2

_ΣZ² = (2)² + (-1)² + (-2)² + (-1)² + (2)² = 14
_λ = 1

Polinomio cúbico:

_Para X=1: Z_{_III} = -1
_Para X=2: Z_{_III} = 2
_Para X=3: Z_{_III} = 0
_Para X=4: Z_{_III} = -2
_Para X=5: Z_{_III} = 1

_ΣZ² = (-1)² + (2)² + (0)² + (-2)² + (1)² = 10
_λ = 5/6

Polinomio cuártico:

_Para X=1: Z_{_III}_{_I} = 1
_Para X=2: Z_{_III}_{_I} = -4
_Para X=3: Z_{_III}_{_I} = 6
_Para X=4: Z_{_III}_{_I} = -4
_Para X=5: Z_{_III}_{_I} = 1

_ΣZ² = (1)² + (-4)² + (6)² + (-4)² + (1)² = 70
_λ = 35/12

n=6:

Polinomio linear:

_Para X=1: Z_{_I} = -5
_Para X=2: Z_{_I} = -3
_Para X=3: Z_{_I} = -1
_Para X=4: Z_{_I} = 1
_Para X=5: Z_{_I} = 3
_Para X=6: Z_{_I} = 5

_ΣZ² = (-5)² + (-3)² + (-1)² + (1)² + (3)² + (5)² = 70
_λ = 2

Polinomio cuadrático:

_Para X=1: Z_{_II} = 5
_Para X=2: Z_{_II} = -1
_Para X=3: Z_{_II} = -4
_Para X=4: Z_{_II} = -4
_Para X=5: Z_{_II} = -1
_Para X=6: Z_{_II} = 5

_ΣZ² = (5)² + (-1)² + (-4)² + (-4)² + (-1)² + (5)² = 84
_λ = 3/2

Polinomio cúbico (Z_{_III}):

_Para X=1: Z_{_III} = -5
_Para X=2: Z_{_III} = 7
_Para X=3: Z_{_III} = 4
_Para X=4: Z_{_III} = -4
_Para X=5: Z_{_III} = -7
_Para X=6: Z_{_III} = 5

_ΣZ² = (-5)² + (7)² + (4)² + (-4)² + (-7)² + (5)² = 180
_λ = 5/3

Polinomio cuártico:

_Para X=1: Z_{_III}_{_I} = 1
_Para X=2: Z_{_III}_{_I} = -3
_Para X=3: Z_{_III}_{_I} = 2
_Para X=4: Z_{_III}_{_I} = 2
_Para X=5: Z_{_III}_{_I} = -3
_Para X=6: Z_{_III}_{_I} = 1

_ΣZ² = (1)² + (-3)² + (2)² + (2)² + (-3)² + (1)² = 28
_λ = 7/12

n=7:

Polinomio linear:

_Para X=1: Z_{_I} = -3
_Para X=2: Z_{_I} = -2
_Para X=3: Z_{_I} = -1
_Para X=4: Z_{_I} = 0
_Para X=5: Z_{_I} = 1
_Para X=6: Z_{_I} = 2
_Para X=7: Z_{_I} = 3

_ΣZ² = (-3)² + (-2)² + (-1)² + (0)² + (1)² + (2)² + (3)² = 28
_λ = 1

Polinomio cuadrático:

_Para X=1: Z_{_II} = 5
_Para X=2: Z_{_II} = 0
_Para X=3: Z_{_II} = -3
_Para X=4: Z_{_II} = -4
_Para X=5: Z_{_II} = -3
_Para X=6: Z_{_II} = 0
_Para X=7: Z_{_II} = 5

_ΣZ² = (5)² + (0)² + (-3)² + (-4)² + (-3)² + (0)² + (5)² = 84
_λ = 1

Polinomio cúbico:

_Para X=1: Z_{_III} = -1
_Para X=2: Z_{_III} = 1
_Para X=3: Z_{_III} = 1
_Para X=4: Z_{_III} = 0
_Para X=5: Z_{_III} = -1
_Para X=6: Z_{_III} = -1
_Para X=7: Z_{_III} = 1

_ΣZ² = (-1)² + (1)² + (1)² + (0)² + (-1)² + (-1)² + (1)² = 6
_λ = 1/16

Polinomio cuártico:

_Para X=1: Z_{_III}_{_I} = 3
_Para X=2: Z_{_III}_{_I} = -7
_Para X=3: Z_{_III}_{_I} = 1
_Para X=4: Z_{_III}_{_I} = 6
_Para X=5: Z_{_III}_{_I} = 1
_Para X=6: Z_{_III}_{_I} = -7
_Para X=7: Z_{_III}_{_I} = -3

_ΣZ² = (3)² + (-7)² + (1)² + (6)² + (1)² + (-7)² + (-3)² = 154
_λ = 7/12

n=8:

Polinomio linear:

_Para X=1: Z_{_I} = -7
_Para X=2: Z_{_I} = -5
_Para X=3: Z_{_I} = -3
_Para X=4: Z_{_I} = -1
_Para X=5: Z_{_I} = 1
_Para X=6: Z_{_I} = 3
_Para X=7: Z_{_I} = 5
_Para X=8: Z_{_I} = 7

_ΣZ² = (-7)² + (-5)² + (-3)² + (-1)² + (1)² + (3)² + (5)² + (7)² = 168
_λ = 2

Polinomio cuadrático:

_Para X=1: Z_{_II} = 7
_Para X=2: Z_{_II} = 1
_Para X=3: Z_{_II} = -3
_Para X=4: Z_{_II} = -5
_Para X=5: Z_{_II} = -5
_Para X=6: Z_{_II} = -3
_Para X=7: Z_{_II} = 1
_Para X=8: Z_{_II} = 7

_ΣZ² = (7)² + (1)² + (-3)² + (-5)² + (-5)² + (-3)² + (1)² + (7)² = 168
_λ = 1

Polinomio cúbico:

_Para X=1: Z_{_III} = -7
_Para X=2: Z_{_III} = 5
_Para X=3: Z_{_III} = 7
_Para X=4: Z_{_III} = 3
_Para X=5: Z_{_III} = -3
_Para X=6: Z_{_III} = -7
_Para X=7: Z_{_III} = -5
_Para X=8: Z_{_III} = 7

_ΣZ² = (-7)² + (5)² + (7)² + (3)² + (-3)² + (7)² + (-5)² + (7)² = 264
_λ = 2/3

Polinomio cuártico:

_Para X=1: Z_{_III}_{_I} = 7
_Para X=2: Z_{_III}_{_I} = -13
_Para X=3: Z_{_III}_{_I} = -3
_Para X=4: Z_{_III}_{_I} = 9
_Para X=5: Z_{_III}_{_I} = 9
_Para X=6: Z_{_III}_{_I} = -3
_Para X=7: Z_{_III}_{_I} = -13
_Para X=8: Z_{_III}_{_I} = 7

_ΣZ² = (7)² + (-13)² + (-3)² + (9)² + (9)² + (-3)² + (-13²) + (7)² = 616
_λ = 7/12

Polinomio quíntico:

_Para X=1: Z_{_III}_{_I}_{_I} = -7
_Para X=2: Z_{_III}_{_I}_{_I} = 23
_Para X=3: Z_{_III}_{_I}_{_I} = -17
_Para X=4: Z_{_III}_{_I}_{_I} = -15
_Para X=5: Z_{_III}_{_I}_{_I} = 15
_Para X=6: Z_{_III}_{_I}_{_I} = 17
_Para X=7: Z_{_III}_{_I}_{_I} = -23
_Para X=8: Z_{_III}_{_I}_{_I} = 7

_ΣZ² = (-7)² + (23)² + (-17)² + (-15)² + (15)² + (17)² + (-23)² + (7)²= 2184
_λ = 7/10

n=9:

Polinomio linear:

_Para X=1: Z_{_I} = -4
_Para X=2: Z_{_I} = -3
_Para X=3: Z_{_I} = -2
_Para X=4: Z_{_I} = -1
_Para X=5: Z_{_I} = 0
_Para X=6: Z_{_I} = 1
_Para X=7: Z_{_I} = 2
_Para X=8: Z_{_I} = 3
_Para X=9: Z_{_I} = 4

_ΣZ² = (-4)² + (-3)² + (-2)² + (-1)² + (0)² + (1)² + (2)² + (3)² + (4)² = 60
_λ = 1

Polinomio cuadrático:

_Para X=1: Z_{_II} = 28
_Para X=2: Z_{_II} = 7
_Para X=3: Z_{_II} = -8
_Para X=4: Z_{_II} = -17
_Para X=5: Z_{_II} = -20
_Para X=6: Z_{_II} = -17
_Para X=7: Z_{_II} = -8
_Para X=8: Z_{_II} = 7
_Para X=9: Z_{_II} = 28

_ΣZ² = (28)² + (7)² + (-8)² + (-17)² + (-20)² + (-17)² + (-8)² + (7)² + (28)² = 2772
_λ = 3

Polinomio cúbico:

_Para X=1: Z_{_III} = -14
_Para X=2: Z_{_III} = 7
_Para X=3: Z_{_III} = 13
_Para X=4: Z_{_III} = 9
_Para X=5: Z_{_III} = 0
_Para X=6: Z_{_III} = -9
_Para X=7: Z_{_III} = -13
_Para X=8: Z_{_III} = -7
_Para X=9: Z_{_III} = 14

_ΣZ² = (-14)² + (7)² + (13)² + (9)² + (0)² + (-9)² + (-13)² + (-7)² + (14)² = 990
_λ = 5/6

Polinomio cuártico:

_Para X=1: Z_{_III}_{_I} = 14
_Para X=2: Z_{_III}_{_I} = -21
_Para X=3: Z_{_III}_{_I} = -11
_Para X=4: Z_{_III}_{_I} = 9
_Para X=5: Z_{_III}_{_I} = 18
_Para X=6: Z_{_III}_{_I} = 9
_Para X=7: Z_{_III}_{_I} = -11
_Para X=8: Z_{_III}_{_I} = -21
_Para X=9: Z_{_III}_{_I} = 14

_ΣZ² = (14)² + (-21)² + (-11)² + (9)² + (18)² + (9)² + (-11)² + (-21)² + (14)² = 2002
_λ = 7/12

Polinomio quíntico:

_Para X=1: Z_{_III}_{_I}_{_I} = -4
_Para X=2: Z_{_III}_{_I}_{_I} = 11
_Para X=3: Z_{_III}_{_I}_{_I} = -4
_Para X=4: Z_{_III}_{_I}_{_I} = -9
_Para X=5: Z_{_III}_{_I}_{_I} = 0
_Para X=6: Z_{_III}_{_I}_{_I} = 9
_Para X=7: Z_{_III}_{_I}_{_I} = 4
_Para X=8: Z_{_III}_{_I}_{_I} = -11
_Para X=9: Z_{_III}_{_I}_{_I} = 4

_ΣZ² = (-4)² + (11)² + (-4)² + (-9)² + (0)² + (9)² + (4)² + (-11)² + (4)² = 468
_λ = 3/20

n=10:

Polinomio linear:

_Para X=1: Z_{_I} = -9
_Para X=2: Z_{_I} = -7
_Para X=3: Z_{_I} = -5
_Para X=4: Z_{_I} = -3
_Para X=5: Z_{_I} = -1
_Para X=6: Z_{_I} = 1
_Para X=7: Z_{_I} = 3
_Para X=8: Z_{_I} = 5
_Para X=9: Z_{_I} = 7
_Para X=10: Z_{_I} = 9

_ΣZ² = (-9)² + (-7)² + (-5)² + (-3)² + (-1)² + (1)² + (3)² + (5)² (7)² + (9)² = 330
_λ = 2

Polinomio cuadrático:

_Para X=1: Z_{_II} = 6
_Para X=2: Z_{_II} = 2
_Para X=3: Z_{_II} = -1
_Para X=4: Z_{_II} = -3
_Para X=5: Z_{_II} = -4
_Para X=6: Z_{_II} = -4
_Para X=7: Z_{_II} = -3
_Para X=8: Z_{_II} = -1
_Para X=9: Z_{_II} = 2
_Para X=10: Z_{_II} = 6

_ΣZ² = (6)² + (2)² + (-1)² + (-3)² + (-4)² + (-4)² + (-3)² + (-1)² + (2)² + (6)² = 132
_λ = 1/2

Polinomio cúbico:

_Para X=1: Z_{_III} = -42
_Para X=2: Z_{_III} = 14
_Para X=3: Z_{_III} = 35
_Para X=4: Z_{_III} = 31
_Para X=5: Z_{_III} = 12
_Para X=6: Z_{_III} = -12
_Para X=7: Z_{_III} = -31
_Para X=8: Z_{_III} = -35
_Para X=9: Z_{_III} = -14
_Para X=10: Z_{_III} = 42

_ΣZ² = (-42)²+(14)²+(35)²+(31)²+(12)²+(-12)²+(-31)²+(-35)²+(-14)²+(42)² = 8580
_λ = 5/3

Polinomio cuártico:

_Para X=1: Z_{_III}_{_I} = 18
_Para X=2: Z_{_III}_{_I} = -22
_Para X=3: Z_{_III}_{_I} = -17
_Para X=4: Z_{_III}_{_I} = 3
_Para X=5: Z_{_III}_{_I} = 18
_Para X=6: Z_{_III}_{_I} = 18
_Para X=7: Z_{_III}_{_I} = 3
_Para X=8: Z_{_III}_{_I} = -17
_Para X=9: Z_{_III}_{_I} = -22
_Para X=10: Z_{_III}_{_I} = 18

_ΣZ² = (18)²+(-22)²+(-17)²+(3)²+(18)²+(18)²+(3)²+(-17)²+(-22)²+(18)² = 2860
_λ = 5/12

Polinomio quíntico:

_Para X=1: Z_{_III}_{_I}_{_I} = -6
_Para X=2: Z_{_III}_{_I}_{_I} = 14
_Para X=3: Z_{_III}_{_I}_{_I} = -1
_Para X=4: Z_{_III}_{_I}_{_I} = -11
_Para X=5: Z_{_III}_{_I}_{_I} = -6
_Para X=6: Z_{_III}_{_I}_{_I} = 6
_Para X=7: Z_{_III}_{_I}_{_I} = 11
_Para X=8: Z_{_III}_{_I}_{_I} = 1
_Para X=9: Z_{_III}_{_I}_{_I} = -14
_Para X=10: Z_{_III}_{_I}_{_I} = 6

_ΣZ² = (-6)²+(14)²+(-1)²+(-11)²+(-6)²+(6)²+(11)²+(1)²+(-14)²+(6)²=780
_λ = 1/10

El ajuste de datos a fórmulas

miércoles, 12 de marzo de 2008

Apéndice: Coeficientes de Polinomios Ortogonales

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