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Fig. 1
Fig. 1

The curve of sine x, and the first-, third- and fifth-order approximations.

Fig. 2
Fig. 2

The curve of tangent x, and the first-, third- and fifth-order approximations.

Equations (79)

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sin u = u - u 3 / 6 + u 5 / 120 , tan u = u + u 3 / 3 + 2 u 5 / 15.
u 1 = 1 2 ( x ¯ 1 2 + x ¯ 2 2 ) ,             u 2 = ( x ¯ 1 x ¯ 3 + x ¯ 2 x ¯ 4 ) ,             u 3 = ( x ¯ 3 2 + x ¯ 4 2 ) .
A i k u i u k = A 11 u 1 2 + 2 A 12 u 1 u 2 + 2 A 13 u 1 u 3 + A 22 u 2 2 + 2 A 23 u 2 u 3 + A 33 u 3 2 ,
E = E 0 + Ā i ū i + 1 2 Ā i k ū i ū k + 1 6 A i k λ u i u k u λ .
ξ 1 = - E / x ¯ 1 , ξ 3 = E / x ¯ 3 , ξ 2 = - E / x ¯ 2 , ξ 4 = E / x ¯ 4 ,
- ξ 1 = E u ˜ 1 x ¯ 1 + E ū 2 x ¯ 3 = ( Ā 1 + Ā 1 i ū i + 1 2 Ā 1 i k ū i ū k ) x ¯ 1 + ( Ā 2 + Ā 2 i ū i + 1 2 Ā 2 i k ū i ū k ) x ¯ 3 , ξ 3 = E ū 2 x ¯ 1 + E ū 3 x ¯ 3 = ( Ā 2 + Ā 2 i ū i + 1 2 Ā 2 i k ū i ū k ) x ¯ 1 + ( Ā 3 + Ā 3 i ū i + 1 2 Ā 3 i k ū i ū k ) x ¯ 3 .
x i = Ā 2 x ¯ i ,             u i = Ā 2 2 ū i
- ξ 1 = ( A 1 + A 1 i u i + 1 2 A 1 i k u i u k ) x 1 + ( 1 + A 2 i u i + 1 2 A 2 i k u i u k ) x 3 , ξ 3 = ( 1 + A 2 i u i + 1 2 A 2 i k u i u k ) x 1 + ( A 3 + A 3 i u i + 1 2 A 3 i k u i u k ) x 3 ,
A i = Ā i / Ā 2 ,             A i k = Ā i k / Ā 2 2 , A i k λ = Ā i k λ / Ā 2 3 .
sin σ = ( ξ 2 1 + ξ 2 2 ) 1 2 , tan σ = ( ξ 2 1 + ξ 2 2 ) 1 2 ( 1 - ( ξ 2 1 + ξ 2 2 ) ) 1 2 , sin σ = ( ξ 2 3 + ξ 2 4 ) 1 2 , tan σ = ( ξ 2 3 + ξ 2 4 ) 1 2 ( 1 - ( ξ 2 3 + ξ 2 4 ) ) 1 2 .
Ξ 1 = ξ 1 ( 1 - ( ξ 2 3 + ξ 2 4 ) ) 1 2 = ( B 1 + B 1 i u i + 1 2 B 1 i k u i u k ) x 1 + ( 1 + B 2 i u i + 1 2 B 2 i k u i u k ) x 3 , Ξ 3 = ξ 3 ( 1 - ( ξ 2 3 + ξ 2 4 ) ) 1 2 = ( 1 + B 2 i u i + 1 2 B 2 i k u i u k ) x 1 + ( B 3 + B 3 i u i + 1 2 B 3 i k u i u k ) x 3 ,
x 1 g = x 3 + λ ξ 3 , x 2 g = x 4 + λ ξ 4 ,             or             x 1 g = x 3 + g Ξ 3 , x 2 g = x 4 + g Ξ 4 , g = λ ( 1 - ( ξ 2 3 + ξ 2 4 ) ) 1 2 .
x 1 g = g ( 1 + B 2 i u i + 1 2 B 2 i k u i u k ) x 1 + g ( 1 / g + B 3 + B 3 i u i + 1 2 B 3 i k u i u k ) x 3 .
- B 3 x - x 1 = ( B 2 i u i + 1 2 B 2 i k u i u k ) x 1 + ( B 3 i u i + 1 2 B 3 i k u i u k ) x 3 .
x = 1 B 3 x 1 = β x 1 ,
x 3 g = x 1 + g Ξ 1 = - g [ B 1 - 1 / g + B 1 i u i + 1 2 B 1 i k u i u k ] x 1 - g [ 1 + B 2 i u i + 1 2 B 2 i k u i u k ] x 3 .
- ( B 1 x 3 * + x 3 ) = [ B 1 i u i + 1 2 B 1 i k u i u k ] x 1 + [ B 2 i u i + 1 2 B 2 i k u i u k ] x 3 .
x 3 = - B 1 x 3 * = B P x 3 * ,
Ξ 1 = ξ 1 ( 1 - ( ξ 2 1 + ξ 2 2 ) ) 1 2 = ξ 1 [ 1 + 1 2 ( ξ 2 1 + ξ 2 2 ) + 3 8 ( ξ 2 1 + ξ 2 2 ) 2 ] , Ξ 3 = ξ 3 ( 1 - ( ξ 2 3 + ξ 2 4 ) ) 1 2 = ξ 3 [ 1 + 1 2 ( ξ 2 3 + ξ 2 4 ) + 3 8 ( ξ 2 3 + ξ 2 4 ) 2 ] .
1 2 ( ξ 2 1 + ξ 2 2 ) = A 1 2 u 1 + A 1 u 2 + u 3 + 2 A 1 A 11 u 1 2 + ( 3 A 1 A 12 + A 11 ) u 1 u 2 + 2 ( A 1 A 13 + A 12 ) u 1 u 3 + ( A 1 A 22 + A 12 ) u 2 2 + ( A 1 A 23 + 2 A 22 + A 13 ) u 2 u 3 + 2 A 23 u 3 2
B 1 = A 1 ,             B 3 = A 3 ,
B 11 = A 11 + A 1 3 , B 12 = A 12 + A 1 2 , B 21 = A 12 + 1 , B 13 = A 13 + A 1 , B 22 = A 22 + A 3 , B 21 = A 12 + A 1 2 , B 23 = A 23 + A 2 3 , B 22 = A 22 + A 1 , B 31 = A 13 + A 3 , B 23 = A 23 + 1 , B 32 = A 23 + A 2 3 , B 33 = A 33 + A 3 3 ,
B 111 = A 111 + 6 A 1 2 A 11 + 3 A 1 5 , B 211 = A 112 + 6 A 12 + 3 , B 112 = A 112 + 4 A 1 2 A 12 + 2 A 1 A 11 + 3 A 1 4 , B 212 = A 122 + 2 A 3 A 12 + 3 A 22 + A 13 + 3 A 3 , B 113 = A 113 + 3 A 1 2 A 13 + 2 A 1 A 12 + A 11 + 3 A 1 3 , B 213 = A 123 + A 3 2 A 12 + 2 A 3 A 13 + 3 A 23 + 3 A 3 2 , B 122 = A 122 + 2 A 1 2 A 22 + 4 A 1 A 12 + 3 A 1 3 , B 222 = A 222 + 4 A 3 A 22 + 2 A 23 + 3 A 3 2 , B 123 = A 123 + A 1 2 A 23 + 2 A 1 ( A 13 + A 22 ) + A 12 + 3 A 1 2 , B 223 = A 223 + A 3 2 A 22 + 4 A 3 A 23 + A 33 + 3 A 3 3 , B 133 = A 133 + 4 A 1 A 23 + 2 A 13 + 3 A 1 , B 233 = A 233 + 2 A 3 2 A 23 + 4 A 3 A 33 + 3 A 3 4 , B 211 = A 112 + 2 A 1 2 A 12 + 4 A 1 A 11 + 3 A 1 4 , B 311 = A 113 + 4 A 3 A 12 + 2 A 13 + 3 A 3 , B 212 = A 122 + A 1 2 A 22 + 4 A 1 A 12 + A 11 + 3 A 1 3 , B 312 = A 123 + A 3 2 A 12 + 2 A 3 ( A 13 + A 22 ) + A 23 + 3 A 3 2 , B 213 = A 123 + A 1 2 A 23 + 2 A 1 A 13 + 3 A 12 + 3 A 1 2 , B 313 = A 133 + 3 A 3 2 A 13 + 2 A 3 A 23 + A 33 + 3 A 3 3 , B 222 = A 222 + 4 A 1 A 22 + 2 A 12 + 3 A 1 2 , B 322 = A 223 + 2 A 3 2 A 22 + 4 A 3 A 23 + 3 A 3 3 , B 223 = A 223 + 2 A 1 A 23 + ( 3 A 22 + A 13 ) + 3 A 1 , B 323 = A 233 + 4 A 3 2 A 23 + 2 A 3 A 33 + 3 A 3 4 , B 233 = A 233 + 6 A 23 + 3 , B 333 = A 333 + 6 A 3 2 A 33 + 3 A 3 5 .
A i κ λ = A κ i λ , B i κ λ not necessarily = B κ i λ .
B 11 = A 11 + B 1 3 , B 12 = A 12 + B 1 2 = B 21 , B 21 = A 12 + 1 , B 13 = A 13 + B 1 , B 31 = A 13 + B 3 , B 22 = A 22 + B 1 , B 22 = A 22 + B 3 , B 23 = A 23 + 1 , B 23 = A 23 + B 2 3 = B 32 , B 33 = A 33 + B 3 3 .
B 23 - B 32 = 0 ,             B 21 - B 12 = 0.
B 23 - B 23 = B 2 3 - 1 = α 1 , B 22 - B 22 = B 13 - B 13 = B 3 - B 1 = α 2 , B 12 - B 12 = 1 - B 1 2 = α 3 .
C 111 = A 111 - 3 B 1 5 , C 112 = A 112 - 3 B 1 4 , C 112 = A 112 - 3 , C 113 = A 113 - 3 B 1 3 , C 122 = A 122 - 3 B 1 3 , C 122 = A 122 - 3 B 3 , C 113 = A 113 - 3 B 3 , C 123 = A 123 - 3 B 1 2 , C 222 = A 222 - 3 B 1 2 , C 222 = A 222 - 3 B 3 2 , C 123 = A 123 - 3 B 3 2 , C 133 = A 133 - 3 B 1 , C 223 = A 223 - 3 B 1 , C 223 = A 223 - 3 B 3 3 , C 133 = A 133 - 3 B 3 3 , C 233 = A 233 - 3 , C 233 = A 233 - 3 B 3 4 , C 333 = A 333 - 3 B 3 5 .
B 111 = C 111 + 6 B 1 2 B 11 , B 112 = C 112 + 4 B 1 2 B 12 + 2 B 1 B 11 , B 211 = C 112 + 2 B 1 2 B 12 + 4 B 1 B 11 , B 113 = C 113 + 3 B 1 2 B 13 + 2 B 1 B 12 + B 11 , B 122 = C 122 + 2 B 1 2 B 22 + 4 B 1 B 12 , B 212 = C 122 + B 1 2 B 22 + 4 B 1 B 12 + B 11 , B 123 = C 123 + B 1 2 B 23 + 2 B 1 ( B 13 + B 22 ) + B 12 , B 213 = C 123 + B 1 2 B 23 + 2 B 1 B 13 + 3 B 12 , B 222 = C 222 + 4 B 1 B 22 + 2 B 12 , B 133 = C 133 + 4 B 1 B 23 + 2 B 13 , B 223 = C 223 + 2 B 1 B 23 + ( 3 B 22 + B 13 ) , B 233 = C 233 + 6 B 23 ,
B 211 = C 112 + 6 B 12 , B 212 = C 122 + 2 B 3 B 12 + 3 B 22 + B 13 , B 311 = C 113 + 4 B 3 B 12 + 2 B 13 , B 222 = C 222 + 4 B 3 B 22 + 2 B 23 , B 213 = C 123 + B 2 3 B 12 + 2 B 3 B 13 + 3 B 23 , B 312 = C 123 + B 2 3 B 12 + 2 B 3 ( B 13 + B 22 ) + B 23 , B 223 = C 223 + B 2 3 B 22 + 4 B 3 B 23 + B 33 , B 322 = C 223 + 2 B 2 3 B 22 + 4 B 3 B 23 , B 313 = C 133 + 3 B 2 3 B 13 + 2 B 3 B 23 + B 33 , B 233 = C 233 + 2 B 2 3 B 23 + 4 B 3 B 33 , B 323 = C 233 + 4 B 2 3 B 23 + 2 B 3 B 33 , B 333 = C 333 + 6 B 2 3 B 33 .
B 311 - B 112 = 2 B 1 ( B 11 - B 1 B 12 ) , B 312 - B 213 = 2 ( B 3 B 22 - B 23 ) , B 212 - B 122 = B 11 - B 1 2 B 22 , B 322 - B 223 = B 2 3 B 22 - B 33 , B 213 - B 123 = 2 ( B 12 - B 1 B 22 ) , B 323 - B 233 = 2 B 3 ( B 3 B 23 - B 33 ) ,
C 112 - C 112 = 3 ( B 1 4 - 1 ) = γ 5 , C 122 - C 122 = C 113 - C 113 = 3 ( B 1 3 - B 3 ) = γ 4 , C 123 - C 123 = C 222 - C 222 = 3 ( B 1 2 - B 2 3 ) = γ 3 , C 223 - C 223 = C 133 - C 133 = 3 ( B 1 - B 3 3 ) = γ 2 , C 233 - C 233 = 3 ( 1 - B 4 3 ) = γ 1 .
z c = 1 2 D 1 ( x c 2 + y c 2 ) + 1 8 D 2 3 ( x c 2 + y c 2 ) 2 = D 1 u c + 1 2 D 2 3 u c 2 , z c = D 3 u c + 1 2 D 4 3 u 2 c .
z ¯ c = ( D 1 / Ā 2 ) ū c + 1 2 ( D 2 / Ā 2 ) 3 ū c 2 , z ¯ c = ( D 3 / Ā 2 ) ū c + 1 2 ( D 4 / Ā 2 ) 3 ū 2 c .
D 1 = D 2             ( D 3 = D 4 ) ,
An ellipsoid for ( D 1 / D 2 ) > 0 ( ( D 3 / D 4 ) > 0 ) , A hyperboloid for ( D 1 / D 2 ) < 0 ( ( D 3 / D 4 ) < 0 ) ,
a = D 1 2 / D 2 3 ,             b = ( ( - ) D 1 / D 2 3 ) 1 2 ,
- B 3 x - x 1 = [ B 2 i u i + 1 2 B 2 i κ u i u κ ] x 1 + [ B 3 i u i + 1 2 B 3 i κ u i u κ ] x 3 ,
- B 3 x c - x c = [ B 2 i c u i c + 1 2 B 2 i κ c u i c u κ c ] x c + [ B 3 i c u i c + 1 2 B 3 i κ c u i c u κ c ] x 3 c .
x c = x 1 + z c Ξ 1 ,             x c = x + z c Ξ 3 .
x c = x 1 - ( D 1 B 1 u 1 x 1 + D 1 u 1 x 3 ) - 1 2 [ ( 2 D 1 B 11 - 4 D 1 2 B 1 2 + D 2 3 B 1 ) u 1 2 + 2 ( D 1 B 12 - D 1 2 B 1 ) u 1 u 2 + 2 D 1 B 13 u 1 u 3 ] x 1 - 1 2 [ ( 2 D 1 B 12 - 4 D 1 2 B 1 + D 2 3 ) u 1 2 + 2 ( D 1 B 22 - D 1 2 ) u 1 u 2 + 2 D 1 B 23 u 1 u 3 ] x 3 ,
u 1 c = u 1 - 2 B 1 D 1 u 1 2 - D 1 u 1 u 2 , u 2 c = u 2 - D 1 B 1 u 1 u 2 - 2 D 1 u 1 u 3 , u 3 c = u 3 .
- B 3 x c - x c = [ ( B 21 - D 3 B 3 ) u 1 + B 22 u 2 + B 23 u 3 ] x 1 + [ ( B 13 - D 3 ) u 1 + B 23 u 2 + B 33 u 3 ] x 3 + { 1 2 [ B 212 - D 3 B 3 ( 6 B 12 - 4 D 3 B 3 ) - D 4 B 2 3 ] u 1 2 + [ B 212 - D 3 B 3 ( 3 B 22 + B 13 - D 3 ) ] u 1 u 2 + [ B 213 - 3 D 3 B 3 B 23 ] u 1 u 3 + 1 2 [ B 222 - 2 D 3 B 3 B 23 ] u 2 2 + [ B 223 - D 3 B 3 B 33 ] u 2 u 3 + 1 2 B 233 u 3 2 } x 1 + { 1 2 [ B 311 - D 3 B 3 ( 2 B 13 + 4 B 3 B 12 - 4 D 3 ) ] u 1 2 + [ B 312 - D 3 B 3 ( B 23 + B 3 ( 2 B 22 + B 13 ) - D 3 ) ] u 1 u 2 + [ B 313 - D 3 B 3 ( B 33 + 2 B 3 B 23 ) ] u 1 u 3 + 1 2 [ B 322 - D 3 B 23 ] u 2 2 + [ B 323 - D 3 B 33 ] u 2 u 3 + 1 2 B 333 u 3 2 ] } x 3 .
B 12 c = B 12 + D 1 B 1 - D 3 / B 3 , B 13 c = B 13 + D 1 - D 3 , B 22 c = B 22 , B 23 c = B 23 , B 33 c = B 33 ,
B 211 c = B 211 + 2 D 1 B 4 + 6 B 12 ( B 1 D 1 - D 3 B 3 ) + ( B 1 D 2 3 - D 4 3 B 3 3 ) + 2 ( B 1 D 1 - D 3 B 3 ) ( B 1 D 1 - 2 D 3 B 3 ) , B 212 c = B 212 + D 1 B 12 + D 1 ( B 12 + 2 B 1 B 22 ) - D 3 B 3 ( B 13 + 3 B 22 ) + D 3 B 3 ( D 3 - D 1 ) , B 311 c = B 311 + 2 D 1 B 12 + 2 D 1 ( B 12 + 2 B 1 B 13 ) - 2 D 3 B 3 ( B 13 + 2 B 3 B 12 ) + ( D 2 3 - D 4 3 B 2 3 ) + 2 ( D 3 B 3 - B 1 D 1 ) ( 2 D 3 - D 1 ) , B 213 c = B 213 + D 1 B 13 + D 1 ( 2 B 22 + B 1 B 23 ) - 3 D 3 B 3 / B 3 , B 222 c = B 222 - 2 D 3 B 23 / B 3 , B 312 c = B 312 + D 1 B 22 + D 1 ( B 13 + B 22 + B 1 B 23 ) - ( D 3 / B 3 ) ( B 23 + B 3 ( B 13 + 2 B 22 ) ) + D 3 ( D 3 - D 1 ) , B 223 c = B 223 - D 3 B 33 / B 3 , B 313 c = B 313 + D 1 B 23 + 3 D 1 B 23 - ( D 3 / B 3 ) ( B 33 + 2 B 3 B 23 ) , B 322 c = B 322 - 2 D 3 B 23 , B 233 c = B 233 , B 323 c = B 323 - D 3 B 33 , B 333 c = B 333 .
z * = 1 2 E 3 ( x * 3 2 + y * 3 2 ) = E 3 u * 3 .
x * 3 = x 3 + E 3 u 3 Ξ 3 = E 3 u 3 x 1 + ( 1 + E 3 B 3 u 3 ) x 3 ) ,             u * 1 = u 1 ,             u * 2 = 2 E 3 u 3 u 1 + ( 1 + E 3 B 3 u 3 ) u 2 ,             u * 3 = u 3 + E 3 u 2 u 3 + 2 E 3 B 3 u 3 2 .
- B 3 x - x 1 = [ B * 2 i u * i + 1 2 B * 2 i κ u * i u * κ ] x 1 + [ B * 3 i u * i + 1 2 B * 3 i κ u * i u * κ ] x * 3 ,
B * 12 = B 12 , B * 23 = B 23 , B * 22 = B 22 , B * 33 = B 33 . B * 13 = B 13 ,
B * 211 = B 211 , B * 212 = B 212 , B * 311 = B 311 , B * 213 = B 213 - E 3 ( 2 B 22 + B 13 ) , B * 222 = B 222 , B * 312 = B 312 , B 223 = B 223 - E 3 ( 2 B 23 + B 3 B 22 ) , B * 313 = B 313 - E 3 ( 2 B 23 + B 3 B 13 ) , B * 322 = B 322 , B * 233 = B 233 - 2 E 3 ( B 33 + 2 B 3 B 23 ) , B * 323 = B 323 - E 3 ( B 33 + 2 B 3 B 23 ) , B * 333 = B 333 - 6 E 3 B 3 B 33 .
x ¯ 1 = x ¯ ˙ 1 ,             x ¯ 3 = x ¯ ˙ 3 + k Ξ 3
x 3 / A 2 = x 3 / A 2 + k Ξ 3 ,             x 1 / A 1 = x 1 / A 2 .
κ 3 = A 2 / A 2 ,             λ 3 = k A 2 ,
x 3 = κ 3 x 3 + λ 3 Ξ 3 = κ 3 x 3 + λ 3 ( 1 + B 2 i u i + 1 2 B 2 i κ u i u κ ) x 1 + λ 3 ( B 3 + B 3 i u i + 1 2 B 3 i κ u i u κ ) x 3 = λ 3 x 1 + ( κ 3 + B 3 λ 3 ) x 3 + λ 3 ( B 2 i u i + 1 2 B 2 i κ u i u κ ) x 1 + λ 3 ( B 3 + B 3 i u i + 1 2 B 3 i κ u i u κ ) x 3 x 1 = κ 3 x 1 .
x 3 = λ 3 x 1 + ( κ 3 + B 3 λ 3 ) x 3 ,             x 1 = κ 3 x 13 ,
Ξ 3 = x 1 + B 3 x 3 = x 1 + B 3 x 3 .
B 3 = B 3 ,             κ 3 + B 3 λ 3 = 1.
- Ξ 1 = B 1 x 1 + x 3 = B 1 x 1 + x 3 ,
B 1 κ 3 + λ 3 = B 1 .
κ 3 = 1 - B 1 B 3 1 - B 1 B 3 ,             λ 3 = B 1 - B 1 1 - B 1 B 3 .
x 1 = κ 3 x 1 ,             x 3 = λ 3 x 1 + x 3 + λ 3 ( B 2 i u i + 1 2 B 2 i κ u i u κ ) x 1 + λ 3 ( B 3 i u i + 1 2 B 3 i κ u i u κ ) x 3
u 1 = κ 3 2 u 1 , u 2 = 2 κ 3 λ 3 u 1 + κ 3 u 2 + κ 3 λ 3 ( 2 B 2 i u i u 1 + B 3 i u i u 2 ) , u 3 = λ 3 2 u 1 + λ 3 u 2 + u 3 + λ 3 [ B 2 i u i ( 2 λ 3 u 1 + u 2 ) + B 3 i u i ( λ 3 u 2 + 2 u 3 ) ] .
Ξ 3 = [ 1 + B 2 i u i + 1 2 B 2 i κ u i u κ ] x 1 + [ B 3 + B 3 i u i + 1 2 B 3 i κ u i u κ ] x 3 = [ 1 + B 2 i u i + 1 2 B 2 i κ u i u κ ] x 1 + [ B 3 + B 3 i u i + 1 2 B 3 i κ u i u κ ] x 3
κ 3 B 12 = κ 3 3 B 12 + κ 3 2 λ 3 ( 2 B 22 + B 13 ) + 3 κ 3 λ 3 2 B 23 + λ 3 3 B 33 , κ 3 B 13 = κ 3 2 B 13 + 2 κ 3 λ 3 B 23 + λ 3 2 B 33 , κ 3 B 22 = κ 3 2 B 22 + 2 κ 3 λ 3 B 23 + λ 3 2 B 33 , κ 3 B 23 = κ 3 B 23 + λ 3 B 33 , κ 3 B 33 = B 33 ,
κ 3 B 211 = κ 3 5 B 211 + κ 3 4 λ 3 ( B 311 + 4 B 212 ) + 2 κ 3 3 λ 3 2 [ B 213 + 2 ( B 222 + B 312 ) ] + 2 κ 3 2 λ 3 3 [ 2 ( B 223 + B 322 ) + B 313 ] + κ 3 λ 3 4 ( B 233 + 4 B 323 ) + λ 3 5 B 333 + 2 κ 3 λ 3 B 12 ( B 13 + 2 B 22 ) , κ 3 B 212 = κ 3 4 B 212 + κ 3 3 λ 3 ( B 213 + 2 B 222 + B 312 ) + κ 3 2 λ 3 2 ( 3 B 223 + B 313 + 2 B 322 ) + κ 3 λ 3 3 ( B 233 + 3 B 323 ) + λ 3 4 B 333 + 2 κ 3 λ 3 [ B 12 B 23 + B 22 ( B 13 + B 22 ) ] , κ 3 B 311 = κ 5 4 B 311 + 4 κ 3 3 λ 3 B 312 + 2 κ 3 2 λ 3 2 ( B 313 + 2 B 322 ) + 4 κ 3 λ 3 3 B 323 + λ 3 4 B 333 + 2 κ 3 λ 3 ( B 2 13 + 2 B 12 B 23 ) , κ 3 B 213 = κ 3 3 B 213 + κ 3 2 λ 3 ( 2 B 223 + B 313 ) + κ 3 λ 3 2 ( B 233 + 2 B 323 ) + λ 3 3 B 333 + κ 3 λ 3 [ B 12 B 33 + B 22 ( 3 B 13 + 2 B 22 ) ] , κ 3 B 222 = κ 3 3 B 222 + κ 3 2 λ 3 ( 2 B 223 + B 322 ) + κ 3 λ 3 2 ( B 233 + 2 B 323 ) + λ 3 3 B 333 + 6 κ 3 λ 3 B 22 B 23 , κ 3 B 312 = κ 3 3 B 312 + κ 3 2 λ 3 ( B 313 + 2 B 322 ) + 3 κ 3 λ 3 2 B 323 + λ 3 3 B 333 + κ 3 λ 3 [ B 12 B 33 + B 23 ( 2 B 22 + 3 B 13 ) ] , κ 3 B 223 = κ 3 2 B 223 + κ 3 λ 3 ( B 233 + B 323 ) + λ 3 2 B 333 + 2 κ 3 λ 3 [ 2 B 2 23 + B 22 B 33 ] , κ 3 B 313 = κ 3 2 B 313 + 2 κ 3 λ 3 B 323 + λ 3 2 B 333 + 2 κ 5 λ 3 ( B 2 23 + 2 B 13 B 33 ) , κ 3 B 322 = κ 3 2 B 322 + 2 κ 3 λ 3 B 323 + λ 3 2 B 333 + 2 κ 3 λ 3 ( 2 B 2 23 + B 22 B 33 ) , κ 3 B 233 = κ 3 B 233 + λ 3 B 333 + 6 κ 3 λ 3 B 23 B 33 , κ 3 B 323 = κ 3 B 323 + λ 3 B 333 + 6 κ 3 λ 3 B 23 B 33 , κ 3 B 333 = B 333 + 6 κ 3 λ 3 B 2 33 .
κ 3 C 112 = κ 3 5 C 112 + κ 3 4 λ 3 ( 3 C 122 + C 113 ) + 2 κ 3 3 λ 3 2 ( 3 C 123 + 2 C 222 ) + 2 κ 3 2 λ 3 3 ( C 133 + 4 C 223 ) + 5 κ 3 λ 3 4 C 233 + λ 3 5 C 333 + 2 κ 3 λ 3 B 12 ( B 13 + 2 B 22 ) , κ 3 C 122 = κ 3 4 C 122 + 2 κ 3 3 λ 3 ( C 123 + C 222 ) + κ 3 2 λ 3 2 ( C 133 + 5 C 223 ) + 4 κ 3 λ 3 3 C 223 + λ 3 4 C 333 + 2 κ 3 λ 3 [ B 12 B 23 + B 22 ( B 13 + B 22 ) ] , κ 3 C 113 = κ 3 4 C 113 + 4 κ 3 3 λ 3 C 123 + 2 κ 3 2 λ 3 2 ( C 133 + 2 C 223 ) + 4 κ 3 λ 3 2 C 233 + λ 3 4 C 333 + 2 κ 3 λ 3 [ B 2 13 + 2 B 12 B 23 ] , κ 3 C 123 = κ 3 3 C 123 + κ 3 2 λ 3 ( 2 C 223 + C 133 ) + 3 κ 3 λ 3 2 C 233 + λ 3 3 C 333 + κ 3 λ 3 [ B 12 B 33 + B 23 ( 3 B 13 + 2 B 22 ) ] , κ 3 C 222 = κ 3 3 C 222 + 3 κ 3 2 λ 3 C 223 + 3 κ 3 λ 3 2 C 233 + λ 3 3 C 333 + 6 κ 3 λ 3 B 22 B 23 , κ 3 C 133 = κ 3 2 C 133 + 2 κ 3 λ 3 C 233 + λ 3 2 C 333 + 2 κ 3 λ 3 [ B 2 23 + 2 B 13 B 33 ] , κ 3 C 223 = κ 3 2 C 223 + 2 κ 3 λ 3 C 233 + λ 3 2 C 333 + 2 κ 3 λ 3 [ 2 B 2 23 + B 22 B 33 ] , κ 3 C 233 = κ 3 C 233 + λ 3 C 333 + 6 κ 3 λ 3 B 23 B 33 , κ 3 C 333 = C 333 + 6 κ 3 λ 3 B 2 33 .
x 1 = κ 1 x 1 - λ 1 Ξ 1 ,             x 3 = κ 1 x 3 .
- Ξ 1 = B 1 x 1 + x 3 = B 1 x 1 + x 3 ,             Ξ 3 = x 1 + B 3 x 3 = x 1 + B 3 x 3 ,
x 1 = ( κ 1 + λ 1 B 1 ) x 1 + λ 1 x 3 , x 3                                                                     κ 1 x 3 .
B 1 = B 1 ,             κ 1 + λ 1 B 1 = 1 ,             λ 1 + κ 1 B 3 = B 3 ,
κ 1 = 1 - B 1 B 3 1 - B 1 B 3 ,             λ 1 = B 3 - B 3 1 - B 1 B 3 .
x 1 = x 1 + λ 1 x 3 + λ 1 ( B 1 i u i + 1 2 B 1 i κ u i u κ ) x 1 + λ 1 ( B 2 i u i + 1 2 B 2 i κ u i u κ ) x 3
x 3 = κ 1 x 3 .
- Ξ 1 = ( B 1 + B 1 i u i + 1 2 B 1 i κ u i u κ ) x 1 + ( 1 + B 2 i u i + 1 2 B 2 i κ u i u κ ) x 3 = ( B 1 + B 1 i u i + 1 2 B 1 i κ u i u κ ) x 1 + ( 1 + B 2 i u i + 1 2 B 2 i κ u i u κ ) x 3 ,
κ 1 B 12 = κ 1 B 12 + λ 1 B 11 , κ 1 B 13 = κ 1 2 B 13 + κ 1 λ 1 ( 2 B 12 - α 3 ) + λ 1 2 B 11 , κ 1 B 22 = κ 1 2 B 22 + κ 1 λ 1 ( 2 B 12 - α 3 ) + λ 1 2 B 11 , κ 1 B 23 = κ 1 3 B 23 + κ 1 2 λ 1 ( B 13 + 2 B 22 - α 2 ) + κ 1 λ 1 2 ( 3 B 12 - 2 α 1 ) + λ 1 3 B 11 , κ 1 B 33 = κ 1 4 B 33 + κ 1 3 λ 1 ( 4 B 23 - α 1 ) + κ 1 2 λ 1 2 [ 2 ( 2 B 22 + B 13 ) - 3 α 2 ] + κ 1 λ 1 5 ( 4 B 12 - 3 α 1 ) + λ 1 4 B 11 .
κ 1 B 111 = B 111 + 6 κ 1 λ 1 B 11 2 , κ 1 B 112 = κ 1 B 112 + λ 1 B 111 + 6 κ 1 λ 1 B 11 B 12 , κ 1 B 211 = κ 1 B 211 + λ 1 B 111 + 6 κ 1 λ 1 B 11 B 12 , κ 1 B 113 = κ 1 2 B 113 + 2 κ 1 λ 1 B 112 + λ 1 2 B 111 + κ 1 λ 1 ( 4 B 11 B 13 + 2 B 12 2 ) , κ 1 B 122 = κ 1 2 B 122 + 2 κ 1 λ 1 B 112 + λ 1 2 B 111 + κ 1 λ 1 ( 2 B 11 B 22 + 4 B 12 2 ) , κ 1 B 212 = κ 1 2 B 212 + κ 1 λ 1 ( B 211 + B 112 ) + λ 1 2 B 111 + κ 1 λ 1 ( 2 B 11 B 22 + 4 B 12 2 ) , κ 1 B 123 = κ 1 3 B 123 + κ 1 2 λ 1 ( B 113 + 2 B 112 ) + 3 κ 1 λ 1 2 B 112 + λ 1 3 B 111 + κ 1 λ 1 [ B 11 B 23 + B 12 ( 3 B 13 + 2 B 22 ) ] , κ 1 B 213 = κ 1 3 B 213 + κ 1 2 λ 1 ( B 113 + 2 B 212 ) + κ 1 λ 1 2 ( B 211 + 2 B 112 ) + λ 1 3 B 111 + κ 1 λ 1 [ B 11 B 23 + B 12 ( 3 B 13 + 2 B 22 ) ] , κ 1 B 222 = κ 1 3 B 222 + κ 1 2 λ 1 ( B 122 + 2 B 212 ) + κ 1 λ 1 2 ( B 211 + 2 B 112 ) + λ 1 3 B 111 + 6 κ 1 λ 1 B 12 B 22 , κ 1 B 133 = κ 1 4 B 133 + 4 κ 1 3 λ 1 B 123 + 2 κ 1 2 λ 1 2 ( B 113 + 2 B 122 ) + 4 κ 1 λ 1 3 B 112 + λ 1 4 B 111 + κ 1 λ 1 [ 2 B 13 2 + 4 B 12 B 23 ] , κ 1 B 223 = κ 1 4 B 223 + κ 1 3 λ 1 ( 2 B 222 + B 213 + B 123 ) + κ 1 2 λ 1 2 ( 3 B 212 + 2 B 122 + B 113 ) + κ 1 λ 1 3 ( B 211 + 3 B 112 ) + λ 1 4 B 111 + 2 κ 1 λ 1 [ B 12 B 23 + B 22 ( B 13 + B 22 ) ] , κ 1 B 233 = κ 1 5 B 233 + κ 1 4 λ 1 ( 4 B 223 + B 133 ) + κ 1 3 λ 1 2 ( 4 B 222 + 2 B 213 + 4 B 123 ) + κ 1 2 λ 1 3 ( 4 B 112 + 4 B 122 + 2 B 113 ) + κ 1 λ 1 4 ( B 211 + 4 B 112 ) + λ 1 5 B 111 + 2 κ 1 λ 1 B 23 ( B 13 + 2 B 22 )
B 211 = λ 1 B 111 + B 211 + 6 λ 1 B 12 B 11 , B 212 = λ 1 B 112 + λ 1 B 211 + κ 1 B 212 + 2 λ 1 B 11 ( κ 1 B 22 + λ 1 B 12 ) + 4 λ 1 B 12 B 12 , B 311 = λ 1 B 211 + λ 1 B 211 + κ 1 B 311 + 4 λ 1 B 11 ( κ 1 B 13 + λ 1 B 12 ) + 2 λ 1 B 12 B 12 , B 213 = λ 1 B 113 + λ 1 2 B 211 + 2 κ 1 λ 1 B 212 + κ 1 2 B 213 + λ 1 B 11 ( κ 1 2 B 23 + 2 κ 1 λ 1 B 22 + λ 1 2 B 12 ) + 2 λ 1 B 12 ( κ 1 B 22 + λ 1 B 12 ) + 3 λ 1 B 13 B 12 , B 222 = λ 1 B 122 + λ 1 2 B 211 + 2 κ 1 λ 1 B 212 + κ 1 2 B 222 + 4 λ 1 B 12 ( κ 1 B 22 + λ 1 B 12 ) + 2 λ 1 B 22 B 12 , B 312 = λ 1 B 212 + λ 1 2 B 211 + κ 1 λ 1 ( B 212 + B 311 ) + κ 1 2 B 312 + λ 1 B 11 [ κ 1 2 B 23 + κ 1 λ 1 ( B 13 + B 22 ) + λ 1 2 B 12 ] + λ 1 B 12 [ κ 1 ( 3 B 13 + B 22 ) + 4 λ 1 B 12 ] + λ 1 B 22 B 12 , B 223 = λ 1 B 123 + λ 1 3 B 211 + 3 κ 1 λ 1 2 B 212 + κ 1 2 λ 1 ( B 213 + 2 B 222 ) + κ 1 3 B 223 + λ 1 B 12 ( κ 1 2 B 23 + 2 κ 1 λ 1 B 22 + λ 1 2 B 12 ) + 2 λ 1 ( B 22 + B 13 ) ( κ 1 B 22 + λ 1 B 12 ) + λ 1 B 23 B 12 , B 313 = λ 1 B 213 + λ 1 3 B 211 + κ 1 λ 1 2 ( 2 B 212 + B 311 ) + κ 1 2 λ 1 ( B 213 + 2 B 312 ) + κ 1 3 B 313 + λ 1 B 12 [ 3 κ 1 2 B 23 + 2 κ 1 λ 1 ( B 13 + 2 B 22 ) + 3 λ 1 2 B 12 ] + 2 λ 1 B 13 ( κ 1 B 13 + λ 1 B 12 ) + λ 1 B 23 B 12 , B 322 = λ 1 B 222 + λ 1 3 B 211 + κ 1 λ 1 2 ( 2 B 212 + B 311 ) + κ 1 2 λ 1 ( B 222 + 2 B 312 ) + κ 1 3 B 322 + 2 λ 1 B 12 [ κ 1 2 B 23 + κ λ ( B 13 + B 22 ) + λ 1 2 B 12 ] + 2 λ 1 B 22 [ κ 1 ( B 13 + B 22 ) + 2 λ 1 B 12 ] , B 233 = λ 1 B 133 + λ 1 4 B 211 + 4 κ 1 λ 1 3 B 212 + 2 κ 1 2 λ 1 2 ( B 213 + 2 B 222 ) + 4 κ 1 3 λ 1 B 223 + κ 1 4 B 233 + 2 λ 1 B 13 ( κ 1 2 B 23 + 2 κ 1 λ 1 B 22 + λ 1 2 B 12 ) + 4 λ 1 B 23 ( κ 1 B 22 + λ 1 B 12 ) , B 323 = λ 1 B 223 + λ 1 4 B 211 + κ 1 λ 1 3 ( 3 B 212 + B 311 ) + κ 1 2 λ 1 2 ( B 213 + 2 B 222 + 3 B 312 ) + κ 1 3 λ 1 ( B 223 + B 313 + 2 B 322 ) + κ 1 4 B 323 + λ 1 B 22 [ 3 κ 1 2 B 23 + 2 κ 1 λ 1 ( B 13 + 2 B 22 ) + 3 λ 1 2 B 12 ] + λ 1 B 23 [ κ 1 ( B 13 + B 22 ) + 2 λ 1 B 12 ] + λ 1 B 13 [ κ 1 2 B 23 + κ 1 λ 1 ( B 13 + B 22 ) + λ 1 2 B 12 ] , B 333 = λ 1 B 233 + λ 1 5 B 211 + λ 1 4 κ 1 ( 4 B 212 + B 311 ) + 2 κ 1 2 λ 1 3 ( 2 B 232 + B 213 + 2 B 312 ) + 2 κ 1 3 λ 1 2 ( 2 B 223 + B 313 + 2 B 322 ) + κ 1 4 λ 1 ( B 233 + 4 B 322 ) + κ 1 5 B 333 + 2 λ 1 B 23 [ 3 κ 1 2 B 23 + 2 κ 1 λ 1 ( B 13 + 2 B 22 ) + 3 λ 1 2 B 12 ] .
κ 1 B 211 = κ 1 B 211 + λ 1 B 111 + 6 κ 1 λ 1 B 11 B 12 , κ 1 B 212 = κ 1 2 B 212 + κ 1 λ 1 ( B 211 + B 112 ) + λ 1 2 B 111 + 2 κ 1 λ 1 [ B 11 B 22 + 2 B 12 B 12 ] , κ 1 B 311 = κ 1 2 B 311 + κ 1 λ 1 ( B 211 + B 211 ) + λ 1 2 B 111 + 2 κ 1 λ 1 [ 2 B 11 B 13 + B 12 B 12 ] , κ 1 B 213 = κ 1 3 B 213 + κ 1 2 λ 1 ( B 113 + 2 B 212 ) + κ 1 λ 1 2 ( 2 B 112 + B 211 ) + λ 1 3 B 111 + κ 1 λ 1 [ B 11 B 23 + 2 B 12 B 22 + 3 B 13 B 12 ] , κ 1 B 222 = κ 1 3 B 222 + κ 1 2 λ 1 ( B 122 + 2 B 212 ) + κ 1 λ 1 2 ( 2 B 112 + B 211 ) + λ 1 3 B 111 + 2 κ 1 λ 1 [ 2 B 12 B 22 + B 22 B 12 ] , κ 1 B 312 = κ 1 3 B 312 + κ 1 2 λ 1 ( B 212 + B 212 + B 311 ) + κ 1 λ 1 2 ( B 211 + B 112 + B 211 ) + λ 1 3 B 111 + κ 1 λ 1 [ B 11 B 23 + B 12 ( B 22 + 3 B 13 ) + B 22 B 12 ] , κ 1 B 223 = κ 1 4 B 223 + κ 1 3 λ 1 ( B 123 + 2 B 222 + B 213 ) + κ 1 2 λ 1 2 ( 2 B 122 + B 113 + 3 B 212 ) + κ 1 λ 1 3 ( 3 B 112 + B 211 ) + λ 1 4 B 111 + κ 1 λ 1 [ B 12 B 23 + 2 ( B 13 + B 22 ) B 22 + B 23 B 12 ] , κ 1 B 313 = κ 1 4 B 313 + κ 1 3 λ 1 ( B 213 + B 213 + 2 B 312 ) + κ 1 2 λ 1 2 ( 2 B 212 + 2 B 212 + B 113 + B 311 ) + κ 1 λ 1 3 ( 2 B 112 + B 211 + B 211 ) + λ 1 4 B 111 + κ 1 λ 1 [ 3 B 12 B 23 + 2 B 13 B 13 + B 23 B 12 ] , κ 1 B 322 = κ 1 4 B 322 + κ 1 3 λ 1 ( B 222 + B 222 + 2 B 312 ) + κ 1 2 λ 1 2 ( B 122 + 2 B 212 + 2 B 212 + B 311 ) + κ 1 λ 1 3 ( 2 B 112 + B 211 + B 211 ) + λ 1 4 B 111 + 2 κ 1 λ 1 [ B 12 B 23 + B 22 ( B 13 + B 22 ) ] , κ 1 B 233 = κ 1 5 B 233 + κ 1 4 λ 1 ( B 133 + 4 B 223 ) + κ 1 3 λ 1 2 ( 4 B 123 + 2 B 213 + 4 B 222 ) + κ 1 2 λ 1 3 ( 4 B 122 + 2 B 113 + 4 B 212 ) + κ 1 λ 1 4 ( 4 B 112 + B 211 ) + λ 1 5 B 111 + 2 κ 1 λ 1 [ B 13 B 23 + 2 B 23 B 22 ] , κ 1 B 323 = κ 1 5 B 323 + κ 1 4 λ 1 ( B 223 + B 223 + B 313 + 2 B 322 ) + κ 1 3 λ 1 2 ( B 123 + 2 B 222 + B 213 + B 213 + 2 B 222 + 3 B 312 ) + κ 1 2 λ 1 3 ( B 113 + 2 B 122 + 3 B 212 + 3 B 212 + B 311 ) + κ 1 λ 1 4 ( 3 B 112 + B 211 + B 211 ) + λ 1 5 B 111 + κ 1 λ 1 [ B 13 B 23 + 3 B 22 B 23 + B 23 ( B 13 + B 22 ) ] , κ 1 B 333 = κ 1 6 B 333 + κ 1 5 λ 1 ( B 233 + B 233 + 4 B 323 ) + κ 1 4 λ 1 2 ( B 133 + 4 B 223 + 4 B 223 + 2 B 313 + 4 B 322 ) + 2 κ 1 3 λ 1 3 ( 2 B 123 + B 213 + 2 B 222 + B 213 + 2 B 222 + 2 B 312 ) + κ 1 2 λ 1 4 ( 4 B 122 + 2 B 113 + 4 B 212 + 4 B 212 + B 311 ) + κ 1 λ 1 5 ( 4 B 112 + B 211 + B 211 ) + λ 1 6 B 111 + 6 κ 1 λ 1 B 23 B 23 .
κ 1 C 112 = κ 1 C 112 + λ 1 C 111 + 6 κ 1 λ 1 B 11 B 12 , κ 1 C 113 = κ 1 2 C 113 + κ 1 λ 1 ( 2 C 112 - γ 5 ) + λ 1 2 C 111 + 2 κ 1 λ 1 ( B 12 2 + 2 B 11 B 13 ) , κ 1 C 122 = κ 1 2 C 122 + κ 1 λ 1 ( 2 C 112 - γ 5 ) + λ 1 2 C 111 + 2 κ 1 λ 1 ( 2 B 12 2 + B 11 B 22 ) , κ 1 C 123 = κ 1 3 C 123 + κ 1 2 λ 1 ( C 113 + 2 C 122 - γ 4 ) + κ 1 λ 1 2 ( 3 C 112 - 2 γ 5 ) + λ 1 3 C 111 + κ 1 λ 1 [ B 11 B 23 + B 12 ( 3 B 13 + 2 B 22 ) ] , κ 1 C 222 = κ 1 3 C 222 + κ 1 2 λ 1 ( 3 C 122 - γ 4 ) + κ 1 λ 1 2 ( 3 C 112 - 2 γ 5 ) + λ 1 3 C 111 + 6 κ 1 λ 1 B 12 B 22 , κ 1 C 133 = κ 1 4 C 133 + κ 1 3 λ 1 ( 4 C 123 - γ 3 ) + κ 1 2 λ 1 2 ( 2 C 113 + 4 C 122 - 3 γ 4 ) + κ 1 λ 1 3 ( 4 C 112 - 3 γ 5 ) + λ 1 4 C 111 + 2 κ 1 λ 1 ( B 13 2 + 2 B 12 B 23 ) , κ 1 C 223 = κ 1 4 C 223 + κ 1 3 λ 1 ( 2 C 123 + 2 C 222 - γ 3 ) + κ 1 2 λ 1 2 ( C 113 + 5 C 122 - 3 γ 4 ) + κ 1 λ 1 3 ( 4 C 112 - 3 γ 5 ) + λ 1 4 C 111 + 2 κ 1 λ 1 [ B 12 B 23 + B 22 ( B 13 + B 22 ) ] , κ 1 C 233 = κ 1 5 C 233 + κ 1 4 λ 1 ( 4 C 223 + C 133 - γ 2 ) + κ 1 3 λ 1 2 ( 8 C 123 + 4 C 222 - 6 γ 3 ) + κ 1 2 λ 1 3 ( 2 C 113 + 8 C 122 - 6 γ 4 ) + κ 1 λ 1 4 ( 5 C 112 - 4 γ 5 ) + λ 1 5 C 111 + 2 κ 1 λ 1 B 23 [ B 13 + 2 B 22 ] , κ 1 C 333 = κ 1 6 C 333 + κ 1 5 λ 1 ( 6 C 233 - γ 1 ) + κ 1 4 λ 1 2 ( 12 C 223 + 3 C 133 - 5 γ 2 ) + κ 1 3 λ 1 3 ( 12 C 123 + 8 C 222 - 10 γ 3 ) + κ 1 2 λ 1 4 ( 12 C 122 + 3 C 113 - 10 γ 4 ) + κ 1 λ 1 5 ( 6 C 112 - 5 γ 5 ) + λ 1 6 C 111 + 6 κ 1 λ 1 B 23 2 .